Eric D’Hoker

Department of Physics and Astronomy University of California, Los Angeles, CA 90095

2004, October 3

Contents

1 Introduction 3 1.1 Relativity and quantum mechanics ...... 4 1.2 WhyQuantumFieldTheory?...... 6 1.3 Further conceptual changes required by relativity ...... 6 1.4 Some History and present signiﬁcance of QFT ...... 8

2 Quantum Mechanics – A synopsis 10 2.1 ClassicalMechanics...... 10 2.2 Principles of Quantum Mechanics ...... 12 2.3 Quantum Systems from classical mechanics ...... 13 2.4 Quantum systems associated with Lie algebras ...... 15 2.5 Appendix I : Lie groups, Lie algebras, representations ...... 16 2.6 Functionalintegralformulation ...... 19

3 Principles of Relativistic Quantum Field Theory 22 3.1 The Lorentz and Poincar´egroups and algebras ...... 23 3.2 ParityandTimereversal ...... 25 3.3 Finite-dimensional Representations of the Lorentz Algebra ...... 26 3.4 Unitary Representations of the Poincar´eAlgebra ...... 29 3.5 The basic principles of quantum ﬁeld theory ...... 32 3.6 Transformation of Local Fields under the Lorentz Group ...... 34 3.7 DiscreteSymmetries ...... 39 3.7.1 Parity ...... 39 3.7.2 Timereversal ...... 39 3.7.3 ChargeConjugation ...... 40 3.8 Signiﬁcance of ﬁelds in terms of particle contents ...... 40 3.9 Classical Poincar´einvariant ﬁeld theories ...... 41 4 Free Field Theory 43 4.1 TheScalarField ...... 43 4.2 The Operator Product Expansion & Composite Operators ...... 51 4.3 TheGaugeorSpin1Field...... 54 4.4 The Casimir Eﬀect on parallel plates ...... 59 4.5 Bose-Einstein and Fermi-Dirac statistics ...... 62 4.6 TheSpin1/2Field ...... 65 4.6.1 The Weyl basis and Free Weyl spinors ...... 66 4.7 Thespinstatisticstheorem...... 71 4.7.1 Parastatistics? ...... 71 4.7.2 Commutation versus anti-commutation relations ...... 72

5 Interacting Field Theories – Gauge Theories 73 5.1 InteractingLagrangians ...... 73 5.2 AbelianGaugeInvariance ...... 74 5.3 Non-AbelianGaugeInvariance...... 74 5.4 Componentformulation ...... 76 5.5 Bianchiidentities ...... 77 5.6 Gaugeinvariantcombinations ...... 77 5.7 Classical Action & Field equations ...... 78 5.8 Lagrangians including fermions and scalars ...... 79 5.9 Examples ...... 80

6 The S-matrix and LSZ Reduction formulas 81 6.0.1 InandOutStatesandFields ...... 82 6.0.2 TheS-matrix ...... 83 6.0.3 Scatteringcrosssections ...... 83 6.1 Relating the interacting and in- and out-ﬁelds ...... 85 6.2 The K¨allen-Lehmann spectral representation ...... 87 6.3 TheDiracField...... 89 6.4 TheLSZReductionFormalism ...... 90 6.4.1 In- and Out-Operators in terms of the free ﬁeld ...... 90 6.4.2 Asymptotics of the interacting ﬁeld ...... 91 6.4.3 Thereductionformulas...... 92 6.4.4 TheDiracField...... 93 6.4.5 ThePhotonField...... 94

2 1 Introduction

Quantum Field Theory (abbreviated QFT) deals with the quantization of ﬁelds. A familiar example of a ﬁeld is provided by the electromagnetic ﬁeld. Classical electromagnetism describes the dynamics of electric charges and currents, as well as electro-magnetic waves, such as radio waves and light, in terms of Maxwell’s equations. At the atomic level, however, the quantum nature of atoms as well as the quantum nature of electromagnetic radiation must be taken into account. Quantum mechanically, electromagnetic waves turn out to be composed of quanta of light, whose individual behavior is closer to that of a particle than to a wave. Remarkably, the quantization of the electromagnetic ﬁeld is in terms of the quanta of this ﬁeld, which are particles, also called photons. In QFT, this ﬁeld particle correspondence is used both ways, and it is one of the key assumptions of QFT that to every elementary particle, there corresponds a ﬁeld. Thus, the electron will have its own ﬁeld, and so will every quark. Quantum Field Theory provides an elaborate general formalism for the ﬁeld–particle correspondence. The advantage of QFT will be that it can naturally account for the cre- ation and annihilation of particles, which ordinary quantum mechanics of the Schr¨odinger equation could not describe. The fact that the number of particles in a system can change over time is a very important phenomenon, which takes place continuously in everyone’s daily surroundings, but whose signiﬁcance may not have been previously noticed. In classical mechanics, the number of particles in a closed system is conserved, i.e. the total number of particles is unchanged in time. To each pointlike particle, one associates a set of position and momentum coordinates, the time evolution of which is governed by the dynamics of the system. Quantum mechanics may be formulated in two stages.

1. The principles of quantum mechanics, such as the deﬁnitions of states, observables, are general and do not make assumptions on whether the number of particles in the system is conserved during time evolution.

2. The speciﬁc dynamics of the quantum system, described by the Hamiltonian, may or may not assume particle number conservation. In introductory quantum mechanics, dynamics is usually associated with non-relativistic mechanical systems (augmented with spin degrees of freedom) and therefore assumes a ﬁxed number of particles. In many important quantum systems, however, the number of particles is not conserved.

A familiar and ubiquitous example is that of electromagnetic radiation. An excited atom may decay into its ground state by emitting a single quantum of light or photon. The photon was not “inside” the excited atom prior to the emission; it was “created” by the excited atom during its transition to the grounds state. This is well illustrated as

3 follows. An atom in a state of suﬃciently high excitation may decay to its ground state in a single step by emitting a single photon. However, it may also emit a ﬁrst photon to a lower excited state which in turn re-emits one or more photons in order to decay to the ground state (see Figure 1, (a) and (b)). Thus, given initial and ﬁnal states, the number of photons emitted may vary, lending further support to the fact that no photons are “inside” the excited state to begin with. Other systems where particle number is not conserved involve phonons and spin waves in condensed matter problems. Phonons are the quanta associated with vibrational modes of a crystal or ﬂuid, while spin waves are associated with ﬂuctuating spins. The number of particles is also not conserved in nuclear processes like fusion and ﬁssion.

1.1 Relativity and quantum mechanics Special relativity invariably implies that the number of particles is not conserved. Indeed, one of the key results of special relativity is the fact that mass is a form of energy. A particle at rest with mass m has a rest energy given by the famous formula

E = mc2 (1.1)

The formula also implies that, given enough energy, one can create particles out of just energy – kinetic energy for example. This mechanism is at work in ﬁre and light bulbs, where energy is being provided from chemical combustion or electrical input to excite atoms which then emit light in the form of photons. The mechanism is also being used in particle accelerators to produce new particles through the collision of two incoming particles. In Figure 1 the example of a photon scattering oﬀ an electron is illustrated. In (c), a photon of low energy ( m c2) is being scattered elastically which results simply e in a deﬂection of the photon and a recoil of the electron. In (d), a photon of high energy ( m c2) is being scattered inelastically, resulting not only in a deﬂection of the photon e and a recoil of the electron, but also in the production of new particles. The particle data table also provides numerous examples of particles that are unstable and decay. In each of these processes, the number of particles is not conserved. To list just a few,

n p+ + e− +¯ν → e π0 γ + γ → π+ µ+ + ν → µ µ+ e+ + ν +¯ν → e µ As already mentioned, nuclear processes such as fusion and ﬁssion are further examples of systems in which the number of particles is not conserved.

4 photon photon photon

initial initial

final final

(a) (b)

scattered photon scattered photon

photon photon new particles

recoiled electron recoiled electron

(c) (d)

Figure 1: Production of particles : (a) Emission of one photon, (b) emission of two photons between the same initial and ﬁnal states; (c) low energy elastic scattering, (d) high energy inelastic scattering of a photon oﬀ a recoiling electron.

5 1.2 Why Quantum Field Theory ? Quantum Field Theory is a formulation of a quantum system in which the number of particles does not have to be conserved but may vary freely. QFT does not require a change in the principles of either quantum mechanics or relativity. QFT requires a diﬀerent formulation of the dynamics of the particles involved in the system. Clearly, such a description must go well beyond the usual Schr¨odinger equation, whose very formulation requires that the number of particles in a system be ﬁxed. Quantum ﬁeld theory may be formulated for non-relativistic systems in which the number of particles is not conserved, (recall spin waves, phonons, spinons etc). Here, however, we shall concen- trate on relativistic quantum ﬁeld theory because relativity forces the number of particles not to be conserved. In addition, relativity is one of the great fundamental principles of Nature, so that its incorporation into the theory is mandated at a fundamental level.

1.3 Further conceptual changes required by relativity Relativity introduces some further fundamental changes in both the nature and the for- malism of quantum mechanics. We shall just mention a few. Space versus time • In non-relativistic quantum mechanics, the position x, the momentum p and the en- ergy E of free or interacting particles are all observables. This means that each of these quantities separately can be measured to arbitrary precision in an arbitrarily short time. By contrast, the accurarcy of the simultaneous measurement of x and p is limited by the Heisenberg uncertainty relations, ∆x ∆p h¯ ∼ There is also an energy-time uncertainty relation ∆E ∆t h¯, but its interpretation is quite ∼ diﬀerent from the relation ∆x ∆p h¯, because in ordinary quantum mechanics, time ∼ is viewed as a parameter and not as an observable. Instead the energy-time uncertainty relation governs the time evolution of an interacting system. In relativistic dynamics, particle-antiparticle pairs can always be created, which subjects an interacting particle always to a cloud of pairs, and thus inherently to an uncertainty as to which particle one is describing. Therefore, the momentum itself is no longer an instantaneous observable, but will be subject to the a momentum-time uncertainty relation ∆p ∆t h/c¯ . As c , ∼ →∞ this eﬀect would disappear, but it is relevant for relativistic processes. Thus, momentum can only be observed with precision away from the interaction region. Special relativity puts space and time on the same footing, so we have the choice of either treating space and time both as observables (a bad idea, even in quantum mechanics) or to treat them both as parameters, which is how QFT will be formulated.

6 “Negative energy” solutions and anti-particles • The kinetic law for a relativistic particle of mass m is

E2 = m2c4 + p2c2

Positive and negative square roots for E naturally arise. Classically of course one may just keep positive energy particles. Quantum mechanically, interactions induce transitions to negative energy states, which therefore cannot be excluded arbitrarily. Following Feynman, the correct interpretation is that these solutions correspond to negative frequencies, which describe physical anti-particles with positive energy traveling “backward in time”. Local Fields and Local Interactions • Instantaneous forces acting at a distance, such as appear in Newton’s gravitational force and Coulomb’s electrostatic force, are incompatible with special relativity. No signal can travel faster than the speed of light. Instead, in a relativistic theory, the interaction must be mediated by another particle. That particle is the graviton for the gravitational force and the photon for the electromagnetic force. The true interaction then occurs at an instant in time and at a point in space, thus expressing the forces in terms of local interactions only. Thus, the Coulomb force is really a limit of relativistic “retarded” and “advanced” interactions, mediated by the exchange of photons. The exchange is pictorially represented in Figure 2.

electron

electron

photon

electron

electron

Figure 2: The Coulomb force results from the exchange of photons.

One may realize this picture concretely in terms of local ﬁelds, which have local cou- plings to one another. The most basic prototype for such a ﬁeld is the electro-magnetic ﬁeld. Once a ﬁeld has been associated to the photon, then it will become natural to asso- ciate a ﬁeld to every particle that is viewed as elementary in the theory. Thus, electrons and positrons will have their (Dirac) ﬁeld. There was a time that a ﬁeld was associated also to the proton, but we now know that the proton is composite and thus, instead, there

7 are now ﬁelds for quarks and gluons, which are the elementary particles that make up a proton. The gluons are the analogs for the strong force of the photon for the electro- magnetic force, namely the gluons mediate the strong force. Analogously, the W ± and Z0 mediate the weak interactions.

1.4 Some History and present signiﬁcance of QFT The quantization of the elctro-magnetic ﬁeld was initiated by Born, Heisenberg and Jor- dan in 1926, right after quantum mechanics had been given its deﬁnitive formulation by Heisenberg and Schr¨odinger in 1925. The complete formulation of the dynamics was given by Dirac Heiseberg and Pauli in 1927. Inﬁnities in perturbative corrections due to high energy (or UV – ultraviolet) eﬀects were ﬁrst studied by Oppenheimer and Bethe in the 1930’s. It took until 1945-49 until Tomonaga, Schwinger and Feynman gave a completely relativistic formulation of Quantum Electrodynamics (or QED) and evaluated the radia- tive corrections to the magnetic moment of the electron. In 1950, Dyson showed that the UV divergences of QED can be systematically dealt with by the process of renormalization. In the 1960’s Glashow, Weinberg and Salam formulated a renormalizable quantum ﬁeld theory of the weak interactions in terms of a Yang-Mills theory. Yang-Mills theory was shown to be renormalizable by ‘t Hooft in 1971, a problem that was posed to him by his advisor Veltman. In 1973, Gross, Wilczek and Politzer discovered asymptotic freedom of certain Yang-Mills theories (the fact that the strong force between quarks becomes weak at high energies) and using this unique clue, they formulated (independently also Weinberg) the quantum ﬁeld theory of the strong interactions. Thus, the elctro-magnetic, weak and strong forces are presently described – and very accurately so – by quantum ﬁeld theory, speciﬁcally Yang-Mills theory, and this combined theory is usually referred to as the STANDARD MODEL. To give just one example of the power of the quantum ﬁeld theory approach, one may quote the experimentally measured and theoretically calculated values of the muon magnetic dipole moment, 1 g (exp) = 1.001159652410(200) 2 µ 1 g (thy) = 1.001159652359(282) (1.2) 2 µ revealing an astounding degree of agreement. The gravitational force, described classically by Einstein’s general relativity theory, does not seem to lend itself to a QFT description. String theory appears to provide a more appropriate description of the quantum theory of gravity. String theory is an extension of QFT, whose very formulation is built squarely on QFT and which reduces to QFT in the low energy limit.

8 The devlopment of quantum ﬁeld theory has gone hand in hand with developments in Condensed Matter theory and Statistical Mechanics, especially critical phenomena and phase transitions. A ﬁnal remark is in order on QFT and mathematics. Contrarily to the situation with general relativity and quantum mechanics, there is no good “axiomatic formulation” of QFT, i.e. one is very hard pressed to lay down a set of simple postulates from which QFT may then be constructed in a deductive manner. For many years, physicists and mathematicians have attempted to formulate such a set of axioms, but the theories that could be ﬁt into this framework almost always seem to miss the physically most relevant ones, such as Yang-Mills theory. Thus, to date, there is no satsifactory mathematical “deﬁnition” of a QFT. Conversely, however, QFT has had a remarkably strong inﬂuence on mathematics over the past 25 years, with the development of Yang-Mills theory, instantons, monopoles, con- formal ﬁeld theory, Chern-Simons theory, topological ﬁeld theory and superstring theory. Some developments is QFT have led to revolutions in mathematics, such as Seiberg-Witten theory. It is suspected by some that this is only the tip of the iceberg, and that we are only beginning to have a glimpse at the powerful applications of quantum ﬁeld theory to mathematics.

9 2 Quantum Mechanics – A synopsis

The basis of QFT is quantum mechanics and therefore we shall begin by reviewing the foundations of this dsicipline. In fact, we begin with a brief summary of Lagrangian and Hamiltonian classical mechanics as this will also be of great value.

2.1 Classical Mechanics Consider a system of N degrees of freedom q (t), i = 1, , N. (For example n particles i ··· in R3 has N =3n.) Dynamics is governed by the Lagrangian function∗ dq (t) L(q , q˙ )q ˙ (t) i (2.1) i i i ≡ dt or by the action functional

t2 S[qi]= dt L(qi(t), q˙i(t)) (2.2) Zt1 The classical time evolution of the system is determined as a solution to the variational problem as follows δS[q ]=0 d ∂L ∂L i =0 (2.3) δqi(t1)= δqi(t1)=0 ⇒ dt ∂q˙i − ∂qi and these equations are the Euler-Lagrange equations. In general, the diﬀerential equations are second order in t, and thus the Cauchy data are qi(t1) andq ˙i(t1) at initial time t1 : namely positions and velocities. Hamiltonian Formulation • The Hamiltonian formulation of classical mechanics proceeds as follows. We introduce the canonical momenta, ∂L pi (qj, q˙j) q˙i(q,p) (2.4) ≡ ∂q˙i ⇒ and introduce the Hamiltonian function N H(p, q) p q˙ (p, q) L(q , q˙ (p, q)) (2.5) ≡ i i − j j Xi=1 The time evolution equation can now be re-expressed as the Hamilton equations ∂H ∂H q˙i = p˙i = (2.6) ∂pi − ∂qi

∗For almost all the problems that we shall consider, L has no explicit time dependence and all time- dependence is implicit through qi(t) andq ˙i(t).

10 The time evolution of any function F (p,q,t) along the classical trajectory may be expressed as follows d ∂F F (p,q,t) = H, F + dt { } ∂t N ∂A ∂B ∂B ∂A A, B (2.7) { } ≡ ∂pi ∂qi − ∂pi ∂qi Xi=1 The Poisson bracket , is antisymmetric and satisﬁes the following relations { } 0 = A, B,C + B, C, A + C, A, B { { }} { { }} { { }} A,BC = A, B C + A, C B { } { } { } p , q = δ (2.8) { i j} ij The second line shows that the Poisson bracket acts as a derivation. Symmetries • A symmetry of a classical system is a transformation on the degrees of freedom qi such that under its action, every solution to the corresponding Euler-Lagrange equations is transformed into a solution of these same equations. Symmetries form a group under successive composition. An important special class consists of continuous symmetries, where the transformation may be labeled by a parameter (or by several parameters) in a continuous way. Denoting a transformation depending on a single parameter α R by ∈ σ(α), the transformation may be represented as follows,

σ(α) : q (t) q (t, α) q (t, 0) = q (t) (2.9) i −→ i i i The transformation σ(α) is a continuous symmetry of the Lagrangian L if and only if the derivative with repect to α satisﬁes the following property

∂ d L(q (t, α), q˙ (t, α)) = X(q , q˙ ) (2.10) ∂α i i dt i i obtained without using the Euler-Lagrange equations. Noether’s Theorem • The existence of a continuous symmetry implies the presence of a time independent charge Q (also sometimes called a ﬁrst integral of motion) by Noether’s theorem. An explicit formula is available for this charge

N ∂qi(t, α) Q = pi X(pi, qi) (2.11) i=1 ∂α α=0 − X

11 The relation Q˙ = 0 follows now by using the Euler-Lagrange equations. In the Hamiltonian formulation, the charge generates the transformation in turn by Poisson bracketting, ∂q (t, α) i = Q, q (t, α) (2.12) ∂α { i } Continuous symmetries of a given L form a Lie group. Lie groups and algebras will be deﬁned generally in the next section.

2.2 Principles of Quantum Mechanics A quantum system may be deﬁned in all generality in terms of a very small number of physical principles. 1. Physical States are represented by rays in a Hilbert space . Recall that a Hilbert H space is a complex (complete) vector space with a positive deﬁnite inner product, H which we denote by : C `ala Dirac, and we have h|i H×H −→ φ ψ = ψ φ ∗ • h | i h | i φ φ 0 • h | i ≥ φ φ =0 φ =0 (2.13) • h | i ⇒ | i A ray is an equivalence class in by φ λ φ , λ C 0 . H | i ∼ | i ∈ −{ } 2. Physical Observables are represented by self-adjoint operators on . For the mo- H ment, we shall denote such operators with a hat, e.g. Aˆ, but after this chapter, we drop hats. The operators are linear, so that Aˆ(a φ + b ψ )= aAˆ φ + bAˆ ψ , for all | i | i | i | i φ , ψ and a, b C. Recall that for ﬁnite-dimensional matrices, Hermiticity | i | i ∈ H ∈ and self-adjointness are the same. For operators acting on an inﬁnite dimensional Hilbert space, self-adjointness requires that the domains of the operator and its ad- joint be the same as well.

3. Measurements : A state ψ in has a deﬁnite measured value α for some observable | i H Aˆ if the state is an eigenstate of Aˆ with eigenvalue α; Aˆ ψ = α ψ . (Since Aˆ is | i | i self-adjoint, states associated with diﬀerent eigenvalues of Aˆ are orthogonal to one another.) In a quantum system, what you can measure in an experiment are the eigenvalues of various observables, e.g. the energy levels of atoms.

4. The transition probability for a given state ψ to be found in one of a set of | i ∈ H orthogonal states φ is given by | ni P ( ψ φ )= ψ φ 2 (2.14) | i→| ni |h | ni| for normalized states ψ ψ = φ φ = 1. The quantities ψ φ are referred to as h | i h n| ni h | ni the transition amplitudes. Notice that their dependence on the state ψ is linear, | i 12 so that these amplitudes obey the superposition principle, while the probability amplitudes are quadratic and cannot be linearly superimposed. Notice that amongst the principles of quantum mechanics, there is no reference to a Hamiltonian, a Schr¨odinger equation and the like. Those ingredients are dynamical and will depend upon the speciﬁc system under consideration. Commuting observables – quantum numbers • If two observables Aˆ and Bˆ commute, [A,ˆ Bˆ] = 0, then they can be diagonalized simultaneously. The associated physical observables can then be measured simultaneously, yielding eigenvalues α and β respectively. On the other hand, if [A,ˆ Bˆ] = 0, the associated 6 physical observables cannot be measured simultaneously. One deﬁnes a maximal set of commuting observales as a set of observables Aˆ , Aˆ , , Aˆ which all mutually commute, [Aˆ , Aˆ ] = 0, for all i, j = 1, , n, and 1 2 ··· n i j ··· such that no other operator (besides linear combinations of the Aˆi) commutes with all

Aˆi. It follows that the eigenspaces for each of the sets of eigenvalues is one-dimensional, and therefore label the states in in a unique manner. Given the set Aˆ , the H { i}i=1,···,n corresponding eigenvalues are the quantum numbers. Symmetries in quantum systems • Symmetries in quantum mechanics are realized by unitary transformations in Hilbert space . Unitarity guarantees that the transition amplitudes and thus probabilities will H be invariant under the symmetry.

2.3 Quantum Systems from classical mechanics A very large and important class of quantum systems derive from classical mechanics systems (for example the Hydrogen atom). Given a classical mechanics system, one can always construct an associated quantum system, using the correspondence principle. Con- sider a classical mechanics system with degrees of freedom p (t), q (t) R, i = 1, , N i i ∈ ··· and with Hamiltonian H(p, q). The associated quantum system is obtained by mapping the reals pi and qi into observables in a Hilbert space . The Hilbert space may be taken 2 N H to be L (R ), i.e. square integrable functions of qi. The Poisson bracket is mapped into a commutator of operators. The detailed correspondence is

pi, qi pˆi, qˆi → i p , q = δ [ˆp , qˆ ]= δ { i j} ij → h¯ i j ij p ,p = q , q =0 [ˆp , pˆ ]=[ˆq , qˆ ]=0 { i j} { i j} → i j i j Under the correspondence principle, the Hamiltonian H(p, q) produces a self-adjoint op- erators Hˆ (ˆp, qˆ). In general, however, this correspondence will not be unique, because in

13 the classical system, p and q are commuting numbers, so their ordering in any expressing was immaterial, while in the quantum system,p ˆ andq ˆ are operators and their ordering will matter. If such ordering ambiguity arises in passing from the classical to the quantum system, further physical information will have to be supplied in order to lift the ambiguity. If the classical system had a continuous symmetry, then by Noether’s theorem, there is an associated time-independnt charge Q. Using the correspondence principle, one may construct an associated quantum operator Qˆ. However, the correspondence principle, in general, does not produce a unique Qˆ and the question arises whether an operator Qˆ can be constructed at all which is time independent. If yes, the classical symmetry extends to a quantum symmetry. If not, the classical symmetry is said to have an anomaly; this eﬀect does not arise in quantum mechanics but only appears in quantum ﬁeld theory. A particularly important symmetry of a system with time-independent Hamiltonian is time translation invariance. Time-evolution is governed by the Heisenberg equation

dAˆ(t) ih¯ = [Aˆ(t), Hˆ ] (2.15) dt

The operatorsp ˆ,q ˆ and Hˆ themselves are observable, and their eigenvalues are the mo- mentum, position and energy of the state. The prime example, and which will be extremely ubiquitous in the practice of quantum ﬁeld theory is the Harmonic Oscillator. We consider here the simplest case with N = 1,

1 1 Hˆ = pˆ2 + ω2qˆ2, ω> 0 (2.16) 2 2 Notice that since there are no terms involving bothp ˆ andq ˆ, the passage from classical to quantum was unambiguous here. Introducing the linear combinations,

1 aˆ (+ipˆ + ωqˆ) ≡ √2ωh¯ 1 aˆ† ( ipˆ + ωqˆ) (2.17) ≡ √2ωh¯ −

The system may be recast in the following fashion,

[ˆa, aˆ†] = 1 1 Hˆ =hω ¯ (ˆa†aˆ + ) (2.18) 2

The operators are referred to as annihilation aˆ and creation aˆ† operators because their ap- plication on any state lowers and raises the energy of the state with precisely one quantum

14 of energyhω ¯ . This is shown as follows,

[H,ˆ aˆ] = hω¯ aˆ − H,ˆ aˆ†] = +¯hωaˆ† (2.19) { Thus, if E is an eigenstate with energy E, so that Hˆ E = E E , then it follows that | i | i | i Hˆ (ˆa E ) = (E hω¯ )(ˆa E ) | i − | i Hˆ (ˆa† E ) = (E +¯hω)(ˆa† E ) (2.20) | i | i Using this algebraic formulation, the sytem may be solved very easily. Note that Hˆ > 0, so there must be a state E = 0, such thata ˆ E = 0, whence it follows that E = 1 hω¯ | 0i 6 | 0i 0 2 and the energies of the remaining states (ˆa†)n E are E =¯hω(n + 1 ). | 0i n 2 2.4 Quantum systems associated with Lie algebras Not all quantum systems have classical analogs. For example, quantum orbital angular momentum corresponds to the angular momentum of classical systems, but spin angular momentum has no classical counterpart. Another example would be the color assignment of quarks, and one may even view ﬂavor assignments (namely whether they are up, down, charm, strange, top or bottom) as a quantum property without classical counterpart. It is natural to understand these quantum systems in terms of the theory of Lie groups and Lie algebras. After all, spin appeared as a special kind of representation of the rotation group that cannot be realized in terms of orbital angular momentum, and color will ultimately be associated with the gauge group of the strong interactions SU(3)c. Even the quantum system of free particles will be thought of in terms of representations of the Poincar´egroup. Consider a Lie group G and its associated Lie algebra . Let ρ be a unitary represen- G tation of acting on a Hilbert space . (For representations of ﬁnite dimension N, we G H have = CN .) This set-up naturally deﬁnes a quantum system associated with the Lie H algebra and the representation ρ. The quantum observables are the linear self-adjoint G operators ρ(T a). The maximal set of commuting observables is the largest set of com- muting generators of . For semi-simple Lie algebras this is the Cartan sub-algebra. We G conclude with a few important examples,

1. Angular momentum is associated with the group of space rotations O(3); its unitary representations are labelled by half integers j =0, 1/2, 1, 3/2, . Only for integer ··· j is there a classical realization in terms of orbital angular momentum.

2. Special relativity states that the space-time symmetry group of Nature is the Poincar´egroup. Elementary particles will thus be associated with unitary repre- sentations of the Poincar´egroup.

15 2.5 Appendix I : Lie groups, Lie algebras, representations

The concept of a group was introduced in the context of polynomial equations by Evariste Galois, and in a more general context by Sophus Lie. Given an algebraic or diﬀerential equation, the set of transformations that map any solution into a solution of the same equation forms a group, where the group operation is composition of maps. As physics is formulated in terms of mathematical equations, groups naturally enter into the under- standing and solution of these equations.

Deﬁnition of a group • Generally, a set G forms a group provided it is endowed with an operation, which we denote , and which satisﬁes the following conditions ∗

1. Closure : g ,g G, then g g G; 1 2 ∈ 1 ∗ 2 ∈ 2. Associativity : g (g g )=(g g ) g for all g ,g ,g G; 1 ∗ 2 ∗ 3 1 ∗ 2 ∗ 3 1 2 3 ∈ 3. There exists a unit I such that I g = g I = g for all g G; ∗ ∗ ∈ 4. Every g G has an inverse g−1 such that g−1 g = g g−1 = I. ∈ ∗ ∗

Additionally, the group G is called commutative or Abelian is g g = g g for all ,∗ 1 ∗ 2 2 ∗ 1 g ,g G. If on the other hand there is at least one pair g ,g such that g g = g g , 1 2 ∈ 1 2 1 ∗ 2 6 2 ∗ 1 then the group is said to be non-commutative or non-Abelian.

Given a group G,a subgroup H is a subset of G that contains I such that h h H for 1 ∗ 2 ∈ all h , h H. An invariant subgroup or ideal H of G is a subgroup such that g h g−1 H 1 2 ∈ ∗ ∗ ∈ for all g G and all h H. For example, the Poincar´egroup has an invariant subgroup ∈ ∈ consisting of translations alone. A group whose only invariant subgroups are e and G { } itself is called a simple group.

Lie groups and Lie algebras • As we have encountered already many times in physics, group elements may depend on parameters, such as rotations depend on the angles, translations depend on the distance, and Lorentz transformations depend on the boost velocity). A Lie group is a parametric group where the dependence of the group elements on all its parameters is continuous (this by itself would make it a topological group) and diﬀerentiable. The number of independent parameters is called the dimension D of G. It is a powerful Theorem of Sophus Lie that if the parametric dependence is diﬀerentiable just once, the additional group property makes the dependence automatically inﬁnitely diﬀerentiable, and thus real analytic.

16 A Lie algebra associated with a Lie group G is obtained by expanding the group G element g around the identity element I. Assuming that a set of local parameters xi, i =1, ,D has been made, we have ··· D g(x )= I + x T i + (x2) (2.21) i i O Xi=1 The T i are the generators of the Lie algebra and may be thought of as the tangent G vectors to the group manifold G at the identity. The fact that this structure forms a group allows us to compose and in particular form,

g(x )g(y )g(x )−1g(y )−1 G (2.22) i i i i ∈

Retaining only the terms linear in xi and linear in yj, we have

D g(x )g(y )g(x )−1g(y )−1 = I + x y [T i, T j]+ (x2,y2) (2.23) i i i i i j O i,jX=1 Hence there must be a linear relation,

D [T i, T j]= f ijk T k (2.24) kX=1 for a set of constants f ijk which are referred to as the structure constants, which satisfy the Jacobi identity,

0 = [[T i, T j], T k] + [[T j, T k], T i] + [[T k, T i], T j] (2.25)

Lie’s Theorem states that, conversely, if we have a Lie algebra , deﬁned by generators G T , i =1, ,D satisfying both (2.24) and (2.25), then the Lie algebra may be integrated i ··· up to a Lie group (which is unique if connected and simply connected). Thus, except for some often subtle global issues, the study of Lie groups may be reduced to the study of Lie algebras, and the latter is much simpler in general ! Examples include,

1. G = U(N), unitary N N matrices (with complex entries); g U(N) is deﬁned by × ∈ g†g = I; D = N 2. Its Lie algebra (often also denoted by U(N)) consists of N N G × Hermitian matrices, (iT )† =(iT ).

2. G = O(N), orthogonal N N matrices (with real entries); g O(N) is deﬁned by × ∈ gtg = I; D = N(N 1)/2. Its Lie algebra (often also denoted by O(N)) consists − G of N N anti-symmetric matrices, T t = T . × − 17 3. Any time S appears before the name, the group elements have also unit determinant. Thus G = SU(N) consists of g such that g†g = I and det g = 1. And G = SO(N) consists of g such that gtg = I and det g = 1. Notice that U(N) has two invariant subgroups : the diagonal U(1) and SU(N). Examples 1, 2 and 3 describe compact Lie groups, which are deﬁned to be compact spaces, i.e. bounded and closed. A famous non-comapct group is described next.

4. G = Gl(N, R) and G = Gl(N, C) are the general linear groups of N N matrices × g with det g = 0 (respectively with real and complex entries). G = Sl(N, R) and 6 G = Sl(N, C) are the special linear groups deﬁned by det g = 1.

Representations of Lie groups and Lie algebras • A (linear) representation R with dimension N of a group G is a map

R : G GL(N, C) or Gl(N, R) (2.26) → such that the group operation is mapped onto matrix multiplication in Gl(N), ∗ R(g g )= R(g ) R(g ) & R(I) = I (2.27) 1 ∗ 2 1 2 N In other words, the representation realizes the abstract group in terms of N N ma- × trices. A representation R : G GL(N, C) is called a complex representation, while → R : G GL(N, R) is called a real representation. Two representations R and R0 are → equivalent if they have the same dimension and if there exists a constant invertible matrix S such that R0(g) = S−1R(g)S for all g G. A representation is said to be reducible if ∈ there exists a constant matrix D, which is not a scalar multiple of the identity matrix, and which commutes with R(g) for all g. If no such matrix exists, the representation is irreducible. A special class of representations that is especially important in physics is that of unitary representations, for which R : G U(N). It is a standard result that → every ﬁnite-dimensional representation of a compact (or ﬁnite) group is equivalent to a unitary representation. A representation of a Lie algebra is a linear map : l(N, C), such that G R G → G D [ (T i), (T j)] = f ijk (T k) (2.28) R R R kX=1 As (T i) are matrices, the Jacobi identity is automatic. R A representation R : G GL(N) naturally acts on an N-dimensional vector space, → which in physics is a subspace of states or wave functions. The linear space is said to transform under the representation R, and the distiction between the representation and the vector space on which it acts is sometimes blurred.

18 2.6 Functional integral formulation

In preparation for the path integral formulation of quantum mechanics, for systems asso- ciated with classical mechanics systems with degrees of freedom p(t), q(t), we introduce two special adapted bases, one for position and one for momentum. Taking the ﬁrst line in the table as a deﬁnition, the remaining lines follow. The evolution operator

Uˆ(t) = exp itH/h¯ (2.29) {− } governs the transition amplitude ψ Uˆ(t) φ , for an intital state φ to evolve into a ﬁnal h | | i | i state ψ after a time t. For systems whose only degrees of freedom are p and q, such | i amplitudes may be expressed in terms of the wave functions q φ and q ψ of these h | i h | i states using the completeness relations,

+∞ +∞ ψ Uˆ(t) φ = dq dq0 q φ q0 ψ ∗ q0 Uˆ(t) q (2.30) h | | i Z−∞ Z−∞ h | ih | i h | | i To evaluate the matrix elements of Uˆ in position basis, we make use of the group property of Uˆ, Uˆ(t0 + t00)= Uˆ(t0)Uˆ(t00) and insert complete sets of position states,

+∞ q0 Uˆ(t0 + t00) q = dq00 q0 Uˆ(t00) q00 q00 Uˆ(t0) q (2.31) h | | i Z−∞ h | | ih | | i To evaluate q0 Uˆ(t) q , we divide the time interval t into N equal segments of length = h | | i t/N and insert N 1 complete sets of position eigenstates (labelled by q , k =1, , N 1) − k ··· − into the formula Uˆ(t)= Uˆ()N . This yields

N−1 +∞ N 0 q Uˆ(t) q = dqk qk Uˆ() qk−1 (2.32) h | | i −∞ h | | i kY=1 Z kY=1 0 where qN = q and q0 = q.

Position Basis Momentum Basis qˆ q = q q pˆ p = p p | i | i | i | i q0 q = δ(q0 q) p0 p = δ(p0 p) h+∞| i − h+∞| i − −∞ dq q q = IH −∞ dp p p = IH +iapˆ −ia| pˆih | −ibqˆ +|ibqˆih | eR qeˆ =q ˆ+ ah¯ eR peˆ =p ˆ+ bh¯ e−iapˆ q = q + ah¯ e+ibqˆ p = p + bh¯ | i | i | i | i Table 1:

19 Each matrix element q Uˆ() q may in turn be evaluated as follows. First, we h k| | k−1i insert a complete set of states in the momentum basis (labelled by p , k =1, , N) k ··· +∞ qk Uˆ() qk−1 = dpk qk Uˆ() pk pk qk−1 (2.33) h | | i Z−∞ h | | ih | i Using the overlaps of position and momentum eigenstates, we have

q p = e+ipkqk/¯h p q = e−ipkqk−1/¯h (2.34) h k| ki h k| k−1i The reason for introducing also the basis of momentum eigenstates is that now the needed matrix elements of the quantum Hamiltonian Hˆ (ˆp, qˆ) can be evaluated. In the limit N , we have 0 and thus the evolution operator over the inﬁnites- →∞ → imal time span may be evaluated by expanding the exponential to ﬁrst order only, Uˆ()= I i Hˆ + (2) (2.35) H − h¯ O

We assume that Hˆ is analytic inp ˆ andq ˆ, so that upon using the commutation relation [ˆp, qˆ]= ih¯, we may rearrange Hˆ so as to put allp ˆ to the right and allq ˆ to the left in any − given monomial. This allows us to deﬁne a classical function H(p, q) by

H(p , q ) q Hˆ (ˆp, qˆ) p (2.36) k k ≡h k| | ki and therefore, we have

q Uˆ() p = e−iH(pk,qk)/¯h q p (2.37) h k| | ki h k| ki Assembling all results, we now ﬁnd the following expression,

N−1 +∞ N +∞ N 0 iSk(p,q)/¯h q Uˆ(t) q = lim dqk dpl e (2.38) h | | i N→∞ −∞ −∞ kY=1 Z lY=1 Z mY=1 where

S (p, q) p (q q ) H(p , q ) (2.39) k ≡ k k − k−1 − k k In the limit N , for t ﬁxed, the label k on p and q becomes continuous, →∞ k k p p(t0) t0 = k = kt/N, = dt0 k → q q(t0) k → 0 0 0 0 0 Sk(p, q) dt p(t )q ˙(t ) H(p(t ), q(t )) (2.40) → − 20 As a result, the product of exponentials converges as follows

N i t lim eiSk(p,q)/¯h = exp dt0 pq˙ H(p, q) (t0) (2.41) N→∞ h¯ 0 − mY=1 Z The measure “converges” to a random walk measure, which is denoted as follows,

N−1 +∞ N +∞ lim dqk dpl = q p (2.42) N→∞ −∞ −∞ D D kY=1 Z lY=1 Z Z Z The ﬁnal result is the path integral formulation of quantum mechanics, which gives an expression for the position basis matrix elements of the evolution operator, i t q0 Uˆ(t) q = q p exp dt0 pq˙ H(p, q) (t0) (2.43) h | | i Z D Z D h¯ Z0 − with the boundary conditions that q(t0 =0)= q and q(t0 = t)= q0. This formulation was ﬁrst proposed by Dirac and rederived by Feynman. The quantity that enters the exponential in the path integral is really the classical action of the system,

S[p, q] dt0 pq˙ H(p, q) (t0) (2.44) ≡ Z − Therefore, in the classical limit where one formally takes the limith ¯ 0, the part integral → will be dominated by the stationnary or saddle points of the integral, which are deﬁned as those points at which the ﬁrst order (functional derivatives of S[p, q] with respect to p and q vanish. It is easy to work out what these equations are δS ∂H 0= = p˙ δq − − ∂q δS ∂H 0= = +q ˙ (2.45) δp − ∂p These are precisely the Hamilton equations of the classical Hamiltonian H. Thus, we recover very easily in the path integral formulation that in the classical limit, quantum mechanics receives its dominant contribution from classical solutions, as expected.

21 3 Principles of Relativistic Quantum Field Theory

The laws of quantum mechanics need to be supplemented with the principle of relativity. We shall ignore gravity and thus general relativity throughout and assume that space-time is ﬂat Minkowski space-time.† The principle of relativity is then that of special relativity, which states that in ﬂat Minkowski space-time, the laws of physics are invariant under the Poincar´egroup. Recall that the Poincar´egroup is the (semi-direct) product of the group of translations R4 and the Lorentz group SO(1, 3),

ISO(1, 3) R4 £< SO(1, 3) (3.1) ≡ Quantum ﬁeld theory is a quantum system, so we have the same deﬁnitions of states in Hilbert space and of self-adjoint operators corresponding to observables. In addition, the requirements that these states and operators transform consistently under the Poincar´e group need to be implemented. In brief,

The states and observables in Hilbert space will transform under unitary represen- • tations of the Poincar´egroup.

The ﬁelds in QFT are local observables and transform under unitary representations • of the Poincar´egroup, which induce ﬁnite dimensional representations of the Lorentz group on the components of the ﬁeld.

Micro-causality requires that any two local ﬁelds evaluated at points that are space- • like separated are causally unrelated and thus must commute (or anti-commute for fermions) with one another.

Before making these principles quantitative, it is helpful to review the deﬁnitions and properties of the Poincar´eand Lorentz groups and to discuss the unitary representations of the Poincar´egroup as well as the ﬁnite-dimensional representations of the Lorentz group.

†Henceforth, we shall work in units where c =¯h = 1.

22 3.1 The Lorentz and Poincar´egroups and algebras In special relativity the speed of light as observed from diﬀerent inertial frames is the same, c. Inertial frames are related to one another by Poincar´etransformations, which in turn may be deﬁned as the transformations that leave the Minkowski distance between two points invariant.‡ This distance is deﬁned by

s2 (x0 y0)2 (~x ~y)2 η (xµ yµ)(xν yν) (3.2) ≡ − − − ≡ µν − − where the Minkowski metric tensor is given by

η diag [1 1 1 1] (3.3) µν ≡ − − − Clearly, the Minkowski distance is invariant under translations by aµ and Lorentz trans- µ formations by Λ ν, xµ x0µ =Λµ xν + aµ R(Λ, a) ν (3.4) yµ → y0µ =Λµ yν + aµ → ν where the Lorentz transformation matrix must satisfy

Λµ η Λν = η ΛT ηΛ= η (Λ−1) ν =Λν (3.5) ρ µν σ ρσ ⇔ ⇔ µ µ The above transformation laws form the Poincar´egroup under the multiplication law,

R(Λ1, a1)R(Λ2, a2)= R(Λ1Λ2, a1 +Λ1a2) (3.6)

Special cases include,

1. a = 0 yields the Lorentz group, which is a subgroup of the Poincar´egroup. The deﬁnition (3.5) is analogous to that of an orthogonal matrix, M tIM = I; for this reason the Lorentz group is denoted by O(1, 3).

2. A further special case of the Lorentz group is formed by the subgroup of rotations, 0 0 i characterized by Λ 0 = 1, Λ i = Λ 0 = 0. Boosts do not by themselves form a subgroup; a boost in the direction 1 leaves x02,3 = x2,3 and

x00 = +x0chτ x1shτ − x01 = x0shτ + x1chτ chτ = √1 v2 (3.7) − − ‡Time and space coordinates are denoted by t and xi, i = 1, 2, 3 respectively. Space coordinates are often regrouped in a 3-vector ~x, while it will be convenient to measure time in distances, using the speed of light constant c via the relation x0 ct. All four coordinates may then be conveniently regrouped into a 4-vector xµ, µ =0, 1, 2, 3. Throughout,≡ we adopt the Einstein convention in which repeated upper and lower indices are summed over and the summation symbol is implicit.

23 3. Λ= I yields the translation group, which is an invariant subgroup of Poincar´e.

The Poincar´ealgebra is gotten by considering inﬁntesimal translations by aµ and in- ﬁnitesimal Lorentz transformations,

Λµ = δµ + ωµ η ωµ + η ων =0 ω + ω =0 (3.8) ν ν ν ⇒ µρ ρ ρν σ ⇒ µν νµ

so that ωµν is antisymmetric, and has 6 independent components. It is customary to deﬁne

the corresponding Lie algebra generators Pµ and Mµν with a factor of i,

i µν µ R(Λ, a) = exp ω Mµν + ia Pµ (3.9) − 2 i I ωµνM + iaµP + (ω2, a2, ωa) ∼ − 2 µν µ O so that the generators Pµ and Mµν are self-adjoint in any unitary representation. The structure relations of the Poincar´ealgebra are given by

[Pµ,Pν] =0 [M ,P ] =+i η P i η P µν ρ νρ µ − µρ ν [M , M ]=+i η M i η M i η M + i η M (3.10) µν ρσ νρ µσ − µρ νσ − νσ µρ µσ νρ

From these relations, it is clear that the generators Mµν form the Lorentz subalgebra, while the generators P µ form the Abelian invariant subalgebra of translations. The Poincar´ealgebra has an immediate and important quadratic Casimir operator, namely a bilinear combination of the generators that commutes with all the generators of the Poincar´ealgebra. This Casimir is the mass square operator

m2 P µP (3.11) ≡ µ which is in a sense the classical deﬁnition of mass. m2 is real, but despite its appearance, it can take positive or negative values.

24 3.2 Parity and Time reversal Parity (or reversal of an odd number of space directions) and time reversal are elements of the Lorentz group with very special signiﬁcance in physics. They are deﬁned as follows,

P : x00 =+x0 x0i = xi x0µ P xµ = x − ≡ µ T : x00 = x0 x0i =+xi x0µ T xµ = x (3.12) − ≡ − µ Notice that the product P T corresponds to x0µ = xµ or Λ= I. − − The role of P and T is important in the global structure (or topology) of the Lorentz group, as may be seen by studying the relation ΛtηΛ = η more closely. By taking the determinant on both sides, we ﬁnd,

(det Λ)2 =1 (3.13) while the µ = 0, ν = 0 component of the equation is

1=(Λ0 )2 (Λi )(Λi ) Λ0 1 (3.14) 0 − 0 0 ⇒| 0| ≥ Transformations with det Λ = +1 are proper while those with det Λ = 1 are improper; − those with Λ0 1 are orthochronous, while those with Λ0 1 are non-orthochronous. 0 ≥ 0 ≤ − Hence the Lorentz group has four disconnected components.

L↑ detΛ=+1 Λo +1 proper orthochronous + o ≥ L↓ detΛ=+1 Λo 1 proper non orthochronous + o ≤ − − L↑ det Λ = 1 Λo +1 improper orthochronous − − o ≥ L↓ det Λ = 1 Λo 1 improper non orthochronous − − o ≤ − − ↑ Since the identity is in L+, this component is the only one that forms a group. Parity and time reversal interchange these components, for example

P : L↑ L↑ + ↔ − T : L↑ L↓ + ↔ − P T : L↑ L↓ (3.15) + ↔ + Each component is itself connected. Any Lorentz transformation can be decomposed as a product of a rotation, a boost, and possibly T and P . It will often be important to consider the complexiﬁed Lorentz group, which is still deﬁned by Λµ η Λν = η , but now with Λ complex. We still have det Λ = 1, but it ρ µν σ ρσ ± is not true that Λo 1. Hence within the complexiﬁed Lorentz group, L↑ and L↓ are | o| ≥ + + connected to one another.

25 3.3 Finite-dimensional Representations of the Lorentz Algebra The representations of the Lorentz group that are relevant to physics may be single-valued, i.e. tensorial or double valued, i.e. spinorial, just as was the case for the representations of its rotation subgroup. In fact, more generally, we shall be interested in single and double valued representations of the full Poincar´egroup, which are deﬁned by

R(Λ , a )R(Λ , a )= R(Λ Λ , a +Λ a ) (3.16) 1 1 2 2 ± 1 2 1 1 2

The representations of the Lorentz group then correspond to having a1 = a2 = 0. To study the representations of O(1, 3), it is convenient to notice that the complexi-

ﬁed algebra factorizes, by taking linear combinations of rotations Ji and boosts Ki with complex coeﬃcients. We deﬁne 1 J M K M (3.17) i ≡ 2 ijk jk i ≡ oi then the structure relations are

[Ji,Jj] = +i ijkJk

[Ji,Kj] = +i ijkKk [K ,K ] = i J (3.18) i j − ijk k Clearly, one should introduce 1 N ± (J i K ) (3.19) i ≡ 2 i ± i so that the commutation relations decouple,

± ± ± + − [Ni , Nj ]= i ijkNk [Ni , Nj ]=0 (3.20)

Thus, each N + generates an SU(2) algebra, considered here with complex coeﬃcients. { i } The ﬁnite dimensional representations of SU(2) are labelled by a half integer j 0, ≥ and will be denoted by (j). It will be useful to have the tensor product formula,

j+j0 (j) (j0)= (l) (3.21) ⊗ 0 l=M|j−j | where the sum runs over integer spacings and the dimension is dim(j)=2j +1. A familiar example is from the addition of angular momentum representations. For example the addition of spin 1/2 to an orbital angular momentum l produces total momentum states with j = l +1/2 and j = l 1/2. − 26 The representation theory of the complexiﬁed Lorentz algebra is obtained as the prod- uct of the representations of both SU(2) factors, each of which may be described by the eigenvalue of the corresponding Casimir operator,

+ + Ni Ni j+(j+ + 1) 1 3 N −N − j (j + 1) j =0, , 1, , 2, (3.22) i i − − ± 2 2 ···

Thus, the ﬁnite-dimensional irreducible representations of O(1, 3) are labelled by (j+, j−). + − Since J3 = N3 + N3 , the highest weight vector of the corresponding rotation group is just the sum j+ + j−, which should be thought of as the spin of the representation. Complex conjugation and parity have the same action on these representations,

∗ (j+, j−) = (j−, j+) ∗ P (j+, j−) = (j−, j+) (3.23)

and thus only the representations (j, j) or the reducible representations (j , j ) (j , j ) + − ⊕ − + can be real – the others are necessarily complex.

Some fundamental examples

1. (0, 0) with spin zero is the scalar; Its parity may be even orodd (scalar-pseudoscalar)

1 1 1 2. ( 2 , 0) is the spin 2 left-handed Weyl 2-component spinor (and (0, 2 )) is the right- handed Weyl 2-component spinor). Both are complex, and transform into one an- other under complex and parity conjugation.

3. ( 1 , 0) (0, 1 ) is the Dirac spinor when the left and right spinors are independent; 2 ⊕ 2 it is the Majorana spinor when the left and right spinors are complex conjugates of one another, so that the Majorana spinor is a real spinor.

4. Tensor products of the Weyl spinors yield higher (j+, j−) representations.

5. ( 1 , 1 )=( 1 , 0) (0, 1 ) with spin 1 and 4 components is a 4-vector, such as the E&M 2 2 2 ⊗ 2 gauge potential and the momentum Pµ.

6. (1, 0) (0, 1) with spin 1, is a real representation under which rank 2 antisymmetric ⊕ tensors transform. Examples are the E& M ﬁeld strength tensor F = F , as µν − νµ well as Mµν themselves.

7. (1, 0) (resp. (0, 1)) with spin 1 and 3 complex components is the so-called self- dual (resp. anti self-dual) antisymmetric rank 2-tensor. To construct it in terms of

27 tensors, one deﬁnes the dual by 1 F˜ F ρσ 0123 = =1 (3.24) µν ≡ 2 µνρσ − 0123

Since F˜˜ = F , the eigenvalues of˜are i, and the eigenvectors are µν − µν ± 1 F ± (F iF˜ ) F˜± = iF ± (3.25) µν ≡ 2 µν ± µν µν ∓ µν The self-dual F + corresponds to the representation (1, 0) (and the anti self-dual F − to (0, 1). Notice that the presence of i in the deﬁnition of the dual, and therefore of ± + − F forces these representations to be complex. In particular Ni and Ni correspond to the self-dual and anti self-dual part of Mµν . The precise relations are, i M ± = M ± = iN ± (3.26) 0j ∓2 jkl kl ∓ j

8. (1, 1) of spin 2 corresponds to the symmetric traceless rank 2 tensor, i.e. the graviton.

9. Generalizing ordinary orbital angular momentum to include time yields another rep- resentation of the Lorentz algebra on functions of the coordinates xµ,

∂ ∂ Lµν i(xµ∂ν xν∂µ) ∂µ , ~ (3.27) ≡ − ≡ ∂xµ ≡ ∂t ∇!

A further generalization is to representations that include spin;

M L + S (3.28) µν ≡ µν µν

where the spin representation Sµν satisﬁes the Lorentz structure relations by itself,

while Sµν commutes with Pµ and Lµν . Supplementing this algebra with Pµ = i∂µ yields a representation of the full Poincar´ealgebra.

28 3.4 Unitary Representations of the Poincar´eAlgebra Next, the unitary representations of the Poincar´ealgebra are reviewed and constructed. Recall the structure relations of the algebra,

[Pµ,Pν] =0 [M ,P ] =+iη P iη P µν ρ νρ µ − µρ ν [M , M ]=+iη M iη M iη M + iη M (3.29) µν ρσ νρ µσ − µρ νσ − νσ µρ µσ νρ In any unitary representation, the generators M and P will have to be realized as self- adjoint operators. Constructing the representations of the Poincar´ealgebra is a bit more tricky than those of the Lorentz algebra, because the Lorentz algebra is semi-simple, but the Poincar´ealgebra has the Abelian invariant subalgebra of translations, and is therefore not semi-simple. + + What are its representations? The Casimirs of the Lorentz algebra NiNi and Ni Ni µ do not commutue with Pµ any longer. However, PµP is a relativistic invariant, and hence a Casimir of Poincar´e. If we can ﬁnd another 4-vector that commute with all P ’s, its square will also be a Casimir. The Pauli-Lubanski vector is deﬁned by 1 W µ µνρσP M (3.30) ≡ 2 ν ρσ

µ ν µ It commutes with momentum, [W ,P ] = 0 and its square W Wµ commutes with the entire Poincar´ealgebra and is therefore a (quartic) Casimir operator for the Poincar´e

algebra. Now remember that the most general expression for Mµν was

M = x P x P + S (3.31) µν µ ν − ν µ µν so that the angular momentum part cancels out and we are left with 1 W µ = µνρσP S (3.32) 2 ν ρσ µ µ The Casimirs of the Poincar´egroup are thus PµP and WµW , and the states will be 3 labelled by Pµ and W . Wµ generates an SU(2) type algebra for ﬁxed Pµ:

µ ν µναβ [W , W ]= i PαWβ (3.33)

which is the so-called little group. The basic idea is to ﬁx P µ and then to obtain the representations of the little group.

29 One-particle state representations • We apply the method above to the construction of the (unitary) one-particle state representations, which are the most basic building blocks of the Hilbert space of a QFT.

We choose to diagonalize the components of the momentum operator Pµ since they mutualy 2 µ commute and commute with the mass operator m = P Pµ. We denote the eigenstates by p, ı , where p denotes the eigenvalues of the operator P and the label i speciﬁes whatever | i µ µ remaining quantum numbers,

P p, i = p p, i (3.34) µ| i µ| i One deﬁnes a one particle state as a representation in which the range of the label i is discrete. If one superimposes two one particle states, the index would include relative momenta, and this index would be continuous. At ﬁxed M 2, the range of i is ﬁnite, both in QFT and in string theory. To obtain the action of the Lorentz group, we proceed as follows. First, we list the transformations of the Poincar´egenerators under Poincar´etransformations,

(Λ, a)M (Λ, a)−1 = (Λ−1) α(Λ−1) β(M a P + a P ) U µν U µ ν αβ − α β β α (Λ, a)P (Λ, a)−1 = (Λ−1) αP (3.35) U µU µ α Applying these relations to one particle states, one establishes that the state (Λ, 0) p, i U | i indeed has momentum Λp, as expected, P µ (Λ, 0) p, i =Λµ pν (Λ, 0) p, i , and therefore U | i ν U | i may be decomposed onto such states,

(Λ, 0) p, i = C (Λ,p) Λp, j (3.36) U | i i,j | i Xj To ﬁnd this action and understand it, we specify a reference momentum kµ, for given M 2, so that pµ is a Lorentz transform of kµ,

µ µ ρ 2 2 p = L ρ(p)k k = m (3.37)

Thus, the states p, i may then be reconstructed as follows, | i p, i = (L(p), 0) k, i (3.38) | i U | i Once this construction has been eﬀected, the action of general Lorentz transformations may be obtained as follows,

(Λ, 0) p, i = (Λ, 0) (L(p), 0) k, i U | i U U | i = (L(Λp), 0) (W, 0) k, i (3.39) U U | i 30 Deﬁning now the composite transformation

(W, 0) = (L(Λp), 0)−1 (λ, 0) (L(p), 0) (3.40) U U U U which represents the composite Lorentz transformation W ,

W = L(λ(p))−1ΛL(p) Wk = L(Λp)−1ΛL(p)k = L(Λp)−1Λp = k (3.41)

Therefore, W leaves the reference momentum kµ invariant. The group of all such trans- formations is the little group. Thus, given the reference momentum k, the unitary repre- sentation of the one particle state is classiﬁed by the representation of the little group.

µ 2 µ 1. P Pµ = m > 0 : These are the massive particle states; k =(m, 0, 0, 0). It follows from the explicit form of W µ that W µW = m2S~2 = m2s(s + 1), where µ − − s is the spin of the representation of the little group, and it takes the usual values 1 s =0, , 1,... (3.42) 2

µ µ 2. PµP = 0 : These are the massless particle states; k =(κ, κ, 0, 0). µ µ µ It follows from the explicit form of W that WµW = 0. Since we also have PµW = 0, it follows that W~ P~ = W~ P~ and hence the vectors W~ and P~ must be colinear, · ±| || | W~ = λP~ and thus

W µ = λP µ (3.43)

µ 0 0 Again, using the explicit form of W , we have W = PiSi = λP and we see that λ is the projection of spin onto momentum, i.e. helicity which takes values, 1 λ =0, , 1,... (3.44) ±2 ± Hence any massless particles of s = 0 has 2 degrees of freedom, photon 1, neutrino 6 ± 1 , graviton 2. ± 2 ± µ µ 3. P Pµ < 0 : These are the tachyonic particle states; k = (0, m, 0, 0). These representations are unitary, and are acceptable as free particles. As soon as interactions are turned on, however, they cause problems with micro-causality and it is generally believed that tachyons are unacceptable in physical theories.

Note that all these unitary representations are inﬁnite dimensional (except for the triv- ial representation). Then there are of course also all the non-unitary representations, which we do not consider here. A standard reference is E.P. Wigner, “Unitary Representations of the Inhomogeneous Lorentz Group,” Ann. Math. 40 (1939) 149.

31 3.5 The basic principles of quantum ﬁeld theory

A summary is presented of the fundamental principles of relativistic quantum ﬁeld theory. By analogy with quantum mechanics, these principles do not refer to any speciﬁc dynamics (except for the fact that the dynamics is invariant under Poincar´esymmetry). Thus, the principles generalize those of quantum mechanics to include the principles of relativity.

1. States of QFT are vectors in a Hilbert space . H 2. States transform under unitary representations of the Poincar´egroup, which will be denoted by (Λ, a), U state state0 = (Λ, a) state (3.45) | i → | i U | i

3. There exists a unique ground state (the vacuum), usually denoted by 0 , which is | i the singlet representation of the Poincar´egroup,

(Λ, a) 0 = 0 (3.46) U | i | i

4. The fundamental observables in QFT are local quantum ﬁelds, which are space- time dependent self-adjoint operators on . They are generically denoted by φ (x); H j speciﬁc notations for the ﬁelds most relevant for physics will be introduced later.§

5. Throughout, the number n of independent ﬁelds φ (x), j =1, , n will be assumed j ··· to be FINITE. Generalizations with inﬁnite numbers of ﬁelds can be constructed (for example string theory), but only at the cost of severe complications, which usually go outside the framework of standard QFT.

6. Poincar´etransformations on the ﬁelds are realized unitarily in , as follows, H n φ0 (x) (Λ, a)φ (x) (Λ, a)† = S (Λ−1)φ (Λx + a) (3.47) j ≡ U j U jk k kX=1 Since the number of ﬁelds is assumed ﬁnite, S(Λ−1) must be a ﬁnite-dimensional representation of the Lorentz group. Except for the trivial one, such representations are always non-unitary. An important special case of this relation is when Λ = 1,

§Note that the operators or observables are thus time-dependent, and the states will be time- independent. This formulation is usually referred to as the Heisenberg formulation of quantum mechanics, as opposed to the Schr¨odinger formulation in which observables are time-independent but states are time- dependent.

32 which yields the behavior of observables under translations (this includes under time- translation, namely dynamics),

µ µ φ0 (x) e−ia Pµ φ (x)eia Pµ = φ (x + a) (3.48) j ≡ j j since (I, a) = exp iaµP . The transformation law when Λ = I will be examined U {− µ} 6 in greater detail later.

7. Microscopic Causality or local commutativity states that any two observables φJ and

φk that have no mutual causal contact must be simultaneously observable and must therefore commute with one another,¶

[φ (xµ),φ (yµ)]=0 if (x y)2 < 0 (3.49) j k − Classically, as no relativistic signal can connect these two observables, it should better be possible to assign independent initial data. Quantum mechanically, it should be possible to measure them simultaneously to arbitrary precision and this independently at xµ and yµ.

8. There is a relation between states in and ﬁelds. From the ﬁelds in QFT, it is H possible to build up the Hilbert space. The simplest states (besides the vacuum) in Hilbert space are the 1-particle states, denoted by p, i . For every 1-particle | i species i, there must be a corresponding ﬁeld φi(x) that produces this species from the vacuum state,

0 φ (x) p, i =0 (3.50) h | i | i 6

9. Symmetries other than the Poincar´egroup will be realized by unitary transformations on Hilbert space which induce ﬁnite-dimensional transformations on the ﬁelds,

φ0 (x) (g)φ (x) (g)† = S(g−1) kφ (gx) (3.51) j ≡ U j U j k Xk When g has no action on x, namely gx = x, the transformations generate internal symmetries; examples are ﬂavor, color, isospin etc. On the other hand, if gx x, the 6 transformations generate space-time symmetries; examples are Poincar´einvariance, scale and conformal symmetries, parity, supersymmetry. Time-reversal is realized by an anti-unitary transformation.

¶Fermionic ﬁelds will anti-commute.

33 3.6 Transformation of Local Fields under the Lorentz Group Fields may be distinguished by the diﬀerent ﬁnite-dimensional representations of the Lorentz group S(Λ−1) under which they transform. It is convenient to decompose these representations into a direct sum of irreducible representations, whose study we now initi- ate. We apply the results of 3.3 to the general transformation rule of (3.47), specialized § to Lorentz transformations. It is convenient to let Λ Λ−1 and x Λx, so that → → n 0 0 k 0 φj(x )= Sj (Λ)φk(x) x =Λx (3.52) kX=1 When the transformation is continuous (we shall treat Parity and Time-reversal separately) we may describe the transformation inﬁnitesimally,

Λµ = δµ + ωµ + (ω2) ν ν ν O φ0 (x) = φ (x)+ δφ (x)+ (ω2) j j j O i S(I + ω) = I ωµνS + (ω2) (3.53) − 2 µν O

Combining these ingredients to compute δφj, we ﬁnd i φ(x)+ δxµ∂ φ(x)+ δφ(x)= φ(x) ωµνS φ(x)+ (ω2) (3.54) µ − 2 µν O whence the transformation law naturally involves the total Lorentz generator L + S, i δφ(x)= ωµν(L + S )φ(x) (3.55) −2 µν µν

1. The scalar ﬁeld The scalar has spin 0, corresponding to the representation (0, 0) in the notation of 3.3. 0 ↑ § Thus, φ (Λx)= φ(x) for all Lorentz transformations in L+. Inﬁnitesimally, we have i δφ(x)= ωµνL φ(x) (3.56) −2 µν All irreducible representations are 1-dimensional; a single scalar ﬁeld is denoted by φ.

2. The vector ﬁeld

Vector ﬁelds are generally denoted by Aµ, such as for example the electro-magnetic potential. The transformation law of a vector ﬁeld is

0 0 ν Aµ(x )=Λµ Aν(x) (3.57)

34 which corresponds to the following inﬁnitesimal transformation law,

i δA (x)= A0 (x) A (x)= ωρσL A (x)+ ω ν A (x) (3.58) µ µ − µ −2 ρσ µ µ ν

(Using the transformation law for a scalar φ, it is easily checked that the derivative of a

scalar ∂µφ transforms as a vector with the above transformation law.) The explicit form of the vector representation matrices may be deduced from this formula,

(S ) β = i η δ β i η δ β (3.59) µν α µα ν − να µ

1 1 which is characteristic of the 2 , 2 representation of the Lorentz group. 3. General tensor representations

These representations may be built up by taking the tensor product of vector repre- sentations using the tensor product formula,

0 j±+j± 0 0 (j+, j−) (j+, j−)= (s+,s−) (3.60) ⊗ 0 s±=X|j±−j±|

The spins s = s+ + s− are all integer. A rank n tensor Bµ1···µn transforms as

B0 (Λx)=Λ ν1 Λ νs B (x) (3.61) µ1···µn µ1 ··· µn ν1···νn It is consistent with Lorentz transformations to symmetrize, anti-symmetrize or take the trace of tensor ﬁelds. For totally symmetric tensor, the spin and the rank coincide, n = s, but upon anti-symmetrization, n>s in general. The representation (1, 1) corresponds to the graviton. Fields of spin higher than 2 will never be needed in QFT, and in fact we shall always specialize to ﬁelds of spin 1. ≤ 4. The Dirac spinor ﬁeld

The Dirac ﬁeld, usually denoted ψ(x), is a 4-component complex column matrix, whose transformation law is deﬁned to be

(Λ, a)ψ(x) (Λ, a)† = S(Λ−1)ψ(Λx + a) (3.62) U U where S(Λ) = R(Λ, 0) is the spinor representation characterized by ( 1 , 0) (0, 1 ) in the 2 ⊕ 2 classiﬁcation of Lorentz representations. To obtain an explicit construction of S(Λ), we proceed as follows.

35 First, recall that the 2-dimensional spinor representation of the rotation group is re- alized in terms of the 2 2 Pauli matrices, σi, i = 1, 2, 3, deﬁned to satisfy the relationk × σi, σj =2δij. It is conventional to take the following basis, { } 0 1 0 i 1 0 σ1 = σ2 = σ3 = (3.63) 1 0 i −0 0 1 − The representation generators are Si = σi/2, and satisfy the standard rotation algebra relations [Si,Sj]= iijkSk. The spinor representations of the Lorentz algebra are realized analogously in terms of the 4 4 Dirac matrices γµ, deﬁned to satisfy the Cliﬀord-Dirac algebra, × γµ,γν =2ηµν (3.64) { } The 4-dimensional Dirac spinor representation is deﬁned by the following generators, i S [γ ,γ ] (3.65) µν ≡ 4 µ ν which satisfy the Lorentz algebra

[S ,S ]=+i η S i η S i η S + i η S (3.66) µν ρσ νρ µσ − µρ νσ − νσ µρ µσ νρ This may be veriﬁed by making use of the Cliﬀord-Dirac algebra relations only. As for any representation of the Lorentz group, (3.35) holds, which in this case may be expressed as

−1 −1 α −1 β S(Λ)SµνS(Λ) = (Λ )µ (Λ )ν Sαβ (3.67)

Furthermore, from the relation

[S ,γ ]= i η γ i η γ (3.68) µν ρ νρ µ − µρ ν −1 −1 ν it follows that S(Λ)γµS(Λ) = (Λ )µ γν or simply that the γµ matrices are invariant, provided both its “vector” and “spinor” indices are transformed,

ν −1 Λµ S(Λ)γνS(Λ) = γµ (3.69)

The γ-matrices are the Clebsch-Gordon coeﬃcients for the tensor product of two Dirac representations onto the vector representation.

k A, B AB + BA is deﬁned to be the anti-commutator of A and B. { }≡ 36 The Dirac spinor representation is reducible, as may be seen from the existence of the 5 chirality matrix γ = γ5, which is deﬁned by

i γ5 = γ = γµγνγργσ = iγ0γ1γ2γ3 (3.70) 5 4! µνρσ

Clearly, we have (γ5)2 = I and tr(γ5) = 0, so that γ5 is not proportional to the identity 5 5 matrix. The matrix γ anti-commutes with all γµ and therefore γ commutes with Sµν,

γ5,γ = [γ5,S ]=0 (3.71) { µ} µν Thus, by Shur’s lemma, the representation S is reducible, since its generators commute with a matrix γ5 that is not proportional to the identity matrix. (Notice that the γµ matrices do form an irreducible representation of the Cliﬀord algebra.) A basis for all 4 4 matrices is given by the following set × I, γµ, γµν, γµνρ, γµνρσ (3.72)

where γµ1···µn is the product γµ1 γµn , completely antisymmetrized in its n indices. Some ··· of these combinations may be reexpressed in terms of already familiar quantities,

γµν = 2i Sµν − γµνρ = i µνρσγ5γ − σ γµνρσ = i µνρσγ5 (3.73) − 5. The Weyl spinor ﬁelds The irreducible components of the Dirac representation are a - chirality or left Weyl 1 1 spinor ( 2 , 0) and an independent + chirality or right Weyl spinor (0, 2 ), corresponding to the 1 and +1 eigenvalues of the chirality matrix γ5. Ina chiral basis where γ5 is diagonal, − we have the following convenient representation of the γ-matrices,

µ 5 I2 0 µ 0 σ γ = − γ = µ (3.74) 0 +I2 σ¯ 0

where I2 is the identity matrix in 2 dimensions, and

σµ =(I , σi)σ ¯µ =(I , σi) (3.75) 2 2 − The Dirac representation matrices S(Λ) as well as the Dirac ﬁeld ψ(x) decompose into the

Weyl spinor representation matrices SL(Λ) and SR(Λ) and the Weyl spinor ﬁelds ψL(x)

37 and ψR(x) as follows, S (Λ) 0 ψ (x) S(Λ) = L ψ(x)= L (3.76) 0 SR(Λ) ψR(x) where the Weyl representation matrices are given by

i + µν + i SL(Λ) = exp (ω ) Mµν = exp ~σ (~ω + i~ν) − 2 2 · i − µν − i SR(Λ) = exp (ω ) Mµν = exp ~σ (~ω i~ν) (3.77) − 2 2 · − Here, ω± and M ± are the self-dual and anti self-dual parts of ω and M respectively, and ~ω and ~ν are real 3-vectors representing rotations and boosts respectively.

6. Equivalent representations

If Sµν in the Dirac representation satisﬁes the structure relations of the Lorentz algebra, then transposition and complex conjugation automatically also produce representations of the same dimensions, with matrices St and S∗ . These representations must be equiv- − µν − µν alent to the Dirac representation and hence related by conjugation. These conjugations reﬂect the discrete symmetries and (to be discussed shortly), C P St = C−1S C C − µν µν S∗ = B−1S B (3.78) P − µν µν 7. The Majorana spinor ﬁeld A Majorana spinor corresponds to a real representation ( 1 , 0) (0, 1 ) where the two 2 ⊕ 2 Weyl spinors are complex conjugates of one another. Complex conjugation by itself de- pends upon the basis chosen, but charge conjugation, to be deﬁned in the subsequent subsection, is a Lorentz invariant operation. Thus, the proper requirement for a spinor to be “real” is provided by the Majorana spinor ﬁeld condition, ψc(x)= ψ(x) (ψc(x))c = ψ(x) (3.79) A Majorana spinor is equivalent to its left ﬁeld component (or its right ﬁeld component).

8. The spin 3/2 ﬁeld 3 3 1 Spin 2 can be either 2 , 0 or 2 , 1 and their charge conjugates. Only the latter is physically signiﬁcant. It can be gotten from 1 1 1 1 1 , , 0 = 1, 0, (3.80) 2 2 ⊗ 2 2 ⊕ 2 The Rarita-Schwinger ﬁeld ψµ precisely corresponds to the above representation plus its complex conjugate. To project out the purely spin (1, 1/2) (1/2, 1) part, the γ-trace ⊕ µ must be removed which may be achieved by suplementary condition γ ψµ = 0.

38 3.7 Discrete Symmetries We group together here the transformations under the discrete symmetries , and P T charge conjugation of the basic ﬁelds. C

3.7.1 Parity

One distinguishes two types of scalars under parity

φ(x) † =+φ(P x) scalar P P φ(x) † = φ(P x) pseudo scalar (3.81) P P − − This distinction is important as the particles π± and π0 are for example pseudo-scalars,

while the He4 nucleus is a scalar. One also distinguished two types of vectors under parity,

A (x) † =+Aµ(P x) vector P µ P A (x) † = Aµ(P x) axial vector (3.82) P µ P − Again, this distinction is very important; the weak interactions are mediated by a super- position of a vector and an axial vector. Finally, under parity, the Dirac spinor transforms as follows,

ψ(x) † = eiθP γ0ψ(x) (3.83) P P which amounts to an interchange of left and right spinor components, and θP is an arbitrary angle. (Note that a theory invariant under parity will have to contain both left and right spinors of each species.)

3.7.2 Time reversal

Time reversal symmetry is the only case which cannot be realized by a unitary transfor- mation in Hilbert space; instead it is realized by an anti-unitary transformation , which T has the property of changing the sign of the number i,

i = i (3.84) T − T Its action on the scalar, vector and Dirac sprinor is given by

φ(x) † = φ(T x) T T A (x) † = Aµ(T x) T µ T − ψ(x) † = iCγ5ψ(T x) (3.85) T T

39 3.7.3 Charge Conjugation From the simple relation (σi)∗ = σ2σiσ2, it follows that − ∗ 2 2 SL(Λ) = σ SR(Λ)σ (3.86)

c As a result, the charge conjugates ψL,R of ψL,R, deﬁned by ψ (x) † = ψc (x) σ2ψ∗ (x) C L C L ≡ − L ψ (x) † = ψc (x) +σ2ψ∗ (x) (3.87) C R C R ≡ R

transforms under SR(Λ) and SL(Λ) respectively. Charge conjugation as deﬁned above, was formulated in a speciﬁc basis of σ-matrices, which is why σ2 played a special role. It is possible to produce a basis independent deﬁnition of charge conjugation as follows. For any basis, the charge conjugation matrix C and the charge conjugate of a Dirac spinor are deﬁned by

γt = C−1γ C ψc Cγt ψ∗ = Cψ¯t (3.88) µ − µ ≡ 0 It follows that [γ5,C] = 0. In the chiral basis, the transposition equations read

γ = C−1γ C 0,2 − 0,2 −1 γ1,3 = +C γ1,3C which may be solved and we have σ2 0 C = − 2 (3.89) 0 +σ up to a multiplicative factor, which one may choose to be unity. With this factor, the preceding deﬁnition of charge conjugation on left and right spinors is recovered.

3.8 Signiﬁcance of ﬁelds in terms of particle contents spin 0 possible Higgs ﬁelds, pions etc. spin 1 (vector ﬁeld) gauge ﬁelds, Aµ, the ρ-ﬁeld (anti-symmetric tensor) not used at present spin 1/2 all the observed fermions Weyl massless neutrinos Dirac massive fermions e−, e+,... Majorana possibly massive neutrino spin 3/2 possibly the gravitino (supergravity) spion 2 graviton

40 3.9 Classical Poincar´einvariant ﬁeld theories The point of departure for the construction of Poincar´einvariant local quantum ﬁeld theories will almost always be a classical action for a set of classical ﬁelds, which will have to be local as well. Maxwell theory is an example of such a classical ﬁeld theory where the fundamental ﬁeld is the gauge vector potential. Bosonic ﬁelds are ordinary functions of space-time; fermionic ﬁelds, however, will have to be represented by anti-commuting or Grassmann valued functions. These will be discussed in the next chapter.

Classical bosonic ﬁelds φi(x) (not necessarily scalars) are real or complex valued or- dinary functions of x. We assume that the number of ﬁelds is ﬁnite, i = 1, , n. The ··· fundamental actions in Nature appear to involve dependence on time-derivatives (and by Lorentz invariance also space-derivatives) only of ﬁrst order. Therefore, the most general bosonic action of this type takes the form,

4 S[φi]= d x = φi(x), ∂µφi(x) (3.90) Z L L L The Euler-Lagrange equations or ﬁeld equations determine the stationnary ﬁeld conﬁgu-

rations of S[φi], and are given by δS[φ] ∂ ∂ = L (x) ∂µ L (x)=0 (3.91) δφi(x) ∂φi − ∂(∂µφi) µ The canonical momentum πj conjugate to φj, and the Hamiltonian are deﬁned by

µ ∂ 3 0 πj (x) L (x) H d x πj ∂0φj (3.92) ≡ ∂(∂µφj) ≡ −L Z Xj As usual, we have Poisson brackets deﬁned by

3 δA δB δA δB A, B = d ~x 0 0 (3.93) { } " δπj (t, ~x) δφj(t, ~x) − δφj(t, ~x) δπj (t, ~x)# Xj Z In particular, we have π0(t, ~x),φ (t, ~x0) = δ δ3(~x ~x0). { j k } jk − A symmetry of the ﬁeld equations is a transformation of the ﬁelds φj such that every solution is transformed into a solution of the same ﬁeld equations. For continuous sym- metries, the usual inﬁnitesimal criterion may be applied. The inﬁnitesimal transformation δφ (x)= φ0 (x) φ (x) is a symmetry provided j j − j δ = ∂ Xµ (3.94) L µ without the use of the ﬁeld equations. If so, Noether’s theorem may be applied, which implies the existence of a conserved current, jµ πµδφ Xµ ∂ jµ =0 (3.95) ≡ j j − µ Xj 41 and thus of a time independent charge

Q d3x j0(t, ~x) (3.96) ≡ Z Furthermore, the transformation is recovered by Poisson bracketing with the charge,

δφ = Q, φ (3.97) j { j} Poincar´ecovariance of the ﬁeld equations and Poincar´einvariance of the action requires that the Lagrangian density transform as a scalar. As a special case, translation invari- L ance requires that have no explicit xµ dependence. Assuming Poincar´einvariance of a L Lagrangian , the conserved currents are as follows. L Translations • δφ = aµ∂ φ Xµ = aµ j µ j ⇒ L µ µ ν j = θ νa

where the conserved canonical energy-momentum-stress tensor – or simply the stress tensor – is deﬁned by

θ π ∂ φ η (3.98) µν ≡ jµ ν j − µν L Xj Lorentz transformations • κλ 1 1 i µ µν δφj = ω xκ∂λφj xλ∂κφj Sκλφj X = ω xν 2 − 2 − 2 ⇒ − L i jµ = πµωκλ(x ∂ φ S φ )+ ωµκx j κ λ j − 2 κλ j κL Xj i = ωκλx θµ π µωκλS φ κ λ − 2 j κλ j Xj = ωκλx µ (3.99) κT λ The conserved and symmetric improved stress tensor is given by

µν µν µνρ T = θ + ∂ρB i Bµνρ = Bνµρ = Bµρν = πµSνρφ (3.100) − − 2 j j Xj A more universal interpretation of the stress tensor is that it is the quantity in a classical (and quantum) ﬁeld theory that couples to variations in the space-time metric and thus to gravity (cfr Einstein’s equations of gravity).

42 4 Free Field Theory

Free ﬁeld theory corresponds to linear equations of motion, possibly with an inhomoge- neous driving source, and thus to an action which is quadratic in the ﬁelds. In itself, free ﬁeld theory is not so interesting, because no interactions take place at all. Free ﬁeld theory oﬀers, however, an excellent testing ground on which to realize the basic principles of QFT, and many of the methods needed to deal with interacting ﬁeld theory may be reduced to problems in a theory that is eﬀectively free. We shall deal successively with scalar, spin 1 and spin 1/2 free ﬁeld theories. As stated from the outset, we restrict to Lagrangians with at most ﬁrst time derivatives on the ﬁelds, and with Poincar´einvariance.

4.1 The Scalar Field The most general classical Poincar´einvariant quadratic Lagrangian density and action of a single real scalar ﬁeld φ(x) is given by

1 µ 1 2 2 4 0(φ, ∂µφ)= ∂µφ∂ φ m φ S0[φ]= d x 0(φ, ∂µφ) (4.1) L 2 − 2 Z L The Euler Lagrange equations are

2φ + m2φ =0 2 ∂ ∂µ (4.2) ≡ µ The conjugate momentum π(x) is the time component of the ﬁeld πµ(x),

∂ πµ L = ∂µφ π = π0 = ∂0φ (4.3) ≡ ∂(∂µφ)

The Hamiltonian is

3 3 1 2 1 2 1 2 2 H0 d x[π∂0φ 0]= d x π + (~ φ) + m φ (4.4) ≡ Z −L Z 2 2 ∇ 2 The classical Poisson brackets are

φ(t, ~x), φ(t, ~y) = 0 { } π(t, ~x), π(t, ~y) = 0 { } π(t, ~x), φ(t, ~y) = δ3(~x ~y) (4.5) { } − It is straightforward to solve the classical ﬁeld equations by using either Fourrier transfor- mation techniques, or by solving the corresponding diﬀerential equations. We shall not do this here and directly pass to the quantization of the system; solving for the Heisenberg ﬁeld equations instead.

43 Canonical quantization • Canonical quantization associates with the ﬁelds φ and π operators that obey the canonical equal time commutation relations,

[φ(t, ~x), φ(t, ~y)] = [π(t, ~x), π(t, ~y)]=0 [π(t, ~x), φ(t, ~y)] = iδ(3)(~x ~y) (4.6) − − The Heisenberg ﬁeld equations for the φ-ﬁeld are obtained as follows

∂φ 1 i (x) = [φ(x),H] = d3~y φ(t, ~x), π2(t, ~y)+(~ φ(t, ~y))2 + m2φ2(y) ∂t 2 Z ∇ 1 h i = d3~y φ(t, ~x), π2(t, ~y) = iπ(x) (4.7) 2 Z h i while for the π-ﬁeld they are derived analogously and found to be

∂π i (x) = [π(x),H]= i∆φ(t, ~x) im2φ(t, ~x) (4.8) ∂t − Therefore, the Heisenberg ﬁeld equations for the quantum ﬁeld φ(x) are the same as the classical ﬁeld equations, but now hold for the operator φ(x),

(2 + m2)φ(x)=0 (4.9)

Although canonical quantization is not a manifestly Poincar´ecovariant procedure, the ﬁeld equation are nonetheless Poincar´einvariant. The Heisenberg equations are solved by Fourrier transformation in ~x only,

3~ d k ~ i~k·~x φ(t, ~x) = 3 a(t, k) e (4.10) Z (2π) where the time-evolution of a(t,~k) is deduced from the ﬁeld equations and is given by

~ 2 ~ ~ 2 2 a¨(t, k)+ ωka(t, k)=0 ωk k + m (4.11) ≡ q which can be solved,

1 a(t,~k)= a(~k)e−iωkt +˜a(~k)eiωkt (4.12) 2ω k The reality condition on the classical ﬁeld translates into the self-adjointness condition on the ﬁeld φ† = φ, which in turn requires thata ˜(~k)= a†( ~k). Once this condition has been − 44 satiﬁed, we may write the complete solution in a manifestly covariant manner,

3 d k ~ −ik·x † ~ ik·x φ(x)= 3 a(k)e + a (k)e (4.13) (2π) 2ωk Z h i µ 0 0 µ where k = (k ,~k), k = ωk, and henceforth, we shall us the notations k x = kµx and 2 µ · k = kµk . While this expression may not look Lorentz covariant, it actually is. This may be seen by recasting the measure in terms of a 4-dimensional integral,

3 4 d k d k 2 2 0 3 = 4 2π δ(k m )θ(k ) (4.14) (2π) 2ωk (2π) − and noticing now that it is invariant under orthochronous Lorentz transformations. Commutation relations on φ(x) and π(x) imply the commutation relations between the a’s, which are found to be given by a:

[a(~k), a(~k0)] = 0 [a†(~k), a†(~k0)] = 0 [a(~k), a†(~k0)] = (2π)3 2ω δ(3)(~k ~k0) (4.15) k − Using these results, an important consistency check of the basic principles announced earlier may be carried out. The commutator of two φ ﬁelds is,

3 0 d k −ik·(x−x0) ik·(x−x0) [φ(x), φ(x )] = 3 e e (4.16) Z (2π) 2ωk − which is a relativistic invariant, and depends only on (x y)2. If(x y)2 < 0, then one − − may make x0 y0 = 0 by a Lorentz transformation. By the equal time commutation − relations, this commutator must thus vanish. This checks out because the integral also vanishes, since the integration measure is even under ~k ~k, while the integrand is odd. → − Thus, we have the relativistic invariant equation,

[φ(x), φ(x0)] = 0 when (x x0)2 < 0 (4.17) − which expresses the property of microcausality of the φ(x) ﬁeld.

A Comment on UV convergence • When the classical ﬁeld equations are solved, the Fourrier coeﬃcients a(~k) are de- termined by the initial conditions on the ﬁeld. If the ﬁelds φ and π at initial time are well-behaved, then the integration over momenta ~k will be convergent. When the Fourrier coeﬃcients a(~k) are operators however, UV convergence of the ~k integral may be problematic, and indeed will be ! In those cases, the integral should be deﬁned with a UV regulator. We shall encounter the need for regulators soon.

45 Fock space construction of the Hilbert space • The states of the QFT associated with the φ ﬁeld are vectors in a Hilbert space, which we shall now construct. To this end, we view the operator a(~k) as the annihilation operator for a φ-particle with momentum ~k; similarly, its adjoint a†(~k) is the creation operator for a φ-particle with momentum ~k. A basis of states is deﬁned as follows,

The ground state 0 or vacuum a(~k) 0 = 0 for all ~k; • | i | i The 1-particle state ~k ~k = a†(~k) 0 ; • | i | i | i The n-particle state ~k , ,~k ~k , ,~k = a†(~k ) a†(~k ) 0 . • | 1 ··· ni | 1 ··· ni 1 ··· n | i The ground state is unique and may be normalized to 1, while particle states form a continuous spectrum. Their continuum normalization follows by the use of the canonical commutation relations, and we have for example,

0 0 =1 ~k ~k0 = (2π)32ω δ(3)(~k ~k0) (4.18) h | i h | i k − We can now check another basic principle : the one-particle state is obtained from the vacuum by applying the ﬁeld φ,

µ ~k φ(x) 0 = eikµx =0 (4.19) h | | i 6 The number operator, N is deﬁned as follows,

3 d k † ~ ~ N 3 a (k)a(k) (4.20) ≡ Z (2π) 2ωk and obeys the relations

[N, a(~k)] = a(~k) N 0 =0 (4.21) † − † |~i ~ ~ ~ ( [N, a (~k)]=+a (~k) ( N k1, , kn = n k1, , kn | ··· i | ··· i As a result of these commutation relations, N counts the number of particles in state.

Energy, Momentum and Angular Momentum • The Hamiltonian, momentum and angular momentum may all be derived from the stress tensor. The classical stress tensor for a scalar is given by

T µν = ∂µφ∂ν φ ηµν (4.22) c − L and hence we have the translation and Lorentz generators

µ 3 0µ µν 3 µ 0ν ν 0µ P = d xTc M = d x(x Tc x Tc ) (4.23) Z Z − 46 Upon quantization, classical ﬁelds are replaced with quantum ﬁeld operators. Carrying out this substitution for example on the Hamiltonian and momentum, 1 1 1 H = d3x π2 + (~ φ)2 + m2φ2 Z 2 2 ∇ 2 P~ = d3xπ ~ φ (4.24) Z ∇ Using the mode expansion for φ and π, d3k π(x)= ia(~k)e−ik·x + ia†(~k)eik·x (4.25) 2(2π)3 − Z h i one ﬁnds d3k ω H = k a†(~k)a(~k)+ a(~k)a†(~k) (2π)32ω 2 Z k h i d3k ~k P~ = a†(~k)a(~k)+ a(~k)a†(~k) (4.26) (2π)32ω 2 Z k h i Both operators commute with the number operator. Therefore, in free ﬁeld theory the number of particles in any state is conserved, as it was in non-relativistic quantum me- chanics. In a fully interacting theory, particle number will of course no longer be conserved.

The energy and momentum of the vacuum • To evaluate H 0 and P~ 0 , we use the deﬁning property of the vacuum state, a(~k) 0 = | i | i | i 0 for the ﬁrst terms of the integrands of H and P~ . The second terms, however, require a re-ordering of the operators before use can be made of a(~k) 0 = 0, | i a(~k)a†(~k) = a†(~k)a(~k) + [a(~k), a†(~k)] 3 (3) = 2ωk(2π) δ (0) (4.27) Here, a problem is encountered as δ(3)(0) = . Actually, the quantity (2π)3δ(3)(0) V ∞ ≡ should be interpreted as the volume of space, as may be seen by considering a Fourrier transform of 1 in ﬁnite volume V , and then letting ~k 0. The fact that the energy of → the vacuum is proportional to the volume of space makes sense because the vacuum is homogeneous and any vacuum energy will be distributed uniformly throughout space. For the momentum, the integrand is now odd in ~k and therefore vanishes, P~ 0 = 0. | i For the vacuum energy, however, a more serious problem is encountered, as may be seen by collecting the remaining contributions, d3k 1 H 0 = V 3 ωk 0 (4.28) | i Z (2π) 2 | i This quantity is quartically divergent. The physical origin of this inﬁnity is that there is one oscillator for each momentum mode, contributing more energy to the vacuum as momentum is increased.

47 Actually, we made a mistake in assuming that the quantum Hamiltonian was uniquely determinedby the classical Hamiltonian. Indeed, to the quantum Hamiltonian, terms of the form [π(t, ~x),φ(t, ~x)] could be added which would vanish in the classical Hamiltonian. Such terms contribute a constant energy density and are immaterial for the time-evolution of the system, as they will commute with any ﬁeld. Thus, it would have been more general to include a constant energy density ρ in the Hamiltonian,

3 1 2 1 2 1 2 2 Hρ = d x π + (~ φ) + m φ + ρ Z 2 2 ∇ 2 d3k ω H = (2π)3δ(3)(0)ρ + k a†(~k)a(~k)+ a(~k)a†(~k) (4.29) ρ (2π)32ω 2 Z k h i and realizing that the constant ρ is a computable quantity that cannot be computed using the ﬁeld quantization methods advocated here. Progress is made by realizing that the energy of the vacuum (in the absence of gravity) is not physically observable. Instead, only diﬀerences in energy between states with diﬀerent particle contents are observable. Therefore, the constant ρ may just as well be chosen so that the vacuum has vanishing energy, or alternatively, 3 d k † ~ ~ H 3 ωk a (k)a(k) (4.30) ≡ Z (2π) 2ωk The process of realizing that certain quantities (such as the vacuum energy) cannot be computed within the quantization procedure of the ﬁelds and subsequently ﬁxing the arbitrariness of these quantities in terms of physical conditions goes under the name of renormalization (indicated with a subscript R on HR). The physical conditions themselves are called renormalization conditions. For renormalization in free ﬁeld theory, it often suﬃces to use the operation of placing all creation operators to the left and all annihilation operators to the right, which is referred to as normal ordering, as was done here.

Energy and momentum of particle states • Given the above results for the Hamiltonian and momentum operators, it is easy to evaluate them on particle states, n ~ ~ ~ ~ H k1, , kn = ωki k1, , kn | ··· i i=1 | ··· i Xn P~ ~k , ,~k = ~k ~k , ,~k (4.31) | 1 ··· ni i| 1 ··· ni Xi=1 Finally, it may be veriﬁed that all the generators of the Poincar´ealgebra, namely P µ and M µν will have the proper commutation relations with φ, and reproduce the basic principle of Poincar´etransformations of the scalar ﬁeld (Λ, a)φ(x) (Λ, a)† = φ(Λx + a). U U 48 Time ordered product of two operators • A key concept in QFT is that of time ordering. For any two scalar ﬁelds φ1 and φ2, one deﬁnes their time-ordered product as follows,∗∗

Tφ (t, ~x)φ (t0, ~x0)= θ(t t0)φ (t, ~x)φ (t0, ~x0)+ θ(t0 t)φ (t0, ~x0)φ (t, ~x) (4.32) 1 2 − 1 2 − 2 1 which can obviously be generalized to the case of the product of any number of ﬁelds. Time diﬀerentiation of the time ordered product follows a simple rule,

∂ Tφ (t, ~x)φ (t0, ~x0)= T ∂ φ (t, ~x)φ (t0, ~x0)+ δ(t t0)[φ (t, ~x),φ (t, ~x0)] (4.33) t 1 2 t 1 2 − 1 2 For a single ﬁeld, the time ordered product is a Klein-Gordon Green function,

(2 + m2) Tφ(x)φ(x0) = ∂2Tφ(x)φ(x0)+ T ( ∆+ m2) φ (x)φ(x0) x t − x 1 = ∂ T ∂ φ(x)φ(x0)+ T [ (∂ )2φ(x)φ(x0)] t t − t = δ(t t0)[π(x), φ(x0)] = iδ(4)(x x0) (4.34) − − − The operator 2 + m2 may be supplemented with a variety of boundary conditions, so care is needed when inverting it.

The Feynman propagator • The Feynman propagator is deﬁned by

G (x x0) 0 Tφ(x)φ(x0) 0 F − ≡ h | | i (2 + m2) G (x x0) = iδ(4)(x x0) (4.35) x F − − − 2 so that GF is the inverse of the diﬀerential operator i(2 + m ). To evaluate the Feynman propagator, we ﬁrst compute

3 3 0 d k d l ~ −ik·x † ~ ik·x 0 φ(x)φ(x ) 0 = 3 3 0 a(k)e + a (k)e h | | i Z (2π) 2ωk Z (2π) 2ωl h | h 0 i 0 a(~l)e−il·x + a†(~l)eil·x 0 × | i 3 3 h i d k d l ~ † ~ −ik·x+il·x0 = 3 3 0 a(k)a (l) 0 e Z (2π) 2ωk Z (2π) 2ωl h | | i 3 d k −ik·(x−x0) = 3 e (4.36) Z (2π) 2ωk

∗∗The Heaviside θ-function, or step-function, is deﬁned to be θ(t)=1 for t> 0 and θ(t)=0 for t< 0, and θ(0) = 1/2. Its derivative, deﬁned in the sense of distributions, is ∂tθ(t)= δ(t).

49 0 0 Making the substitution k = ωk, l = ωl, and assembling both time orderings yields

3 0 0 0 0 d k 0 −iωk(t−t )+i~k(~x−~x ) 0 −iωk(t −t)+i~k(~x −~x) GF (x y)= 3 θ(t t )e + θ(t t)e − Z (2π) 2ωk − − Next, we make use of the following integral representation, valid for any 0 < 1, 0 −ik0(t−t0) 1 0 0 dk e θ(t t0)e−iωk(t−t ) + θ(t0 t)e+iωk(t−t ) = (4.37) 2ω − − − 2πi (k0)2 ω2 + i k h i Z − k to recast the Feynman propagator as follows,

4 d k i 0 G (x x0)= eik·(x−x ) (4.38) F − (2π)4 k2 m2 + i Z − The Klein-Gordon equation with an extrenal (φ-independent) source j(x) may be solved in terms of GF ,

2 4 0 0 0 (2 + m )φ(x)= j(x) φ(x)= i d x GF (x x )j(x ) (4.39) ⇔ Z − Note that for m2 (x x0)2 1, we have | − | i G (x x0)= − (4.40) F − 4π2(x x0)2 − Feynman introduced a graphical notation for the propagator as a line joining the points x and x0. This is the simplest case of a Feynman diagran, and is depicted in Fig 3(a).

Time ordered product of n operators • The time-ordered product of n canonical ﬁelds φ will play a fundamental role in scalar ﬁeld theory. First, notice that the theory has a discrete symmetry a(~k) † = a(~k) φ0(x)= φ(x) † = φ(x) U U − (4.41) U U − ⇔ ( a†(~k) † = a†(~k) U U − Under this symmetry, the vacuum is invariant ( 0 = 0 ). By inserting I = † between U| i | i U U each pair of operators and the vacuum, we have

0 Tφ(x ) φ(x ) 0 = 0 †T ( φ(x ) †) ( φ(x ) †) 0 h | 1 ··· n | i h |U U 1 U ··· U n U U| i = ( )n 0 Tφ(x ) φ(x ) 0 (4.42) − h | 1 ··· n | i When n is odd, we ﬁnd that 0 Tφ(x ) φ(x ) 0 = 0. When n =2p is even, we proceed h | 1 ··· n | i by applying the kinetic operator in one variable,

n−1 (2 + m2) 0 Tφ(x ) φ(x ) 0 = iδ(4)(x x ) 0 Tφ(x ) φˆ(x ) φ(x ) 0 xn h | 1 ··· n | i − n − k h | 1 ··· k ··· n−1 | i kX=1 50 1 4 1 4 1 4 x x© + + 2 3 2 3 2 3

(a) (b)

0 Figure 3: Free Scalar Correlation Functions; (a) The Feynman propagator GF (x x ), (b) the 4-point function 0 Tφ(x )φ(x )φ(x )φ(x ) 0 . − h | 1 2 3 4 | i

Here, φˆ stands for the fact that the operators φ(xk) is to be omitted. The solution is again obtained in terms of the Feynman propagator G (x x0) as follows, F − n−1 0 Tφ(x ) φ(x ) 0 = G (x x ) 0 Tφ(x ) φˆ(x ) φ(x ) 0 (4.43) h | 1 ··· n | i F n − k h | 1 ··· k ··· n−1 | i kX=1 The solution to this recursive equation is manifest,

0 Tφ(x ) φ(x ) 0 = G (x x ) G (x x ) (4.44) h | 1 ··· n | i F k1 − l1 ··· F kp − kp disjointX pairs(k,l) where the sum is over all possible pairs without common points. The simple example of the 4-point correlation function is given in Fig 3(b).

4.2 The Operator Product Expansion & Composite Operators

It is possible to re-analyze the issues of the vacuum energy of the free scalar ﬁeld theory in terms of a more general problem. The canonical commutation relations provide expressions for the commutators of the canonical ﬁelds. In the construction of the Hamiltonian, mo- mentum and Lorentz generators, composite operators enter, namely the operators φ(x)2, π(x)2 and (~ φ(x))2. These operators are obtained by putting two canonical ﬁelds at the ∇ same space-time point. From the short distance behavior of the Feynman propagator, (4.40), it is clear that putting operators at the same point produces singularities, which ultimately result in the inﬁnite expression for the energy density of the vacuum. So, one very important problem in QFT is to properly deﬁne composite operators. One of the most powerful ways is via the use of the Operator Product Expansion (or OPE), a method that was introduced by Ken Wilson in 1969. (Another way, which is mostly restricted to free ﬁeld theory or at best perturbation theory, is normal ordering.)

51 If it were not for the fact that the product of local ﬁelds at the same space-time point is singular, one would naively be led to Taylor expand the product of two operators at nearly points, ∞ 1 ? φ(x)φ(y)= (x y)µ1 (x y)µn (∂ ∂ φ(y)) φ(y) (4.45) n! − ··· − µ1 ··· µn nX=0 Instead, the expansion is rather of the Laurent type, beginning with a singular term. In free ﬁeld theory, the operators may be characterized by their standard dimension, and ordered according to increasing dimension. In the scalar free ﬁeld case, reﬂection symmetry φ φ requires the operators on the rhs to be even in φ and of degree less or equal to 2. → − A list of possible operators for dimension less than 6 is given in table4.2.

Dimension spin 0 spin 1 spin 2 spin 3 spin 4 0 I 2 φ2 3 φ∂µφ 4 φ2φ φ∂µ∂ν φ µ ∂µφ∂ φ ∂µφ∂ν φ 5 φ2∂µφ φ∂µ∂ν∂ρφ ∂µφ2φ ∂µφ∂ν∂ρφ ν ∂µ∂ν φ∂ φ 2 6 φ2 φ φ∂µ∂ν 2φ φ∂µ∂ν ∂ρ∂σφ µ ∂µφ2∂ φ ∂µφ∂ν 2φ ∂µφ∂ν∂ρ∂σφ 2φ2φ ∂µ∂ν φ2φ ∂µ∂νφ∂ρ∂σφ µ ν ρ ∂µ∂νφ∂ ∂ φ ∂µ∂ρφ∂ν∂ φ

Table 2:

If no further selection rules apply, the OPE of the two canonical ﬁelds will involve all of these operators and the coeﬃcient functions may be calculated. Thus, we obtain an expansion of the type,

φ(x)φ(y)= C (x, y) (y) (4.46) φφO O XO The most singular part is proportional to the operator of lowest dimension, i.e. the identity, i I φ(x)φ(y)= − + [φ2](y)+ (4.47) 4π2 (x y)2 ··· − where the terms vanish as x y. The coeﬃcient function of the identity operator ··· → is the same as the limiting behavior as (x y)2 0 in the Feynman propagator. The − → 52 operator [φ2] refers to the ﬁnite (or renormalized) operator associated with φ2. It may be deﬁned by the OPE in the following manner,

i I [φ2](y) lim φ(x)φ(y)+ (4.48) ≡ x→y 4π2 (x y)2 ! − This limit is well-deﬁned and unique. More generally, but still in the scalar free ﬁeld theory, the OPE of any two operators will be a Laurent series in terms of all the operators of the theory (not neccessarily of order 2 in the ﬁeld φ), and we have

(x) (y)= C (x, y) (y) (4.49) O1 O2 O1O2O O XO The coeﬃcient functions may be calculated exactly as long as we are in free ﬁeld theory, and they will have integer powers. For an interacting theory, a similar expansion exists, but fractional expansion powers will occur, refelecting the fact that in an interacting theory, the dimensions of the operators will received quantum corrections.

53 4.3 The Gauge or Spin 1 Field

The simplest free spin 1 ﬁeld is the gauge vector potential Aµ in Maxwell theory of E&M. This ﬁeld is real and its classical Lagrangian, in the absence of sources, is given by 1 = F F µν F ∂ A ∂ A (4.50) L −4 µν µν ≡ µ ν − ν µ

The customary electric Ei and magnetic Bi ﬁelds are components of Fµν, 1 E F B F (4.51) i ≡ i0 i ≡ 2 ijk jk By construction, the Lagrangian is invariant under gauge transformations by any ar- L bitary real scalar function θ(x), given by A (x) A0 (x)= A (x) ∂ θ(x) (4.52) µ → µ µ − µ Two gauge ﬁelds related by a gauge transformation have the same electric and magnetic ﬁelds and are thus physically equivalent.†† The canonical momenta are deﬁned as usual, ∂ πµ L (4.53) ≡ ∂(∂0Aµ) and are found to be π0 =0 πi = Ei (4.54) The fact that π0 = 0 implies that we are dealing with a constrained system, namely the canonical momenta and canonical ﬁelds are not independent of one another. The ﬁeld A0 has vanishing canonical momentum and is therefore non-dynamical. This is an immediate consequence of the gauge invariance of the Lagrangian and is also reﬂected in the structure of the ﬁeld equations,

µν ~ E~ =0 ∂µF =0 ~ ∇·~ ~ (4.55) ⇔ ( ∂tE B =0 − ∇ × Indeed, ~ E~ = 0 is a non-dynamical equation since it gives a condition that the canonical ∇· momenta must satisfy at any time, including on the initial data. Yet a further manifesta- tion of this phenomenon is that the second order diﬀerential operator, in terms of which the Lagrangian may be re-expressed (up to a total derivative term) is not invertible, 1 = A (2ηµν ∂µ∂ν )A (4.56) L 2 µ − ν The operator 2ηµν ∂µ∂ν is not invertible; it kernel are precisely gauge ﬁelds of the − form Aµ = ∂µθ, and gauge equivalent to 0. Clearly, the vanishing of one of the canonical momenta causes complications for the quantization of the gauge ﬁeld.

††Except on non-simply connected domains when there can be glovbal eﬀects.

54 Transverse gauge quantization (Lorentz non-covariant) A ﬁrst approach in dealing with the complications that ensue from gauge invariance and the constraints on the momenta is to make a gauge choice in which the constraint can be solved explicitly. This will leave only dynamical ﬁelds, on which canonical quantization may now be carried out. The drawback of this approach is that it cannot be Lorentz covariant, as a preferred direction is chosen in space-time. With the help of a gauge transformation, any gauge ﬁeld conﬁguration may be trans- formed into a transverse gauge ~ A~ = 0. As a result of the constraint ~ E~ = 0, one ∇ ∇ automatically has A0 = 0 and the Lagrangian and canonical momenta reduce to 1 1 = + (A˙ )2 B2 L 2 i − 2 i ∂ πi = L = A˙ i ∂iAi =0 (4.57) ∂(∂0Ai) The last equation reﬂects the gauge choice. The Euler-Lagrange equations are

(2ηµν ∂µ∂ν )A =0 (4.58) − µ and, in transverse gauge, reduce to

2Ai =0 ∂iAi =0 (4.59)

The ﬁeld equations are solved ﬁrst by Fourier transform,

3 d k ~ −ik·x † ~ ik·x Ai(x)= 3 ai(k)e + ai (k)e (4.60) Z (2π) 2ωk after which we must further impose the radiation gauge conditions, which result in the following transversality conditions on the oscillators, ~ † ~ kiai(k)= kiai (k)=0 (4.61) It is convenient to introduce basis vectors for the two-dimensional space of transverse polarization vectors ~ε(~k, α), α =1, 2, which satisfy

kiεi(~k, α)=0 ~ ∗ ~ εi(k, α) εi(k, β)= δαβ (4.62) kikj ε (~k, α)ε (~k, α)= δ 2 α i j ij − k In terms of this basis, we mayP introduce independent ﬁeld oscillators a(~k, α) in terms of which the old oscillators may be decomposed as

ai(~k)= a(~k, α)εi(~k, α) (4.63) α X 55 and the ﬁeld takes the form 3 d k ~ ~ −ik·x † ~ ∗ ~ ik·x Ai(x)= 3 a(k, α)εi(k, α)e + a (k, α)εi (k, α)e (4.64) (2π) 2ωk Z αX=1,2

Clearly, the photon has two degrees of freedom, even though the ﬁeld Aµ has 4 components. The canonical commutation relations amongst the a(~k, α) and a†(~k, α) are just as for the scalar theory, but now with two independent components,

[a(~k, α), a(~k0, β)] = [a†(~k, α), a†(~k0, β)]=0 [a(~k, α), a†(~k0, β)] = δ 2ω (2π)3δ3(~k ~k0) (4.65) αβ k −

The equal time commutators may be expressed in terms of the ﬁelds Ai as follows, ∂ ∂ [A˙ (t, ~x), A (t, ~y)] = iP δ(3)(~x ~y) P˜ = δ i j (4.66) i j − ij − ij ij − ∆ from which it is clear again that only two degrees of freedom are independent. The Hamiltonian is standard, 1 1 H = d3x E~ 2 + B~ 2 (4.67) Z 2 2 The Hilbert space is constructed very much as in the case of the single scalar ﬁeld. The vacuum 0 is deﬁned by | i a(~k, α) 0 =0 forall ~k, α (4.68) | i The n–particle states are found to be

k , α ; k , α ; ; k , α a†(k , α )a†(k , α ) a†(k , α ) 0 (4.69) | 1 1 2 2 ··· n ni ≡ 1 1 1 1 ··· 1 1 | i The (renormalized) Hamiltonian and momentum operators may be worked out,

3 d k † ~ ~ H = 3 ωka (k, α)a(k, α) Z (2π) 2ωk α 3 X ~ d k ~ † ~ ~ P = 3 ka (k, α)a(k, α) (4.70) (2π) 2ωk α Z X Finally, the number operator may also be introduced,

3 d k † ~ ~ N = 3 a (k, α)a(k, α) (4.71) (2π) 2ωk α Z X Since this is a free theory, we have [H, N] = [P~ , N] = 0, so the number of photons is conserved.

56 Gupta-Bleuler Quantization (Lorentz Covariant) The second approach is to keep Lorentz invariance manifest at all times, possibly at the cost of giving up manifest gauge invariance. To see exactly what goes on, we introduce a conserved external source jµ, which we assume is independent of A. The Lagrangian is, 1 λ = F F µν (∂ Aµ)2 A jµ ∂ jµ =0 (4.72) L −4 µν − 2 µ − µ µ In this Lagrangian, for λ = 0, the time component A does have a non-vanishing conjugate 6 0 momentum, and the quadratic part of the Lagrangian involves an operator 2η (1 µν − − λ)∂ ∂ which is invertible. The Lagrangian is, however, no longer gauge invariant, so in µ ν L the end we will have to deal with non-gauge invariant states. The canonical relations are

0 ∂ ν i ∂ io π = L = λ∂ν A π = L = F ∂(∂0A0) − ∂(∂0Ai) [πµ(t, ~x), Aν(t, ~x0)] = i ηµν δ3(~x ~x0) (4.73) − The sign in the canonical commutation relation is chosen such that the canonical quan- tization of the space components agree with the one derived previously. The sign for the time-components is then forced by Lorentz invariance.

Specializing to the case λ = 1 (Feynman gauge), the ﬁeld equations reduce to 2Aµ = 0.

The decomposition of Aµ(x) into modes then gives

3 d k ~ −ik·x † ~ ik·x Aµ(x)= 3 aµ(k)e + aµ(k)e (4.74) Z (2π) 2ωk and the canonical commutation relations on the oscillators become,

~ ~ 0 † ~ † ~ 0 [aµ(k), aν(k )] = [aµ(k), aν(k )]=0 [a (~k), a† (~k0)] = η 2ω (2π)3δ(3)(~k ~k0) (4.75) µ ν − µν k − We can still deﬁne the vacuum by the requirement that a (~k) 0 = 0, and one-particle µ | i states by

~k,µ = a† (~k) 0 (4.76) | i µ | i The normalization of these states may be worked out in terms of the canonical commutation relations as usual,

~k,µ ~k0, ν = η 2ω (2π)3δ(3)(~k ~k0) (4.77) h | i − µν k −

While the oscillators ai create states with positive norm, as was the case in the scalar theory, the operators a0 create states with negative norm, which cannot be physically

57 acceptable in a quantum theory. These states are unphysical, and arise because we insisted on maintaining manifest Lorentz invariance, which forced us to carry along more degrees of freedom than physically needed or allowed. We now investigate how these unphysical degrees of freedom are purged from the theory. To see how this can be done, we notice that there is a free ﬁeld in the theory that is a pure gauge artifact. Look at the equations for Aµ:

∂ F µν λ∂ν ∂ Aµ + jν =0 (4.78) µ − µ µ µ µ Conservation of j implies that λ2∂µA = 0; hence ∂µA is a free ﬁeld with no coupling to the source jµ. In view of the canonical quantization relations, we cannot simply set µ µ ∂µA = 0. However, we had way too many states created by A , so we may set it be zero on the physical states. In fact this is truly beneﬁcial, because it implies that on the physical states, the Heisenberg equation of motion is gauge invariant:

∂ Aµ phys =0 (4.79) µ | i It is very important to remark that we work with two types of spaces: (1) The Fock space

for the full Aµ ﬁeld, which contains states of negative norm; (2) The physical Hilbert space, obeying the restrictions above. To solve for the ﬁeld in terms of the source,

µν µ ν µ 2η +(λ 1)∂ ∂ Aν = j (4.80) − we introduce the photon propagator,

2 κµ κ µ κ 4 η +(λ 1)∂ ∂ Gµν(x y)= iδν δ (x y) (4.81) − x − − − The propagator is related to the time-ordered product of the quantum ﬁelds,

G (x y)= 0 T A (x)A (y) 0 (4.82) µν − h | µ ν | i It may be obtained in terms of a Fourrier integral,

4 d k 1 λ kµkν i −ik·(x−y) Gµν (x y)= 4 ηµν + − 2 2− e (4.83) − Z (2π) " λ k + i # k + i Clearly, the propagator in Feynman gauge (λ = 1) simpliﬁes considerably; in Landau gauge (λ = ) it becomes transverse. ∞

58 4.4 The Casimir Eﬀect on parallel plates Thus far, we have dealt with the quantization of the free electromagnetic ﬁelds E~ and B~ , Already at this stage, it is possible to infer measurable predictions, such as the Casimir eﬀect. The Casimir eﬀect is a purely quantum ﬁeld theory phenomenon in which the quantization of the electromagnetic ﬁelds in the presence of a electric conductors results in a net force between these conductors. The most famous example involves two conducting parallel plates, separated by a distance a, as represented in Fig4(a).

Ep =0 L

L

L

-- a --

(a) (b)

Figure 4: The Casimir eﬀect : (a) electromagnetic vacuum ﬂuctuations produce a force between two parallel plates; (b) the problem in a space box.

The assumptions entering the set-up are as follows.

1. For a physical conductor, the electric ﬁeld E~p parallel to the plates will vanish for

frequencies much lower than the typical atomic scale ωc. At frequencies much higher than the atomic scale, the conductor eﬀectively becomes transparent and no con- straint is to be imposed on the electric ﬁeld. In the frequency region comparable to

the atomic scale ωc, the conductivity is a complicated function of frequency, which we shall simply approximate by a step function θ(ω ω). c − 2. The separation a should be taken much larger than interatomic distances, or aω 1. c 3. In order to regularize the problems associated with inﬁnite volume, we study the problem in a square box in the form of a cube, with periodic boundary conditions, as depicted in Fig4(b).

59 Calculation of the frequencies • There are three distinct regimes in which the frequencies of the quantum oscillators need to be computed. The momenta parallel to the plates are always of the following form, 2πn 2πn k = x k = y n , n Z x L y L x y ∈ Along the third direction, we distinguish three diﬀerent regimes, so that the frequency has three diﬀerent branches,

(i) 2 2 (i) 2 ω = kx + ky +(kz ) i =1, 2, 3 (4.84) r The three regimes are as follows,

1. ω>ωc : the frequencies are the same as in the absence of the plates, since the plates are acting as transparent objects, 2πn k(1) = z n Z z L z ∈

2. ω<ωc & between the two plates. πn k(2) = z 0 n Z z a ≤ z ∈

3. ω<ωc & outside the two plates. πn k(3) = z 0 n Z z L a ≤ z ∈ − Summing up the contributions of all frequencies • Finally, we are not interested in the total energy, but rather in the enrgy in the presence of the plates minus the energy in the absence of the plates. Thus, when the frequencies

exceed ωc, all contributions to the enrgy cancel since the plates act as transparent bodies. Thus, we have

1 (2) 1 (3) 1 (1) E(a) E0 = ω (nx, ny, nz)+ ω (nx, ny, nz) ω (nx, ny, nz) (4.85) − n ,n ,n 2 2 − 2 xXy z When n = 0, two polarization modes have E~ with 2 components along the plates, while z 6 for n = 0, n , n = 0, there is only one. Therefore, we isolate n = 0, z x y 6 z 1 E(a) E0 = ω(nx, ny, 0) (4.86) − 2 n ,n Xx y ∞ (2) (3) (1) + ω (nx, ny, nz)+ ω (nx, ny, nz) 2ω (nx, ny, nz) n ,n − Xx y nXz=1 Next, we are interested in taking the size of the box L very large compared to all other scales in the problem. The levels in nx and ny then become very closely spaced and the

60 sums over nx and ny may be well-approximated by integrals, with the help of the following conversion process, Ldk Lk dn = x dn = y (4.87) x 2π y 2π so that L2 +∞ +∞ 1 E(a) E0 = 2 dkx dky ωθ(ωc ω) − 4π Z−∞ Z−∞ 2 − ∞ + ω(2)θ(ω ω(2))+ ω(3)θ(ω ω(3)) 2ω(1)θ(ω ω(1)) c − c − − c − nXz=1 (1) (3) Now, for ω and ω , the frequency levels are also close together and the sum over nz may also be approximated by an integral,

∞ ∞ (2) (2) L a 2 2 2 2 2 2 ω θ(ωc ω ) = − dkz kx + ky + kz θ(ωc kx + ky + kz ) − π 0 − nXz=1 Z q q ∞ ∞ (1) (1) L 2 2 2 2 2 2 2ω θ(ωc ω ) = 2 dkz kx + ky + kz θ(ωc kx + ky + kz ) − 2π 0 − nXz=1 Z q q 2 2 2 Introducing k = kx + ky, and in the last integral the continuous variable n = akz/π, we have

2 ∞ ∞ 2 2 2 2 L 1 2 π n 2 2 π n E(a) E0 = dk k kθ(ωc k)+ k + θ(ωc k ) − 2π 0 2 − s a2 − − a2 Z nX=1 ∞ π2n2 π2n2 dn k2 + θ(ω2 k2 ) s 2 c 2 − Z0 a − − a Introducing the function

∞ dk π2n2 π2n2 f(n) k k2 + θ(ω2 k2 ) (4.88) s 2 c 2 ≡ Z0 2π a − − a we have E(a) E 1 ∞ ∞ − 0 = f(0) + f(n) dnf(n) (4.89) L2 2 − 0 nX=1 Z Now, there is a famous formula that relates a sum of the values of a function at integers to the integral of this function,

∞ 1 ∞ ∞ B dnf(n)= f(0) + f(n)+ 2p f (2p−1)(0) (4.90) 0 2 (2p)! Z nX=1 pX=1 61 where the Bernoulli numbers are deﬁned by x ∞ xn 1 1 = B B = , B = , (4.91) ex 1 n n! 2 −6 4 −30 ··· − nX=0 Assuming a sharp cutoﬀ, so that θ is a step function, we easily compute f(n),

3 3 1 3 π n f(n)= ωc 3 (4.92) 6π − a Thus, f (2p−1)(0) = 0 as soon as p> 2, while f (3)(0) = π2/a3, and thus we have − E(a) E π2 B π2 L2 − 0 = 4 = hc¯ (4.93) L2 a3 4! −720 a3 This represents a universal, attractive force proportional to 1/a4. To make their depen- dence explicit, we have restored the factors ofh ¯ and c.

4.5 Bose-Einstein and Fermi-Dirac statistics In classical mechanics, particles are distinguishable, namely each particle may be tagged and this tagging survives throughout time evolution since particle identities are conserved. In quantum mechanics, particles of the same species are indistinguishable, and cannot be tagged individually. They can only be characterized by the quantum numbers of the state of the system. Therefore, the the operation of interchange of any two particles of the same species must be a symmetry of the quantum system. The square of the interchange operation is the identity.‡‡ As a result, the quantum states must have deﬁnite symmetry properties under the interchange of two particles. All particles in Nature are either bosons or fermions;

BOSONS : the quantum state is symmetric under the interchange of any pair of • particles and obey Bose-Einstein statistics. Bosons have integer spin. For example, photons, W ±, Z0, gluons and gravitons are bosonic elementary particles, while the

Hydrogen atom, the He4 and deuterium nuclei are composite bosons.

FERMIONS : the quantum state is anti-symmetric under the interchange of any pair • of particles and obey Fermi-Dirac statistics. Fermions have integer plus half spin. For example, all quarks and leptons are fermionic elementary particles, while the

proton, neutron and He3 nucleus are composite fermions.

‡‡This statement holds in 4 space-time dimensions but it does not hold generally. In 3 space-time dimensions, the topology associated with the interchange of two particles allows for braiding and the square of this operation is not 1. The corresponding particles are anyons.

62 Remarkably, the quantization of free scalars and free photons carried out in the pre- ceeding subsections has Bose-Einstein statistics built in. The bosonic creation and an- † ~ ~ nihilation operators are denoted by aσ(k) and aσ(k) for each species σ. The canonical commutation relations inform us that all creation operators commute with one another † ~ † ~ 0 [aσ(k), aσ0 (k )] = 0. As a result, two states diﬀerening only by the interchange of two particle of the same species are identical quantum mechanically,

a† (~k ) a† (~k ) a† (~k ) a† (~k ) 0 σ1 1 ··· σi i ··· σj j ··· σn n | i = a† (~k ) a† (~k ) a† (~k ) a† (~k ) 0 σ1 1 ··· σj j ··· σi i ··· σn n | i The famous CPT Theorem states that upon the quantization of a Poincar´eand CPT invariant Lagrangian, integer spin ﬁelds will always produce particle states that obey Bose-Einstein statistics, while integer plus half ﬁelds always produce states that obey Fermi-Dirac statistics.

Fermi-Dirac Statistics • It remains to establish how integer plus half spin ﬁelds are to be quantized. This will be the subject of the subsection on the Dirac equation. Here, we shall take a very simple approach whose point of departure is the fact that fermions obey Fermi-Dirac statistics. First of all, a free fermion may be expected to be equivalent to a collection of oscillators, just as bosonic free ﬁelds were. But they cannot quite be the usual harmonic oscillators, because we have just shown above that harmonic operators produce Bose-Einstein statis- tics. Instead, the Pauli exclusion principle states that only a single fermion is allowed to occupy a given quantum state. This means that a fermion creation operator ˆb† for given quantum numbers must square to 0. Then, its repeated application to any state (which would create a quantum state with more than one fermion particle with that species) will produce 0. The smallest set of operators that can characterize a fermion state is given by the fermionic oscillators ˆb and ˆb†, obeying the algebra

ˆb, ˆb = ˆb†, ˆb† =0 ˆb, ˆb† =1 (4.94) { } { } { } In analogy with the bosonic oscillator, we consider the following simple Hamiltonian, ω 1 Hˆ = ˆb†ˆb ˆbˆb† = ω ˆb†ˆb (4.95) 2 − − 2 Naturally, the ground state is deﬁned by ˆb 0 = 0 and there are just two quantum states | i in this system, namely 0 and ˆb† 0 with energies 1 ω and + 1 ω respectively. A simple | i | i − 2 2 63 representation of this algebra may be found by noticing that it is equivalent to the algebra of Pauli matrices or Cliﬀord-Dirac algebra in 2 dimensions, ω σ1 = ˆb + ˆb† σ2 = iˆb iˆb† H = σ3 (4.96) − 2 Vice-versa, the γ-matrices in 4 dimensions are equivalent to two sets of fermionic oscillators ˆ ˆ† bα and bα, α = 1, 2. The above argumentation demsonstrates that its only irreducible representation is 4-dimensional, and spanned by the states 0 , ˆb† 0 , ˆb† 0 , ˆb†ˆb† 0 . | i 1| i 2| i 1 2| i In terms of particles, we should aﬃx the quantum number of momentum ~k, so that ~ † ~ we now have oscillators bσ(k) and bσ(k), for some possible species index σ. Postulating anti-commutation relations, we have

0 † † 0 b (~k), b 0 (~k ) = b (~k), b 0 (~k ) =0 { σ σ } { σ σ } † 0 3 (3) 0 b (~k), b 0 (~k ) =2ω δ 0 (2π) δ (~k ~k ) (4.97) { σ σ } k σσ −

2 2 where again ωk = ~k + m . The Hamiltonian and momentum operators are naturally q 3 d k † ~ ~ H = 3 ωk bσ(k)bσ(k) Z (2π) 2ωk σ 3 X ~ d k ~ † ~ ~ P = 3 k bσ(k)bσ(k) (4.98) (2π) 2ωk σ Z X The vacuum state 0 is deﬁned by b (~k) 0 = 0 and multiparticle states are obtained by | i σ | i applying creation operators to the vacuum,

b† (~k ) b† (~k ) 0 (4.99) σ1 1 ··· σn n | i Using the fact that creation operators always anti-commute with one another, it is straight- forward to show that

b† (~k ) b† (~k ) b† (~k ) b† (~k ) 0 σ1 1 ··· σi i ··· σj j ··· σn n | i = b† (~k ) b† (~k ) b† (~k ) b† (~k ) 0 − σ1 1 ··· σj j ··· σi i ··· σn n | i so that the state obeys Fermi-Dirac statistics.

64 4.6 The Spin 1/2 Field

In view of the preceeding discussion on Fermi-Dirac statistics, it is clear that the fermionic ﬁeld oscillators and thus the fermionic ﬁelds themselves should obey canonical anti- commutation relations, such as

ψ (t, ~x), ψ (t, ~y) = 0 { α β } ψ†α(t, ~x), ψ†β(t, ~y) = 0 { } ψ (t, ~x), ψ†β(t, ~y) = ihδ¯ βδ(3)(~x ~y) (4.100) { α } α − Here, α and β stand for spinor indices. The classical limit of commutation relations (without dividing byh ¯) conﬁrms that classical bosonic ﬁelds are ordinary functions. The classical limit of anti-commutation relations (also without dividing byh ¯) for fermionic ﬁelds informs us that classical fermionic ﬁelds must be Grassmann-valued functions, and obey

ψ (t, ~x), ψ (t, ~y) = ψ∗ (t, ~x), ψ∗ (t, ~y) = ψ (t, ~x), ψ∗ (t, ~y) =0 (4.101) { α β } { α β } { α β } Grassmann numbers are well-deﬁned by the above anticommutation relations. It is also useful, however, to obtain a concrete representation in terms of matrices. Let bj, j = 1, , N be N Grassmann numbers. These algebraic objects may be realized in terms of ··· γ-matrices in dimension 2N, deﬁned to obey the Cliﬀord-Dirac algebra γ ,γ =2δ I, { m n} mn with m, n, =1, , 2N. The following combinations ··· 1 b (γ + iγ ) j =1, , N (4.102) j ≡ 2 j j+N ··· will form N Grassmann numbers. Note that b , b† = δ , so Grassmann numbers may { j k} ij be viewed as half of the γ-matrices.

The Free Dirac equation • The ﬁeld equations for the free Dirac ﬁeld must be Poincar´ecovariant and linear in ψ(x). Of course, one might postulate (2 + m2)ψ(x) = 0, but it turns out that this is neither the simplest nor the correct equation. The Dirac equation is a ﬁrst order partial diﬀerential equation,

(iγµ∂ mI)ψ(x)=0 (4.103) µ − Here, γµ are the Dirac matrices in 4 dimensional Minkowski space-time and I is the identity matrix in the Dirac algebra. Often, I is not written explicitly. We begin by showing that

65 this equation is Poincar´einvariant. Translation invariance is manifest, and we recall the Lorentz transformation properties of the various ingredients,

0 ν ∂µ = Λµ ∂ν µ µ ν −1 γ = Λ νS(Λ)γ S(Λ) ψ0(x0) = S(Λ)ψ(x) (4.104) Combining all, we may establish the covariance of the equation, (iγµ∂0 mI)ψ0(x0) = (iΛµ S(Λ)γνS(Λ)−1Λ ρ∂ mI)S(Λ)ψ(x) µ − ν µ ρ − = (iS(Λ)γµS(Λ)−1∂ mI)S(Λ)ψ(x) µ − = S(Λ)(iγµ∂ mI)ψ(x) (4.105) µ − Group-theoretically, the structure of the Dirac equation may be understood as follows, 1 1 1 1 ψ ( , 0) (0, ) ∂ ( , ) ∼ 2 ⊕ 2 ∼ 2 2 1 1 1 1 1 1 1 1 ∂ψ ( , ) ( , 0) (0, ) = (0, ) (1, ) ( , 0) ( , 1) ∼ 2 2 ⊗ 2 ⊕ 2 2 ⊕ 2 ⊕ 2 ⊕ 2 The role of the γ-matrices is to project ∂ψ onto the spin 1/2 representations. If ψ(x) obeys the Dirac equation, then ψ(x) automatically also obeys the Klein Gordon equation, as may be seen by multiplying the Dirac equation to the left by ( iγµ∂ mI), − µ − ( iγµ∂ mI)(iγµ∂ mI)ψ(x)=(γµγν∂ ∂ u + m2)ψ(x)=(2 + m2)ψ(x) − µ − µ − µ n But the converse is manifestly not true.

4.6.1 The Weyl basis and Free Weyl spinors In the Weyl basis, we have µ 5 I 0 µ 0 σ ψL γ = − γ = µ ψ = (4.106) 0 +I σ¯ 0 ψR where σµ =(I, σi) andσ ¯µ =(I, σi). In this basis, the Dirac equation becomes − iσ¯µ∂ ψ mψ =0 µ L R (4.107) iσµ∂ ψ − mψ =0 µ R − L The mass term couples left and right spinors. When the mass vanishes, the two Weyl

spinors ψL and ψR are decoupled. It now becomes possible to set one of the chiralities to zero and retain a consistent equation for the other. The left Weyl equation is

µ σ¯ ∂µψL =0 (4.108)

The spinors ψL and ψR transform under the representations (1/2, 0) and (0, 1/2) respec- tively. The Weyl equation is Poincar´einvariant by itself.

66 The Free Weyl and Dirac actions • The combination ψ¯ ψ†γ0 has a simple Lorentz transformation property in view of † 0 ≡0 the fact that γµ = γ γµγ , and we have

ψ0(x0)= S(Λ)ψ(x) (ψ0)† = ψ†(x)S(Λ)† ψ¯0(x)= ψ¯(x)S(Λ)−1 (4.109) ⇒ ⇒ Therefore, we readily construct an invariant action,

4 µ S[ψ, ψ¯]= d x = ψ¯(iγ ∂µ m)ψ (4.110) Z L L − Viewing ψ and ψ¯ as independent ﬁelds, it is clear that the Dirac equation follows from ¯ 0 the variation of these ﬁelds. The canonical momenta are πψ = iψγ and πψ¯ = 0 and the Hamiltonian is given by

H = d3x [π ∂ ψ ]= d3xψ¯ i~γ ~ + m ψ (4.111) ψ 0 −L − · ∇ Z Z h i In the Weyl basis, the action becomes

S[ψ , ψ¯ , ψ , ψ¯ ]= d4x ψ† iσ¯µ∂ ψ + ψ† iσµ∂ ψ mψ† ψ mψ† ψ (4.112) L L R R L µ L R µ R − L R − R L Z h i

Majorana spinors • Recall that a Majorana spinor is a Dirac spinor which equals its own charge conjugate. In the Weyl basis, it takes the form,

ψ ψ = L ψc = ψ (4.113) σ2ψ∗ − L ! Hence, a Majorana spinor in 4 space-time dimensions is equivalent to a Weyl spinor. All the properties can be deduced from this equivalence. Note that there exists an interesting Majorana mass term, 1 1 1 = ψ¯ i ∂ψ + mψT σ2ψ = ψi¯ ∂ψ mψψ¯ (4.114) L L 6 L 2 L L 2 6 − 2 The Majorana mass term violates U(1) symmetry and so a spinor with Majorana mass must be electrically neutral.

Solutions to the Free Dirac equation • The free Dirac equation may be solved by a superposition of Fourrier modes with 4- momentum kµ. Since any solution to the Dirac equation is also a solution to the Klein Gordon equation, we must have k2 = m2, and there will be solutions with k0 < 0 as well

67 as k0 > 0. Just as we did in the case of bosonic ﬁelds, it is useful to separate these two branches, and we organize all solutions in the following manner, both with k0 > 0,

ψ (x)= u(k,r)e−ik·x (k γµ m)us(k)=0 + µ − +ik·x µ s ψ−(x)= v(k,r)e (kµγ + m)v (k)=0 (4.115)

The superscript r =1, 2 labels the linearly independent solutions. That there are precisely two solutions for each equation may be shown as follows. When m = 0, we introduce the following projection operators 6 1 Λ (k) ( k γµ + m) (4.116) ± ≡ 2m ± µ which satisfy

2 (Λ±) =Λ± Λ+ +Λ− = I Λ+Λ− =0 trΛ± =2 (4.117)

Hence the rank of both Λ± is 2, whence there are two independent solutions. The spinors u and v may be normalized as follows,

u(~k,r)¯u(~k,r) = k/ + m r X v(~k,r)¯v(~k,r) = k/ m (4.118) r − X Any bilinear in u and v may be calculated from this formula. For example, we have

µ µ u¯(~k,r)u(~k,s)=2mδrs u¯(~k,r)γ u(~k,s)=2k δrs u¯(~k,r)v(~k,s)=0 u¯(~k,r)γµv(~k,s) =? (4.119) v¯(~k,r)v(~k,s)= 2mδ v¯(~k,r)γµv(~k,s)=2kµδ − rs rs Explicit forms of the solutions may be obtained as follows. +√k σ ξ(r) +√k σ η(r) u(k,r)= v(k,r)= (4.120) +√k · σ¯ ξ(r) √k · σ¯ η(r) · − · with

† † ξ (r)ξ(s)= η (r)η(s)= δrs (4.121)

so that we may choose for example, 1 0 ξ(1) = η(1) = ξ(2) = η(2) = (4.122) 0 1 When m = 0, there are still two solutions u and two solutions v, but the preceeding normalization becomes degenerate. Choose kµ = (k0, 0, 0, k0), so that the equations ± 68 become (γ0 γ3)u(k,r) = 0 and (γ0 + γ3)v(k,r) = 0, each of which has again two − solutions. It suﬃces to take k σξ¯ (r) = k ση(r) = 0 to obtain normalized left and right · · handed spinors. The above normalizations still hold good.

Quantization of the Dirac Field • We are now ready to quantize the Dirac ﬁeld. Since the ﬁeld is complex valued, we must introduce two sets of operators in the decomposition of the ﬁeld solution,

3 d k −ik·x † +ik·x ψ(x)= 3 u(k,r) b(k,r)e + v(k,r) d (k,r)e (4.123) (2π) 2ωk r Z X The anti-commutation relations for ψ and ψ† translate into anti-commutation relations between the b and d oscillators,

† 0 0 3 2 0 b(~k,r), b (~k ,r ) = 2ω δ 0 (2π) δ (~k ~k ) { } k rr − † 0 0 3 3 0 d(~k,r), d (~k ,r ) = 2ω δ 0 (2π) δ (~k ~k ) (4.124) { } k rr − while all others vanish. The time-ordered product is deﬁned by

T ψ (x)ψ¯ (y)= θ(xo yo)ψ (x)ψ¯ (y) θ(yo xo)ψ¯ (y)ψ (x) (4.125) α β − α β − − β α It is straightforward to compute the Green function:

S (x y) = 0 T ψ(x)ψ¯(y) 0 (4.126) F − h | | i d4k i = (i ∂ + m) G (x y)= e−ik(x−y) 6 x F − (2π)4 k m + i Z 6 −

Mode expansion of energy and momentum • These expansions are obtained as soon as the quantized ﬁels is available and we ﬁnd,

3 µ d k µ † ~ ~ † ~ ~ P = 3 k [b (k,r)b(k,r)+ d (k,r)d(k,r)] (4.127) (2π) 2ωk r Z X 0 where k = ωk is always positive. Using canonical anti-commutation relations, we ﬁnd the commutators of P µ with the oscillators,

[P µ, b(~k,r)] = kµb(~k,r) [P µ, b†(~k,r)]=+kµb†(k,r) − [P µ, d(~k,r)] = kµd(~k,r) [P µ, d†(~k,r)]=+kµd†(~k,r) (4.128) − Note that the positive signs on the right hand side for creation operators conﬁrms indeed that b† and d† create particles with positive energy only, while b and d annihilate particles with positive energy.

69 Internal symmetries • The massive or massless Dirac action is invariant under phase rotations of the Dirac ﬁeld, which form a U(1) group of transformations,

U(1) ψ(x) ψ0(x)= eiθV ψ(x) (4.129) V → The associated conserved Noether current jµ and the time independent charge Q are obtained as usual,

jµ(x)= ψ¯(x)γµψ(x) Q = d3xj0 (4.130) Z In the quantum theory, the operator Q may be expressed in terms of the oscillators,

3 d k † ~ ~ † ~ ~ Q = 3 [b (k,r)b(k,r) d (k,r)d(k,r)] (4.131) (2π) 2ωk r − Z X Thus, we have the following commutation relations of Q with the individual oscillators,

[Q, b(~k,r)] = b(~k,r) [Q, b†(~k,r)]=+b†(k,r) − [Q, d(~k,r)] = +d(~k,r) [Q, d†(~k,r)] = d†(~k,r) (4.132) − Thus the particles created by b† and by d† have opposite charge. It will turn out that this quantity is electric charge and that b corresponds to electrons, while d corresponds to positrons. The massless Dirac action has a chiral symmetry, which consists of a phase rotation independently on the left and right handed chirality ﬁelds,

0 iθL ψL(x) ψL(x) = e ψL(x) U(1)L → ψ (x) ψ0 (x) = ψ (x) R → R R (4.133) 0 ψL(x) ψL(x) = ψL(x) U(1)R → ψ (x) ψ0 (x) = eiθR ψ (x) R → R R By taking linear combinations of the phases, θ = θ + θ and θ = θ θ , we may V L R A L − R equivalently rewrite the transformations in terms of Dirac spinors,

0 iθV U(1)V ψ(x) ψ (x)= e ψ(x) → 5 U(1) ψ(x) ψ0(x)= eiθAγ ψ(x) (4.134) A →

The U(1)A is referred to as axial symmetry, and the associated conserved current is the axial (vector) current and the and time independent charge is the axial charge, given by

µ ¯ m 5 3 0 j5 (x)= ψ(x)γ uγ ψ(x) Q5 = d xj5 (4.135) Z

70 4.7 The spin statistics theorem The spin statistics theorem states that in any QFT invariant under CPT symmetry, integer spin ﬁelds must be quantized with commutation relations, while half plus integer spin ﬁelds must be quantized with anti-commutation relations. This is in fact what we have assumed so far. We now show that this is necessary by studying some counterexamples.

4.7.1 Parastatistics ? The ﬁrst question to be answered is why we should have commutation or anticommutation relations at all, instead of some mixture of the two. To see this concretely, consider the case of Dirac spinors. Time translation invariance requires that [H, b(~k, α)] = k0b(~k, α) − and thus for free ﬁeld theory, we must have

3 0 ~ d k 1 + ~ 0 0 ~ 0 0 ~ [H, b(k, α)] = 3 [b (k , α )b(k , α ), b(k, α)] (4.136) 0 (2π) 2 Xα Z The only way to realize this for all ~k is to have the following relation amongst the b’s,

1 † 0 0 0 0 3 3 0 [b (~k , α )b(~k , α ), b(~k, α)] = 2ωkb(~k, α)(2π) δ (~k ~k ) (4.137) 0 2 − − Xα Let us now postulate not the usual commutation or anti-commutation relations, but gen- eralization thereof which allows for some mixture of the two. For simplicity, we shall retain a structure that is bilinear in the operators,

† 0 0 † 0 0 0 b(~k, α)b (~k , α )+ F+ b (~k , α )b(~k, α) = G+(~k,~k ) 0 0 0 0 0 b(~k, α)b(~k , α )+ F− b(~k , α )b(~k, α) = G−(~k,~k ) (4.138)

We always require these relations to be local in space-time so that F± are independent of ~k and ~k0. When F = 1, these relations are some-times referred to as parastatistics. ± 6 ± Working out the cubic equation now yields

G =0 F = F G = (2π)32ω δ3(~k ~k0) (4.139) − + − + k − Now look at the last equation and interchange ~k and ~k0,

0 0 b(~k)b(~k )+ F−b(~k )b(~k)=0 (4.140) 0 0 b(~k )b(~k)+ F−b(~k)b(~k )=0 (4.141)

2 Combining the two, we ﬁnd F− = +1. Therefore, the only possibilities are commutation relations or anticommutation relations. This can be done similarly for any ﬁeld.

71 4.7.2 Commutation versus anti-commutation relations The next question to be addressed is how one decides between commutation relations and anticommutation relations. For a change, let’s work with a complex scalar ﬁeld and attempt quantization with anticommutation relations:

d3x φ(t, ~x) = a(~k)e−ikx + c†(~k)eikx (2π)32ω Z k h i 3 + d k ~ −ikx † ~ +ikx φ (t, ~x) = 3 c(k)e + a (k)e (4.142) (2π) 2ωk Z h i and postulate anticommutation relations between the ﬁelds φ and φdagger,

φ˙†(t, ~x), φ(t, ~y) = i δ3(~x ~y) etc. (4.143) { } − − Anticommutation relations also follow on the oscillators,

a+(~k), a(~k0) = (2π)3ω δ(3)(~k ~k0) { } − k − c†(~k), c(~k0) = +(2π)32ω δ(3)(~k ~k0) (4.144) { } k − The fact that the two signs are opposite means that particle and antiparticle together cannot have both positive norm. If we attempt to quantize the fermion theory with equal time commutation relations, we get similar violations of known and well-established physical principles.

72 5 Interacting Field Theories – Gauge Theories

Free ﬁeld theories were described by quadratic Lagrangians, so that the equations of motion are linear. Interactions will occur when the ﬁelds are coupled to external sources or to one another. As for free ﬁeld theory, we shall insist on the interactions being Poincar´e invariant (when the source is external, it should be also transformed under the Poincar´e transformatiions) and local. In a fundamental theory of Nature, there is no room for external sources, since the descriptions of all phenomena is to be given in a single theory. Often, however, the eﬀects of a certain sector of the theory may be summarized in the form of external sources, thereby simplifying the practical problems involved.

5.1 Interacting Lagrangians We begin by presenting a brief list of some of the most important fundamental interacting ﬁeld theories for elementary particles whose spin is less or equal to 1. These interacting ﬁeld theories are described by local Poincar´einvariant Lagrangians. 1. Scalar ﬁeld theory The simplest case involves a free kinetic term plus an interaction potential. 1 = ∂ φ∂µφ V (φ) (5.1) L 2 µ − 2. The linear σ-model This is a theory of N real scalar ﬁelds with quartic self-couplings,

N N 1 a µ a λ a a 2 2 = ∂µσ ∂ σ σ σ v ) (5.2) L 2 − 4 − ! aX=1 aX=1 3. Electrodynamics This theory governs most of the world that we see, 1 = F + ψ¯(i ∂ m)ψ eA ψγ¯ µψ (5.3) L −4 µν 6 − − µ 4. Yang-Mills theory Yang-Mills theory describes the electro-weak and strong interactions. It is associated with a simple Lie algebra , with generators T a, a = 1, 2, , dim and structure G ··· G constants f abc. The Lagrangian is

1 a aµν a a a abc b c = Fµν F Fµν = ∂µAν ∂ν Aµ + g f AµAν (5.4) L −4 a − c X X It is also possible to couple these various theories to one another.

73 5.2 Abelian Gauge Invariance Consider a charged complex ﬁeld ψ(x), which transforms non-trivially under U(1) gauge transformations: ψ(x) ψ0(x)= U(x) ψ(x) U(x)= eiqθ(x) (5.5) −→ Bilinears, such as the current, are invariant under these transformations ψ¯Γψ(x) ψ¯Γψ(x) Γ= I, Γµ, Γµν . . . Γ5 (5.6) −→ Derivatives of the ﬁeld, however, do not transform well, ∂ ψ(x) U(x) ∂ ψ(x)+ ∂ U(x) ψ(x) (5.7) µ −→ µ µ so that the standard kinetic term ψ¯ ∂ψ is not invariant. 6 In QED, we have also a gauge ﬁeld Aµ, whose transformation law involves an in-

homogeneous piece in U, in such a way that the covariant derivative Dµψ transforms homogeneously, D ∂ + iqA µ ≡ µ µ (5.8) 0 0 Dµψ (x) = U(x) Dµψ(x)

Working out the required transformation law of Aµ, we ﬁnd i A0 = A + U −1∂ U = A ∂ θ (5.9) µ µ q µ µ − µ We are now guaranteed that ψ¯ Dψ is invariant. 6 In additiion,

[Dµ,Dν]ψ = iqFµν ψ (5.10)

which may be used to deﬁne the ﬁeld strength Fµν . (Comparing with general relativity, one has A Γk is a connection, while F R is a curvature). µ ∼ µν µν ∼ µνκλ 5.3 Non-Abelian Gauge Invariance

On a multiplet of ﬁelds ψi i = 1,...,N, we are free to consider a larger set of transfor- mations ψ(x) ψ0(x) U(x)ψ(x) matrixform −→ ≡ ψ (x) ψ0(x) U j(x)ψ (x) componentform (5.11) i −→ i ≡ i j where U(x) G`(N, R) or G`(N,C) depending on whether the ﬁelds ψ are real or com- ∈ i plex.

74 While it is possible (and sometimes useful, such as in supergravity) to consider ab- solutely general gauge groups, for our purposes, we shall always be interested in groups with ﬁnite-dimensional unitary representations, i.e. compact groups. We denote these generically by G, and so U(x) G, for any x. This guarantees that bilinears of ψ are ∈ gauge-invariant

ψ¯Γψ ψ¯Γψ Γ= I,γµ,γµν,γ5 . . . (5.12) → Just as in QED, covariant derivatives are again deﬁned so that their transformation law is homogeneous. This requires introducing a non-Abelian gauge ﬁeld Aµ(x), which is an N N matrix so that × D ∂ + iA µ ≡ µ µ (5.13) D0 ψ0(x) U(x) D ψ(x) µ ≡ µ To work out the proper transformation law of Aµ, we need to be careful and take into account that the ﬁelds are now all (non-commuting) matrices.

0 (∂µ + igAµ)U(x)ψ(x)= U(x)(∂µ + iAµ)ψ(x) (5.14)

or

−1 0 U (x)(∂µ + iAµ(x))U(x)ψ(x)=(∂µ + iAµ)ψ(x) (5.15)

0 Working this out, and expressing Aµ in terms of Aµ and U, we have

0 −1 −1 Aµ(x)= U(x)Aµ(x)U (x)+ i(∂µU)(x)U (x) (5.16)

We proceed to compute the analogue of the ﬁeld strength:

[D ,D ]ψ = [∂ + iA , ∂ + iA ]ψ = i(∂ A ∂ A + i[A , A ])ψ (5.17) µ ν µ µ ν ν µ ν − ν µ µ ν and therefore the object analogous to the ﬁeld strength is

F ∂ A ∂ A + i[A , A ] F 0 (x)= U(x)F (x)U −1(x) (5.18) µν ≡ µ ν − ν µ µ ν µν µν Notice that while for QED, the relation between ﬁeld strength and ﬁeld was linear, here it is non-linear. Thus, the requirement of non-Abelian gauge invariance has produced a novel interaction.

75 5.4 Component formulation

We assume that the gauge transformation belong to a Lie group G (compact), and that ψ transforms under a representation T of G, which is unitary. At the level of the Lie algebra, we have the expansion

U(ω )= I + iω T a + O(ω2) a =1,..., dim (5.19) a a G where ω are the generators of the Lie group and T a are the representation matrices of a G in the representation T . They satisfy (T a)† = T a and the structure relations

[T a, T b]= i f abcT c (5.20) where f abc is totally antisymmetric and real.

The transformation laws in components now take the form

δψ(x)= ψ0(x) ψ(x)= iω (x)T aψ(x) − a (5.21) δA (x)= A0 (x) A (x)= T a∂ ω + iω [T c, A ] ( µ µ − µ − µ a c µ Therefore A transforms under the adjoint representation of . The matrix A itself acts µ G µ in the representations T on ψ. Thus, we may decompose Aµ as follows:

A = Aa T a a =1,..., dim (5.22) µ µ G The transformation law of the component is

δAa = ∂ ωa f cbaω Ab (5.23) µ − µ − c µ = (∂ ωa f abcAb ωc) (5.24) − µ − µ Recalling that f abc are the representation matrices for the adjoint representation, and that ωa itself transforms under the adjoint representation, we have

δAa = (D ω)a (D ω)a ∂ ωa f abcAnωc (5.25) µ − µ µ ≡ µ − µ The ﬁeld strength in components is

F F a T a µν ≡ µν (5.26) F a = ∂ Aa ∂ Aa f abcAb Ac ( µν µ ν − ν µ − µ ν Of course, F also transforms under the adjoint rep. of . µν G 76 5.5 Bianchi identities

Dµ is an operator on the space of ﬁelds and therefore must satisfy the Jacobi identity

[Dµ, [Dν,Dρ]] + [Dν, [Dρ,Dµ]] + [Dρ, [Dµ,Dν]]=0 (5.27)

In 4-dimension, this equality is equivalent to

µνρσ [Dν, [Dρ,Dσ]]=0 (5.28)

Using [Dρ,Dσ]= iFρσ, we get the Bianchi identity,

µνρσ µν DνFρσ = Dν F˜ =0 (5.29)

µν It generalizes the case of QED: ∂ν F˜ = 0.

5.6 Gauge invariant combinations

µν tr Fµν F ˜µν ˜µν 1 µναβ (5.30) ( tr Fµν F F Fαβ ≡ 2 To pass to components, use the fact that the Cartan-Killing form is positive-deﬁnite and use the normalization 1 tr T aT b = δab (5.31) 2 We then have

µν 1 a aµν tr FµνF = 2 Fµν F ˜µν µ tr FµνF = ∂µK (5.32) Kµ µαβγ ∂ F 1 A A A = “Chern-Simons term” ∼ α βγ − 3 α β γ When G is a simple group, these invariants are the only ones possible. When G is not simple, it must be of the form of a product of simple groups times a product of U(1) factors.

G = G1 x ... xGp x U(1)1 x ... xU(1)q A ... A B ... B 1µ pµ 1µ qµ (5.33) F1µν ... Fpµν G1µν ... Gqµν g1 ... gp e1 ... eq and we have independent invariants for each:

iµν tr Fiµν F ˜iµν (5.34) ( tr Fiµν F

77 The Abelian factors in general may mix; they can however be diagolized,

q q µν 1 µν CijGiµνGj 2 Giµν Gi (5.35) −→ ei i,jX=1 Xi=1 and for the simple parts

p 1 µν 2 tr FjµνFj (5.36) gj jX=1 5.7 Classical Action & Field equations A suitable action for the Yang-Mills ﬁeld is given by the expression (dimension 4), for each simple group component

1 4 µν θ 4 ˜µν 2 4 S[A, J] 2 d x tr Fµν F + 2 2 d x tr Fµν F d x(JAµ) (5.37) ≡ −2g Z 8π g Z − g Z where J µ J µaT a is a gauge current coupling to the Yang-Mills ﬁeld A and g is a coupling ≡ µ constant. It is often convenient to change normalization of the ﬁeld by factoring out the coupling constant g.

Aµ gAµ 1 −→ F F = ∂ A ∂ A + ig[A , A ] (5.38) g µν −→ µν µ ν − ν µ µ ν The action is then

1 4 µν θ 4 ˜µν 4 µ S[A, J]= d x tr FµνF + 2 d x tr FµνF 2 d x tr J Aµ (5.39) −2 Z 8π Z − Z and its variation is given by

4 µν µ δS = d x tr(F δFµν +2J δAµ) (5.40) − Z To evaluate this variation, one makes use of the following relations,

igFµν = [Dµ,Dν]

δDµ = igδAµ

igδFµ = ig[δAµ,Dν]+ ig[Dµ, δAν] (5.41)

which imply that the variation of the ﬁeld strength is very simple,

δF = D δA D δA (5.42) µν µ ν − ν µ 78 4 µν ν δS = 2 d x tr(F DµδAν + J δAν ) (5.43) − Z

We’d like to integrate by parts, but Dµ is not quite a derivative in the usual sense. Non- theless, all proceeds as if it were.

µν µν tr F DµδAν = tr F (∂µδAν + ig[Aµ, δAν]) = ∂ (tr F µνδA ) tr ∂ F µνδA + ig tr(F µνA δA F µνδA A ) µ ν − µ ν µ ν − ν µ = ∂ (tr F µνδA ) tr δA D F µν (5.44) µ ν − ν µ Hence, we have

4 µν ν δS = 2 d x tr( DµF + J )δAν (5.45) − Z − Thus, the ﬁeld equations are

µν ν DµF = J (5.46)

As a consequence, the current J ν is covariantly conserved. 1 1 D D F µν = [D ,D ]F µν = [F , F µν]=0 (5.47) ν µ 2 ν µ 2 νµ

ν ν ν Dν J = ∂ν J + ig[Aν,J ]=0 (5.48)

5.8 Lagrangians including fermions and scalars

The basic Lagrangian for a gauge ﬁeld Aµ associated with a simple gauge group G, in the presence of “matter ﬁelds”

fermions in representations T and T of G; • L R scalars in representation T of G. • S Then the Lagrangians of relevance are of the form 1 = tr F F µν + ψ¯ γµ(i∂ gAa T a)ψ + ψ¯ γµ(i∂ gAa T a)ψ L −2g2 µν L µ − µ L L R µ − µ R R +D φ+Dµφ V (φ) (5.49) µ − where V (φ) is invariant under G and

D φ ∂ φ + igAa T aφ (5.50) µ ≡ µ µ s 79 Actually, when T ? T T contains one or more singlets, one may also have Yukawa L ⊗ R ⊗ S terms

+ λ(ψ¯LφψR + ψ¯Rφ ψL) (5.51)

In addition, if T ? T contains singlets, one may also have a “Dirac mass term” L ⊗ R

M(ψ¯LψR + ψ¯RψL) (5.52)

Finally, if TL = TR, then the theory is parity conserving, while it is parity violating when T = T . L 6 R 5.9 Examples

1. QCD (ignoring the weak interactions); G = SU(3)c

ψt =(udcstb)c (5.53)

c = color SU(3) • each ﬂavor is in fundamental representation SU(3) • T = T =3 . . . 3 = T • L R ⊕ ⊕ 6 times | {z } 6 6 1 µν µ a a = 2 tr Fµν F + q¯f γ (i∂µ g3AµT )qf mf q¯f qf (5.54) L −2g3 − − fX=1 fX=1

2. SU(2)L weak interactions U(1)Y (no QCD);

u e− q = ` = ` = e− (5.55) d L ν R R ! e !L

1 µν µ a a q µ = 2 tr FµνF +¯qLγ (i∂µ g2AµTL g1BµYL )qL +¯qRγ (i∂µ g1BµYR)qR L −2g2 − − − +D φ+D φ + `¯ γµ(i∂ g Aa T a g Y `B )` + `¯ γµ(i∂ g B )` µ µ L µ − 2 µ − 1 L µ L R µ − 1 µ R V (φ+φ) q¯ φq `¯ φ` q¯ φq˜ (5.56) − − L R − L R − L R

80 6 The S-matrix and LSZ Reduction formulas

So far we have only discussed ﬁelds, but we are interested in the interaction of particles. However remember from the introduction that the positions and the momenta of the particles are not directly observable. In particular the position can never be observed to arbitrary precision due to the occurrence of anti-particles. On the other hand, momenta of particles are observable provided one observes the particles asymptotically as ∆t , →∞ then ∆p 0. → Hence we will be interested in observing the dependence on the momenta of incoming and outgoing particles. To do this we idealize the interaction.

Interaction regio n In Out

Figure 5: Idealized view of the process of free initial, interacting and free ﬁnal particles

The basic assumption is that as t , the particles behave as free particles. Is this → ±∞ true for the interactions that we know of ?

1. Strong nuclear forces at internuclear distances are governed by the Yukawa potential, 1 exp r m (6.1) ∼ r {− π} These are so-called short-ranged and eﬀectively vanish beyond the size of the nucleus. This is why eﬀectively, in day to day life, we do not see their eﬀects. For short ranged forces, the idealized view should hold.

2. The weak forces are also short ranged, 1 1 exp r M ± or exp r M 0 (6.2) ∼ r {− W } ∼ r {− Z }

81 3. The electro-magnetic force falls oﬀ at inﬁnity as well, 1 (6.3) ∼ r2 By contrast with the weak and strong forces however, this fall-oﬀ is only power-like and such forces are referred to as long-ranged. This means that electro-magnetic ﬁelds can extend over long distances and will not be conﬁned to some small interac- tion region. For example, the galactic magnetic ﬁelds extend over many parsecs. The idealized picture may or may not hold. We should foresee diﬃculties at low energies and low momenta (so-called IR propblems) when we apply the idealized view here.

4. The force of gravity behaves in the same manner as the electro-magnetic force and is also long-ranged. Fortunately, the force of gravity is not on the menu for this course.

5. The most problematic of all – and the most interesting – is QCD which describes the strong forces at a fundamental level. The force between two quarks is constant, even for large distances (ignoring quark pair creation) and the quarks are conﬁned, as it would require an inﬁnite amount of energy to separate two quarks that are subject to a law of attraction that is constant. This eﬀect is so dramatic that the spectrum is altered to a hadronic spectrum.

6.0.1 In and Out States and Fields

If at t the particles are free, then we better introduce a free ﬁeld for them, which → −∞ I are the so-called in-ﬁeld, denoted generically by Φin(x). The Fock space associated with Φ is , and the states are labelled exactly as the free ﬁeld states were. If at t + in Hin → ∞ the particles are free, we introduce free out-ﬁelds ΦJ (x) and the Fock space is . out Hout Of course we also have the Hilbert space of the full quantum ﬁeld theory, which we denote by . Axiomatic ﬁeld theory requires = = . Here, however, we shall H Hin Hout H attempt to deﬁne these diﬀerent theories, to link them and then to check the nature of the various Hilber spaces. Obviously, we should have the property = , which is called Hin Hout asymptotic completeness. The n-particle in-states created by in ﬁelds are denoted by

in, p , ,p ; I , , I (6.4) | 1 ··· n 1 ··· ni with particle momenta p , ,p and internal indices I , , I . The out-states created 1 ··· n 1 ··· n by the out ﬁels are of the form

out, q ...q , J J (6.5) | 1 m 1 ··· mi 82 where the m-particle out state has particle momenta q1 ...qm, and internal indices

J1 ...Jm. Note that it makes sense to label states in this way since momenta are ob- servable to arbitrary precision in the t = to limit. ±∞

6.0.2 The S-matrix

The fundamental question in scattering theory and in quantum ﬁeld theory is going to be as follows. Given an initial in-state (the state of the system at t ), → −∞ in p ,...p ; I ,...I (6.6) | 1 n 1 ni what is the probability (given the dynamics of the theory) for the ﬁnal out-state (the state at t + ) to be → ∞ out q ...q ; J , ,J (6.7) | 1 m 1 ··· mi The answer is given by the inner product of the two states, as always

S = out q ...q ; J J in p ,...p , I I (6.8) f←i h 1 m 1 ··· m| 1 n 1 ··· ni and the probability is

W = S 2 (6.9) f←i | f←i| Clearly, this question can be asked for every state in onto every state in . Assuming Hin Hout asymptotic completeness, = , then there should exist a matrix S that relates the Hin Hout in-states to the out-states,

out = S in (6.10) | i | i The equivalence of the two Hilbert spaces requires that S be unitary. This may also be seen from the fact that for a given initial state, a summation over all possible ﬁnal states must give unit probability. Thus, the S-matrix must be unitary ! Consequently, we may describe all these overlap probabilities in terms of in or out states and the S-matrix. | i | i

6.0.3 Scattering cross sections

From the probability, one can compute the thing the experimental high energy physicist talks about all day and dreams about: the scattering cross section. We shall limit ourselves to an initial state with only 2 particles.

83 The incoming state is now in general a wave packet with distribution ρ, so 3 3 d p1 d p2 in = 3 o 3 o ρ1(p1)ρ2(p2) in p1,p2 (6.11) | i Z (2π) 2p1 Z (2π) 2p2 | i This wave packet is associated with a distributionρ ˜(x) (for a free particle) and positive energy 3 d p −ip·x ρ˜(x)= 3 e ρ(p) (6.12) Z (2π) 2ωp The ﬂux is given by 3 d p 2 # of particles Φρ 3 o ρ(p) = (6.13) ≡ Z (2π) 2p | | unit of time To isolate the interaction part from the free part, it is useful to decompose the S-matrix into an identity part and a T matrix part, describing interaction S = I + iT (6.14) Energy momentum conservation then implies that f T p ,p = (2π)4δ4(P p p ) f p ,p (6.15) h | | 1 2i f − 1 − 2 h |T | 1 2i In general, one will be interested in scattering cross sections for which f i = 0. (If h | i i = f , one deals with so-called forward scattering, and we shall not deal with it here.) | i | i In the above case f S i = f i i f T i = i f T i (6.16) h | | i h | i − h | | i − h | | i and W = f S i 2 = f T i 2 (6.17) f←i |h | | i| |h | | i|

0 0 ∗ 0 ∗ 0 4 Wf←i = dp1 dp2 dp1 dp2 ρ1(p1)ρ1(p1)ρ2(p2)ρ2(p2)(2π) δ(p1 + p2 Pf ) Z Z Z Z − (2π)4δ(p + p p0 p0 ) f p ,p ∗ f p0 ,p0 (6.18) × 1 2 − 1 − 2 h |T | 1 2i h |T | 1 2i Using now the fact that

(2π)4δ4(k)= d4xe−ikx (6.19) Z we may represent the second momentum conservation δ-function as a Fourrier integral,

4 0 0 ∗ 0 ∗ 0 Wf←i = d x dp1 dp2dp1 dp2ρ1(p1)ρ1(p1)ρ2(p2)ρ2(p2) Z Z Z Z 0 0 (2π)4δ4(p + p P )e−ip1+p2−p1−p2)·x f p ,p ∗ f τ p0 ,p0 (6.20) × 1 2 − f h |T | 1 2i h | | 1 2i Now we assume that the distributions ρ1 and ρ2 are very narrow, and sharply peaked

about momentap ¯1 andp ¯2. We assume that the variation of the ρ’s is so small over this

84 distribution that the matrix elements of are essentially constant over that variation. We T then have

f p ,p f p0 ,p0 f p¯ , p¯ (6.21) h |T | 1 2i∼h |T | 1 1i'h |T | 1 2i Hence

4 2 2 4 4 2 Wf←i = d x ρ˜1(x) ρ˜2(x) (2π) δ (¯p1 +¯p2 Pf ) f p¯1, p¯2 (6.22) Z | | | | − |h |T | i| Now to get the cross section, we want the transition probability per unit volume and per unit time: dW p←i = ρ˜ (x) 2 ρ˜ (x) 2(2π)4δ4(¯p +p ¯ P ) f p¯ , p¯ 2 (6.23) dtdV | 1 | | 2 | 1 2 − f |h |T | 1 2i|

Now the relative ﬂux between distributions ρ1 and ρ2 is given by

4 m γ p m γ p f˜ (x) 2 f˜ (x) 2 (6.24) | 2 2 1 − 1 1 2|| 1 | | 2 | and therefore, the diﬀerential cross section is given by

1 dσ = (2π)4δ4(P p¯ p¯ ) f p¯ , p¯ 2 (6.25) f − 1 − 2 2E 2E ~v ~v |h |T | 1 2i| 1 2| 1 − 2| The normalization holds for scalar, photons and spin 1/2 particles all the same!

6.1 Relating the interacting and in- and out-ﬁelds

The quantum ﬁeld theory is formulated in terms of interacting ﬁelds φ(x) instead of the free ﬁelds φin(x) and φout(x). To compute elements of the S matrix, we need to connect the states in and out to the interacting ﬁelds in the theory. We carry this out here for | i | i the case of a single scalar ﬁeld φ. Other cases are completely analogous.

The in and out-ﬁelds satify the free ﬁeld equations

2 (2 + m )φin(x) = 0 2 (2 + m )φout(x)=0 (6.26)

as well as the canonical commutation relations

[φ˙ (x), φ (y)]δ(x0 y0) = i δ(x y) in in − − − 85 [φ˙ (x), φ (y)]δ(x0 y0) = i δ(x y) (6.27) out out − − − The in and out states are now built with the help of the Fourier mode operators:

in p , ,p = a† (~p )a† (~p ) a† (p ) 0 | 1 ··· ni in 1 in 2 ··· in n | i out q , , q = a† (~q )a† (~q ) a† (q ) 0 (6.28) | 1 ··· mi out 1 out 2 ··· out m | i Recall the normalizations of the free in and out-ﬁelds with the 1-particle states,

in p φ (x) 0 = eip·x h | in | i out q φ (x) 0 = eiq·x (6.29) h | out | i Note that the normalization on the rhs is 1.

As remarked above, the dynammics of the theory is described by neither φin nor φout,

but rather by the interacting ﬁeld φ. In some sense, the ﬁeld φ should behave like φin or φ according to whether t or t + . out → −∞ → ∞ To see concretely how this happens, consider the overlap of the ﬁeld φ with the 1- particle states. Accoding to general principles, the ﬁeld φ should have non-zero overlap with in p . The x- and p-dependence of this overlap is determined by Poincar´esymmetry | i in the following manner,

in p φ(x) 0 = in p eix·P φ(0)e−ix·P 0 = eip·x in p φ(0) 0 h | | i h | | i h | | i = √Zeip·x (6.30)

Z is indepedent of p. We therefore conclude that

in p φ(x) 0 = √Z in p φ (x) 0 h | | i h | in | i out p φ(x) 0 = √Z out p φ (x) 0 (6.31) h | | i h | in | i Therefore, we have – in a somewhat formal sense – the following limits

lim φ(x) = √Zφin(x) t→−∞

lim φ(x) = √Zφout(x) (6.32) t→+∞

This limit cannot really be understood in the sense of a limit of operators, but rather must be viewed as a limit that will hold when matrix elements are taken with states containing a ﬁnite number of particles.

86 6.2 The K¨allen-Lehmann spectral representation To evaluate Z, we construct the K¨allen-Lehmann representation for the interacting theory. The starting point is the commutator,

i ∆(x, x0)= 0 [φ(x), φ(x0)] 0 (6.33) h | | i Let us collectively denote all the states of the theory n . Then, using completeness of the | i states n , we insert a complete set of normalized states, | i i ∆(x, x0)= 0 φ(x) n n φ(x0) 0 0 φ(x0) n n φ(x) 0 (6.34) n h | | ih | | i−h | | ih | | i X Translation symmetry of the theory allows us to pull out the x-dependence using φ(x)= eix·P φ(0)e−ix·P so that

0 0 i ∆(x, x0)= 0 φ(0) n 2 e−ipn·(x−x ) eipn·(x−x ) (6.35) n |h | | i| − X Using 1 = d4qδ(q p ), we have − n R 4 0 0 d q −iq(x−x0) iq(x−x0) i ∆ (x, x )= 3 ρ(q) e e (6.36) Z (2π) − where the spectral density is deﬁned to be

3 4 2 ρ(q) (2π) δ (q pn) 0 φ(0) n (6.37) ≡ n − |h | | i| X Notice that ρ is positive, and vanishes when either q2 < 0 or q0 < 0. Also it is Lorentz invariant, and therefore can only depend upon q2. Thus, it is convenient to introduce the function σ(q2) as follows,

ρ(q)= θ(q0)σ(q2) σ(q2)=0 if q2 < 0 (6.38)

Recall the free commutator,

4 2 d q 0 2 2 −iq(x−y) iq(x−y) i ∆0(x y, m )= 3 θ(q )δ(q m ) e e (6.39) − Z (2π) − − Hence we may express the commutator of the fully interacting ﬁelds as,

∞ 0 2 2 0 2 ∆(x x )= dµ σ(µ )∆0(x x ; µ ) (6.40) − Z0 − This is the K¨all´en-Lehmann representation for the commutator.

87 0 m 2 m 2 t

Figure 6: Support of the spectral function σ(µ2)

The contribution to the spectral density due to the one particle state is given in terms of the constant Z. The remaining contributions come from states with more than 1 particle, and may be summarized as follows,

∞ ∆(x x0)= Z∆ (x x0, m2)+ dµ2σ(µ2)∆ (x x0; µ2) (6.41) 0 2 0 − − Zmt −

2 Here, mt is the mass-squared at which the remaining spectrum starts. If the interactions in the theory are very weak, the particles will behave almost as free particles. If we further assume that no bound states occur in the full spectrum (but even at weak coupling this assumption may be false, see the hydrogen atom), then we can give a general characterization of the spectral function and the nature of the thresholds of σ(µ2). A state with n (nearly) free particles starts contributing to σ(µ2) provided

2 2 µ = (p1 + p2 . . . + pn) = (po + po + . . . + po )2 (~p + ~p + . . . + ~p )2 n2m2 (6.42) 1 2 n − 1 2 n ≥ Thus, under the assumptions made here, there will be successive thresholds starting at 4m2, 9m2, 16m2 etc. In most theories however, the spectrum may be much more complicated. To obtain Z, it suﬃcies to take the x0 derivative at both sides and set x0 = x00. This produces the canonical commutators, as follows,

∞ 3 0 3 0 2 2 3 0 δ (~x ~x )= Z δ (~x ~x )+ 2 dµ σ(µ )δ (~x ~x ) (6.43) − − Zmt − and identifying the coeﬃcients of δ(~x ~x0), we get − ∞ 1= Z + dµ2σ(µ2) (6.44) 2 Zmt Since σ(µ2) 0, any interacting theory will have ≥ 0

In particular, Z = 1 is possible only when φ = φin is a free ﬁeld. On the other hand, Z =0 would mean that the ﬁeld φ does not create the 1-particle state, which is in contradiction with one of our basic assumptions.

88 One may also obtain a K¨allen-Lehmann representation for the two-point function in terms of similar methods, ∞ 0 2 2 2 0 Tφ(x)φ(x ) 0 = dµ σ(µ )GF (x y,µ ) (6.46) h | | i Z0 − Notice that the ﬁeld φ which governs the dynammics and which obeys canonical com- mutation relations is not the ﬁeld that obeys a canonical normalization on one-particle states. There is a multiplicative factor between the two ﬁelds. This is the ﬁrst non-trivial example of the phenomenon of renormalization: the parameter that govern the dynamics in the theory like the ﬁeld, the mass, the coupling constants are not the parameters that are observable in the asymptotic regime. Later on it will be convenient to work directly with this renormalized ﬁeld 1 φR(x) φ(x) (6.47) ≡ √Z

Of course the ﬁeld φR will no longer satisfy canonical commutation relations.

6.3 The Dirac Field A similar treatment is available for the Dirac ﬁeld ψ(x). We restrict to quoting the results here. The normalizations of the in and out ﬁelds and of the interacting ﬁeld are as follows, 0 ψ (x) in p,λ = u(p,λ)e−ip·x h | in | i 0 ψ(x) in p,λ = Z 0 ψ (x)in p,λ h | | | i 2h | in i q = Z 0 ψ (x) out p,λ (6.48) 2h | out | i q The K¨allen-Lehmann representation is obtained by introducing the spectral density β 3 4 ¯β ρα(q)=(2π) δ (pn q) 0 ψα(0) n ψ (0) 0 (6.49) n − h | | ih | i X so that, using Lorentz invariance, one gets

β 0 2 β 2 β ρα(q)= θ(q ) σ1(q )( q)α + σ2(q )δα (6.50) 6 The result is most easily expressed in terms of the free scalar propagator, ∞ β 0 2 2 2 β 0 2 0 ψα(x)ψ¯ (x ) 0 = dµ iσ1(µ ) ∂x + σ2(µ ) GF (x x ; µ ) (6.51) h { }| i Z0 6 α − Properties of σ and σ are as follows, σ (µ2) 0, σ (µ2) 0, and µσ (µ2) σ (µ2) 0. 1 2 1 ≥ 2 ≥ 1 − 2 ≥ The spectral representation for Z2 is obtained again via the equal time commutators, ∞ i 0 ψ (x)ψ¯β(x0) 0 = Z S(x x0)+ dµ2 iσ (µ2) ∂ + σ (µ2) βG (x x0; µ2) α 2 2 1 x 2 F h |{ }| i − Zmt 6 α − and we get ∞ 1= Z + dµ2σ (µ2) (6.52) 2 2 1 Z4m

89 6.4 The LSZ Reduction Formalism

It is now our task to evaluate the S-matrix elements in terms of quantities which can be expressed in terms of the dynammics of the QFT. We shall treat in detail here the case of the real scalar ﬁeld; the other cases may be handled analogously. The key object of interest is the transition amplitude

out q q in p p (6.53) h 1 ··· m| 1 ··· ni

We shall work out a recursion formula which decreases the number of particles in the in and out states and replaces the overlap with expectation values of the fully interacting

quantum ﬁeld φ. To this end, we isolate the in-particle with momentum pn, and write

out q q in p p = out β in α p h 1 ··· m| 1 ··· ni h | ni in α = in p p | i | 1 ··· n−1i out β = out q q | i | 1 ··· mi in p p = a† (p ) in α (6.54) | 1 ··· ni in n | i

6.4.1 In- and Out-Operators in terms of the free ﬁeld

The decomposition into creation and annihilation operators of a free scalar ﬁeld ϕ(x) with

mass m and its canonical momentum ∂0φ(t, ~x) are given in terms by

3 d k ~ −k·x † ~ ik·x ϕ(t, ~x) = 3 a(k)e + a (k)e (2π) 2ωk Z d3k ∂ ϕ(t, ~x) = ia(~k)e−ik·x + ia†(~k)eik·x (6.55) 0 (2π)32 − Z The creation and annihilation operators may be recovered by by taking the space Fourier transforms

↔ † ~ 3 −ik·x a (k) = i d x e ∂ 0 ϕ(x) − Z ↔ ~ 3 +ik·x a(k) = +i d x e ∂ 0 ϕ(x) (6.56) Z

↔ ↔ Here, the symbol ∂ stands for the antisymmetrixed derivative u ∂ 0 v u∂0v ∂0uv. The ≡ − left hand side is independent of the time coordinate x0 at which the ﬁeld ϕ(x) is evaluated on the left hand side as long as the momentum satisﬁes k2 = m2.

90 6.4.2 Asymptotics of the interacting ﬁeld Next, we make use of the fact that, when evaluated between states, where it is to play the role of creation operator, the ﬁeld tends towards free ﬁeld expectation values, 1 − 2 0lim out φ(x) in Z out φout(x) in x →+∞h | | i ∼ h | | i 1 lim out φ(x) in Z− 2 out φ (x) in (6.57) x0→−∞h | | i ∼ h | in | i Combining this result with the free ﬁeld formulas for the creation operators for in- and out-operators, we obtain (omitting now the in- and out-states matrix elements), i ↔ a† (~k) = lim d3x e−ik·x φ(x) in 0 ∂ 0 −√Z x →−∞ Z i ↔ a† (~k) = lim d3x e−ik·x φ(x) out 0 ∂ 0 −√Z x →+∞ Z i ↔ a (~k) = + lim d3x e+ik·x φ(x) in 0 ∂ 0 √Z x →−∞ Z ↔ ~ i 3 +ik·x aout(k) = + 0lim d x e ∂ 0 φ(x) (6.58) √Z x →+∞ Z Using the fact that the diﬀerence between the same quantity evaluated at x0 may → ±∞ be expressed as an integral, +∞ 0 0lim ψ(x) 0lim ψ(x)= dx ∂0ψ(x) (6.59) x →+∞ − x →−∞ Z−∞ we obtain the following diﬀerence formulas between in- and out-operators, ↔ † ~ † ~ i 4 −ik·x ain(k) aout(k) = + d x∂0 e ∂ 0 φ(x) − √Z Z ↔ ~ ~ i 4 +ik·x ain(k) aout(k) = d x∂0 e ∂ 0 φ(x) (6.60) − −√Z Z These combinations admit further simpliﬁcations which may be seen by working out the integrand and using the fact that k2 = m2, ↔ 4 ±ik·x 4 0 2 2 ±ik·x d x∂0 e ∂ 0 φ(x) = d x (k ) φ(x)+ ∂0 φ(x) e Z Z = d4xe±ik·x(2 + m2)φ(x) (6.61) Z Thus, we obtain the following formulas, † ~ † ~ i 4 −ik·x 2 2 ain(k) aout(k) = + d x e ( + m )φ(x) − √Z Z ~ ~ i 4 +ik·x 2 2 ain(k) aout(k) = d x e ( + m )φ(x) (6.62) − −√Z Z Clearly, for a free ﬁeld, satisfying (2 + m2)φ = 0, there is no diﬀerence between the in- and out-operators, but as soon as interactions are turned on, they are no longer equal.

91 6.4.3 The reduction formulas The next step is the evaluation of the operator diﬀerence (6.62) between the states in α = | i in p p and out β = out q q , | 1 ··· n−1i | i | 1 ··· mi

† † i 4 −iqm·x 2 out β a (~qm) a (~qm) in α = d x e (2 + m ) out β φ(x) in α (6.63) h | in − out | i √Z h | | i Z Thus, we have

out p p in q q h 1 ··· n| 1 ··· mi = out p p a† (~q ) in q q (6.64) h 1 ··· n| out m | 1 ··· m−1i i 4 −iqm·x 2 + d x e (2 + m ) out p1 pn φ(x) in q1 qm−1 √Z Z h ··· | | ··· i The ﬁrst term on the rhs vanishes, unless it happens to be that out p p contain h 1 ··· n| precisely the particle with momentum ~qm, in which case the amplitude contributes with strength unity. Therefore, this term corresponds to a disconnected contribution to the scattering process (see Fig.7). The disconnected contributions simply reduce to S-matrix elements with fewer particles and may be handled recursively.

=

Figure 7: Contribution to the S-matrix with disconnected part.

Henceforth, we concentrate on the connected contributions, denoted with the corre- sponding suﬃx. The reduction formula for peeling oﬀ a single particle thus reduces to

out p1 pn in q1 qm conn (6.65) h ··· | ··· i i 4 −iqm·x 2 = d x e (2 + m ) out p1 pn φ(x) in q1 qm−1 √Z Z h ··· | | ··· i Clearly, the process may be repeated on all incoming particles, until all incoming particles have been removed,

out p1 pn in q1 qm conn (6.66) h ··· | ··· i i m m 4 −iqj ·yj 2 2 = d yj e ( + m )yj out p1 pn φ(y1) φ(ym) in 0 √Z ! h ··· | ··· | i jY=1 Z where the state in 0 stands for the in-state with zero in-particles. | i 92 It remains to remove also the out-particles. However, now there is a new issue to be dealt with. The removal of the out particles must always occur to the left of the removal of the in-particles, since the two operator types will not have simple commutation relations. This ordering is achieved by ordering the later times to the left of the earlier times, since out-states correspond to x0 + while in-states correspond to x0 . Since out- → ∞ → −∞ states will have to be removed using annihilation operators, there is also an adjoint to be taken. Thus, the ﬁnal formula is

out p1 pn in q1 qm conn (6.67) h ··· | ··· i im( i)n n 4 +ipk·xk 2 2 = −m+n d xk e ( + m )xk √Z kY=1 Z m d4y e−iqj ·yj (2 + m2) out 0 Tφ(x ) φ(x )φ(y ) φ(y ) in 0 × j yj h | 1 ··· n 1 ··· m | i jY=1 Z The key lesson to be learned from this formula is that the calculation of any scattering matrix element requires the calculation in QFT of only a single type of object, namely the time ordered correlation function, or correlator,

0 Tφ(x ) φ(x ) 0 (6.68) h | 1 ··· n | i Much of the remainder of the subject of quantum ﬁeld theory will be concerned with the calculation of these quantities.

6.4.4 The Dirac Field

The derivation for the Dirac ﬁeld is completely analogous, though the precise 1-particle wave functions are of course those associated with the Dirac equation instead of with the Klein-Gordon equation. Furthermore, one will have to be careful about taking the anti-commutative nature of the ﬁeld ψ into account. For example, for in-ﬁelds we have

3 d k −ik·x † ik·x ψin(x) = 3 bin(k,λ)u(k,λ) e + din(k,λ)v(k,λ) e (6.69) (2π) 2ωk Z λX=± which gives rise to the following expressions for the in- and out-operators,

3 ik·x 0 bin(k,λ) = d xu¯(k,λ) e γ ψin(x) Z † 3 −ik·x 0 din(k,λ) = d xv¯(k,λ) e γ ψin(x) Z † 3 ¯ 0 −ik·x bin(k,λ) = d xψ(x)γ e u(k,λ) Z 3 ¯ 0 −ik·x din(k,λ) = d xψ(x)γ e v(k,λ) (6.70) Z 93 The 1-particle state normalizations with the in-ﬁeld ψin and with the fully interacting ﬁeld ψ are as follows, 0 ψ (x) in p λ = u(p,λ)e−ip·x h | in | i 0 ψ(x) in p λ = Z u(p,λ)e−ip·x (6.71) h | | i 2 q You see that one can get a similar formula as for the bosonic scalar case, but now you have to be careful about where the spinor indices are contracted. Clearly, everything is again expressed in terms of time-ordered correlators for the fully interacting fermion ﬁelds, 0 T ψ (x )ψ (x ) ...ψ (x )ψ¯ (y ) . . . ψ¯ (y ) 0 (6.72) h | α1 1 α2 2 αn n β1 1 βp p | i 6.4.5 The Photon Field Since the lontgitudinal photons are free, we are only interested in scattering transverse µ photons. kµ (k)=0 β(k )out α in = β out α (k) in h 1 | i | − i + β out aout(k) aout(k) ain(k) α in ↔ h 3 |ikx· − · T −out · | Riin = i d xe ∂ o β out (k) A (x) (k)A (x) α in − t h | · − | i Z ↔ 1/2 3 ikx T = i Z3 d xe ∂ o β out (k) A (x) α in − t→∞ h | · | i Z ↔ −1/2 3 ikx T + i Z3 d xe ∂ o β out (k)A (x) α in Zt→−∞ h | | i −1/2 4 ikx 2 T = i Z3 d x e ∂o β out (k) A (x) α in − Z h | · | i (∂2eikx) β outn (k) AT (x) α in − o h | · | i = i Z−1/2 d4x eikx(2 + µ2) β outo (k) AT (x)α in (6.73) − 3 xh | · i Z n o Now: 1 AT (x)= A (x)+ ∂ ∂ A (6.74) µ µ m2 µ ·

1 µ2 (2 + µ2) AT (x) = (2 + µ2)A + 2∂ (∂ A)+ ∂ ∂ A x µ µ m2 µ · m2 µ · = (2 + µ2)A ∂ (∂ A)+ λ∂ (∂ A) µ − µ · µ · = jµ , our best friend! (6.75) Hence, in the non-forward regime, we have the simple formula:

1/2 4 ikx µ β(k, ) out α in == i Z3 d xe β out (k)jµ(x) α in (6.76) h | i Z h | | i Now let’s play this game over with two photons: β(k , )out α(k , )in h f f | i i i 94 = Z−1 d4x d4yei(kf x−kiy)(2 + µ2) − 3 y Z Z 1 β out T f j(x) i (A(y)) + 2 ∂(∂A)(y) α in (6.77) h | · · m | i You see that this is not just the time ordered product of the currents, there will be some additional terms, arising however only at xo = yo. By locality, the contribution must now be entirely localized at ~x = ~y = ~y as well. One can show, after a lot of work that one exactly gets the T ∗ product. What else could it be?

β(k , ) out α(k , ) in h f f | i i i = Z−1 d4x d4yei(kf x−kiy) − 3 β outZT ∗ Zj(x) j(y) α in (6.78) h | f · i · | i It is straightforward to generalize this formula.

95 QUANTUM FIELD THEORY – 230B

Eric D’Hoker

Department of Physics and Astronomy University of California, Los Angeles, CA 90095

Contents

1 Functional Methods in Scalar Field Theory 2 1.1 Functional integral formulation of quantum mechanics ...... 2 1.2 The functional integral for scalar ﬁelds ...... 3 1.3 Time ordered correlation functions ...... 4 1.4 The generating functional of correlators ...... 5 1.5 The generating Functional for free scalar ﬁeld theory ...... 5

2 Perturbation Theory for Scalar Fields 8 2.1 Scalar Field Theory with potential interactions ...... 8 2.2 Feynman rules for φ4-theory ...... 10 2.3 The cancellation of vacuum graphs...... 11 2.4 General Position Space Feynman Rules for scalar ﬁeld theory ...... 12 2.5 Momentum Space Representation ...... 13 2.6 Does perturbation theory converge ? ...... 14 2.7 The loop expansion ...... 15 2.8 Truncated Diagrams ...... 15 2.9 Connected graphs ...... 16 2.10 One-Particle Irreducible Graphs and the Eﬀectice Action ...... 17 2.11 Schwinger Dyson Equations ...... 19 2.12 The Background Field Method for Scalars ...... 20

3 Basics of Perturbative Renormalization 22 3.1 Feynman parameters ...... 22 3.2 Integrations over internal momenta ...... 23 3.3 The basic idea of renormalization ...... 24 3.3.1 Mass renormalization ...... 25 3.3.2 Coupling Constant Renormalization ...... 26 3.3.3 Field renormalization ...... 28 3.4 Renormalization point and renormalization conditions ...... 30 3.5 Renormalized Correlators ...... 31 3.6 Summary of regularization and renormalization procedures ...... 33

4 Functional Integrals for Fermi Fields 35 4.1 Integration over Grassmann variables ...... 36 4.2 Grassmann Functional Integrals ...... 37 4.3 The generating functional for Grassmann ﬁelds ...... 38 4.4 Lagrangian with multilinear terms ...... 39

5 Quantum Electrodynamics : Basics 40 5.1 Feynman rules in momentum space ...... 40 5.2 Verifying Feynman rules for one-loop Vacuum Polarization ...... 41 5.3 General fermion loop diagrams ...... 43 5.4 Pauli-Villars Regulators ...... 43 5.5 Furry’s Theorem ...... 43 5.6 Ward-Takahashi identities for the electro-magnetic current ...... 44 5.7 Regularization and gauge invariance of vacuum polarization ...... 46 5.8 Calculation of vacuum polarization ...... 47 5.9 Resummation and screening eﬀect ...... 49 5.10 Physical interpretation of the eﬀective charge ...... 50 5.11 Renormalization conditions and counterterms ...... 51 5.12 The Gell-Mann – Low renormalization group ...... 52

6 Quantum Electrodynamics : Radiative Corrections 54 6.1 A ﬁrst look at the Electron Self-Energy ...... 54 6.2 Photon mass and removing infrared singularities ...... 56 6.3 The Electron Self-Energy for a massive photon ...... 57 6.4 The Vertex Function : General structure ...... 57 6.5 Calculating the oﬀ-shell Vertex ...... 59 6.6 The Electron Form Factor and Structure Functions ...... 61 6.7 Calculation of the structure functions ...... 62 6.8 The One-Loop Anomalous Magnetic Moment ...... 64

2 7 Functional Integrals For Gauge Fields 66 7.1 Functional integral for Abelian gauge ﬁelds ...... 66 7.2 Functional Integral for Non-Abelian Gauge Fields ...... 68 7.3 Canonical Quantization ...... 69 7.4 Functional Integral Quantization ...... 71 7.5 The redundancy of integrating over gauge copies ...... 72 7.6 The Faddeev-Popov gauge ﬁxing procedure ...... 74 7.7 Calculation of the Faddeev-Popov determinant ...... 77 7.8 Faddeev-Popov ghosts ...... 78 7.9 Axial and covariant gauges ...... 78

8 Anomalies in QED 80 8.1 Massless 4d QED, Gell-Mann–Low Renormalization ...... 80 8.2 Scale Invariance of Massless QED ...... 81 8.3 Chiral Symmetry ...... 82 8.4 The axial anomaly in 2-dimensional QED ...... 83 8.5 The axial anomaly and regularization ...... 85 8.6 The axial anomaly and massive fermions ...... 87 8.7 Solution to the massless Schwinger Model ...... 88 8.8 The axial anomaly in 4 dimensions ...... 89

3 1 Functional Methods in Scalar Field Theory

The essential information of a quantum ﬁeld theory is contained in its time ordered cor- relation functions. From these, the scattering matrix elements, the transition amplitudes and probabilities for any process may be computed. In the preceding sections we have evaluated the correlation functions exactly for the free ﬁeld theories containing scalar, spin 1/2 and spin 1 ﬁelds. The evaluation of the correlation functions in any interacting theory is a very diﬃcult problem that can only rarely be carried out exactly. Frequently, one will have to resort to an approximate or perturbative treatment, for example based on an expansion in powers of the coupling constants. Even the perturbative series is quite complicated because the combinatorics as well as the space-time properties give rise to involved problems. Historically, perturbative calculations were ﬁrst carried out in the operator formalism of QFT, by recasting each of the expansion coeﬃcients in terms of free ﬁeld theory correlation functions. For scalar ﬁeld theory problems, these calculations were involved but manageable. For gauge theories, however, the operator methods very quickly become unmanageable. For non-Abelian Yang-Mills theory, operator methods are seldom useful. The functional integral formulation (already derived for quantum mechanics in SA2) provides an alternative way of dealing with QFT. Historically, the functional integral for- mulation of QFT was regarded foremost merely as an eﬃcient way of organizing perturba- tion theory. Its indispensible value was established permanently, however, with the advent of non-Abelian Yang-Mills theory. Its direct connection with statistical mechanics led Wil- son and others to use the functional integral formulation as a non-perturbative formulation of QFT, and to a much deeper understanding of the properties of renormalization.

1.1 Functional integral formulation of quantum mechanics In SA.2.6, the functional integral formulation of quantum mechanics was derived for a Hamiltonian H(p, q) and the following formula was obtained, Z Z Z t hq0|e−itH |qi = Dp Dq exp i dt0 pq˙ − H(p, q) (t0) (1.1) 0 with the boundary conditions q(t0 = 0) = q and q(t0 = t) = q0, and free boundary conditions on p(t0). Recall that the measure arose from a discretisation of the time interval [0, t], for t > 0, into N segments [ti, ti+1], t0 = 0, tN = t, i = 0,N − 1, and was formally deﬁned by N−1 dp(t )dq(t ) DpDq ≡ lim Y i i (1.2) N→∞ i=1 2π

4 Notice that the intermediate times are ordered, t = t0 < t1 < ··· < tN−1 < tN = t, so that the integral is really over paths rather than over functions.

1.2 The functional integral for scalar ﬁelds We now generalize the above discussion to the case of one real scalar ﬁeld φ, thus to an inﬁnite number of degrees of freedom. We denote the momentum canonically conjugate to φ by π and the Hamiltonian by H[φ, π]. The analogue of the position basis of states |qi and |pi is given by |φ(~x)i and by |π(~x)i. Remark that these states do not depend on ~x; instead they are functionals of φ. The completeness relations are now Z I = Dφ(~x) |φ(~x)ihφ(~x)| Z I = Dπ(~x) |π(~x)ihπ(~x)| (1.3)

Clearly, the derivation is formal and we shall postpone all issues of cutoﬀ and renormal- ization until later. The derivation then follows identically the same path as we had for quantum mechanics and therefore, we have the following result, Z φ(t, ~x) = φ0(~x) hφ0|e−itH |φi = DφDπ exp{iI[φ, π]} (1.4) φ(0, ~x) = φ(~x) where the action is deﬁned by

Z t Z I[φ, π] ≡ dt0 d3xπφ˙ − H[π, φ] (t0) (1.5) 0 In the special case of the following simple but useful Hamiltonian, Z 1 1 H[φ, π] = d3x π2 + (∇~ φ)2 + V (φ) (1.6) 2 2 one gets after performing the Gaussian integration over π Z Z t Z φ(t, ~x) = φ0(~x) hφ0|e−itH |φi = Dφ exp i dt0 d3xL(φ, ∂φ) (1.7) 0 φ(0, ~x) = φ(~x) Notice that, as an added bonus, this formulation is now manifestly Lorentz covariant, as it is given in terms of the Lorentz invariant Lagrangian 1 L(φ, ∂φ) = ∂ φ∂µφ − V (φ) (1.8) 2 µ Preserving Lorentz invariance will prove invaluable later on.

5 1.3 Time ordered correlation functions

The above result may be applied directly to the calculation of the vacuum expectation value of the time ordered product of a string of ﬁelds φ,

h0|T φ(x1)φ(x2) ··· φ(xn)|0i (1.9)

Notice that this object is a completely symmetric function of x1, ··· xn. Hence, without o o o o loss of generality, the points xi may be time-ordered, x1 ≥ x2 ≥ x3 · · · ≥ xn so that

h0|T φ(x1)φ(x2) ··· φ(xn)|0i = h0|φ(x1)φ(x2) ··· φ(xn)|0i (1.10)

Next, one uses the evolution operator to bring out the time-dependence of the ﬁeld oper-

ators φ(xi) given by the following expression,

o o o ixi H −ixi H φ(xi , ~xi) = e φ(0, ~xi)e (1.11)

Using this formula for the time-dependence of every inserted operator, we obtain,

h0|φ(x1)φ(x2) ··· φ(xn)|0i (1.12) o o o o o o o o ix H −i(x −x )H −i(x −x )H −i(x −xn)H −ixnH = h0|e 1 φ(0, ~x1)e 1 2 φ(0, ~x2)e 2 3 ··· e n−1 φ(0, ~xn)e |0i

Next, we insert the identity operator exactly n times and express the identity as a com- pleteness relation in the position basis of states |φ(~x)i Since the insertions have to be o carried out n times, we introduce n integration ﬁelds , which we denote by φ(xi , ~xi), i = 1, ··· , n. The operator insertions φ(0, ~xi) are diagonal in this basis and we obtain,

Z Z o o o ix1H o o = Dφ(x1, ~x1) ··· Dφ(xn, ~xn)h0|e |φ(x1, ~x1)iφ(x1, ~x1) (1.13) o o o −i(x1−x2)H o o ×hφ(x1, ~x1)|e |φ(x2, ~x2)iφ|x2, ~x2i o o o o −i(x2−x3)H o o −ixnH ×hφ(x2, ~x2)|e |φ(x3, ~x3)i · · · hφ(xn, ~xn)|e |0i

At this stage, we use the functional integral form for the matrix elements in the ﬁeld basis of the evolution operator, as derived in (). It is immediately realized that the boundary o conditions at each integration over φ(xi , ~xi) simply requires integration over a path in time that is continuous. Furthermore, since the Hamiltonian acts on the ground state, assumed to have zero energy, the time integrations may be taken to ±∞ on both ends. Finally, there is a normalization factor Z[0] that will guarantee that the ground state is properly

6 normalized to h0|0i = 1; 1 Z Z h0|φ(x ) ··· φ(x )|0i = Dφ φ(x ) ··· φ(x ) exp i d4xL (1.14) 1 n Z[0] 1 n It is clear that the right hand side is automatically a symmetric function of the space-time points x1, ··· , xn. Therefore, the rhs in fact equals the time ordered expectation value 1 Z Z h0|T φ(x ) ··· φ(x )|0i = Dφ φ(x ) ··· φ(x ) exp i d4xL 1 n Z[0] 1 n Z Z Z[0] ≡ Dφ exp i d4xL (1.15) for any time ordering of the points xi.

1.4 The generating functional of correlators Now it is very easy to write down a generating functional for the time-ordered expectation values. Deﬁne Z Z Z[J] ≡ Dφ exp i d4xφ(x)J(x) eiS[φ]

∞ n n Z Z X i Y 4 iS[φ] = d xjJ(xj) Dφ φ(x1) ··· φ(xn) e (1.16) n=0 n! j=1 so that ∞ n n Z Z[J] X i Y 4 = d xjJ(xj) h0|T φ(x1) ··· φ(xn)|0i Z[0] n=0 n! j=1 Z = h0|T exp i d4xJ(x)φ(x) |0i (1.17)

Equivalently, the n-point correlator may be expressed as an n-fold functional derivative with repsect to the source J,

! n δ δ Z[J] h0|T φ(x1) ··· φ(xn)|0i = (−i) ··· (1.18) δJ(x1) δJ(xn) Z[0] J=0

1.5 The generating Functional for free scalar ﬁeld theory It is instructive to see what these objects are in free ﬁeld theory: 1 1 L = ∂ φ∂µφ − m2φ2 (1.19) o 2 µ 2

7 We may directly compute the free generating functional. Since the integral is Gaussian, all J-dependence may be obtained by shifting the integration as follows,

Z Z 1 1 Z [J] = Dφ exp i d4x ∂ φ∂µφ − m2φ2 + Jφ (1.20) o 2 µ 2 Z Z 1 = Dφ exp i d4x − φ(2 + m2)φ + Jφ 2 i Z Z 1 = exp + J(2 + m2)−1J Dφ exp i d4x − φ˜(2 + m2)φ˜ 2 2 where we have used the notation φ˜ = φ − (2 + m2)−1J. By shifting the φ˜ back to φ, it is shown that the integration factor in the last line is equal to Z[0]. Furthermore, the inverse of the operator i(2 + m2) is nothing but the scalar Green function deﬁned previously by the Feynman propagator G(x − y; m2) = G(x − y),

2 2 (4) (2 + m )xG(x − y; m ) = −iδ (x − y) (1.21) so that the ﬁnal answer may be put in the form

1 Z Z Z [J] = Z [0] exp − d4x d4yJ(x)G(x − y)J(y) (1.22) o o 2 In particular, the two point function is given by

! δ δ Z [J] 0 o 0 h0|T φ(x)φ(x )|0i = − 0 = G(x − x ) (1.23) δJ(x) δJ(x ) Zo[0] J=0 Similarly, the 4-point function is obtained by

! δ δ δ δ Z [J] o h0|T φ(x1)φ(x2)φ(x3)φ(x4)|0i = δJ(x1) δJ(x2) δJ(x3) δJ(x4) Zo[0] J=0 δ δ 1 1 Z Z 2 = ··· − d4u d4vJ(u)G(u − v)J(v) δJ(x1) δJ(x4) 2 2

= G(x1 − x2)G(x3 − x4) + G(x1 − x3)G(x2 − x4) (1.24)

+G(x1 − x4)G(x2 − x3)

Recall that this may be represented graphically as follows:

Figure 1: The scalar 2- and 4-point function in free ﬁeld theory

8 More generally, the n-point function vanishes when n is odd (by φ → −φ symmetry), while 2n-point functions are given by

1 X h0|T φ(x1) ··· φ(x2n)|0i = n G(xσ(1) − xσ(2)) ··· G(xσ(2n−1) − xσ(2n)) (1.25) 2 n! σ The summation is eﬀectively over all possible inequivalent pairings of the 2n points, in which each inequivalent term then appears with coeﬃcient 1.

9 2 Perturbation Theory for Scalar Fields

For a general interacting theory, none of the Green functions can be computed exactly. Recall that in ordinary quantum mechanics, one can do this only in the cases where extra symmetries decouple the degrees of freedom, which is exceedingly rare. It is even more rare in quantum ﬁeld theory and is known to take place in some 1+1-dimensional space-time ﬁeld theories. Short of exact solutions, one needs to make approximations. In perturbation theory, one regards one or several parameters as small and such that the theory can be solved exactly (in practice this is always free ﬁeld theory) when all these parameters are set to 0. Next, one expands in powers of these small parameters.

2.1 Scalar Field Theory with potential interactions The prototypical scalar ﬁeld theory we shall be interested in has an action of the form, Z 1 1 S[φ] = d4z ∂ φ∂µφ − m2φ2 − λV (φ) (2.1) 2 µ 2 where the potential V (φ) is an analytic function of φ admitting a powers series expansion in φ, and λ is a small parameter. It is convenient to isolate from V (φ) the terms that are linear and quadratic in φ, to shift φ so as to cancel the linear terms and to regroup the quadratic terms into a mass term, which may be treated exactly. Thus, we may assume that V 0(0) = V 00(0) = 0. We consider the example of the quartic potential ﬁrst, and set V (φ) = φ4/4!. The correlation functions of interest are R Dφ φ(x ) ··· φ(x ) eiS[φ] h0|T φ(x ) ··· φ(x )|0i = 1 n (2.2) 1 n R Dφ eiS[φ] The simplest thing to compute is the denominator assuming that λ is small, Z Z ( iλ Z # Dφ eiS[φ] = Dφ exp iS [φ] − d4zφ4(z) (2.3) o 4! ∞ !p Z Z Z X 1 −iλ 4 4 4 4 iSo[φ] = Dφ d z1 ··· d zpφ (z1) ··· φ (zp) e p=0 p! 4!

This is now a problem of free φ-ﬁelds, governed by the free action So[φ]. Therefore, the expansion may be expressed in terms of free ﬁeld vacuum expectation values,

R iS[φ] ∞ !p Z Dφ e X 1 −iλ 4 4 4 4 = d z1 ··· d zph0|T φ (z1) ··· φ (zp)|0io (2.4) R iSo[φ] Dφ e p=0 p! 4!

10 Similarly, the numerator is expanded in powers of λ as well, and we have

R iS[φ] Dφ φ(x1) ··· φ(xn) e R (2.5) Dφ eiSo[φ] ∞ !p Z X 1 −iλ 4 4 4 4 = d z1 ··· d zph0|T φ(x1) ··· φ(xn)φ (z1) ··· φ (zp)|0io p=0 p! 4!

Again this has become a problem in free ﬁeld theory. Therefore, the problem that we set out to solve is basically solved by this formula. It is useful to generalize this formula to any potential. It is also convenient to recast the result in terms of the generating functional, from which any correlator may be obtained. Thus, the quantity we are interested in is as follows,

Z Z Z[J] ≡ Dφ exp iS[φ] + i d4xJ(x)φ(x) (2.6)

Just as before, we expand the action into its free part and the interacting part, Z 4 S[φ] = So[φ] − λ d xV (φ) (2.7)

This allows us to expand also the generating functional in powers of λ,

∞ Z Z p Z X 1 4 4 Z[J] = Dφ −iλ d xV (φ) exp iSo[φ] + i d xJ(x)φ(x) (2.8) p=0 p! ∞ ( Z !)p Z Z X 1 4 δ 4 = −iλ d xV −i Dφ exp iSo[φ] + i d xJ(x)φ(x) p=0 p! δJ(x)

Hence we have a very neat shorthand for the expansion, as follows,

( Z δ !) Z[J] = exp −iλ d4xV −i Z [J] (2.9) δJ(x) o and of course the generating functional for the free theory was known exactly in terms of the scalar Feynman propagator G, 1 Z Z Z [J] = exp − d4x d4yJ(x)G(x − y)J(y) (2.10) o 2 Again, the problem of perturbation theory has been completely reformulated in terms of problems in free scalar ﬁeld theory.

11 2.2 Feynman rules for φ4-theory

Fermi and Feynman introduced a graphical expansion for perturbation theory that com- monly goes under the name of Feynman rules. We begin by working this out for the case of V (φ) = φ4/4!. Because of φ → −φ symmetry, correlators with an odd number of ﬁelds vanish identically, in the free as well as in the interacting theory. To order λ, the following correlators are required,

1 h0|T φ(x ) ··· φ(x ) φ4(z)|0i 1 2n 4! o 1 = lim h0|T φ(x1) ··· φ(x2n)φ(z1)φ(z2)φ(z3)φ(z4)|0io (2.11) 4! zi→z

The free ﬁeld correlator of a string of canonical scalar ﬁelds is a quantity that we have learned how to evaluate previously. There are 3 diﬀerent types of contributions.

1. All contractions of the φ(zi) are carried out onto φ(zj) (this is the only possibility

if n = 0), and all contractions of φ(xk) are carried out onto φ(xl), which yields the graphical contribution of Fig 2 (a). There are 3 distinct double pairings amongst 4 points, thus the remaining combinatorial factor is 1/8.

2. Two φ(zi) are contracted with one another, while all others are contracted with

φ(xj), which yields the graphical representation of Fig 2 (b). There are 6 possible pairs amongst 4 points, but the remaining 2 points not in the pair may be permuted freely yielding another factor of 2. Thus, the combinatorial factor is 1/2.

3. Finally, all φ(zi) may be contracted with φ(xj), yielding combinatorial factors of 1, as represented in Fig 2 (c).

Figure 2: First order contributions to h0|T φ(x1) ··· φ(x4)|0i

12 In equations, we have

1 h0|T φ(x ) ··· φ(x ) φ4(z)|0i (2.12) 1 2n 4! o 1 = G(0)2h0|T φ(x ) ··· φ(x )|0i 8 1 2n o 2n 1 X + G(0) G(z − xi)G(z − xj)h0|T φ(x1) ···ˆiˆj ··· φ(x2n)|0io 2 i6=j=1 2n X + G(z − xi)G(z − xj)G(z − xk)G(z − xl) i6=j6=k6=l=1 ˆˆ ×h0|T φ(x1) ···ˆiˆjkl ··· φ(x2n)|0io

Notice that the contribution from Fig 2 (a) is disconnected, namely the double bubble is disconnected from any of the external points xi. But this bubble is common to the numerator and denominator in the functional integral and therefore cancels out. Thus, the only contributions arise from Fig 2 (b) and (c), or analytically,

h0|T φ(x1) ··· φ(x2n)|0i (2.13)

= h0|T φ(x1) ··· φ(x2n)|0io 2n Z iλ X 4 − d zG(z − z)G(z − xi)G(z − xj)h0|T φ(x1) ···ˆiˆj ··· φ(x2n)|0io 2 i6=j=1 2n Z X 4 −iλ d zG(z − xi)G(z − xj)G(z − xk)G(z − xl) i6=j6=k6=l=1 ˆˆ ×h0|T φ(x1) ···ˆiˆjkl ··· φ(x2n)|0io

Clearly, it becomes very cumbersome to write out these formulas completely, and a graph- ical expansion turns out to capture all the relevant information concisely.

2.3 The cancellation of vacuum graphs.

Let us deﬁne a graph without vacuum parts as a graph where all interaction vertices are connected to an external φ by at least one line. It is clear that

X X X = × (2.14) all graphs no vac parts bubble graphs

13 Hence X h0|T φ(x1) ··· φ(xn)|0i = (2.15) no vac parts Thus in practice, one never has to compute the bubble graphs in computing the time ordered products. To illustrate this, take the 2-point function in λφ4-theory. To order λ2, we have 1 Z h0|T φ(x)φ(y)|0i = G(x − y) + (−iλ) d4zG(0)G(x − z)G(y − z) (2.16) 2 1 Z Z + (−iλ)2 d2z d2z G(x − z )G(0)G(z − z )G(0)G(z − y) 4 1 2 1 1 2 2 1 Z Z + (−iλ)2 d2z d2z G(x − z )G(z − z )2G(0)G(z − y) 4 1 2 1 1 2 1 1 Z Z + (−iλ)2 d2z d2z G(x − z )G(z − z )3G(z − y) + O(λ3) 6 1 2 1 1 2 2 The Feynman representation is given in Fig 3.

Figure 3: First and second order contributions to h0|T φ(x1)φ(x2)|0i

2.4 General Position Space Feynman Rules for scalar ﬁeld theory The starting point is the formula for the generating functional ( Z δ !) Z[J] = exp −i d4xV −i Z [J] δJ(x) o 1 Z Z Z [J] = exp − d4x d4yJ(x)G(x − y)J(y) Z [0] (2.17) o 2 o From the combinatorics of these formulae one deduces the following rules

1. Let x1, x2 ··· xn be the external φ’s in h0|T φ(x1) ··· φ(xn)|0i;

2. Let z1 ··· zp be the internal interaction vertices (order p in V );

3. From every external φ(xi) departs one line, which can connect to an external φ

(except itself) or to an interaction vertex zj;

1 N 4. From an interaction vertex of the type N! φ , there depart N lines, which join an external φ, another interaction vertex or itself;

14 5. To every line is associated GF (x − y) between points x and y;

1 N 6. Every interaction vertex N! φ has a coupling constant factor of igN ;

R 4 7. Every interaction vertex is integrated over d zi;

8. There is an overall symmetry factor.

2.5 Momentum Space Representation We begin by deﬁning the Fourier transform of the n-point function,

n 4 ! Z d pj Y ipj ·xj ˜ h0|T φ(x1) ··· φ(xn)|0i = 4 e G(p1, ··· , pn) (2.18) j=1 (2π) as well as the inverse Fourier transform relation, n Z ˜ Y 4 −ipj ·xj G(p1, ··· , pn) = d xj e h0|T φ(x1) ··· φ(xn)|0i (2.19) j=1 Recall, however, that the scalar ﬁeld theory action S[φ], as well as the vacuum |0i are translation invariant. Therefore, the time ordered expectation value h0|T φ(x1) ··· φ(xn)|0i only depends on the diﬀerences of the arguments. As a consequence, the Fourier transform ˜ G(p1, ··· , pn) is non-vanishing only when overall momentum is conserved, p1 +···+pn = 0 and is proportional to an overall δ-function,

˜ 4 4 G(p1, ··· , pn) = (2π) δ (p1 + ··· + pn)G(p1, ··· , pn) (2.20)

The Feynman rules in momentum space are easily derived, and are customarily formulated for the quantities G(p1, ··· , pn).

1. Draw all topologically distinct connected diagrams with n external legs of incoming

momenta p1, ··· pn. Denote by k1, ··· k` the momenta of internal lines.

2. To the j-th external line, assign the value of the propagator, i GF (pj) = 2 2 , j = 1, ··· , n (2.21) pj − m + i

3. To every internal line, assign

4 Z d k` i 4 2 2 (2.22) (2π) k` − m + i

15 4. To each interaction vertex, assign

4 4 −igN (2π) δ (q) (2.23) where q is the sum of incoming momenta to the vertex.

5. Integrate over the internal momenta, and multiply by the combinatorial factor.

2.6 Does perturbation theory converge ? Having produced a perturbative expansion in the coupling constant, the question arises as to whether the corresponding power series is convergent or not. As a caricature of the perturbative expansion of the functional integral is to take an ordinary integral of the same type. Thus, we consider Z +∞ Z(m, λ) ≡ dφ exp −m2φ2 − λ2φ4 (2.24) −∞ There are now two possible expansions; the ﬁrst in power of m, the second in powers of λ. The expansion in powers of m yields ∞ 1 Z +∞ Z(m, λ) = X (−m2)n dφ φ2n exp −λ2φ4 n=0 n! −∞ ∞ 1 Γ( n + 1 ) m2 !n = X √ 2 4 − (2.25) n=0 2 λ Γ(n + 1) λ while the expansion in powers of λ yields, ∞ 1 Z +∞ Z(m, λ) = X (−λ2)n dφ φ4n exp −m2φ2 n=0 n! −∞ ∞ 1 2 !n X 1 Γ(2n + 2 ) λ = − 4 (2.26) n=0 m Γ(n + 1) m Clearly, from analyzing the n-dependence of the ratios of Γ functions, the expansion in m is convergent (with ∞ radius of convergence) while the expansion in power of λ has zero radius of convergence. This comes as no surprise. If we had ﬂipped the sign of m2, the integral itself would still be ﬁne and convergent. On the other hand, if we had ﬂipped the sign of λ2, the integral badly diverges, and therefore, it cannot have a ﬁnite radius of convergence around 0. The moral of the story is that perturbation theory around free ﬁeld theory is basically always given by an asymptotic series which has zero radius of convergence. Nonetheless, and it is one of the most remarkable things about QFT, perturbation theory gives astoundingly accurate answers when it is applicable.

16 2.7 The loop expansion We now derive theh ¯-dependence of the various orders of perturbation theory: Z i S[φ] h0|T φ(x1) ··· φ(xn)|0i = Dφφ(x1) ··· φ(xn)e h¯ (connected) (2.27)

A general diagram has

E external legs I internal lines (2.28) V vertices

The number of loops L is equal to the number of independent internal momenta, after all comservatioins have been imposed on vertices, minus one overall momentum conservation L = I − (V − 1). Now calculate to what order inh ¯ this graph contributes:

• each propagator contributes one power ofh ¯;

• each vertex contributes one power of 1/h¯.

Hence a graph contributes

h¯E+I−V =h ¯(E+L−1) (2.29)

For ﬁxed number of external lines, theh ¯ expansiion is a loop expansion, provided all the couplings in the classical actioin are independent ofh ¯.

2.8 Truncated Diagrams To simplify and systematize the enumeration of all diagrams that contribute to a given process, it is useful to identify special sub-categories of diagrams. We have already dis- cussed the connected diagrams, from which all others may be recovered. We now introduce truncated and 1-particle irreducible diagrams which allow to further cut down the number of key diagrams. In the calculation of the S-matrix, the full Green function is not required; instead we only use it in the combination (scalar ﬁeld)

n Z Y 4 −ipi·xi 2 −1/2 d xie (2 + m )xi Z h0|T φ(x1) ··· φ(xn)|0i i=1 n −n/2 Y 2 2 ˜(n) = Z (−pi + m )G (p1, ··· , pn) (2.30) i=1

17 Here pi are on-shell external momentum. As we have deduced from the spectral represen- tation, the two-point function behaves like Z G(2)(p, −p) = + reg. as p2 → m2 (2.31) p2 − m2 Hence what really comes in the reduction formula is

n n/2 Y h (2) i−1 (n) lim Z Gc (pi, −pi) G (p1, p2 ··· pn) (2.32) p2→m2 i i=1 One deﬁnes the truncated correlator by

n (n) Y (2) −1 (n) Gtrun(p1, ··· , pn) = Gc (pi, −pi) G (p1, ··· , pn) (2.33) i=1 So that the S-matrix elements are given by

hq1 ··· qm out|p1 ··· pn inic 1/2 m+n 4 4 (n+m) = (Z ) (2π) δ (q − p)Gtrun (q1, ··· , qm; p1, ··· , pn) (2.34)

where δ(q − p) stands for an overall momentum conservation δ-function. Truncation of the n-point correlator reduces the n-point function in that the entire set of diagrams that corresponds to the self-energy corrections of the external legs is to be omitted. Therefore, it is much simpler to enumerate the truncated diagrams than it is to enumerate all connected diagrams.

2.9 Connected graphs A connected diagram is such that its graph is connected. The generating functional for connected graphs is the log of Z[J];

Z[J] = Z[0] exp{iGc[J]} (2.35) and its expansion in powers of J yields the connected n-point functions,

∞ n Z X i 4 4 (n) Gc[J] = −i d x1 ··· d xnJ(x1) ··· J(xn)Gc (x1, ··· , xn) (2.36) n=0 n! Often, connected vacuum expectation values are simply denoted by a subscript c as well, and thus we have

(n) Gc (x1, ··· , xn) = h0|T φ(x1) ··· φ(xn)|0ic (2.37)

18 The proof of the relation between Z[J] and Gc[J] is purely combinatorial. Denoting each

separate connected contribution of degree i in J by Xi, i = 1, ··· , ∞, we have

∞ ∞ X X 1 X p! exp X = Xn1 Xn2 ··· Xnp (2.38) i p! n ! ··· n ! 1 2 p i=1 p=0 n1+2n2···+pnp=p 1 p which indeed reproduces the correct combinatorics for the full expansion order by order p in the power of J, appropriate for Z[J].

2.10 One-Particle Irreducible Graphs and the Eﬀectice Action A one-particle irreducible diagram is a connected truncated graph which remains connected after any one internal line is cut. The Legendre transform of the generating functional for connected graphs Gc[J] is the generating functional for 1PIR’s. The Legendre transform is deﬁned as follows. One begins by deﬁning the vacuum expectation value ϕ(x; J) of the ﬁeld φ in the presence of the source J,

δ ϕ(x; J) ≡ G (J) = h0|φ(x)|0i (2.39) δJ(x) c J

Now we assume that the relationship deﬁning ϕ is invertible, and that J may be eliminated in terms of ϕ in a unique manner,

ϕ(x; J) = ϕ(x) ⇐⇒ J(x) = J(x; ϕ) (2.40)

The Legendre transform Γ[ϕ] of Gc[J] is then deﬁned by

Z 4 Γ[ϕ] ≡ Gc[J] − d xJ(x)ϕ(x) (2.41) J=J(x;ϕ)

where it is understood that J is eliminated in terms of ϕ. As always, it follows that

δΓ[ϕ] = −J(x) (2.42) δϕ(x)

Now J(x) should be thought of as a source which must be applied from the outside in order that the vacuum expectation value of φ be ϕ. When J = 0, we recover the original theory, without source, so that

δΓ[ϕ] = 0 at J(x; ϕc) = 0 (2.43) δϕ(x) ϕc

19 Next we investigate which diagrams are generated by Γ[ϕ]: For simplicity, we assume for

the moment that ϕc = 0 (the case of no spontaneous symmetry breaking)

∞ Z Z X 1 4 4 (n) Γ[ϕ] = d x1 ··· d xn Γ (x1, ··· xn)ϕ(x1) ··· ϕ(xn) (2.44) n=1 n!

The eﬀective ﬁeld equation (2.43) then implies that Γ(1)(x) = 0. In other words, the 1PIR 1-point function vanishes. To obtain relations for more general Γ(n)’s, we diﬀerentiate the relation deﬁning ϕ with respect to ϕ. We use the functional chainrule to express the derivation with respect to ϕ in terms of the diﬀerentiation with respect to J,

δ δG [J]! Z δJ(z) δ2G [J] δ4(x − y) = c = d4z c (2.45) δϕ(y) δJ(x) δϕ(y) δJ(z)δJ(x)

However, from (2.42), we have

δJ(z) δ2Γ[ϕ] − = (2.46) δϕ(y) δϕ(z)δϕ(y)

Evaluating this relation at ϕ = J = 0, we ﬁnd a relation between the connected and 1PIR 2-point functions, Z 4 4 (2) (2) (2) n (2)o−1 −δ (x − y) = d zΓ (x, z)Gc (z, y) or Γ = − Gc (2.47)

It is useful to illustrate this relation by working out schematically the corrections to the 2-point function. If we denote the 1PIR corrections to the self-energy by Σ(p), then the full 2-point function is given by 1 1 1 1 1 1 G(2)(p) = + Σ(p) + Σ(p) Σ(p) + ··· c p2 − m2 p2 − m2 p2 − m2 p2 − m2 p2 − m2 p2 − m2 1 = = Γ(2)(p)−1 (2.48) p2 − m2 − Σ(p)

which conﬁrms our assertion that Γ generates 1PIR only. A graphical representation is given in Fig 5.

Figure 4: Relations between connected and 1PIR graphs in a theory with cubic and quartic couplings, and vanishing 1-point function.

20 By diﬀerentiating once more, and setting ϕ = J = 0, we obtain information on the 3-point function, Z ZZ 4 (3) (2) 4 4 (2) (2) 3 d uΓ (u, y, z)Gc (x, u) = d ud vΓ (u, y)Γ (v, z)Gc (x, u, v) (2.49)

By multiplying by G(2)(y, ·)G(2)(z, ·), we obtain Z G(3)(x, y, z) = d4ud4vd4wG(2)(x, u)G(2)(y, v)G(2)(z, w)Γ(3)(u, v, w) (2.50)

So, for the three-point function, the 1PIR part arises from truncating the graph. One may repeat the exercise for 4 legs. Graphical representations are given in Fig 4. Clearly, the expansion is in terms of the tree-level graphs, Γ(n) used as generalized vertices. This is the skeleton expansion of any graph in terms of 1PIR’s. Γ[ϕ] is the so-called eﬀective action of the system. Why action? Here, it is useful to make a loop expansion Z i Z i Z[J] = Dφ exp S[φ] + Jφ = exp G [J] (2.51) h¯ h¯ c (note, here we have kept the J-independent term in G[J]). Now, keep only the leading term inh ¯ → 0, then the integral will be dominated by its saddle points. Let’s assume that the saddle point is unique and denote it by φs, then we have δS[φ ] s + J(x) = 0 (2.52) δφ(x) But this is precisely the same equation as was satisﬁed by Γ itself. Thus, to leading order inh ¯, we have Γ[ϕ] = S[ϕ]. All Green functions can be reconstructed from it by the skeleton expansion so that Γ[ϕ] is the essential quantity in a quantum ﬁeld theory.

2.11 Schwinger Dyson Equations We concentrate on a scalar ﬁeld theory, with Lagrangian, 1 1 L = ∂ φ∂µφ − m2φ2 − V (φ) (2.53) 2 µ 2 Now we use the fact that the functional integral of a functional derivative vanishes,

Z δ n o Dφ φ(x ) ··· φ(x )eiS[φ] = 0 (2.54) δφ(z) 1 n

21 Working out the actual derivatives under the integral, we obtain an hierarchy of equations, involving the correlators,

n X 4 δ (xi − z)h0|T φ(x1) ··· φd(xi) ··· φ(xn)|0i (2.55) i=1 ∂V −i(2 + m2) h0|T φ(x ) ··· φ(x )φ(z)|0i − ih0|T φ(x ) ··· φ(x ) (z)|0i = 0 z 1 n 1 n ∂φ providing a recursive type of relationship that expresses the quantum equations of motion. These are the Schwinger-Dyson equations.

2.12 The Background Field Method for Scalars Feynman diagrams and standard perturbation theory do not always provide the most eﬃcient calculational methods. This is especially true for theories with complicated lo- cal invariances, such as gauge and reparametrization invariances. But the method also provides important short-cuts for scalar ﬁeld theories. The background ﬁeld method is introduced ﬁrst for a theory with a single scalar ﬁeld

φ. The classical action is denoted S0[φ], and the full quantum action, including all renor- malization counterterms is denoted by S[φ]. The generating functional for connected correlators is given by i Z i i Z exp G [J] ≡ Dφ exp S[φ] + d4xJ(x)φ(x) (2.56) h¯ c h¯ h¯

By the very construction of the renormalized action S[φ], Gc[J] is a ﬁnite functional of J. The Legendre transform is deﬁned by Z 4 Γ[ϕ] ≡ d xJ(x)ϕ(x) − Gc[J] (2.57)

where J and ϕ are related by δG [J] δΓ[ϕ] ϕ(x) = c J(x) = − (2.58) δJ(x) δϕ(x) Recall that Γ[ϕ] is the generating functional for 1-particle irreducible correlators. To obtain a recursive formula for Γ, one proceeds to eliminate J from (2.56) using the above relations, i Z Z i Z δΓ[ϕ] exp Γ[ϕ] + Jϕ = Dφ exp S[φ] − dxφ(x) (2.59) h¯ h¯ δϕ(x)

22 Moving the J-dependence from the left to the right side, and expressing J in terms of Γ in the process, we obtain, i Z i Z δΓ[ϕ] exp Γ[φ¯] = Dφ exp S[φ] − dx(φ − ϕ)(x) (2.60) h¯ h¯ δϕ(x) √ The ﬁnal formula is obtained by shifting the integration ﬁeld φ → ϕ + hφ¯ , i Z i √ √ Z δΓ[ϕ] exp Γ[ϕ] = Dφ exp S[ϕ + hφ¯ ] − h¯ dxφ(x) (2.61) h¯ h¯ δϕ(x) At ﬁrst sight, it may seem that little has been gained since both the left and right hand sides involve Γ. Actually, the power of this formula is revealed√ when it is expanded in powers ofh ¯, since on the rhs, Γ enters to an order higher by h¯ than it enters on the lhs. Theh ¯-expansion is performed as follows,

2 3 S[φ] = S0[φ] +hS ¯ 1[φ] +h ¯ S2[φ] + O(¯h ) 2 3 Γ[ϕ] = Γ0[ϕ] +h ¯Γ1[ϕ] +h ¯ Γ2[ϕ] + O(¯h ) (2.62)

The recursive equation to lowest order clearly shows that Γ0[ϕ] = S0[ϕ]. The interpretation

of the terms Si[φ] for i ≥ 1 is clearly in terms of counterterms to loop order i, while that

of Γi[ϕ] is that of i-loop corrections to the eﬀective action. The 1-loop correction is readily evaluated for example, 2 Z i Z Z δ S0[ϕ] exp iΓ [ϕ] = eiS1[ϕ] Dφ exp d4x φ(x) d4y φ(y) (2.63) 1 2 δϕ(x)δϕ(y)

For example, if S0 describes a ﬁeld φ with mass m and interaction potential V (φ), we have Z 1 1 S [φ] = dx ∂ φ∂µφ − m2φ2 − V (φ) (2.64) 0 2 µ 2 then we have Z i Z eiΓ1[ϕ] = eiS1[ϕ] Dφ exp d4xφ 2 + m2 + V 00(ϕ) φ 2 i Γ [ϕ] = S [ϕ] + ln Det 2 + m2 + V 00(ϕ) (2.65) 1 1 2 Upon expanding this formula in perturbation theory, the standard expressions may be recovered, i 1 Γ [ϕ] = S [ϕ] + Γ [0] + ln Det 1 + V 00(ϕ) (2.66) 1 1 1 2 2 + m2 which clearly reproduces the standard 1-loop expansion. Therefore, for scalar theories, the gain produced by the use of the background ﬁeld method is usually only minimal.

23 3 Basics of Perturbative Renormalization

In this section, we begin by presenting some useful techniques for the calculation of Feyn- man diagrams. Next, we introduce some of the basic ideas and techniques of renormal- ization in perturbation theory. Finally, we study the analytic behavior of the 4-point diagrams to one loop order in λφ4 as a function of external momenta.

3.1 Feynman parameters One will often be confronted with loop integrals over internal momenta that involve a product of momentum space propagators. It would not be totally straightforward to do these integrals, if it were not for the use of Feynman parameters. The simplest type of integrals of this type may be taken to be Z ddk 1 1 I (p) = × (3.1) d (2π)d (k + p)2 − m2 + i k2 − m2 + i We have considered this integral in any dimension of space-time d. Feynman’s trick for dealing with the product of propagators is to represent the product in terms of an integral. The basic formula is as follows, 1 Z 1 1 = dα (3.2) ab 0 [aα + b(1 − α)]2

Applying this formula to the integral Id(p), we obtain Z 1 Z ddk 1 Id(p) = dα (3.3) 0 (2π)d [(k + αp)2 + α(1 − α)p2 − m2 + i]2 For d ≥ 4, this integral actually diverges for any p due to the contributions at large k. We shall temporarily ignore this problem and consider d < 4. In a convergent integral, one may now safely shift k → k − αp, so that we end up with Z 1 Z ddk 1 Id(p) = dα (3.4) 0 (2π)d [k2 + α(1 − α)p2 − m2 + i]2 The great advantage of this form of the integral is that all momenta enter through k2 and p2, which are Lorentz scalars. More generally, the product of several propagator denominators may be treated with multiple integrals. The easiest is to use the exponential representation ﬁrst,

1 Z ∞ Z ∞ λµ1−1 ··· λµn−1 1 n −λ1a1−···−λnan µ1 µn = dλ1 ··· dλn e (3.5) a1 ··· an 0 0 Γ(µ1) ··· Γ(µn)

24 Putting λi = λαi, we recover the Feynman parametrization,

1 1 1 Γ(µ1 + µ2 + ··· + µn) Z Z µ1 µ2 µn = dα1 ··· dαnδ(α1 + ··· + αn − 1) a1 a2 ··· an Γ(µ1)Γ(µ2) ··· Γ(µn) 0 0 µ1−1 µn−1 α1 ··· αn × µ +···+µ (3.6) [α1a1 + ··· + αnan] 1 n

It is of course possible to integrate out one of the αi by using the δ-function, but the result

is then less symmetrical in the ai. For n = 2 and µ1 = µ2 = 1, we recover the earlier result this way.

3.2 Integrations over internal momenta It remains to evaluate the integrations over internal momenta. First, we shall assume that Feynman parameters have been employed to put the integration into standard form,

Z ddk 1 I(M; d, n) ≡ (3.7) (2π)d (k2 − M 2 + i)n We shall now evaluate this integral for all values of d and n where it converges and then analytically continue the result throughout the complex planes for both d and n. As k → ∞, the integrals converge as long as d < 2n, and we assume this bound satisﬁed.

o o The ﬁrst step is to analytically continue to Euclidean internal momenta, k → ikEm leaving M alone. This is possible, because the singularities in the complex ko plane may be completely avoided during this process thanks to the i prescription of the propagator. 2 2 o o As a result, we have k → −kE and dk → idkE, so that

Z d 2n+1 d kE 1 I(M; d, n) = i d 2 2 n (3.8) (2π) (kE + M )

2 where kE is the Euclidean momentum square. To carry out this integration, we use d- dimensional spherical coordinates and the formula for the volume of the sphere Sd−1,

2πd/2 V (Sd−1) = (3.9) Γ(d/2) This formula is most easily proven as follows. Consider the Gaussian inetgral in d dimen- sions, which is just a product of d Gaussian inetgrals in 1 dimension, Z dd~k e−~k2 = πd/2 (3.10)

25 Working out the same formula in spherical coordinates, we have ∞ Z 2 Z 2 1 dd~k e−~k = V (Sd−1) dk kd−1 e−k = V (Sd−1)Γ(d/2) (3.11) 0 2 which upon combination with the ﬁrst yields the desired expression for V (Sd−1). It is easy to check that V (S1) = 2π, V (S2) = 4π and V (S3) = 2π2 as expected. The above formula 2 2 is, however, valid for any d. Introducing real positive k such that kE = k , we have 2 i2n+1 Z ∞ kd−1 I(M; d, n) = dk (3.12) (4π)d/2Γ(d/2) 0 (k2 + M 2)n The remaining integral may be carried out as follows. First, we let k2 = M 2t, so that Z ∞ kd−1 1 1 Z ∞ td/2−1 dk = dt (3.13) 0 (k2 + M 2)n 2 (M 2)n−d/2 0 (t + 1)n Next, we let x = 1/(t + 1), so that Z ∞ kd−1 1 1 Z 1 dk = dx (1 − x)d/2−1xn−d/2−1 0 (k2 + M 2)n 2 (M 2)n−d/2 0 1 1 Γ(d/2)Γ(n − d/2) = (3.14) 2 (M 2)n−d/2 Γ(n) Therefore, we ﬁnd i2n+1Γ(n − d/2) 1 I(M; d, n) = (3.15) (4π)d/2Γ(n) (M 2)n−d/2 The calculation was carried out there where the integral was convergent. We see now how- ever, that the analytic continuation of the expression that we have obtained is straightfor- ward since the analytic continuation of the Γ-function is well-known. Thus, we view the result obtained above as THE answer !

3.3 The basic idea of renormalization As soon as loops occur, the diagram is potentially divergent when the internal momen- tum integration fails to converge in the limit of large momenta. It would seem that the appearance of divergences renders the quantum ﬁeld theory meaningless. Actually, the procedure of renormalization is required to organize and properly interpret the loop cor- rections that arise. We shall illustrate the salient features and ideas of renormalization within the context of λφ4 theory, with Lagrangian, 1 1 λ L = ∂ φ∂µφ − m2φ2 − φ4 (3.16) 2 µ 2 4!

26 3.3.1 Mass renormalization We begin by considering the 2-point function to 1-loop order, given by the insertion of a single bubble graph, as in Fig 5 (a). Two loop contributions are also shown for later use.

Figure 5: 1PIR contributions to mass (a-b-c) and ﬁeld (c) renormalization in λφ4 theory up to order O(λ2)

The self-energy to this order is given by

(2) 2 2 2 Γ (p) = p − m + λGo(m) + O(λ ) Z d4k i G (m) ≡ (3.17) o (2π)4 k2 − m2 + i

Clearly, the integral Go for the bubble graph is UV divergent. The ﬁrst thing we need to do is to regularize the integral. We shall later on give a long list of fancy regulators. For the time being, let’s be simple minded and cutoﬀ the Euclidean continuation of this 2 2 integral by a cutoﬀ kE < Λ . We then have Z 4 2 2 ! d kE 1 1 2 2 Λ + m Go(m, Λ) = = Λ − m ln (3.18) 2 2 4 2 2 2 2 kE <Λ (2π) kE + m (4π) m

The momentum independent Go will have the eﬀect of changing the mass of the partice, albeit by a divergent amount. The ﬁrst key insight oﬀered by renormalization theory is that the mass parameter m that enters the Lagrangian is not equal to the mass of the physical particle created by the φ-ﬁeld. The mass in the Lagrangian equals the mass of the physical φ particle only in the free ﬁeld limit when λ = 0. If λ 6= 0, however, the relation between the mass in the Lagrangian and the physical mass will depend on the coupling. The second key insight oﬀered by renormalization theory is that the mass parameter appearing in the Lagrangian is not a physically observable quantity. Only the mass of the physical φ particle is observable – as a pole in the S-matrix. Therefore, the mass parameter in the Lagrangian, or bare mass mo, should be left unspeciﬁed at ﬁrst, to be determined later by the mass m of the physical particle. The third insight oﬀered by renormalization theory is that it is possible – in so-called renormalizable quantum ﬁeld theories – to determine the bare mass mo as a function of the physical mass m, the coupling λ and the cutoﬀ Λ in such a way that m may be kept ﬁxed as the cutoﬀ is removed Λ → ∞.

27 To illustrate the points made above on φ4 theory to order λ, we are thus instructed to take as a starting point the Lagrangian with bare mass m0, 1 1 λ L = ∂ φ∂µφ − m2φ2 − φ4 (3.19) 2 µ 2 o 4! and to recompute all of the above up to order λ. The slef-energy corrections are given by

(2) 2 2 2 Γ (p) = p − mo + λGo(mo) + O(λ ) 2 2 ! 1 2 2 Λ + mo Go(mo, Λ) = 2 Λ − mo ln 2 (3.20) (4π) mo In order for the mass of the φ particle to be m – to this order λ – it suﬃces to set

2 2 ! 2 2 λ 2 2 Λ + mo 2 m = mo − 2 Λ − mo ln 2 + O(λ ) (3.21) (4π) mo Actually, this relation may be solved to this order by workin iteratively,

λ Λ2 + m2 ! m2 = m2 + Λ2 − m2 ln + O(λ2) (3.22) o (4π)2 m2

The key lesson learned from this simple example is that as the cutoﬀ Λ is sent to ∞, it is necessary to keep on adjusting mo so that the physical mass m can remain constant. As

Λ → ∞, this actually requires mo → ∞ as well.

3.3.2 Coupling Constant Renormalization

The lowest 1PIR corrections to the 4-point function arise at order O(λ2) and are depicted in Fig 6. The three diagrams are permutations in the external legs of the single basic diagram. In the study of any 4-point, it is very useful to introduce the Mandelstam variables, s, t, u, which are the Lorentz invariant combinations one can form out of the (all

incoming) external momenta pi, i = 1, 2, 3, 4. They are deﬁned by

2 2 s ≡ (p1 + p2) = (p3 + p4) 2 2 t ≡ (p1 + p4) = (p2 + p3) 2 2 u ≡ (p1 + p3) = (p2 + p4) (3.23)

2 2 2 2 The Mandelstam relations obey s + t + u = p1 + p2 + p3 + p4, and if the external particles are on-shell then the rhs equals 4m2.

28 Figure 6: 1PIR O(λ2) contributions to coupling constant renormalization in λφ4 theory

The 4-point 1PIR contribution to this order is therefore obtained as follows, (4) 2 Γ (pi) = λ I(s) + I(t) + I(u) (3.24)

The remaining integrals is readily computed using Feynman rules and

Z 4 1 3 d k i i I(s) = (−i) 4 2 2 2 2 (3.25) 2 (2π) k − m + i (k + p1 + p2) − m + i The integral I(s) receives a logarithmically divergent contribution from k → ∞. Remark- ably, this divergence is the same for all external momenta, so that the diﬀerence I(s)−I(s0) is ﬁnite and the divergence is entirely contained in I(0) for example. The momentum independent divergent contribution may be regularized using an Eu- clidean momenta cutoﬀ, and we ﬁnd,

Z d4k 1 1 Λ2 + m2 Λ2 ! I(0) = E = ln − (3.26) 2 2 4 2 2 2 2 2 2 2 kE <Λ (2π) [kE + m ] 32π m Λ + m This contribution is a constant and therefore of the same form as the original λφ4 coupling in the Lagrangian. Thus the total 1PIR 4-point function is given as follows, (4) 2 2 3 Γ (pi) = −λ + 3λ I(0) + λ I(s) + I(t) + I(u) − 3I(0) + O(λ ) (3.27)

Now we repeat verbatim from mass renormalization, the instructions given by renormal- ization theory for the renormalization of the coupling constant. The ﬁrst key insight from renormalization theory is that the coupling parameter λ that enters the Lagrangian is not equal to the coupling of the physical φ particles. The coupling in the Lagrangian equals the coupling of the physical φ particle only in the free ﬁeld limit when λ = 0. If λ 6= 0, however, the relation between the coupling in the Lagrangian and the coupling of the physical φ particles will depend on the coupling itself. The second key insight from renormalization theory is that the coupling parameter appearing in the Lagrangian is not a physically observable quantity. Only the coupling of the physical φ particles is observable – as the transition probability in the S-matrix.

Therefore, the coupling parameter in the Lagrangian, or bare coupling λo, should be left unspeciﬁed at ﬁrst, to be determined later by the coupling λ of the physical φ particles.

29 The third insight from renormalization theory is that it is possible – in so-called renor- malizable quantum ﬁeld theories – to determine the bare coupling λo as a function of the physical coupling λ, the mass m and the cutoﬀ Λ in such a way that λ may be kept ﬁxed as the cutoﬀ is removed Λ → ∞. To illustrate the points made above on φ4 theory to order λ2, we are thus instructed to take as a starting point the Lagrangian with bare mass m0 and bare coupling λo, 1 1 λ L = ∂ φ∂µφ − m2φ2 − o φ4 (3.28) 2 µ 2 o 4! and to recompute all of the above up to order λ2. The 4-point corrections are given by

(4) 2 3 Γ (pi) = −λo + 3λoI(0) + O(λo) (3.29) In order for the coupling of the φ particles to be λ – to this order λ2 – it suﬃces to set

2 3 −λ = −λo + 3λoI(0) + O(λo) (3.30) Actually, this relation may be solved to this order by working iteratively, (it is customary to omit terms that vanish in the limit Λ → ∞, and we shall also do so here) 3λ2 Λ2 λ = λ + ln + O(λ3) (3.31) o 32π2 m2 The key lesson learned from this simple example is that as the cutoﬀ Λ is sent to ∞, it

is necessary to keep on adjusting λo so that the physical coupling λ can remain constant.

As Λ → ∞, this actually requires λo → ∞ as well. Notice that to higher orders, the renormalization of the mass and of the coupling will be intertwined and greater care will be required to extract the correct procedure.

3.3.3 Field renormalization Considering the 2-point function still in λφ4 theory, but now to order O(λ2), we have the additional 1PIR diagrams of Fig 5 (b), (c). We encounter further mass renormalization eﬀects, which may be dealt with by the methods explained above as well. We also en- counter, through the diagram (c) in Fig 5, also a new type of divergence. This may be seen from the explicit expression of the diagram, (the i prescription is understood) λ2 Z d4k Z d4` 1 1 1 Γ(2)(p) = (3.32) 2 6 (2π)4 (2π)4 (k + p)2 − m2 (k + `)2 − m2 `2 − m2 This integral is quadratically divergent, but the quadratic divergence is p-independent. (2) (2) 0 Thus, the diﬀerences Γ2 (p)−Γ2 (p ) are less divergent. By Lorentz invariance, we expect

30 (2) Γ2 (p) to vanish to second order as p → 0. The second derivative is easily computed, 2 (2) 2 Z 4 Z 4 2 ∂ Γ2 (p) λ d k d ` 8m 1 1 µ = 4 4 2 2 3 2 2 2 2 (3.33) ∂pµ∂p 6 (2π) (2π) [(k + p) − m ] (k + `) − m ` − m This integral is indeed logarithmically divergent because the sub-integration over ` is loga- rithmically divergent. Recognizing the integration over ` as the integral I(k2) encountered at the time of coupling constant renormalization, we may write the above as 2 (2) 2 Z 4 2 ∂ Γ2 (p) λ d k 8m 2 µ = i 4 2 2 3 I(0) + {I(k ) − I(0)} (3.34) ∂pµ∂p 3 (2π) [(k + p) − m ] The integral over {I(k2)− I(0)} is convergent. The integral multiplying I(0) is convergent and yields 2 (2) 2 2 2 ∂ Γ2 (p) λ λ Λ µ = 2 I(0) = 4 ln 2 (3.35) ∂pµ∂p I(0) 6π 3(4π) m Thus, we see that a divergent contribution arises which is proportional to p2. At ﬁrst sight, it would not appear that there is a coupling in the Lagrangian capable of absorbing this divergence. But in fact there is : the kinetic term is customarily normalized to 1/2 by choosing an appropriate normalization of the canonical ﬁeld. The divergent correction found here indicates that quantum eﬀects change this nromalization, and give rise to ﬁeld renormalization, also sometimes improperly referred to as wave-function renormlization. Therefore, the starting point ought to be the bare Lagrangian in which the ﬁeld nor- malization, the mass and the coupling have all been left at their bare values. One often uses the following convention for the bare Lagrangian 1 1 λ L = Z∂µφ∂ φ − Zm2φ2 − o Z2φ4 (3.36) B 2 µ 2 o 4! which, when re-expressed in terms of the bare ﬁeld √ φo(x) ≡ Zφ(x) (3.37) takes the form of the classical Lagrangian, but now with bare mass and coupling, 1 1 λ L = ∂µφ ∂ φ − m2φ2 − o φ4 (3.38) B 2 o µ o 2 o o 4! o The constant Z, to this order is given by λ2 Λ2 Z = 1 − ln + O(λ3) (3.39) 3(4π)4 m2 Unlike mass and coupling renormalization in λφ4 theory, ﬁeld renormalization has no contributions to ﬁrst order in the coupling.

31 3.4 Renormalization point and renormalization conditions

Now that we have understood how to extract ﬁnite answers from calculations involving divergent cutoﬀs, we need to make precise how to extract physical answers as well. This is achieved by imposing renormalization conditions on some of the correlators in the theory. To completely deteremine the theory, there are to be exactly as many independent renor- malization conditions as there are bare parameters in the bare Lagrangian. For scalar λφ4 theory, there are 3 parameters and thus there must be three independent renormalization conditions. Mass and ﬁeld renormalization will require two independent renormalization condi- tions of the 2-point function Γ(2)(p) while coupling constant renormalization will require (4) one renormalization condition on the 4-point function Γ (p1, p2, p3, p4). The resulting correlators will depend upon these renormalization condition and therefore they will have been renormalized, a fact that is shown by suﬃxing the letter R. Clearly, there is freedom in choosing the momenta at which the renormalization con- ditions are to be imposed. This choice of momenta is referred to as the renormalization point. It may seem most natural to relate the renormalization conditions to the properties of the physical particles. For example, if m is to be the mass of the physical φ particle (the position of the pole in the S-matrix corresponding to the exchange of a s ingle particle), then the appropriate condition is an on-shell renormalization condition,

(2) ΓR (p) = 0 (3.40) p2=m2

But Lorentz invariance allows for a more general choice, such as for example this oﬀ-shell renormalization condition,

(2) 2 2 ΓR (p) = µ − m (3.41) p2=µ2

The same choice of renormalization points arise when dealing with ﬁeld renormalization. Coupling constant renormalization, which involves the 4-point function, and thus 3 independent momenta, oﬀers even more freedom of choice of renormalization points. Since 2 2 2 2 s+t+u = p1 +p2 +p3 +p4, and on-shell renormalization consition with maximum symmetry amongst the momenta would be

(4) ΓR (p1, p2, p3, p4) = −λ (3.42) s=t=u=4m2/3

32 but it is equally easy to take this condition oﬀ-shell to the point µ,

(4) ΓR (p1, p2, p3, p4) = −λ (3.43) s=t=u=4µ2/3

On-shell versus oﬀ-shell condition may be of use in turn in a variety of circumstances as we shall see later.

3.5 Renormalized Correlators Let us impose the on-shell renormalization conditions

(2) ΓR (p) = 0 p2=m2 (2) ∂ΓR (p) 2 = 1 ∂p p2=m2

(4) ΓR (p1, p2, p3, p4) = −λ (3.44) s=t=u=4m2/3

The full expressions of the renormalized correlators are then found to be given by

(2) 2 2 ΓR (p) = p − m + 4m2 Γ(4)(p ) = −λ + λ2 I(s) + I(t) + I(u) − 3I (3.45) R i 3

It remains to evaluate more explicitly the function I(s) − I(so).

The quantity I(s) − I(so) may be evaluated using a single Feynman parameter, and 2 may be recovered from setting so = 0. It is convenient to set s = p , and to omit the i symbols, so that

i Z 1 Z d4k 1 1 I(s) − I(0) = − dα − (3.46) 2 0 (2π)4 [(k + αp)2 + α(1 − α)s − m2]2 [k2 − m2]2

This integral is convergent. We would like to shift the k-integration in the ﬁrst inte- gral. Since this integral is divergent, however, it is unclear that this procedure would be legitimate. Thus, we begin by proving a lemma. Lemma 1 One may shift under logarithmically divergent integrations;

Z d4k 1 1 − = 0 (3.47) (2π)4 [k2 − m2 + i]2 [(k + p)2 − m2 + i]2

33 To prove this, notice that the equality holds at p = 0. Then, diﬀerentiating with respect to pµ for general p, we obtain Z d4k 4(k + p) µ (3.48) (2π)4 [(k + p)2 − m2 + i]3 But this integral is convergent and after shifting k +p → k is odd in k → −k and therefore vanishes automatically. This proves the lemma for all p. o o Using lemma 1, carrying out the analytic continuation k → ikE, and carrying out the kE integration, we ﬁnd

1 4 1 Z Z d kE 1 1 I(s) − I(0) = dα 4 2 2 2 − 2 2 2 2 0 (2π) [kE − α(1 − α)s + m ] [kE + m ] 1 Z 1 m2 = dα ln (3.49) 32π2 0 m2 − α(1 − α)s We shall now study this integral as a function of s. It is convenient to change integration variables to 2α = 1 + x, so that 1 4m2 1 Z 1 I(s) − I(0) = ln − dx ln(x2 + σ2) (3.50) 32π2 s 32π2 0 where 4m2 σ2 ≡ − 1 (3.51) s Using the relation Z 1 dx ln(x + iσ) = (1 + iσ) ln(1 + iσ) − iσ ln(iσ) − 1 (3.52) 0 we ﬁnd

q 2 s 2 4m 1 iσ − 1 i 4m i s − 1 − 1 I(s) − I(0) = iσ ln = − 1 ln q (3.53) 32π2 iσ + 1 32π2 s 4m2 i s − 1 + 1 The integral is real as long as m2 and s are real and s < 4m2. When s > 4m2, however, there is a branch cut, whose discontinuity is actuually most easily computed from the integral representation that we started from, 1 Z 1 Im I(s) − I(0) = 2πi dαθ(α(1 − α)s − m2) 32π2 0 s i 4m2 = θ(s − 4m2) 1 − (3.54) 16π s

34 3.6 Summary of regularization and renormalization procedures Quantum ﬁeld theories expanded in perturbation theory, will develop UV divergences. The ﬁrst step in the process of deﬁning the theory properly – without UV divergences – is to regularize it. Over the 80 years of dealing with quantum ﬁeld theories, physicists have introduced and invented myriad regulators. Of course, it will be advantageous to regularize theories by preserving as many symmetries of the original theory as possible (e.g. Poincar´einvariance, gauge invariance, conformal invariance). Here, we give a brief list. Regulators may be grouped according to whether they respect Poincar´einvariance. • Poincar´enon-invariant regulators

1. The lattice formulation of quantum ﬁeld theory is an example of a short space-time distance cutoﬀ. As we shall see later, the lattice is best formulated for Euclidean QFT, and provides a perturbative and non-perturbative regularization of the theory. There is a gauge invariant formulation.

2. Cutoﬀ on momentum space at high momenta. This regulator is perhaps the most intuitive one for qualitative studies in perturbation theory. It provides a perturbative or non-perturbative regularization. Gauge invariance is not maintained in general.

• Poincar´einvariant Regulators

1. Dimensional regularization in which the dimension d of space-time is analytically continued throughout the complex d plane. This regularization is only applicable to Feynman diagrams and is thus in essence perturbative. It is very convenient and preserves gauge invariance and reparametrization invariance. νµ νµ • Tensor algebra, δµνδ = ηµνη = d. • Spinor algebra There are some diﬃculties with fermions, essentially because spinor representations cannot naturally be continued to fractional dimensions. Therefore, the rules for spinors are a bit ad hoc. Here are some of the key rules,

{γµ, γν} = 2ηµνI µ γµγ = d · I µ γµγνγ = (2 − d)γν µ γµγνγκγ = 2γκγν + (d − 2)γνγκ tr I = f(d) as long as f(4) = 4 (3.55)

35 There is no good deﬁnition of γ5. Deeper signiﬁcance for these problems will be derived when discussing the chiral anomaly.

2. Pauli-Villars regulators require introducing extra degrees of freedom with high masses (and the wrong causality kinetic term). This works well to one-loop order and preserves gauge invariance.

3. Heat-kernel and ζ-function regularizations are very powerful methods applicable mostly only to 1-loop, but which preserve gauge invariance as well as reparametriza- tion invariance when formulated on general manifolds !

• Renormalization Once the theory has been regularized, we have a UV ﬁnite, but regulator dependent theory. The question that renormalization theory addresses is whether one can make sense out of this theory, so as to get a predictive theory. The key to renormalization theory is that in renormalizable theories, all the divergences are of a special form, namely proportional to terms already present in the Lagrangian. Even when one has renormalized the theory to all orders in perturbatiion theory, the question still remains as to whether the full theory is renormalizable (non-perturbatively). Most of these theories are not resolved to this day, such as φ4-theory, theories with broken symmetries with monopoles and so on. The next steps are those of renormalization.

1. Keeping the bare couplings and masses in the Lagrangian undetermined as a function of the physical parameters and the cutoﬀ.

2. Imposition of certain regulator independent renormalization conditions at a renor- malization point. There should be as many renormalization conditions as indepen- dent couplings in the Lagrangian.

3. Derive the bare coupling constants in the Lagrangian as a function of the cutoﬀ and the physical couplings and masses.

4. Take the limit where the cutoﬀ is removed.

36 4 Functional Integrals for Fermi Fields

Consider a theory of Dirac fermions whose Lagrangian of the Dirac type, i.e. it is bilinear in the Dirac ﬁelds and given in terms of a ﬁrst order diﬀerential operator D by L = ψDψ¯ D = i∂/ − M (4.1) 0 0 0 M(x) = mI + φ(x)I + φ (x)γ5 + gA/(x) + g A/ (x)γ5 Here, M is a matrix-valued function that may include mass terms m, scalar ﬁelds φ and φ0 and gauge ﬁelds amongst others. The operator D will be required to be self-adjoint. Can one construct a functional integral representation for the time ordered Green functions, in analogy to the construction we had given for the scalar ﬁeld ?

¯β1 ¯βn h0|T ψα1 (x1) ··· ψαn (xn)ψ (y1) ··· ψ (yn)|0i (4.2) There is a fundamental diﬀerence with the bosonic case: the correlation function is anti- symmetric in the ψ ﬁelds. Any functional integral based on an integration over commuting

ﬁelds will yields a result that is symmetric in the points xi. The natural way out is via the use of anti-commuting or Grassmann variables, already encountered previously. A set of n anti-commuting variables may be deﬁned as follows,

θ1, θ2 ··· θn so that {θi, θj} = 0 (4.3)

In particular the most general analytic functiion of one θ is F (θ) = Fo + θF1. One may now deﬁne diﬀerentiation ∂ ( ∂ ) F (θ) ≡ F , θ = 1 (4.4) ∂θ 1 ∂θ

Grassmann numbers i.e. functions of ordinary x1 ··· xp and θn, form a graded ring, (1) Even grading : those which commute with all θ’s. Example are the bosonic variables

x1, ··· , xp, the bilinears θiθj, the quadrilinears θiθjθkθl etc.

(2) Odd grading : those which anti-commute with all θ’s. Examples are the θi, the

trilinears θiθjθk, etc. Leibnitz’s rule of diﬀerentiation has to be adapted. Let F and G be two functions of grading f and g, then ∂ ∂F ∂G (FG) = G + (−)f F (4.5) ∂θ ∂θ ∂θ ∂ so that ∂θ should be viewed as odd grading.

37 4.1 Integration over Grassmann variables Integration over Grassmann variables is deﬁned by the following rules: (1) linearity Z Z Z dθF (θ) = dθ 1 Fo + dθ θ F1 (4.6)

(2) integral of a total derivative vanishes Thus, only two integrals are required, Z Z ∂ Z dθ 1 = dθ θ = 0 dθ θ = 1 (4.7) ∂θ We may now state things more generally, and introduce a space of n Grassmann variable

θi, i = 1, ··· , n, ( ∂ ∂ ) ( ∂ ) {θi, θj} = , = 0 , θj = δij (4.8) ∂θi ∂θj ∂θi ∂ So one may view the set of θi’s and ’s as a set of Cliﬀord generators. ∂θi ¯ ¯ Consider θ1 ··· θn and their complex conjugates θ1, ··· θn and calculate the following Gaussian integral, Z Z ¯ Z Z ¯ n n ¯ θiMij θj ¯ ¯ θiMij θj d θ d θ e = dθ1 ··· dθn dθ1 ··· dθn e Z Z 1 n = dθ ··· dθ dθ¯ ··· dθ¯ θ¯ M θ (4.9) 1 n 1 n n! i ij j To calculate this, we use the anti-symmetry of the product of θi, and we obtain

θi1 ··· θin = i1···in θ1θ2 ··· θn (4.10) Therefore, the integrations may be carried out explicitly and we have

Z ¯ 1 dnθdnθ¯ eθiMij θj = M M ··· M = detM (4.11) n! i1···in j1···jn i1j1 i2j2 injn Compare this result with the analogous Gaussian integrals for commuting variables, Z Z z¯iMij zj −1 dz1 ··· dzn dz¯1 ··· dz¯ne = (detM) (4.12) Similarly, correlation functions may also be evaluated explicitly, Z ¯ n n ¯ ¯ θiMij θj −1 d θd θ θkθ` e = (detM)(M )k` (4.13)

Z ¯ h i n n ¯ ¯ ¯ θiMij θj −1 −1 −1 −1 d θd θ θk1 θk2 θ`1 θ`2 e = (detM) (M )k1`1 (M )k2`2 − (M )k1`2 (M )k2`1 So we have constructed time ordered products that are antisymmetric in these indistin- guishable arguments.

38 4.2 Grassmann Functional Integrals

With the above tools in hand, functional integrals over Grassmann ﬁelds may be computed. The ﬁrst object of interest is the 2-point function,

Z R 4 ¯ ¯β i d xL DψDψ ψα(x)ψ (y) e (4.14)

The arguments used to calculate this object are identical for any Lagrangian that is of the Dirac type, so we shall treat the general case. To calculate (4.14), we diagonalize D, which is possible in terms of real eigenvalues since D† = D. We denote the eigenvalues by λn and the corresponding ortho-normalized eigenfunctions by ψn. The index n may have continuous and discrete parts and degenerate eigenvalues will be labelled by distinct n. The ﬁeld may then be decomposed in terms of the basis of eigenfunctions ψn with

Grassmann odd coeﬃcients cn as follows,

X ψ(x) = cnψn(x) n Z 4 X d xL = c¯ncnλn (4.15) n

Next, we change variables in the functional integral from the ﬁeld ψ(x) to the coeﬃcients cn. Since the eigenfunctions satisfy (ψm, ψn) = δmn, the Jacobian of this change of variables is unity. Actually, the expression in terms of the coeﬃcients cn may be viewed as the proper deﬁnition of the measure,

Z Z ¯ Y DψDψ ≡ dcndc¯n (4.16) n

Calculating the vacuum amplitude is now starightforward,

( ) Z R 4 Z ¯ i d xL Y X DψDψ e = dcndc¯n exp i c¯ncnλn = Det(iD) (4.17) n n

The 2-point function may be computed by the same methods,

( ) Z R 4 Z ¯ ¯β i d xL X Y ¯β X DψDψψα(x)ψ (y) e = dcpdc¯p ψmα(x)ψn(y) cmc¯n exp i c¯ncnλn m,n p n ψ (x)ψ¯β(y) = (DetiD) X nα n (4.18) n iλn

39 The object multiplying Det(iD) on the rhs is recognized to be the Dirac propagator, namely the inverse of the operator iD,

¯β X ψnα(x)ψn(y) SF (x, y; M) ≡ (4.19) n iλn β X ¯β β (4) DSF (x, y; M) = −i ψnα(x)ψn(y) = −iδα δ (x − y) α n

This is as should be expected. Since the ﬁeld equations for ψ are (i∂/ − M)ψ = 0, the 2-point function satisﬁes

γ ¯β β (4) (i∂/ − M)α h0|T ψγ(x)ψ (y)|0i = −iδα δ (x − y) (4.20)

For the higher point-functions, the same techniques may be employed and we have

¯β1 ¯βn h0|T ψα1 (x1) ··· ψαn (xn)ψ (y1) ··· ψ (yn)|0i (4.21) 1 Z = DψDψψ¯ (x ) ··· ψ (x )ψ¯β1 (y ) ··· ψ¯βn (y )eiS(ψ) R DψDψe¯ iS[ψ] α1 1 αn n 1 n

4.3 The generating functional for Grassmann ﬁelds In particular, one may again derive a generating functional. The source term cannot how- ever be taken to be an ordinary function, since the time ordered product is antisymmetric. One has to use a new set of Grassmann valued functions: Z Z Z[η, η¯] ≡ DψDψ¯ exp iS[ψ] + d4x(ψη¯ +ηψ ¯ ) (4.22)

Note that we used Lorentz invariant combinations. By expanding the generating functional in powers of the source terms, we have

∞ 1 1 Z Z X 4 4 4 4 α1 αn Z[η, η¯] = d x1 ··· d xm d y1 ··· d ynη¯ (x1) ··· η¯ (xm)ηβ1 (y1) ··· ηβn (yn) m,n=0 m! n! Z ¯ ¯β1 ¯βn iS[ψ] × DψDψψα1 (x1) ··· ψαm (xm)ψ (y1) ··· ψ (yn)e (4.23)

If the action is purely quadratic in ψ, then the number of ψ and ψ¯’s must be equal

Z[η, η¯] ∞ 1 Z Z X 4 4 4 4 α1 αn = 2 d x1 ··· d xn d y1 ··· d ynη¯ (x1) ··· η¯ (xn)ηβ1 (y1) ··· ηβn (yn) Z[0, 0] n=0 (n!) ¯β1 ¯βn ×h0|T ψα1 (x1) ··· ψαn (xn)ψ (y1) ··· ψ (yn)|0i (4.24)

40 For the free theory, the generating functional Zo may be computed by completing the square in the Grassmann functional integral,

Z Z iS[ψ] + d4x(ψη¯ +ηψ ¯ ) = d4x ψiDψ¯ + ψη¯ +ηψ ¯ (4.25) Z n o = d4x (ψ − iD−1η)iD(ψ − iD−1η) +η ¯(iD)−1η

Hence the generating functional may be evaluated in terms of the Feynman propagator, Z η, η¯] Z Z [ = exp d4x d4y η¯(x)S (x, y; M)η(y) (4.26) Z[0, 0] F The time-ordered expectation value of the n-point functions are recovered analogously.

4.4 Lagrangian with multilinear terms When the Lagrangian is not just quadratic in ψ one proceeds as follows. Say it also has quartic terms, e.g. 1 L = ψ¯(i∂/ − m)ψ − g2(ψψ¯ )2 (4.27) 2 It is always possible to introduce a non-dynamical bosonic “auxiliary ﬁeld” σ as well as a new Lagrangian L0, such that elimination of the ﬁeld σ using the ﬁeld equations for σ yields the old Lagrangian. In the quartic case, 1 1 L0 = ψ¯(i∂/ − m)ψ + (σ − gψψ¯ )2 − g2(ψψ¯ )2 2 2 1 = ψ¯(i∂/ − m)ψ + σ2 − gσψψ¯ (4.28) 2 When we compute a vacuum expectatiion value with the ﬁrst Lagrangian

¯β1 ¯βn h0|T ψα1 (x1) ··· ψαn (xn)ψ (y1) ··· ψ (yn)|0i (4.29) 1 Z R ¯ ¯β1 ¯βn i L = R DψDψ ψα1 (x1) ··· ψαn (xn)ψ (yn) ··· ψ (yn)e R DψDψ¯ ei L we may clearly replace it with

1 Z R 0 ¯ ¯β1 ¯βn i L = R DψDψDσ ψα1 (x1) ··· ψαn (xn)ψ (y1) ··· ψ (yn)e (4.30) DψDψ¯Dσ ei L0 The Lagrangian L0 is now quadratic in ψ and can be treated with the preceding methods.

41 5 Quantum Electrodynamics : Basics

This is the theory of an Abelian gauge ﬁeld, coupled to charged Dirac spinor ﬁelds. Here, we take the case of a single spinor ﬁeld ψ, which may be thought of as the ﬁeld decribing the electron and positron. We include the usual gauge ﬁxing term. The action is as follows,

S[A, ψ] = So[A, ψ] + SI [A, ψ] Z 1 λ ! S [A, ψ] = d4x − F F µν − (∂ Aµ)2 + ψ¯(i∂/ − m)ψ o 4 µν 2 µ Z 4 ¯ µ SI [A, ψ] = − d x eAµψγ ψ (5.1)

The electric charge e enters via the ﬁne structure constant

e2 1 α = ∼ (5.2) 4πhc¯ 137.04 which may be viewed as small. Thus, we expand directly in powers of the interaction SI . The free propagators were computed previously,∗

Z 4 ! β β ¯β d k i −ik·(x−y) Sα (x − y) ≡ h0|T ψα(x)ψ (y)|0io = 4 e (2π) k/ − m α (5.3) Z d4k −i k k ! G (x − y) ≡ h0|TA (x)A (y)|0i = η + ξ µ ν e−ik(x−y) µν µ ν o (2π)4 k2 µν k2

It is convenient to use the abbreviation ξ ≡ (1 − λ)/λ when dealing with the gauge choice parameter λ.

5.1 Feynman rules in momentum space

The Feynman rules are completely analogous to (and much simpler than) the ones for the µ β scalar ﬁeld. The vertex is simply the multiplicative factor −ie(γ )α . Propagators and vertex are represented in Fig 4. Note that ψ and ψ¯ are distinct ﬁelds (their change is opposite), so that the counting factors are modiﬁed. In addition, for closed fermion loops,

∗Henceforth, the i contributions to the propagators will be understood in all expressions.

42 there is a factor of -1 for each loop. In momentum space, the Feynman rules are as follows,

i ! β Incoming fermion p/ − m α i ! β Outgoing fermion −p/ − m α −i p p ! Photon η + ξ µ ν (5.4) p2 µν p2

The direction of the arrow indicates the ﬂow of electric charge, taken from ψ¯ to ψ.

α 4 4 Vertex − ie(γµ)β (2π) δ (total incoming momentum) (5.5)

Internal lines have an additional factor of momentum integration,

d4p i ! β Fermion 4 (2π) p/ − m α d4p −i p p ! Photon η + ξ µ ν (5.6) (2π)4 p2 µν p2

A minus sign for each closed fermion loop.

Figure 7: Feynman rules for QED

5.2 Verifying Feynman rules for one-loop Vacuum Polarization

Z Z R 4 ¯ i d xL(A,ψ,ψ¯) h0|TAµ(x)Aν(y)|0i = DA DψDψ e Aµ(x)Aν(y) (5.7) novac 2 Z Z 1 Z R 4 = DA Dψ¯DψA (x)A (y) (−ie)2 d4zψγ¯ µψA ei d xLo µ ν 2 µ

43 The purely fermionic part that has to be computed here is

Z R 4 Z R 4 ¯ ¯ µ ¯ ν i d xLo ¯ ¯ i d xLo DψDψ ψγ ψ(z1) ψγ ψ(z2) e DψDψ e Z Z δ µ α δ δ ν δ δ = (γ )β (γ )γ δ exp η¯(x)S(x − y)η(y) δηβ(z1) δη¯α(z1) δηγ(z2) δη¯ (z2) x y Z Z Z δ µ α δ δ ν δ ρ = (γ )β (γ )γ Sδ (z2 − u)ηρ(u) η¯(x)S(x − y)η(y) δηβ(z1) δη¯α(z1) δηγ(z2) u x y Z Z δ µ α δ ν δ γ = (γ )β (γ )γ Sδ (0) η¯(x)S(x − y)η(y) δηβ(z1) δη¯α(z1) x y Z Z δ µ α δ ν δ ρ σ γ + (γ )β (γ )γ Sδ (z2 − y)ηρ(y) η¯ (x)Sσ (x − z2) (5.8) δηβ(z1) δη¯α(z1) y x

The ﬁrst term on the rhs of the last expression above must vanish by Lorentz invariance. The cancellation may also be seen directly from the expression for the propagator,

Z d4k tr{γµ(6 k + m)} (γν) δS γ(0) = = 0 (5.9) γ δ (2π)4 k2 − m2 + i

The remainder of the calculation proceeds as follows,

Z δ µ α ν δ ρ γ = − (γ )β (γ )γ Sδ (z2 − y)ηρ(y) Sα (z1 − z2) δηβ(z1) y µ α ν δ β γ = −(γ )β (γ )γ Sδ (z2 − z1) Sα (z1 − z2) µ ν = −tr γ S(z1 − z2)γ S(z2 − z1) (5.10)

Hence 1 Z Z h0|TA (x)A (y 0i = e2 d2z d2z h0|TA (x)A (y)A (z )A (z )|0i µ ν | 2 1 2 µ ν κ 1 λ 2 o κ λ ×tr(γ S(z1 − z2)γ S(z2 − z1)) (5.11)

Of the three possible contributions to the tree level 4-point function for the gauge ﬁelds, one is disconnected and corresponds to a vacuum graph which does not contribute to the correlator. The remaining two connected contributions are equal and yield

Z Z 2 4 4 h0|TAµ(x)Aν(y)|0i = Gµν(x − y) + e d z1 d z2Gµκ(x − z1)Gνλ(y − z2) κ λ ×tr(γ S(z1 − z2)γ S(z2 − z1)) (5.12)

44 5.3 General fermion loop diagrams The ﬁrst class of diagrams, to one loop order, consist of a single fermion loop with an arbitrary number of external Aµ ﬁeld insertions. In 1PIR graphs, the external photon propagators are truncated away, a fact that is indicated by using short photon propagator lines. The graphs up to 5 external insertions are depicted in Fig 8.

Figure 8: One-Fermion-Loop graphs

These diagrams represent the perturbative expansion in powers of eAµ of the fermion eﬀective action, deﬁned as follows, Γ[Aµ] = −i ln Det i∂/ − eA/ − m (5.13)

5.4 Pauli-Villars Regulators By power counting, all diagrams with n ≤ 4 are expected to be UV divergent and therefore need to be regularized. It is convenient to use a regulator that preserves Poincar´eas well as gauge invariance. Pauli-Villars regulators introduces extra fermions ﬁelds with identical gauge couplings, but very large masses, on the order of a regulator scale Λ. Since the Pauli- Villars regulator masses are very large there is no modiﬁcation to the physics at distance scales large compared to the regulator scale Λ. We denoted the Pauli-Villars regulators by

ψs and their masses by Ms; the Lagrangian is then

S X ¯ LPV = L + Csψs(i∂/ − eA/ − Ms)ψs (5.14) s=1

The number S of regulators, and the constants Cs and Ms are to be determined by the requirement that the regularized correlators should be ﬁnite. These data are not unique; a change in choice leads to a ﬁnite change in the deﬁnition of the regularized correlators.

5.5 Furry’s Theorem Furry’s Theorem states that the 1-fermion loop diagrams for QED with an odd number of current insertions vanishes identically in view of charge conjugation symmetry. Recall that the charge conjugate ψc of the ﬁeld ψ is deﬁned by

c ¯t o T ∗ ψ = ηcCψ = ηcC(γ ) ψ (5.15)

45 where the charge conjugation matrix is deﬁned to satisfy

t −1 † CγµC = −γµ C C = I (5.16)

As a result, the charge conjugate of the electric current reverses its sign, as might be expected. The fermion propagator in momentum space transforms as follows,

1 !t 1 C C−1 = (5.17) k/ − m −k/ − m and this implies that its transformation in position space is given by Hence

CS(x, y)tC−1 = S(y, x) (5.18)

For a given ordering of momenta and µ labels, two graphs with opposite charge ﬂow (opposite orientation of the arrow) contribute and must be added up. They are given by (n) µ1 µ2 µ3 µn Γ = tr γ S(x1, x2)γ S(x2, x3)γ ··· γ S(xn, x1) (5.19) µ1 µn µn−1 µ2 +tr γ S(x1, xn)γ S(xn, xn−1)γ ··· γ S(x2, x1)

Taking the transpose inside the trace leaves this expression invariant, and re-expressing the transposes with the help of the charge conjugation matrix C yields (n) t µn t µ3 t t µ2 t t µ1 t Γ = tr S(xn, x1) (γ ) ··· (γ ) S(x2, x3) (γ ) S(x1, x2) (γ ) (5.20) t µ2 t µn−1 t t µn t t µ1 t +tr S(x2, x1) (γ ) ··· (γ ) S(xn, xn−1) (γ ) S(x1, xn) (γ ) n µn µ3 µ2 µ1 = (−) tr S(x1, xn)γ ··· γ S(x3, x2)γ S(x2, x1)γ (5.21) n µ2 µn−1 µn µ1 +(−) tr S(x1, x2)γ ··· γ S(xn−1, xn)γ S(xn, x1)γ = (−)nΓ(n) (5.22)

This completes the proof of Furry’s theorem. Therefore, diagrams (b) and (d) in Fig 8 vanish identically, leaving only those with even n.

5.6 Ward-Takahashi identities for the electro-magnetic current

µ µ Whenever there exists a current j (x) which is satisﬁes the conservation equation ∂µj = 0, certain special relations, generally called Ward identities, will hold between certain

46 correlation functions of local operators. In the case at hand, the relevant conserved current is the electro-magnetic current jµ(x) ≡ ψ¯(x)γµψ(x). Our derivation of the Ward identities will, however, be more general than this case. The conservation of the current jµ implies the existence of a time-independent charge Q. Local observables may be organized in eigenspaces of deﬁnite Q-charge, and it suﬃces to examine correlators of operators of deﬁnite charges Z 3 0 [Q, Oi(y)] = qiOi(y) Q = d ~xj (t, ~x) (5.23)

0 Since the commutator [j (t, ~x), Oi(t, ~y)] can receive contributions only from ~x = ~y in view of the locality of the operators, the above commutator with the charge requires the following local commutator to hold,

0 0 0 (4) [j (x), Oi(y)] δ(x − y ) = qiδ (x − y)Oi(y) (5.24)

In speciﬁc theories, this relation will in fact result from the explicit form of the current µ j and the local operators Oi(y) in terms of canonical ﬁelds and the use of the canonical commutation relations of the canonical ﬁelds. For example in QED, the relation may be 0 0 0 worked out for the commutator [j (x), ψα(y)]δ(x − y ). Next, consider a correlator of a product of local operators and the conserved current jµ(x). The interesting quantity to consider is the divergence of this correlator,†

(x) µ ∂µ h∅|T j (x)O1(y1) ···On(yn)|∅i (5.25) Were it not for the time-ordering instruction, this divergence would be identically 0 as 0 0 long as the current is conserved. From the time ordering Heaviside functions on x − yi , however, one picks up non-vanishing terms. This may be seen ﬁrst from the time-ordering on two ﬁelds,

(x) µ 0 0 0 ∂µ T j (x)Oi(yi) = δ(x − yi )[j (x), Oi(y)] = qiδ(x − y)Oi(yi) (5.26) and readily generalizes to n ﬁelds. Inserting this result in the correlator of interest yields,

n (x) µ X ∂µ h∅|T j (x)O1(y1) ···On(yn)|∅i = qi δ(x − yi)h∅|T O1(y1) ···On(yn)|∅i (5.27) i=1 This is a very important equation, as it states that the longitudinal part of the current correlator is expressed in terms of correlators without the insertion of the current.

†Here and elsewhere, the superscript (x) will indicate that the derivatve is applied in the variable x.

47 We give two examples in QED. The ﬁrst applies to the correlator of electro-magnetic currents only. Since the electric charge of the current itself vanishes, one has

(x) µ ν1 νn ∂µ h∅|T j (x)j (y1) ··· j (yn)|∅i = 0 (5.28)

In other words, the current correlators are transverse in each external leg. The second example involves one current and two fermions,

(x) µ ¯β ¯β ∂µ h∅|T j (x)ψα(y)ψ (z)|∅i = +δ(x − y)h∅|T ψα(y)ψ (z)|∅i ¯β −δ(x − z)h∅|T ψα(y)ψ (z)|∅i (5.29)

Feynman diagrammatic representations of both equations are given in Fig.

5.7 Regularization and gauge invariance of vacuum polarization The vacuum polarization graph corresponds to n = 2 and contributes to Γ(2). The unreg- ularized Feynman diagram for vacuum polarization is given by

Z d4p i i ! Γµν(k, m) ≡ (−)(−i)(−ie)2 tr γµ γν (5.30) o (2π)4 p/ − m p/ + k/ − m

The overall − sign is for a closed fermion loop; the factor of (−i) from thedeﬁnition of Γ; the factors of (−ie) from two vertex insertions. This integral is quadratically divergent.

We introduce s = 1, ··· ,S Pauli-Villars regulators with masses Ms ∼ Λ and deﬁne the regularized vacuum polarization tensor by

Z 4 S ! µν 2 d p X µ 1 ν 1 Γ (k, m; Λ) ≡ ie 4 Cstr γ γ (5.31) (2π) s=0 p/ − Ms p/ + k/ − Ms

where M0 = m and C0 = 1. In practice, it suﬃces to take S = 3, but we shall leave this number arbitrary at this stage. Gauge invariance requires conservation of the electro-magnetic current and by the Ward identities derived in the preceding subsection, transversality of the 2-point function, µν kµΓ (k, m; Λ) = 0. This property is readily checked using the following simple, but very useful general formula, 1 1 1 1 k/ = (p/ + k/ − Ms) − (p/ − Ms) p/ − Ms p/ + k/ − Ms p/ − Ms p/ + k/ − Ms 1 1 = − (5.32) p/ − Ms p/ + k/ − Ms

48 As a result,

Z 4 S ! µν 2 d p X µ 1 1 kνΓ (k, m; Λ) = ie 4 Cstr γ k/ (2π) s=0 p/ − Ms p/ + k/ − Ms Z 4 S ! 2 d p X µ 1 1 = ie 4 Cstr γ − (5.33) (2π) s=0 p/ − Ms p/ + k/ − Ms Since the integrals have been rendered convergent by the Pauli-Villars regulators, one may shift the momentum integration by k in the second term so that the diﬀerence of the two integrals vanishes, conﬁrming gauge invariance. Gauge invariance thus amounts to transversality of the 2-point function. This property in turn leads to a dramatic simpliﬁcation of the expression for vacuum polarization. Since Γµν is a Lorentz tensor which depends only on the vector kµ, its general form must be

Γµν(k, m; Λ) = Aηµν + Bkµkν (5.34)

Transversality requires A + k2B = 0, so that vacuum polarization is proportional to a single scalar function,

Γµν(k, m; Λ) = (k2ηµν − kµkν)Γ(¯ k2, m; Λ) (5.35)

The fact that two powers of momentum are extracted by gauge invariance implies, just on dimensional grounds, that the remaining vacuum polarization functionω ¯ will be only logarithmically divergent. Thus, we encounter for the ﬁrst time, a general paradigm that gauge invariance improves UV convergence.

5.8 Calculation of vacuum polarization The ﬁrst step is to rationalize the fermion propagators.

Z 4 S µ ν µν 2 d p X tr γ (p/ + Ms)γ (k/ + p/ + Ms) Γ (k, m; Λ) = ie 4 Cs 2 2 2 2 (5.36) (2π) s=0 (p − Ms )((k + p) − Ms ) The second step is to introduce a Feynman parameter,

Z 1 Z 4 S µ ν µν 2 d p X tr γ (p/ + Ms)γ (k/ + p/ + Ms) Γ (k, m; Λ) = ie dα 4 Cs 2 2 2 2 (5.37) 0 (2π) s=0 (p − Ms + 2αk · p + αk ) The third step is to combine the denominator as follows,

p2 + 2αk · p + αk2 = (p + αk)2 + α(1 − α)k2 (5.38)

49 The fourth step is to shift p+αk → p, which is permitted because the combined integrand leads to a convergent integral, Z 1 Z 4 S µ ν µν 2 d p X tr γ (p/ − αk/ + Ms)γ ((1 − α)k/ + p/ + Ms) Γ (k, m; Λ) = ie dα 4 Cs 2 2 2 2 (5.39) 0 (2π) s=0 (p − Ms + α(1 − α)k ) The ﬁfth step is to work out the trace, discard terms odd in p since those cancel by p → −p symmetry of the remaining integrand, and replace 1 pµpν → p2ηµν (5.40) 4 by using Lorentz symmetry of the integrand, tr γµp/γνp/ → − 2p2ηµν (5.41) Combining these result, Z 1 Z 4 S 2 µν 2 µν µ ν 2 µν µν 2 d p X 2p η − α(1 − α)[4k η − 8k k ] − 4Ms η Γ (k, m; Λ) = −ie dα 4 Cs 2 2 2 2 0 (2π) s=0 (p + α(1 − α)k − Ms ) (5.42) This result is not manifestly transverse and we shall now show that the longitudinal piece vanishes by intgeration by part. To do so, we split this up as follows, Γµν(k, m; Λ) = (ηµνk2 − kµkν)Γ(¯ k2, m; Λ) (5.43) Z 1 Z 4 S 2 2 2 2 µν d p X 2p + 4k α(1 − α) − 4Ms −ie η dα 4 Cs 2 2 2 2 0 (2π) s=0 (p + α(1 − α)k − Ms ) The integrand of the second term is a total derivative, 2 2 2 µ 2p + 4k α(1 − α) − 4Ms ∂ p 2 2 2 2 = µ 2 2 2 (5.44) (p + α(1 − α)k − Ms ) ∂p p + α(1 − α)k − Ms and therefore the p-integral vanishes since the sum over all Pauli-Villars regulator contri- butions is convergent. The remaining scalar integral is given as follows, Z 1 Z 4 S ¯ 2 2 d p X α(1 − α) Γ(k , m; Λ) = 8ie dα 4 Cs 2 2 2 2 (5.45) 0 (2π) s=0 (p + α(1 − α)k − Ms )

It suﬃces to take S = 1,M1 = Λ. The p-integral encountered are scalar and may be carried out easily with the help of the formulas derived earlier, 8e2 Z 1 m2 − α(1 − α)k2 Γ(¯ k2, m, Λ) = dα α(1 − α) ln (5.46) (4π)2 0 Λ2 Next, we shall derive a physical interpretation for this result and carry out the required renormalizations.

50 5.9 Resummation and screening eﬀect

Repeated insertion of the 1-loop vacuum polarization eﬀect into the photon propagator is easily carried out. The tree level photon propagator is given by,

−i 1 − λ k k ! Go (k) = η + µ ν (5.47) µν k2 µν λ k2

Figure 9: Resummation of 1-loop vacuum polarization

Since the vacuum polarization eﬀect is transverse, it is clear that the longitudinal part of the photon propagator is unmodiﬁed to all orders. A single insertion therefore produces a purely transverse eﬀect,

−i2 G1 (k) = (k2η − k k )iΓ(¯ k2, m; Λ) µν k2 µν µ ν −i k k ! = η − µ ν Γ(¯ k2, m; Λ) (5.48) k2 µν k2

Hence the full propagator becomes

−i k k ! 1 G (k) = η − µ ν (5.49) µν k2 µν k2 1 − Γ(¯ k2, m; Λ)

The eﬀect of this correction is to modify Coulomb’s law. In the limit where k → 0, for example, we have

e2 e2 e2 1 → eﬀ ≡ × ~k2 ~k2 ~k2 1 − Γ(0¯ , m; Λ) e2 Λ2 1 − Γ(0¯ , m; Λ) = 1 + ln + O(e4) (5.50) 12π2 m2 Starting out with the charge e in the Lagrangian, we see that the eﬀective charge observed in the Coulomb law is reduced, at distances long compared to 1/Λ. Quantitatively, the eﬀective charge is given by

2 2 e eeﬀ = e2 Λ2 (5.51) 1 + 12π2 ln m2

51 5.10 Physical interpretation of the eﬀective charge

Vacuum polarization is produced by the screening eﬀects of virtual electron-positron pairs. We now explain how this can come about. As is shown schematically by the vacuum polarization Feynman diagram itself, the presence of the photon gives rise to an electro- magnetic ﬁeld which creates a virtual pair. The QED vacuum is therefore not quite an empty space. Instead, in the QED vacuum, electron positron pairs are continuously beeing created and annihilated. Each pair acts like a small short-lived electric dipole. The presence of these electric dipoles makes the QED vacuum into a di-electric medium, analogous to the di-electric media created by matter, such as water, air etc. It is a well-known eﬀect of a di-electric medium that an electric charge is screened by the presence of the electric dipoles in the medium. Qualitatively, in the presence of a + charge, the dipoles will align their negative ends towards the + charge and the net eﬀect of this is that the eﬀective charge decreases with distance. This is schematically depicted in Fig 10.

Figure 10: Polarization of a charge by virtual electron-positron pairs

The electric charge observed in the Coulomb law is the eﬀective charge at long distances,

much larger than 1/me. If the charge at long distances is to be e, then the charge placed at

the center must be larger. We shall denote that charge by e0. The relation now becomes

2 4 2 2 eo 2 2 e Λ 6 e = 2 2 or eo = e + ln + O(e ) (5.52) eo Λ 12π2 m2 1 + 12π2 ln m2

The k-dependent part of Γ(¯ k, m, Λ) also aﬀects the Coulomb interaction in a computable way. The value of the charge will now depend on the momentum. For example in the approximation of momenta large compared to m, we shall have

e2 e2 e2 (k) = o = o eﬀ ¯ 2 ¯ 2 1 − Γ(k , m, Λ) 1 − Γ(0, m, Λ) + −ΓR(k , m, Λ) e2 = (5.53) ¯ 2 1 − ΓR(k , m; Λ) with

2 Z 1 2 2 ¯ 2 e m − α(1 − α)k ΓR(k , m, Λ) = dα α(1 − α) ln (5.54) 2π2 0 m2

52 In the regime −k2 m2, referred to as the deep inelastic scattering region, we have e2 −k2 Γ¯ (k2, m, Λ) = ln + O(e4) (5.55) R 12π2 m2 and the eﬀective charge as a function of k2 is given by

2 2 e eeﬀ (k) = e2 −k2 (5.56) 1 − 12π2 ln m2 This formula conﬁrms yet once more that as one probes smaller and smaller distances (large −k2), the eﬀective charge increases; this is again consistent with our physical picture for vacuum polarization.

5.11 Renormalization conditions and counterterms Our key conclusion is that vacuum polarization modiﬁes the large distance Coulomb law behavior of the photon propagator. But the Coulomb law provides the standard deﬁnition of the electric charge. Thus, we shall reverse the order in which we impose known data.

1. We normalize the Coulomb law at long distances to be governed by the standard electric charge e;

2. Therefore, the bare parameters in the Lagrangian will not be the ones observed at long distance;

3. Imposing the condition that the Coulomb law at long distances is standard amounts to a renormalization condition, imposed at the zero momentum renormalization point. The modiﬁed Lagrangian used as a starting point is the bare Lagrangian. Its derivation from the physical Lagrangian is given by counterterms.

In the bare Lagrangian Lo, the charge is replaced by the bare charge eo, the fermion mass is replaced by the bare mass m and the ﬁelds A and ψ are subject to ﬁeld renormal- √ √ o µ ization factors Z3 and Z2 respectively. It is custormary to introduce a renormalization

constant Z1 for the vertex as well, 1 λ L = − Z F F µν − (∂ Aµ)2 + Z ψ¯(i∂/ − m )ψ − eZ ψA/ψ¯ (5.57) o 4 3 µν 2 µ 2 o 1 1 L = − (Z − 1)F F µν + (Z − 1)ψi∂/ψ¯ − (Z m − m)ψψ¯ − e(Z − 1)ψA/ψ¯ ct 4 3 µν 2 2 o 1

From vacuum polarization the only term we have encountered so far is Z3.

53 The renormalization condition suitable to recovering the physical Coulomb law is that at long distance (i.e. zero momentum k), the full truncated 2-point function

µν µν 2 µ ν ¯ 2 ΓR (k, m) = (η k − k k )ΓR(k , m) ¯ ΓR(0, m) = 1 (5.58)

¯ 2 We know the contributions to ΓR(k, m). The bare coeﬃcient is Z3 = 1 + O(e ); the one-loop correction of vacuum polarization Γ(¯ k, m, Λ)

¯ 2 4 Γ(k, m) = Z3 − Γ(k , m, Λ) + O(e ) (5.59)

The renormalization condition implies the requirement e2 Λ2 Z = 1 + Γ(0¯ , m, Λ) = 1 − ln + O(e4) (5.60) 3 12π2 m2 Notice that the Ward-Takahashi identities relating the electron propagator and the vertex function imply the relation Z1 = Z2. In other words, the combination eAµ = e0A0µ is not renormalized.

5.12 The Gell-Mann – Low renormalization group We shall deal in detail with the renormalization group later on, but historically, the subject started with the following key observation by Gell-Mann and Low. One really is not forced to renormalize the photon 2-point function at zero photon momentum. In fact, when one imagines a theory with vanishing fermion mass m = 0, it is impossible to renormzalize at zero momentum. To solve this problem, Gell-Mann and Low propose to choose an arbitrary scale, called the renormalizatiion scale µ such that

¯ 2 ΓR(k , m; µ) = 1 (5.61) k2=−µ2

Using the explicit form of Z3:

2 Z 1 2 2 e m + α(1 − α)µ 4 Z3 = 1 + dα α(1 − α) ln + O(e ) (5.62) 2π2 0 Λ2

Now if the bare charge is eo, we have

1/2 e(µ) = Z3 (µ, Λ, m)eo(Λ) (5.63)

The charge e(µ), for given bare charge eo, and given cutoﬀ, now depends upon µ. The limit m → 0 is now regular.

54 The dependence of e(µ) on µ deﬁnes the Gell-Mann–Low β-function,

∂e(µ) β(e) ≡ (5.64) ∂ ln µ eo,Λ ﬁxed

The β-function may be evaluated to one loop order with the help of the results on Z3 that we have just derived and one ﬁnds,

1/2 3 ∂Z3 1 ∂ ln Z3 e 5 β(e) = e(Λ) = e(µ) = 2 + O(e ) (5.65) ∂ ln µ eo Λ 2 ∂ ln µ eo,Λ 12π

55 6 Quantum Electrodynamics : Radiative Corrections

In the previous section, one-loop contributions due to dynamical fermions only have been calculated. In these problems, the gauge potential may eﬀectively be treated as an external ﬁeld, since the photon was not quantized there. In this section, the gauge ﬁeld is also quantized. Two important special cases are studied in detail : the fermion propagator and the vertex function, from which the one-loop anomalous magnetic moment is also deduced. A discussion of infrared singularities is also given.

6.1 A ﬁrst look at the Electron Self-Energy By Lorentz and parity invariance of QED, the electron self-energy must be of the following general form, Σ(p, m) = A(p2)p/ + mB(p2)I (6.1) The coeﬃcients A, B may, of course, also depend on the mass and on any regulators, which will not be exhibited. Note that terms of the type C(p2)p/γ5 + D(p2)mγ5 are allowed by Lorentz invariance, but violate parity. The presence of the extra factor of m in the B-term results from approximate chiral symmetry of the theory with small m.

Figure 11: 1-loop 1PIR contribution to the electron self-energy

The 1-loop contribution to the electron self-energy is shown in Fig 11, and takes the form, Z d4k −i k k ! i −i(−ie)2 η + ξ µ ν γµ γν (6.2) (2π)4 k2 µν k2 p/ + k/ − m Contrarily to vacuum polarization, the electron self-energy is dependent upon the gauge parameter ξ. For simplicity, choose Feynman gauge ξ = 0 ﬁrst. The graph has a linear superﬁcial UV divergence. By Lorentz symmetry, however, this divergence is reduced to logarithmic. A convenient UV regulator is by replacing the photon propagator `ala Pauli-Villars, (another choice could be dimensional regularization), 1 1 1 → − (6.3) k2 k2 k2 − Λ2 where Λ is a large regulator mass scale. To evaluate Σ(1), we start by rationalizing the µ α µ α fermion propagator, and using the γ-matrix identities γµγ = 4I, γµγ γ = −2γ , Z d4k −2p/ − 2k/ + 4m 1 1 Σ(1)(p, m, Λ) = ie2 − (6.4) (2π)4 [(p + k)2 − m2] k2 k2 − Λ2

56 Introducing one Feynman parameter α, shifting k → k − αp, recognizing the cancellation

of a k/ term in the numerator, Wick rotating to Euclidean momenta kE, and decomposing according to its Lorentz properties, we ﬁnd

Z 1 Z 4 ! 2 2 d kE 1 − α 1 − α A(p ) = +2e dα 4 2 2 2 − 2 2 2 2 0 (2π) [kE + M ] [kE + M + (1 − α)Λ ] Z 1 Z 4 ! 2 2 d kE 1 1 B(p ) = −4e dα 4 2 2 2 − 2 2 2 2 (6.5) 0 (2π) [kE + M ] [kE + M + (1 − α)Λ ] 2 2 2 where M ≡ −α(1 − α)p + αm . The kE-integrations are ﬁnite and may be evaluated using, Z 4 ! 2 d kE 1 1 1 M1 4 2 2 2 − 2 2 2 = 2 ln 2 (6.6) (2π) [kE + M0 ] [kE + M1 ] (4π) M0 As a result, we have (making use of the fact that m2, p2 Λ2), 2e2 Z 1 (1 − α)Λ2 ! A(p2) = + dα(1 − α) ln (4π)2 0 −α(1 − α)p2 + αm2 4e2 Z 1 (1 − α)Λ2 ! B(p2) = − dα ln (6.7) (4π)2 0 −α(1 − α)p2 + αm2 The UV divergences are easily extracted, and we have

e2 e2 A(p2) = − ln Λ2 B(p2) = + ln Λ2 (6.8) div 16π2 div 4π2 Both are independent of the momentum p and therefore require counterterms precisely of

the same form as already present in the Lagrangian, namely renormalization of Z2 and m. It remains to specify suitable renormalization point and renormalization conditions. Proceeding to impose renormalization conditions when the electrons are on-shell,

(1) (1) ∂ΣR (p, m) Σ (p, m) = 0 = 1 (6.9) R p/=m ∂p/ p/=m we encounter what at ﬁrst might seem like a surprising diﬃculty, Indeed, the derivative of (1) ΣR (p) with respect to p produces (amongst other things) the p-derivative of the function A(p2), which is given by ∂A(p2) 2e2 Z 1 (1 − α)2 = + dα (6.10) ∂p2 (4π)2 0 −(1 − α)p2 + m2 The integral is singular when we let p2 → m2, as the integrand behaves like 1/α.

57 This divergence has nothing to do with UV behavior. Its origin may be traced to the analytic structure of A(p2) as a function of p2. The argument of the logarithm becomes negative when (1 − α)p2 > m2 for at least some α ∈ [0, 1]. This will happen as soon as p2 > m2. It implies that as soon as the CM energy exceeds m, it is possible to create a real electron and a real photon. The reason the branch cut for this process starts right at p2 = m2 is that the photon is massless. The massless nature of the photon allows its creation at any positive energy, no matter how small. Thus, the divergence found here is associated with a low energy or infrared (IR) problem. Its resolution is not through renormalization, because its signiﬁcance is physical. The solution to this IR problem will have to await our study of the vertex function. It is possible to choose a renormalization point where the electrons are oﬀ-shell. Actually, this is not so natural and it is inconvenient to do so while maintaining Lorentz invariance. Instead, we shall introduce a small photon mass to regularize the IR problems and defer to later a proper understanding of the zero photon mass limit.

6.2 Photon mass and removing infrared singularities

Does gauge invariance prevent having a massive photon? One might think yes. Actually Abelian gauge symmetry, such as in QED, allows for a non-zero photon mass. Here is why. Assume the mass is being introduced as a simple term in the Lagrangian,

1 λ µ2 L = − F F µν − (∂ Aµ)2 + A Ak − A jµ (6.11) 4 µν 2 µ 2 k µ The classical ﬁeld equations read

µν ν 2 ν ν ∂µF + λ∂ (∂ · A) + µ A − j = 0 (6.12)

ν Combining current conservation, ∂νj = 0 with the ﬁeld equations, we obtain a ﬁeld equation purely for the longitudinal component of the gauge ﬁeld,

λ2(∂ · A) + µ2(∂ · A) = 0 (6.13)

Remarkably, the longitudinal part of the photon ﬁeld is still a free ﬁeld, as it had been in the massless case, and thus it decouples from the true dynamics of the theory. By contrast, in non-Abelian gauge theories the longitudinal part will not be a free ﬁeld and therefore will not decouple. Masses for non-Abelian gauge ﬁelds cannot be given in this simple manner.

58 The associated massive photon propagator is considerably more complicated than the massless one,

−i k k ! η + ξ µ ν (6.14) k2 − µ2 µν k2 − µ2/λ

Here, we continue to use the abbreviation ξ = (1 − λ)/λ. Clearly, for the massive theory, it is very advantageous to choose Feynman gauge λ = 1 and thus ξ = 0.

6.3 The Electron Self-Energy for a massive photon The Lorentz decomposition of the electron self-energy in (6.1) is unmodiﬁed by the intro- duction of a photon mass. The functions A, B become,

2e2 Z 1 (1 − α)Λ2 ! A(p2) = + dα(1 − α) ln (4π)2 0 −α(1 − α)p2 + αm2 + (1 − α)µ2 4e2 Z 1 (1 − α)Λ2 ! B(p2) = − dα ln (6.15) (4π)2 0 −α(1 − α)p2 + αm2 + (1 − α)µ2

It is readily veriﬁed that ∂A(p2)/∂p2 now has a ﬁnite limit as p2 → m2. The branch cut is now supported at p2 > m2/(1−α)+µ2/α, (with α ∈ [0, 1]), which starts at p2 = (m+µ)2. This is as expected : physical particles can be produced as soon as the rest energy m + µ is available.

6.4 The Vertex Function : General structure The 1-loop 1PIR contribution to the vertex function is shown in Fig 12. Its unregularized Feynman integral expression is given by

Z d4k 1 k k ! 1 1 Γ(1)(p0, p) = −ie3 η + ξ ρ σ γρ γ γσ (6.16) µ (2π)4 k2 ρσ k2 p/0 − k/ − m µ p/ − k/ − m

The k-integral has a logarithmic UV divergence, which will be regularized by a Pauli Villars regulator on the photon propagator, just as we had used already for the electron- self-energy. The divergence itself is clearly proportional to the bare interaction vertex γµ, consistent with 1-loop renormalizability.

Figure 12: 1-loop 1PIR contribution to the vertex function

59 Assuming that this regularization has been carried out, it is instructive to verify ex- plicitly the Ward identity between the vertex function and the electron self-energy Σ(1). The Ward identity takes the form,

0 µ (1) 0 (1) 0 (1) (p − p) Γµ (p , p) = +e Σ (p ) − Σ (p) (6.17)

0 µ 0 It is proven to one-loop using the simple fact that (p − p) γµ = (p/ + k/ − m) − (p/ + k/ − m). Substituting this relation into the integrand of the vertex function one ﬁnds,

1 1 1 1 ! (p0 − p)µ γ = −e − (6.18) p/0 + k/ − m µ p/ + k/ − m p/ + k/ − m p/0 + k/ − m

Actually, the result holds to all orders in e. To see how this works, let S(p) be the full fermion propagator, whose inverse is related to the regularized electron self-energy,

S(p)−1 = −i(p/ − m) − iΣ(p) (6.19)

0 and Γµ(p , p) be the full vertex function, then we have

0 µ 0 −1 −1 0 (p − p) Γµ(p , p) = −ie[S (p) − S (p )] (6.20)

(1) 0 (1) The Ward identities imply that the renormalizations of Γµ (p , p) and Σ (p) are related to one another. Actually, the mass renormalization does not involve the momenta, so only (1) 0 Z2 should aﬀect Γµ (p , p). We can make this connection explicit to one-loop order, where we have already estab- lished the structure of the divergences and required renormalization counterterms,

(1) 0 (1) 0 Γµ (p , p) = −eγµZ1 + ΓRµ(p , p) (1) (1) Σ (p) = Z2(p/ − m) + (Z2mo − m) + ΣR (p) (6.21)

Recall the standard on-shell renormalization conditions on the electron self-energy,

(1) (1) ∂ΣR ΣR (p) = 0 (p) = 1 (6.22) p/=m ∂p/ p/=m For the vertex, the standard renormalization condition is also on-shell for the electrons, R(1) and imposes the vanishing of Γµ at vanishing photon momentum,

0 (1) 0 u¯(p )ΓRµ(p , p)u(p) = 0 (6.23) p0=p;p2=m2

60 Next, letting p0 → p in the Ward identity (6.17) provides the following relation,

∂Σ(1)(p) Γ(1)(p, p) = −e (6.24) µ ∂pµ In terms of the renormalized quantities, we have

∂Σ(1)(p) −eγ Z + Γ(1) (p, p) = −eγ Z − e R (6.25) µ 1 Rµ µ 2 ∂pµ Evaluating this relation on-shell for the electrons, at zero momentum for the photon and (1) using the renormalization conditions, it is immediate that the contributions of ΓRµ and (1) ΣR cancel, leaving the famous Ward identity

Z1 = Z2 (6.26) a result which is valid to all orders in e.

6.5 Calculating the oﬀ-shell Vertex We begin by calculating the vertex function for general oﬀ-shell momenta p, p0. For the sake of generality, and because the result will be needed later as well, we shall include a non-zero photon mass µ in the calculation from the start, Z 4 (1) 0 3 d k 1 kρkσ ρ 1 1 σ Γµ (p , p) = −ie ηρσ + ξ µ2 γ γµ γ (6.27) Λ (2π)4 k2 − µ2 2 p/0 − k/ − m p/ − k/ − m k − λ It will always be assumed that the vertex integral has been UV regularized by using a Pauli Villars regulator Λ on the photon, just as was used for the electron self-energy. This fact will be indicated by the subscript Λ on the k-integral. To begin, we work in Feynman gauge with ξ = 0; we shall later also discuss the contributions from ξ 6= 0. Rationalizing the fermion propagators gives,

Z 4 σ 0 (1) 0 3 d k γ (p/ − k/ + m)γµ(p/ − k/ + m)γσ Γµ (p , p) = −ie (6.28) Λ (2π)4 (k2 − µ2)((p0 − k)2 − m2)((p − k)2 − m2)

Introducing two Feynman parameters α, β for the 3 factors in the denominator, and using the following formula, 1 Z 1 Z 1−α 1 = 2 dα dβ (6.29) ABC 0 0 [(1 − α − β)A + αB + βC]3

61 we obtain Z 1 Z 1−α Z 4 σ 0 (1) 0 3 d k γ (p/ − k/ + m)γµ(p/ − k/ + m)γσ Γµ (p , p) = −2ie dα dβ (6.30) 0 0 Λ (2π)4 ((k − αp0 − βp)2 − M 2(µ))3 where we have deﬁned the function M 2(µ) = (α + β)m2 + (1 − α − β)µ2 − αp02 − βp2 + (αp0 + βp)2 (6.31) Upon shifting k → k + αp0 + βp, we ﬁnd Z 1 Z 1−α Z 4 (1) 0 3 d k Nµ Γµ (p , p) = −2ie dα dβ (6.32) 0 0 Λ (2π)4 (k2 − M 2(µ))3 where the numerator is abbreviated by σ 0 0 Nµ = γ (1 − α)p/ − βp/ − k/ + m γµ (1 − β)p/ − αp/ − k/ + m γσ (6.33) The numerator may be simpliﬁed by using the following identities, σ γ γµγσ = −2γµ σ γ γµγνγσ = +4ηµνI σ γ γµγνγργσ = −2γσγνγµ (6.34) 2 After regrouping and using the result of angular averaging kµkν → 1/4k ηµν, one ﬁnds the following numerator 2 Nµ = −k γµ + Nµ (6.35) 2 0 Nµ = −2m γµ + 4m(1 − 2β)pµ + 4m(1 − 2α)pµ 0 0 +2β(1 − β) p/γµp/ + 2α(1 − α) p/ γµp/ 0 0 −2(1 − α)(1 − β) p/γµp/ − 2αβ p/ γµp/ The k-integrals are readily evaluated, Z d4k 1 i = − Λ (2π)4 (k2 − M 2(µ))3 32π2M 2(µ) Z d4k k2 i M 2(Λ) = ln (6.36) Λ (2π)4 (k2 − M 2(µ))3 16π2 M 2(µ) In the last integral, one may use m, p, p0 Λ to carry out the following simpliﬁcation, M 2(Λ) ∼ (1 − α − β)Λ2. In summary, the full oﬀ-shell vertex function is given by 3 Z 1 Z 1−α 2 ! (1) 0 e M (Λ) Nµ Γµ (p , p) = − dα dβ 2γµ ln + (6.37) (4π)2 0 0 M 2(µ) M 2(µ) The integration over the Feynman parameters is fairly involved. Fortunately, it will not be needed here in full.

62 It remains to make a suitable choice of renormalization conditions. It is natural to take the fermions to be on-shell and the photon to have zero momentum, which implies p0 = p,

(1) 5 u¯(p) −eZ1γµ + Γµ (p, p) u(p) =u ¯(p)(−eγµ)u(p) + O(e ) (6.38) p2=m2 p2=m2 Substituting the data of the renormalization point in the function M 2, we ﬁnd,

M 2(µ) = (1 − α − β)µ2 + (α + β)2m2 (6.39)

It is clear that the integration over α, β would fail to converge for zero photon mass µ = 0, where an IR divergence occurs. As is the case of the electron self-energy, the physical origin of this IR divergence resides in the diverging probability for producing ultra-low energy photons. With µ 6= 0, however, the integrals are convergent and the regulator dependent part of Z1 may be deduced, (it arises only from the ln term), e2 Z = 1 − ln Λ2 + O(e4) (6.40) 1 16π2 The ﬁnite renormalization parts may also be computed by carrying out the α, β-integrals, and the dependence on the gauge parameter may be incorporated as well, (Itzykson and Zuber)

e2 1 Λ2 µ2 1 µ2 9! Z = 1 − ln + 3 ln − ln + + O(e4) (6.41) 1 (4π)2 λ m2 m2 λ λm2 4

6.6 The Electron Form Factor and Structure Functions The renormalized vertex function resulting from the preceding subsection is quite a com- plicated object. In particular, the numerator Nµ generates 7 diﬀerent spinor/tensor struc- tures. Depending on the ultimate purpose of the calculations, not all of this information may actually be required. Such is the case for the calculation of the anomalous magnetic moment of the electron, on which we shall now focus attention. A simpler object than the full vertex function is obtained when both electrons are restricted to being speciﬁc on-shell electron states. Equivalently, one is then evaluating µ ¯β not the full correlator h∅|j (x)ψα(y)ψ (z)|∅i but rather the electron form factor,

0 µ 0 0 hu(p )|j (q)|u(p)i =u ¯(p )Γµ(p , p)u(p) (6.42)

Here, |u(p)i stands for an (on-shell) electron state with wave function u(p), and we have q = p0 −p and p2 = p02 = m2. The photon is not on-shell, i.e. q2 is left general. The reason

63 this object is called a form factor is because it parametrizes the distributions of charge and current “inside” the electron. Even though the electron is assumed to be a “point-like particle”, renormalization eﬀects spread out its charge and gives the electron structure. Despite these restrictions, the form factor actually still contains a wealth of information, such as the anomalous magnetic moment, and yet its form is much simpler than that of the full vertex. This may be seen by listing all possible Lorentz invariant structures that may occur. The Lorentz structure is determined by the Dirac matrix insertion between u¯(p0) and u(p), as well as by the explicit dependence on the external momenta. The 0 5 0 5 matrix elementsu ¯(p )γ u(p) andu ¯(p )γ Sµnuu(p) violate parity symmetry and are not 0 0 0 allowed in QED. This leaves the matrix elementsu ¯(p )u(p),u ¯(p )γµu(p) andu ¯(p )Sµνu(p) only. The dependence on momenta also signiﬁcantly simpliﬁes, since p2 = p02 = m2 and 2p · p0 = 2m2 − q2. Thus, the general Lorentz structure of the form factor is

0 0 0 2 ν 2 0 2 u¯(p )Γµ(p , p)u(p) =u ¯(p ) γµf1(q ) + Sµνq f2(q ) + (pµ + pµ)f3(q ) u(p) (6.43)

Actually, the Gordon identity provides a relation between all three coeﬃcients,

0 0 0 ν 0 (p + p )µu¯(p )u(p) = 2mu¯(p )γµu(p) − 2iq u¯(p )Sµνu(p) (6.44)

the identity may be proven using

i γ p/ = p − 2iS pν S = [γ , γ ] µ µ µν µν 4 µ ν 0 0 0ν p/ γµ = pµ + 2iSµνp [γµ, γν] = −4iSµν (6.45)

It is convenient to eliminate the term inu ¯(p0)u(p) in favor of the other two. It is customary 2 2 to deﬁne the form factor in terms of the two structure functions F1(q ) and F2(q ),

iqν u¯(p0)Γ (p0, p)u(p) =u ¯(p0)γ u(p)F (q2) + u¯(p0)S u(p)F (q2) (6.46) µ µ 1 m µν 2 In the following subsection, it will be shown that the anomalous magnetic moment can be

computed solely from the value of F2(0).

6.7 Calculation of the structure functions

(ξ) 2 The ξ-dependent part of the form factor gives a contribution F1 (q ) to the structure 2 2 function F1(q ) and does not contribute to F2(q ). Indeed, carrying out the contraction of

64 kρkσ, we have the following simpliﬁcation,

1 1 u¯(p0)k/ γ k/u(p) =u ¯(p0)γ u(p) (6.47) p/0 − k/ − m µ p/ − k/ − m µ

This is established using the Dirac equations (p/ − m)u(p) = 0 andu ¯(p0)(p/0 − m) = 0, with the help of which we have k/u(p) = −(p/ − k/ − m)u(p) andu ¯(p0)k/ = −u¯(p0)(p/0 − k/ − m). As a result, we have,

Z 4 3 2 (ξ) 2 3 d k 1 e ξ Λ F1 (q ) = −ie ξ = ln (6.48) Λ (2π)4 (k2 − µ2)(k2 − µ2/λ) (4π)2 µ2

The remainder of the form factor may be evaluated for ξ = 0.

To calculate the remaining form factor, one evaluates the full oﬀ-shell vertex on shell and sorts the various Lorentz invariants according to the decomposition (??). The function M 2(µ) simpliﬁes considerably by putting the electrons on-shell and we have

M 2(µ) = (1 − α − β)µ2 + (α + β)2m2 − αβq2 (6.49)

0 This requires evaluation of the matrix elementu ¯(p )Nµu(p), which follows from the follow- ing simple results,

0 0 2 ν u¯(p ) p/γµp/u(p) =u ¯(p ) m γµ − mqµ − 2imq Sµν u(p) 0 0 0 0 2 ν u¯(p ) p/ γµp/ u(p) =u ¯(p ) m γµ + mqµ − 2imq Sµν u(p) 0 0 0 2 u¯(p ) p/ γµp/u(p) =u ¯(p ) m γµ u(p) 0 0 0 2 2 ν u¯(p ) p/γµp/ u(p) =u ¯(p ) m γµ + q γµ − 4miq Sµν u(p) (6.50)

Therefore, the matrix elements of Nµ are found to be,

0 0 2 u¯(p )Nµu(p) =u ¯(p ) 2mqµ {α(1 − α) − β(1 − β)} − 2(1 − α)(1 − β)q γµ

2 n 2o +m γµ 4 − 4(α + β) − 2(α + β) ν −4i(α + β)(1 − α − β)mq Sµν u(p) (6.51)

Since the function M 2(µ) is a symmetric function of α and β and since the integration measure is symmetric as well, the numerator is eﬀectively symmetrized, which cancels the

65 term in mqµ. The remainder is readily decomposed onto the structure functions F1 and

F2, and we have 3 Z 1 Z 1−α 2 2 e M (Λ) F1(q ) = − dα dβ ln 8π2 0 0 M 2(µ) m2{2 − 2α − 2β − (α + β)2} − q2(1 − α)(1 − β)! + (1 − α − β)µ2 + (α + β)2m2 − αβq2 3 Z 1 Z 1−α 2 2 e m (α + β)(1 − α − β) F2(q ) = dα dβ (6.52) 4π2 0 0 (1 − α − β)µ2 + m2(α + β)2 − αβq2

The function F1 is singular as µ → 0 signaling an IR divergence. If the renormalization conditions are taken on-shell for the electrons and at q2 = 0 for the photons (which is actually oﬀ-shell since µ 6= 0), then the renormalized structure function is given by 3 Z 1 Z 1−α 2 2 2 2 e (1 − α − β)µ + (α + β) m F1(q ) = − dα dβ ln 8π2 0 0 (1 − α − β)µ2 + (α + β)2m2 − αβq2 m2{2 − 2α − 2β − (α + β)2} − q2(1 − α)(1 − β) + (1 − α − β)µ2 + (α + β)2m2 − αβq2 m2{2 − 2α − 2β − (α + β)2}! − (6.53) (1 − α − β)µ2 + (α + β)2m2 The interpretation of the infrared divergence will be given later. The physical interpreta-

tion of F1 is to describe the charge distribution of the electron. 2 2 In F2(q ), the limit µ → 0 is ﬁnite and may be taken. This quantity is properly renormalized all by itself. Important is the q2 → 0 limit, corresponding to low energy photons and homogeneous static ﬁelds, e3 Z 1 Z 1−α 1 ! e3 F2(0) = dα dβ − 1 = (6.54) 4π2 0 0 α + β 8π2

The physical interpretation of F2 is to describe the distribution of magnetic dipole moments of the electron. Next, this result is applied to the calculation of the anomalous magnetic moment of the electron, viewed customarily in response to a constant magnetic ﬁeld, i.e. at q = 0.

6.8 The One-Loop Anomalous Magnetic Moment The form of the renormalized vertex function, evaluated on-shell for the electrons, is in the small q approximation, e3 iqν ! u¯(p0)Γ (p0, p)u(p) =u ¯(p0) eγ + S u(p) (6.55) µ µ 8π2 m µν

66 To compute the anomalous magnetic moment, we need to incorporate the eﬀect of the above correction in the Dirac equation and extract the proper magnetic moment term.

First, multiply by the gauge ﬁeld Aµ. Translating to position space, we use i 1 iqνAµS = (qνAµ − qµAν)S → F µνS (6.56) µν 2 µν 2 µν The modiﬁed Dirac equation is therefore,

e3 1 ! i∂/ − eA/ − F µνS − m ψ = 0 (6.57) 8π2 2m µν

To extract the magnetic moment, we use a standard spinor formula,

2 µ µ µν (i∂/ − eA/) = (i∂µ − eAµ)(i∂ − eA ) − eF Sµν (6.58) which may be derived by decomposing the product of two γ-matrices on the lhs into a sum of their commutator and anti-commutator. Multiplying the eﬀective Dirac equation to the left by (i∂/ − eA/ + m) and using the fact that in the absence of A we have ip/ = m, we have e3 1 ! (i∂/ − eA/ + m) i∂/ − eA/ − F µνS − m ψ 8π2 2m µν e3 ! ! = (i∂ − eA )2 − m2 − e + F µνS ψ = 0 (6.59) µ µ 8π2 µν

Thus, the electron magnetic moment receives a ﬁnite correction:

e α e2 µ = 1 + + O(α2) α = m 2π 4π α = 0.0011614 ··· 2π expt. : 0.0011597 : other eﬀects; QCD, weak ···

67 7 Functional Integrals For Gauge Fields

The functional methods appropriate for gauge ﬁelds present a number of complications, especially for non-Abelian gauge theories. In this chapter, we begin with a rapid and some- what sketchy discussion of the Abelian case and then move onto a systematic functional integral quantization of the Abelian and nonAbelian cases.

7.1 Functional integral for Abelian gauge ﬁelds

The starting point will be taken to be the Hamiltonian, supplemented by Gauss’ law. It is easiest to describe the system in temporal gauge A0 = 0, where we have,

Z 1 1 H = d3x E~ 2 + B~ 2 − A~ · J~ (7.1) 2 2 where J~ is a conserved external current and we the following expressions for electric and magnetic ﬁelds,

∂A~ E~ = B~ = ∇~ × A~ (7.2) ∂t

Furthermore, conﬁguration space is described by all Ai(~x), hence a “position basis” of Fock space consists of all states of the form

F ≡ |Ai(~x)i for all Ai(~x) (7.3)

The Hilbert space of physical states (recall the diﬀerence between the Fock space and the physical Hilbert space, as discussed for free spin 1 ﬁelds) is obtained by enforcing on the states the constraint of Gauss’ law

G(A) ≡ ∇~ · E~ − J o (7.4)

Thus, we have

Hphys ≡ |ψi ∈ F such that G(A)|ψi = 0 (7.5)

Of course, we are only interested in the physical states. In particular, we are interested only in time-evolution on the subspace of physical states. Therefore, we introduce the

68 projection operator onto Hphys as follows,

P ≡ X |φihψ| (7.6)

|ψi∈Hphys

and the projected evolution operator may be decomposed as follows,

−itH −itH/N N e = lim P e (7.7) N→∞ Hphys We always assume that the full Hamiltonian, using the dynammics of j is used, so that it is time independent. Otherwise T -order! The projection operator onto physical states may also be constructed as a functionla intgeral, though admittedly, this construction is a bit formal, 1 Z Z P = DΛ exp i d3xΛ(~x)(∇~ · E~ − J o)(~x) (7.8) R DΛ

Clearly P is the identity on Hphys, and zero anywhere else! Summing over only physical

states is equivalent to summing over all states |Ai(~x)i and using the projector P to project the sum onto physical states only.

Z Z Z −itH 1 e = DA DE DΛ (7.9) R i i Hphys DΛ Z t Z 3 ~ ~ 1 ~ 2 1 ~ 2 ~ ~ o ~ ~ exp i dt d x E · ∂0A − E − B + Λ(∇ · E − J ) + A · J 0 2 2 where it is understood that the usual boundary conditions on the functional integral vari- ables are implemented at time 0 and t. Now rebaptize Λ as Ao, and express the result in terms of a generating functional for the source J µ, 1 Z Z 1 Z[J] ≡ DA exp i d4x − F F µν − J Aµ (7.10) R DΛ µ 4 µν µ However, this is really only a formal expression. In particular, the volume R DΛ of the space of all gauge transformation is wildly inﬁnite and must be dealt with before one can truly make sense of this expression. Recall that the expressions that came in always assumed the form Z[J µ]/Z[0] so that an overall factor of the volume of the gauge transformations cancels out. Thus in the expression for Z[J µ] itself, we may ab initio ﬁx a gauge, so that the volume of the gauge orbit is eliminated. We shall give a full discussion of this phenomenon when dealing with

69 the non-Abelian case in the next subsection; here we take an informal approach. One of the most convenient choices is given by a linear gauge condition,

Z Z Y µ 4 1 µν µ DAµ δ(∂ Aµ − c) exp i d x − FµνF − J Aµ (7.11) x 4 Since no particular slice is preferred, it is natural to integrate with respect to c, say with a Gaussian damping factor,

Z R 4 2 R 4 µ 2 −iλ/2 d xc (x) Y µ −iλ/2 d x(∂ Aµ) Dc e δ(∂ Aµ(x) − c(x)) = e (7.12) x and we recover our good old friend: Z Z 1 1 Z[J] = DA exp i d4x − F F µν − λ(∂ · A)2 − J µA (7.13) µ 4 µν 2 µ The J-dependence of Z[J µ] is independent of λ, so we can formally take λ → 0 and regain our original gauge invariant theory. It is straightforward to evaluate the free ﬁeld result,

Z d4k 1 − λ k k ! e−ik(x−y) h0|TA (x)A (y)|0i = i η + µ ν µ ν (2π)4 µν λ k2 + i k2 + i

= Gµν(x − y) (7.14)

So that Z[j] 1 Z Z = exp d4x d4yJ µ(x)G (x − y)J ν(y) (7.15) Z[0] 2 µν

which, provided J µ is conserved, is independent of λ. In particular when any gauge independent quantity is evaluated, the dependence on λ will disappear. When λ → ∞ one recovers Landau gauge, which is manifestly transverse; when λ → 1, one recovers another very convenient Feynman gauge.

7.2 Functional Integral for Non-Abelian Gauge Fields We consider a Yang-Mills theory based on a compact gauge group G with structure con- stants f abc, a, b, c = 1, ··· , dimG. As discussed previously, if G is a direct product of n simple and n0 U(1) factors, then there will be n + n0 independent coupling constants de- scribing n + n0 diﬀerent gauge theories. Here, we shall deal with only one such component. Thus, we assume that G is simple or that it is just an Abelian theory with G = U(1).

70 a The basic ﬁeld in the theory is the Yang-Mills ﬁeld Aµ. The starting point is the classical action,

1 Z Z S[A; J] = − d4xF a F aµν − d4xJ aµAa 4 µν µ a a a abc b c Fµν = ∂µAν − ∂νAµ + gf AµAν (7.16)

where summation over repeated gauge indices a is assumed. For this action to be gauge invariant, the current must be covariantly conserved µ µ DµJ = 0, which may render the J dependent on the ﬁeld Aµ itself. This compli- cates the discussion, as compared to the Abelian case. Actually, we shall make one of two possible assumptions.

1. The current J µ arises from the coupling of other ﬁelds (such as fermions or scalars) in a complete theory that is fully gauge invariant.

2. The current J µ is an external source, but will be used to produce correlation functions µν of gauge invariant operators, such as trFµνF .

In both cases, the full theory will really be gauge invariant.

7.3 Canonical Quantization

Canonical quantization is best carried out in temporal gauge with A0 = 0. To choose this gauge, the equation ∂0U = igUA0 must be solved, which may always be done in terms of the time-ordered exponential

Z t 0 0 U(t, ~x) = T exp ig dt A0(t , ~x) U(t0, ~x) (7.17) t0

In ﬂat space-time with no boundary conditions in the time direction, this solution always exists. If time is to be period (as in the case of problems in statistical mechanics at ﬁnite temperature) then one cannot completely set A0 = 0. It is customary to introduce Yang-Mills electric and magnetic ﬁelds by

1 Ba ≡ F a i 2 ijk jk a a a Ei ≡ F0i = ∂0Ai (7.18)

71 a a and Ei is the canonical momentum conjugate to Ai . The Hamiltonian and Gauss’ law are given by

Z 1 1 H[E,A; J] = d3x EaEa + BaBa − J aAa 2 i i 2 i i i i a a abc b c G (E,A) = ∂iEi + gf Ai Ei (7.19)

The canonical commutation relations are

a b 0 0 ab (4) [Ei (x),Aj(y)]δ(x − y ) = −iδ δijδ (x − y) (7.20)

The operator Ga commutes with the Hamiltonian (for a covariantly conserved J µ). The action of Ga on the ﬁelds is by gauge transformation, as may be seen by taking the relevant commutator,

a b 0 0 (4) ab abc c (4) [G (x),Ai (y)]δ(x − y ) = ∂iδ (x − y)δ + f Ai δ (x − y) (7.21) or better even in terms of the the integrated form of Ga, Z 3 a Gω ≡ d xG (x)ωa(x) b b [Gω,Ai (y)] = (Diω) (y) (7.22)

Gauge transformations form a Lie algebra as usual,

[Ga(x),Gb(y)]δ(x0 − y0) = if abcGc(x)δ(4)(x − y) (7.23)

Thus, the operator Ga generates inﬁnitesimal gauge transformations. As in the case of the Abelian gauge ﬁeld, Gauss’ law cannot be imposed as an operator

equation because it would contradict the canonical commutation relations between Ai and

Ei. Thus, it is a constraint that should be imposed as an initial condition. In quantum ﬁeld theory, it must be imposed on the physical states. The Fock and physical Hilbert spaces are deﬁned in turn by

a F ≡ |Ai (~x)i a Hphys ≡ |ψi ∈ F such that G (E,A)|ψi = 0 (7.24)

It is interesting to note the physical signiﬁcance of Gauss’ law.

72 7.4 Functional Integral Quantization This proceeds very similarly to the case of the quantization of the Abelian gauge ﬁeld.

The starting points are the Hamiltonian in A0 = 0 gauge and the constraint of Gauss’ law which commuted with the Hamiltonian. We construct the evolution operator projected onto the physical Hilbert space by repeated (and redundant) insertions of the projection operator. Recall the deﬁnition and functional integral representation of this projector, P ≡ X |ψihψ|

|ψi∈Hphys 1 Z Z = DΛa(~x) exp i d3xΛa(~x) D Ea − J a) (7.25) R DΛa i i 0 As in the case of the Abelian gauge ﬁelds, we construct the evolution operator projected onto the physical Hilbert space by repeated insertions of P, N −itH −i t H e = lim P e N (7.26) phys N→∞ 1 Z Z Z = DΛa DAa DEa R DΛa i i Z t Z 0 3 a a 1 a a 1 a a × exp i dt d x Ei ∂0Ai − Ei Ei − Bi Bi 0 2 2 a a a a a +Ai Ji + Λ (DiEi − J0 )

a together with the customary boundary conditions on the ﬁeld Ai and free boundary con- a ditions on the canonically conjugate momentum Ei . Next, we recall that Gauss’ law resulted from the variation of the non-dynamical ﬁeld a A0, which had been set to 0 in this gauge. But here, Λ plays again the role of a non- dynamical Lagrange multiplier and we rebaptize it as follows,

a a Λ (x) → A0(x) (7.27)

a The integration over the non-dynamical canonical momentum Ei may be carried out in an algebraic way by completing the square of the Gaussian, 1 1 Ea∂ Aa − EaEa − BaBa + AaJ a − D AaEa − AaJ a (7.28) i 0 i 2 i i 2 i i i i i 0 i 0 0 1 1 1 = − (Ea − ∂ Aa + D A )2 + (∂ Aa − D A )2 − BaBa − Aa J aµ 2 i 0 i i 0 2 0 i i 0 2 i i µ a a The intgeration measure over DEi is invariant under (functional) shifts in Ei , and we use a this property to realize that the remaining Gaussian integral over Ei is independent of

73 a a Aµ and Jµ, and therefor produces a constant. The remainder is easily recognized as the Lagrangian density with source that was used as a starting point. As a concrete formula, we record the matrix element of the evolution operator taken between (non-physical) states in the Fock space,

Z Z t Z −itH a 0 3 1 a aµν a aµ hA2(~x)|e |A1(~x)i = c DAµ exp i dt d x − FµνF − AµJ 0 4 A(0, ~x) = A (~x) boundary conditions 1 (7.29) A(t, ~x) = A2(~x) Time ordered correlation functions are deduced as usual and may be summarized by the following generating functional, Z 1 eiG[J] ≡ DAa exp − F a F aµν − J aµAa (7.30) µ 4 µν µ Of course, this integral is formal, since gauge invariance leads to the integration of equiva- a lent copies of Aµ. Therefore, we need to gauge ﬁx, a process outlined in the next subsection.

7.5 The redundancy of integrating over gauge copies We wish to make sense out of the functional intgeral of any gauge invariant combination of operators which we shall collectively denote by O. For example, the following operators are gauge invariant,

a aµν O+(x) ≡ FµνF (x) a ˜aµν O−(x) ≡ FµνF (x) (7.31)

An inﬁnite family of operators O may be formed out of a time ordered product of a string

of O+ and O− opertors,

O = T O+(x1) ···O+(xm)O−(y1) ···O+(yn) (7.32)

We have the general formula

R a iS[A] DAµ O e h0|O|0i = R a iS[A] (7.33) DAµ e The key characteristic of this formula is that O is a gauge invariant operator.

a The measure DAµ is also gauge invariant. The way to see this is to notice that a measure results from a metric on the space of all gauge ﬁelds. Once this metric has been

74 found and properly deﬁned, the measure will forllow. The metric may be taken to be given by the following norm, Z a 2 4 a aµ || δAµ || ≡ d x δAµ δA (7.34)

The variation of a gauge ﬁeld transforms homogeneously and thus the metric is invariant. Therefore, we can construct an invariant measure. Thus, we realize now that all pieces of the functional integral for h|O|0i are invariant under local gauge transformations. But this renders the numerator and the denominator inﬁnite, since in each the integration extends over all gauge copies of each gauge ﬁeld. However, since the integrations are over gauge invariant integrands, the full integrations will factor out a volume factor of the space of all gauge transformations times the integral over the quotient of all gauge ﬁelds by all gauge transformations. Concretely,

a A ≡ { all Aµ(x) } G ≡ { all Λa(x) } (7.35)

Thus we have symbolically, Z Z a iS[A] a iS[A] DAµ O e = Vol(G) DAµ O e (7.36) A A/G

and inparticular, the volume of the space of all gauge transformations cancels out in the formula for h0|O|0i,

R a iS[A] R a iS[A] A DAµ O e A/G DAµ O e h0|O|0i = R a iS[A] = R a iS[A] (7.37) A DAµ e A/G DAµ e

The second formula does not have the inﬁnite redundancy integrating over all gauge trans- formations of any gauge ﬁeld. In the case of the Abelian gauge theories, the problem was manifested through the µν fact that the longitudinal part of the gauge ﬁeld does not couple to the FµνF term in the action, a fact that renders the kinetic operator non-invertible. A gauge ﬁxing term µ 2 λ(∂µA ) was added; its eﬀect on the physical dynamics of the theory is none since the current is conserved. For the non-Abelian theory, things are not as simple. First of all there a aµν are now interaction terms in the FµνF term, but also, the current is only covariantly conserved. The methods used for the Abelian case do not generalize to the non-Abelian case and we have to invoke more powerful methods.

75 7.6 The Faddeev-Popov gauge ﬁxing procedure

Geometrically, the problem may be understood by considering the space of all gauge ﬁelds A and studying the action on this space of the gauge transformations G, as depicted in Fig 13. The meaning of the ﬁgures is as follows

Figure 13: Each square represents the space A of all gauge ﬁeld; each slanted thin line represents an orbit of G; each fat line represents a gauge slice S. Every gauge ﬁeld A0 may be brought back to the slice S by a gauge transformation, shown in (c). A gauge slice may intersect a given orbit several times, shown in (d) – this is the Gribov ambiguity.

• (a) The space A of all gauge ﬁelds is schematically represented as a square and the action of the group of gauge transformations is indicated by the thin slanted lines. (Mathematically, A is the space of all connections on ﬁber bundles with structure group G over R4.)

• (b) There is no canonical way of representing the coset A/G. Therefore, one must choose a gauge slice, represented by the fat line S. To avoid degeneracies, the gauge slice should be taken to be transverse to the orbits of the gauge group G. (Mathematically, the gauge slice is a local section of the ﬁber bundle.)

• (c) Transversality guarantees – at least locally – that every point A0 in gauge ﬁeld space A is the gauge transformation of a point A on the gauge slice S.

• (d) It is a tricky problem whether – globally – every gauge orbit intersects the gauge slice just once or more than once. In physics, this problem is usually referred to as the Gribov ambiguity. (Mathematically, it can be proven that interesting slices produce multiple intersections.) Fortunately, in perturbation theory, the ﬁelds remain small and the Gribov copies will be inﬁnitely far away in ﬁeld space and may thus be ignored. Non-perturbatively, one would have to deal with this issue; fortunately, the best method is lattice gauge theory, where one never is forced to gauge ﬁx.

The choice of gauge slice is quite arbitrary. To remain within the context of QFT’s governed by local Poincar´einvariant dynamics, we shall most frequently choose a gauge compatible with these assumptions. Speciﬁcally, we choose the gauge slice S to be given by a local equation F a(A)(x) = 0, where F(A) involves A and derivatives of A in a local

76 manner. Possible choices are

a aµ a a F (A) = ∂µA F = A0 (7.38) But many other choices could be used as well. To reduce the functional integral to an integral along the gauge slice, we need to a factorize the measure DAµ into a measure along the gauge slice S times a measure along the orbits of the gauge transformations G. There is a nice gauge-invariant measure DU on G resulting from the following gauge invariant metric on G, Z || δU ||2 ≡ − d4xtr U −1δU U −1δU (7.39)

If the orbits G were everywhere orthogonal to S, then the factorization would be easy to carry out. In general it is not possible to make such a choice for S and a non-trivial Jacobian factor will emerge. To compute it, we make use of the Faddeev-Popov trick.

Denote a gauge transformation U of a gauge ﬁeld Aµ by i AU ≡ UA U −1 + ∂ UU −1 (7.40) µ µ g µ

Notice that the action of gauge transformations on Aµ satisﬁes the group composition law of gauge transformations, i i (AV )U = U VA V −1 + ∂ VV −1 U −1 + ∂ UU −1 µ µ g µ g µ i = (UV )A (UV )−1 + ∂ (UV )(UV )−1 µ g µ UV = Aµ (7.41) which will be useful later on. Let F(A) denote the loval gauge slice function, the gauge slice being determined by F a(A) = 0. The quantity F a(AU ) will vanish when AU belongs to S, and will be non-zero otherwise. We deﬁne the quantity J [A] by Z 1 = J [A] DU Y δ F a(AU )(x) (7.42) x The functional δ-function looks a bit menacing, but should be understood as a Fourier integral, Z Z Y δ(F a(AU )(x) ≡ DΛa exp i d4xΛa(x)F a(AU )(x) (7.43) x

77 U V UV Using the composition law (Aµ ) = Aµ and the gauge invariance of the measure DU, it follows that J is gauge invariant. This is shown as follows,

Z J [AV ]−1 = DU Y δ F a((AV )U )(x) x Z = DU Y δ F a(AUV )(x) x Z = D(U 0V −1) Y δ F a(AU 0 )(x) x = J [A]−1 (7.44)

The deinition of the Jacobian J readily allows for a factorization of the measure of A into a measure over the splice S times a measure over the gauge orbits G. This is done as follows,

Z Z Z iS[A] Y a U iS[A] DAµ O e = DAµ DU F (A )(x) J [A] O e A A G x Z Z Y a U iS[A] = DU DAµ F (A )(x) J [A] O e G A x Z Z Y a iS[A] = DU DAµ F (A)(x) J [A] O e (7.45) G A x

In passing from the ﬁrst to the second line, we have simply inverted the orders of integration over A and G; from the second to the third, we have used the gauge invariance of the measure DAµ, of O and S[A]; from the third to the fourth, we have used the fact that the functional δ-function restricts the integration over all of A to one just over the slice S. Notice that the integration over all gauge transformation function R DU is independent of A and is an irrelevant constant.

The gauge ﬁxed formula is independent of the gauge function F.

Therefore, we end up with the following ﬁnal formula for the gauge-ﬁxed functional intgeral for Abelian or non-Abelian gauge ﬁelds,

R Q a iS[A] A DAµ x F (A)(x) J [A] O e h0|O|0i = (7.46) R Q a iS[A] A DAµ x F (A)(x) J [A] e

It remains to compute J and to deal with the functional δ-function.

78 7.7 Calculation of the Faddeev-Popov determinant At ﬁrst sight, the Jacobian J would appear to be given by a very complicated functional integral over U. The key observation that makes a simple evaluation of J possible is that F a(A) only vanishes (i.e. its functional δ-function makes a non-vanishing contribution) on the gauge slice. To compute the Jacobian, it suﬃces to linearlize the action of the gauge transformations around the gauge slice. Therefore, we have

U = I + iωaT a + O(ω2) U 2 Aµ = Aµ − Dµω + O(ω ) a a abc b c Dµω = ∂µω − gf Aµω (7.47)

and the gauge function also linearizes,

a U a 2 F (A ) = F (Aµ − Dµω)(x) + O(ω ) Z a ! a 4 δF (A)(x) b 2 = F (A)(x) − d y b Dµω (y) + O(ω ) δAµ(y) Z = F a(A)(x) + d4yMab(x, y)ωb(y) + O(ω2) (7.48)

where the operator M is deﬁned by

a ! ab y δF (A)(x) M (x, y) ≡ Dµ b (7.49) δAµ(y) Now the calculation of J becomes feasible. Z J [A]−1 = DU Y δ F a(AU )(x) G x Z Z = Dωa Y δ F a(A)(x) + d4yMab(x, y)ωb(y) x = (DetM)−1 (7.50)

Inverting this relation, we have ! y δF(A)(x) J [A] = DetM = Det Dµ (7.51) δAµ(y) Given a gauge slice function F, this is now a quite explicit formula. In QFT, however, we were always dealing with local actions, local operators and these properties are obscured in the above determinant formula.

79 7.8 Faddeev-Popov ghosts We encountered a determinant expression in the numerator once before when delaing with intgerations over fermions. This suggests that the Faddeev-Popov determinant may also be cast in the form of a functional intgeration over two complex conjugate Grassmann odd ﬁelds, which are usually referred to as the Faddeev-Popov ghost ﬁelds and are denoted by ca(x) andc ¯a(x). The expression for the determinant is then given by a functional integral over the ghost ﬁelds, Z a a DetM = Dc Dc¯ exp iSghost[A; c, c¯] Z Z 4 4 a ab b Sghost[A; c, c¯] ≡ d x d y c¯ (x)M (x, y)c (y) (7.52)

ab If the function F is a local function of Aµ, then the support of M (x, y) is at x = y and the above double integral collapses to a single and thus local intgeral involving the ghost ﬁelds.

7.9 Axial and covariant gauges First of all, there is a class of axial gauges where no ghosts are ever needed ! The gauge a function must be linear in Aµ and involve no derivatives on Aµ. They involve a constant µ a a µ vector n , and are therefore never Lorentz covariant, F (A)(x) = Aµ(x)n , a δF (A)(x) ab (4) µ b = δ δ (x − y)n δAµ(y)

y δF (A)(x) y (4) µ Dµ = ∂µδ (x − y)n = M(x, y) (7.53) δAµ(y) Therefore, in axial gauges, the operator M is independent of A and thus a constant which may be ignored. Thus, no ghost are required as the Faddeev-Popov formula would just inform us that they decouple. Actually, since axial gauges are not Lorentz covariant, they are much less useful in practice than it might appear. It turns out that Lorentz covariant gauges are much more useful. The simplest such gauges are speciﬁed by a function f a(x) as follows,

a aµ a F (A)(x) ≡ ∂µA (x) − f (x) (7.54) This choice gives rise to the following functional integral, Z Y aµ a iS[A] ZO = DAµ δ(∂µA − f ) DetMO e (7.55) A x

80 Since this functional integral is independent of the choice of f a, we may actually integrate over all f a with any weight factor we wish. A convenient choice is to take the weight factor to be Gaussian in f a, R a i R 4 a a R Q aµ a iS[A] Df exp − 2 d xf f A DAµ x δ(∂µA − f ) DetMO e ZO = (7.56) R a i R 4 a a Df exp − 2 d xf f

The denominator is just again a Aµ-independent constant which may be omitted. The f a-intgeral in the numerator may be carried out since the functional δ-function instructs a aµ us to set f = ∂µA , so that Z Z Z a a ZO = DAµ Dc Dc¯ O exp iS[A] + iSgf [A] + iSghost[A; c, c¯] (7.57) A where the gauge ﬁxing terms in the action are given by Z 1 S [A] = d4x − (∂ Aaµ)2 (7.58) gf 2ξ µ It remains to obtain a more explicit form for the ghost action. To this end, we ﬁrst compute M. ν ab ab y δ∂x Aν(x) M (x, y) = Dµ δAµ(y) y ab µ (4) = (Dµ) ∂x δ (x − y) (7.59) Inserting this operator into the expression for the ghost action, we obtain, Z Z 4 4 a ab b Sghost[A; c, c¯] ≡ d x d y c¯ (x)M (x, y)c (y) Z Z 4 4 y µ = d x d yc¯(x)Dµ∂x δ(x − y)c(y) Z 4 a a = d x∂µc¯ (x)(Dµc) (7.60) Thus, the total action for these gauges is given as follows, 1 1 S [A; c, c¯] = − F a F aµν − (∂ Aaµ)2 + ∂ c¯a(Dµc)a (7.61) tot 4 µν 2ξ µ µ and we have the following general formula for vacuum expectation values of gauge invariant operators,

R R iStot[A;c,c¯] DAµ DcDc¯ O e h0|O|0i = R R iS [A;c,c¯] (7.62) DAµ DcDc¯ e tot In the section on perturbation theory, we shall show how to use this formula in practice.

81 8 Anomalies in QED

Initial exposure to renormalization may give the impression that the procedure is an ad hoc remedy for a putative ill of QFT, that should not be required in some better formulation of the theory. This is not true. The presence of anomalies and the existence of the renormalization group will prove that the renormalization procedure has deep, physically relevant meaning. Questions that are often posed in this context are 1. Are UV divergences a consequence of the perturbative expansion ?

2. If so, is it possible to re-sum them to ﬁnite results in the full theory?

3. Are there any physical consequences of renormalization ? To answer these questions, we are taking a detour to anomalies.

8.1 Massless 4d QED, Gell-Mann–Low Renormalization Recall the form of the 2-point vacuum polarization tensor to one-loop order 2 Πµν(k, m, Λ) = (k ηµν − kµkν)Π(k, m, Λ) (8.1) e2 Z 1 m2 − α(1 − α)k2 Π(k, m, Λ) = 1 − dαα(1 − α) ln + O(e4) 2π2 0 Λ2 When the electron mass m 6= 0, it is natural (though not necessary) to renormalize at 0

photon momentum, and to require ΠR(0, m) = 1, so that

ΠR(k, m) = 1 − Π(0, m, Λ) + Π(k, m, Λ) (8.2) However, when m = 0, Π(0, m, Λ) = ∞, and this renormalization is not possible. This led Gell-Mann–Low to renormalize “oﬀ-shell”, at a mass-scale µ that has no direct relation with the electron mass or photon mass,

ΠR(k, m, µ) = 1 (8.3) k2=−µ2 So that 2 Z 1 2 2 e m − α(1 − α)k 4 ΠR(k, m, µ) = 1 − dα(1 − α)α ln + O(e ) (8.4) 2π2 0 m2 + α(1 − α)µ2 Now, we may safely let m → 0, e2 −k2 ! Π (k, 0, µ) = 1 − ln + O(e4) (8.5) R 12π2 µ2 The cost of renormalizing the massless theory is the introduction of a new scale µ, which was not originally present in the theory.

82 8.2 Scale Invariance of Massless QED

But now think for a moment about what this means in terms of scaling symmetry. When m = 0, classical electrodynamics is scale invariant,

1 ξ L = − F F µν + (∂ Aµ)2 + ψ¯(i∂/ − eA/)ψ (8.6) 4 µν 2 µ

µ 0µ µ A scale transformation x → x = λx , transforms a ﬁeld φ∆(x) of dimension ∆ as follows

( 0 0 −∆ Aµ : ∆ = 1 φ∆(x) → φ∆(x ) = λ φ∆(x) 3 (8.7) ψ : ∆ = 2

The classical theory will be invariant since the eﬀects of scale transformations on the various terms in the Lagrangian are as follows,

µν −4 µν FµνF (x) → λ FµνF (x) µ 2 −4 µ 2 (∂µA ) → λ (∂µA ) ψ¯(i∂/ − eA/)ψ → λ−4ψ¯(i∂/ − eA/)ψ (8.8) where we have assumed that e and ξ do not transform under scale transformations. Thus L → λ−4L, and the action S = R d4xL is invariant. Actually, you can show that L is invariant under all conformal transformations of xµ; inﬁnitesimally δxµ = δλxµ for a scale transformation; δxµ = x2δcµ − xµ(δc · x) for a special conformal transformation. Thus, classical massless QED is scale invariant; nothing in the theory sets a scale.

A fermion mass term, however, spoils scale invariance since ψψ¯ → λ−3ψψ¯ , assuming that m is being kept ﬁxed. Similarly, a photon mass would also spoil scale invariance.

Now, think afresh of the one-loop renormalization problem: when m = 0, one-loop renormalization forced us to introduce a scale µ in order to carry out renormalization. One cannot quantize QED without introducing a scale, and thereby breaking scale invariance.

What we have just identiﬁed is a ﬁeld theory, massless QED, whose classical scale symmetry has been broken by the process of renormalization, i.e. by quantum eﬀects.

Whenever a symmetry of the classical Lagrangian is destroyed by quantum eﬀects caused by UV divergences, this classical symmetry is said to be anomalous. In this case, it is the scaling anomaly, also referred to as the conformal anomaly.

83 8.3 Chiral Symmetry In case the above result is insuﬃciently convincing at this time, we give an example of another symmetry that has an anomaly, seen in a calculation that is ﬁnite. Take again massless QED, 1 L = − F F µν + ψ¯(i∂/ − eA/)ψ (8.9) 4 µν Setting m = 0 has enlarged the symmetry to include conformal invariance (classically). It also produces another new symmetry: chiral symmetry. To see this, decompose ψ into left and right spinors, in a chiral basis where γ5 is diagonal,

5 5 ψ = ψL ⊕ ψR γ ψL = +ψL, γ ψR = −ψR (8.10)

The Lagrangian then becomes, 1 L = − F F µν + ψ¯ (i∂/ − eA/)ψ + ψ¯ (i∂/ − eA/)ψ (8.11) 4 µν L L R R

Clearly, ψL and ψR may be rotated by two independent phases,

( iθL ψL → e ψL U(1)L × U(1)R iθR (8.12) ψR → e ψR

and the Lagrangian is invariant.‡ Now, we re-arrange these transformations as follows

( iθ+ ψL → e ψL U(1)V : iθ+ 2θ+ = θL + θR (8.13) ψR → e ψR

( iθ− ψL → e ψL U(1)A : −θ− 2θ− = θL − θR (8.14) ψR → e ψR or, in 4-component spinor notation

( iθ+ U(1)V : ψ → e ψ iγ5θ− (8.15) U(1)A : ψ → e ψ

As symmetries, they have associated conserved currents.

• the vector current jµ ≡ ψγ¯ µψ

µ ¯ µ • the axial vector current j5 ≡ ψγ γ5ψ

‡Any symmetry that transforms left and right spinors diﬀerently is referred to as chiral.

84 Conservation of these currents may be checked directly by using the Dirac ﬁeld equations, in the following form,

µ µ iγ ∂µψ = eγ Aµψ ¯ µ ¯ µ −i∂µψγ = eψγ Aµ (8.16)

One then has,

µ ¯ µ ¯ µ ¯ µ ¯ µ ∂µj5 = ∂µψγ γ5ψ − ψγ5γ ∂µψ = ieψγ Aµγ5ψ + ieψγ5γ Aµψ = 0 (8.17)

8.4 The axial anomaly in 2-dimensional QED It is very instructive to consider the fate of chiral symmetry in the much simpler 2- dimensional version of QED, given by the same Lagrangian, 1 L = − F F µν + ψ¯(i∂/ − eA/)ψ (8.18) 4 µν Contrarily to 4-dim QED, this theory is not scale invariant, even for vanishing fermion mass, since the dimensions of the various ﬁelds and couplings are given as follows, [e] = 1, [A] = 0, [ψ] = 1/2. But it is chiral invariant, just as 4-dim QED was, and we have two classically conserved currents,

µ ¯ µ µ ¯ µ j ≡ ψγ ψ j5 ≡ ψγ γ5ψ (8.19)

A considerable simpliﬁcation arises from the fact that the Dirac matrices are much simpler: 0 1 0 1 1 0 γ0 = σ1 = γ1 = iσ2 = γ5 = σ3 = (8.20) 1 0 −1 0 0 −1 As a result of this simplicity, the axial and vector currents are related to one another. This µ fact may be established by computing the products γ γ5 explicitly,

( 0 1 3 2 1 γ γ5 = σ σ = −iσ = −γ = +γ1 1 2 3 1 0 (8.21) γ γ5 = iσ σ = −σ = −γ = −γ0

Introducing the rank 2 antisymmetric tensor µν = −νµ, normalized to 01 = 1, the above relations may be recast in the following form,

µ µν µ µν γ γ5 = γν =⇒ j5 = jν (8.22)

Consider the calculation of the vacuum polarization to one-loop, but this time in 2-dim QED. The vacuum polarization diagram is just the correlator of the two vector currents

85 in the free theory, Πµν(k) = eh∅|T jµ(k)jν(−k)|∅i, and is given by,

Z d2p i i ! Πµν(k) = −e tr γµ γν (8.23) (2π)2 k/ + p/ p/

A priori, the integral is logarithmically divergent, and must be regularized. We choose to regularize in a manner that preserves gauge invariance and thus conservation of the current jµ. As a result, we have Πµν(k) = (k2ηµν − kµkν)Π(k2). Gauge invariant regularization may be achieved using a single Pauli-Villars regulator with mass M, for example. Just as in 4-dim QED, current conservation lowers the degree of divergence and in 2-dim, vacuum polarization is actually ﬁnite.

Z 1 Z d2p " α(1 − α) α(1 − α) # Π(k2) = 4e dα − (8.24) 0 (2π)2 (p2 + α(1 − α)k2)2 (p2 + α(1 − α)k2 − M 2)2

Carrying out the p-integration,

ie Z 1 " α(1 − α) α(1 − α) # Π(k2) = dα − (8.25) π 0 −α(1 − α)k2 −α(1 − α)k2 + M 2

Clearly, as M → ∞, the limit is ﬁnite and the contribution from the Pauli-Villars regulator actually cancels. One ﬁnds the ﬁnite result,

ie Π(k2) = − (8.26) πk2 Notice that, even though the ﬁnal result for vacuum polarization is ﬁnite, the Feynman integral was superﬁcially logarithmically divergent and requires regularization. It is during this regularization process that we ensure gauge invariance by insisting on the conservation of the vector current jµ. Later, we shall see that one might have chosen NOT to conserve the vector current, yielding a diﬀerent result.

µν µ ν Next, we study the correlator of the axial vector current, Π5 (k) = eh∅|j5 (k)j (−k)|∅i. µν µν Generally, Π5 and Π would be completely independent quantities. In 2-dim, however, the axial current is related to the vector current by (8.22), and we have

µν µρ ν µν 2 µρ ν 2 Π5 (k) = Πρ (k) = ( k − kρk )Π(k ) (8.27)

µν Conservation of the vector current requires that kνΠ5 (k) = 0, which is manifest in view of the transversality of Πµν(k). The divergence of the axial vector current may also be

86 calculated. The contribution from the correlator with a single vector current insertion is ie k Πµν(k) = k µνk2Π(k2) = − k µν (8.28) µ 5 µ π µ which is manifestly non-vanishing. As a result, the axial vector is not conserved and chiral symmetry in 2-dim QED is not a symmetry at the quantum level : chiral symmetry suﬀers an anomaly. Translating this into an operator statement: e ∂ jµ = − F µν (8.29) µ 5 2π µν Remarkably, this result is exact (Adler-Bardeen theorem). Higher point correlators are convergent and classical conservation valid.

8.5 The axial anomaly and regularization The above calculation of the axial anomaly was based on our explicit knowledge of the vector current two-point function (or vacuum polarization) to one-loop order. Actually, the anomaly equation (8.29) is and exact quantum ﬁeld theory operator equation, which in particular is valid to all orders in perturbation theory. To establish this stronger result, there must be a more direct way to proceed than from the vacuum polarization result. First, consider the issue of conservation of the axial vector current when only the fermion is quantized and the gauge ﬁeld is treated as a classical background. The correlator µ h∅|j5 |∅iA is then given by the sum of all 1-loop diagrams shown in ﬁgure xx. Consider µ ﬁrst the graphs with n ≥ 2 external A-ﬁelds. Their contribution to h∅|j5 |∅iA is given by Z d2p 1 1 1 ! µ;ν1···νn n µ ν1 νn Π5 (k; k1, ··· , kn) = −e 2 tr γ γ5 γ ··· γ (8.30) (2π) p/ + `/1 p/ + `/n p/ + `/n+1

where we use the notation li+1 − li = ki, As we assumed n ≥ 2, this integral is UV convergent, and may be computed without regularization. In particular, the conservation of the axial vector current may be studied by using the following simple identity, (and the

fact that ln+1 − l1 = −k by overall momentum conservation k + k1 + ··· + kn = 0), 1 1 1 1 ν1 k/γ5 γ = −γ5 + γ5 (8.31) p/ + `/n+1 p/ + `/1 p/ + `/1 p/ + `/n+1 Hence the divergence of the axial vector current correlator becomes, Z d2p 1 1 ! µ;ν1···νn n ν1 νn kµΠ5 (k; k1, ··· , kn) = e 2 tr γ5 γ ··· γ (8.32) (2π) p/ + `/1 p/ + `/n Z d2p 1 1 1 ! n ν2 νn ν1 −e 2 tr γ5 γ ··· γ γ (2π) p/ + `/2 p/ + `/n p/ + `/n+1

87 When n ≥ 3, each term separately is convergent. Using boson symmetry in the external

A-ﬁeld (i.e. the symmetry under the permutations on i of (ki, νi)), combined with shift symmetry in the integration variable p, all contributions are found to cancel one another. When n = 2, each term separately is logarithmically divergent, and shift symmetry still µ applies. Thus, we have shown that ∂µj5 receives contributions only from terms ﬁrst order in A, which are precisely the ones that we have computed already in the preceding subsection.

To see how the contribution that is ﬁrst order in A arises, we need to deal with a contribution that is superﬁcially logarithmically divergent, and requires regularization. One possible regularization scheme is by Pauli-Villars, regulators. The logarithmically divergent integral requires just a single regulator with mass M, and we have

Z d2p ( 1 1 1 1 )! Πµν(k, M) = −e tr γµγ γν − γν (8.33) 5 (2π)2 5 p/ + k/ p/ p/ + k/ − M p/ − M

The conservation of the vector current, i.e. gauge invariance, is readily veriﬁed, using the identity

1 1 1 1 1 1 1 1 k/ − k/ = − − + (8.34) p/ + k/ p/ p/ + k/ − M p/ − M p/ p/ + k/ p/ + k/ − M p/ − M and the fact that one may freely shift p in a logarithmically divergent integral. The divergence of the axial vector current is investigated using a similar identity,

( 1 1 1 1 ) trk/γ γν − γν 5 p/ + k/ p/ p/ + k/ − M p/ − M 1 1 1 1 ! = trγ γν − − + 5 p/ + k/ p/ p/ + k/ − M p/ − M 1 1 ! +2Mtr γ γν (8.35) 5 p/ + k/ − M p/ − M

The ﬁrst line of the integrand on the rhs may be shifted in the integral and cancels. The second part remains, and we have

Z d2p 1 1 ! k Πµν(k, M) = −2eM tr γ γν (8.36) µ 5 (2π)2 5 p/ + k/ − M p/ − M

Of course, this integral is superﬁcially UV divergent, because we have used only a single Pauli-Villars regulator. To obtain a manifestly ﬁnite result, at least two PV regulators

88 should be used; it is understood that this has been done. To evaluate (8.36), introduce a single Feynman parameter, α and shift the internal momentum p accordingly,

1 2 ν µν Z Z d p trγ5(p/ + (1 − α)k/ + M)γ (p/ − αk/ + M) kµΠ5 (k, M) = −2eM dα (8.37) 0 (2π)2 (p2 − M 2 + α(1 − α)k2)2

µ1 µn µ ν µν The trace trγ5γ ·γ vanishes whenever n is odd, while for n = 2, we have trγ5γ γ = 2 . In the numerator of (8.37), the contributions quadratic in p, those quadratic in k and those quadratic in M cancel because they involve traces with odd n; those linear in p cancel because of p → −p symmetry of the denominator. This leaves only terms linear in M and ν µν linear in k. Finally, using trγ5k/γ = 2 kµ and the familiar integral

Z d2p 1 i 1 = (8.38) (2π)2 (p2 − M 2 + α(1 − α))2 4π M 2 − α(1 − α)k2

we have

Z 1 2 µν ie µν M kµΠ5 (k, M) = − kµ dα (8.39) π 0 M 2 − α(1 − α)k2

As M is viewed as a Pauli-Villars regulator, we consider the limit M → ∞, which is smooth, and we ﬁnd,

µν ie µν lim kµΠ5 (k, M) = − kµ (8.40) M→∞ π which is in precise agreement with (8.28). Clearly, the anomaly contribution arises here solely from the Pauli-Villars regulator.

8.6 The axial anomaly and massive fermions If a fermion has a mass, then chiral symmetry is violated by the mass term. This may be seen directly from the Dirac equation for a massive fermion in QED,

µ µ iγ ∂µψ = mψ + eγ Aµψ ¯ µ ¯ ¯ µ −i∂µψγ = mψ + eψAµγ (8.41)

Chiral transformations may still be deﬁned on the fermion ﬁelds and the axial vector current may still be deﬁned by the same expression. For massive fermions, however, chiral transformations are not symmetries any more and the axial vector current fails to be conserved. Its divergence is easy to compute classically from the Dirac equation,

89 µ ¯ µ ∂µj5 = 2imψγ5ψ. Hence the Pauli-Villars regulators violate the conservation of j5 , and this eﬀect survives as M → ∞. In fact, one readily establishes the axial anomaly equation for a fermion of mass m, e ∂ jµ = 2imψγ¯ ψ − µνF (8.42) µ 5 5 2π µν In fact, it is impossible to ﬁnd a regulator of UV divergences which is gauge invariant and preserves chiral symmetry. Dimensional regularization makes 2 → d, but now we have a problem deﬁning γ5 = γ0γ1.

8.7 Solution to the massless Schwinger Model Two-dimensional QED is a theory of electrically charged fermions, interacting via a static Coulomb potential, with no dynamical photons. The potential is just the Green function 2 G(x, y) for the 1-dim Laplace operator, satisfying ∂xG(x, y) = δ(x, y), and given by 1 G(x, y) = |x − y| (8.43) 2 The force due to this potential is constant in space. It would take an inﬁnite amount of energy to separate two charges by an inﬁnite distance, which means that the charges will be conﬁned. Interestingly, this simple model of QED in 2-dim is a model for conﬁnement ! Remarkably, 2-dim QED was completely solved by Schwinger in 1958. To achieve this solution, one proceeds from the equations for the conservation of the vector current, the anomaly of the axial vector current, and Maxwell’s equations,

µ ∂µj = 0 e ∂ jµ = µν∂ j = − µνF µ 5 µ ν 2π µν µν ν ∂µF = ej (8.44) Of course, it is crucial to the general solution that these equations hold exactly. Notice that this is a system of linear equations, a fact that is at the root for the existence of an exact solution. We recast the axial anomaly equation in the following form, e ∂ j − ∂ j = − F (8.45) µ ν ν µ π µν µ Finally, taking the divergence in ∂µ, using ∂µj = 0 and Maxwell’s equations, we obtain, e e2 2jν = − ∂ F µν = − jν (8.46) 2π µ π which is the ﬁeld equation for a free massive boson, with mass m2 = e2/π. Actually, this boson is a single real scalar φ, which may be obtained by solving explicitly the conservation

90 µ µ µν equation ∂µj = 0 in terms of φ by j = ∂νφ. Notice that the gauge ﬁeld may also be expressed in terms of this scalar, (or the current), by solving the chiral anomaly equation, π π A = − j + ∂ θ = − ∂νφ + ∂ θ (8.47) µ e µ µ e µν µ where θ is any gauge transformation. This is just the London equation of superconductiv- ity, expressing the fact that the photon ﬁeld is massive now. This last equation is at the origin of the boson-fermion correspondence in 2-dim. Remarkably, the physics of the (massless) Schwinger model is precisely what we would expect from a theory describing conﬁned fermions. The salient features are as follows,

• The asymptotic spectrum contains neither free quarks, nor free gauge particles;

• The spectrum contains only fermion bound states, which are gauge neutral;

• The spectrum has a mass gap, even though the fermions were massless.

It is also interesting to include a mass to the fermions, but this model is much more complicated, and no longer exactly solvable. Its bosonized counterpart is the massive Sine-Gordon theory.

8.8 The axial anomaly in 4 dimensions

µ ν In d=4, the 2-point function h∅|T j5 j |∅i vanishes. However, there are now triangle dia- grams, which do not vanish. The same problems arise with regularization and insisting on gauge invariance, one derives the anomaly equation for the axial vector current,

e2 1 ∂ jµ = F F˜µν + 2imψγ¯ ψ F˜ ≡ F κλ (8.48) µ 5 16π2 µν 5 µν 2 µνκλ

Its calculation is analogous to that in two dimensions.

91 QUANTUM FIELD THEORY – 230C

Eric D’Hoker Department of Physics and Astronomy University of California, Los Angeles, CA 90095

Version QFT-2016v11.tex 22 May 2016

Contents

1 Classical Yang-Mills Theory 4 1.1 InteractingLagrangians ...... 4 1.2 Abeliangaugeinvariance...... 5 1.3 Non-Abeliangaugeinvariance ...... 7 1.4 General gauge groups and general representations ...... 10 1.5 Yang-Mills theory for a simple gauge group ...... 12 1.6 Gauge invariant Lagrangians for simple gauge groups ...... 15 1.7 Bianchi identities and gauge ﬁeld equations ...... 18 1.8 Gauge-invariant Lagrangians for general gauge groups ...... 21

2 Functional Integrals For Gauge Fields 22 2.1 Functional integral for Abelian gauge ﬁelds ...... 22 2.2 Functional Integral for Non-Abelian Gauge Fields ...... 24 2.3 CanonicalQuantization...... 25 2.4 FunctionalIntegralQuantization ...... 26 2.5 The redundancy of integrating over gauge copies ...... 28 2.6 The Faddeev-Popov gauge ﬁxing procedure ...... 29 2.7 Calculation of the Faddeev-Popov determinant ...... 32 2.8 Faddeev-Popovghosts ...... 33 2.9 Axialandcovariantgauges...... 34

3 Perturbative Non-Abelian Gauge Theories 36 3.1 Feynman rules for Non-Abelian gauge theories ...... 37 3.2 Renormalization, gauge invariance and BRST symmetry ...... 40 3.3 Slavnov-Taylor identities ...... 41 4 Anomalies 43 4.1 Massless 4d QED, Gell-Mann–Low Renormalization ...... 43 4.2 ScaleInvarianceofMasslessQED ...... 44 4.3 ChiralSymmetry ...... 45 4.4 The axial anomaly in 2-dimensional QED ...... 46 4.5 The axial anomaly and regularization ...... 48 4.6 The axial anomaly and massive fermions ...... 50 4.7 Solution to the massless Schwinger Model ...... 51 4.8 Theaxialanomalyasanoperatorequation ...... 52 4.9 The non-Abelian axial anomaly in 2 dimensions ...... 53 4.10 The Polyakov - Wiegmann calculation of the eﬀective action ...... 55 4.11 TheWess-Zumino-Wittenterm ...... 57 4.12 The Abelian axial anomaly in 4 dimensions ...... 58 4.13 Anomaliesingaugecurrents ...... 60 4.14 Non-Abelian axial anomalies ...... 62 4.15 Non-Abelian chiral anomaly ...... 63 4.16 Properties of the d-symbols ...... 64

5 One-Loop renormalization of Yang-Mills theory 66 5.1 Calculation of Two-Point Functions ...... 66 5.2 IndicesandCasimirOperators...... 69 5.3 Summaryoftwo-pointfunctions...... 70 5.4 Three-PointFunctions ...... 71 5.5 Four-PointFunctions ...... 71 5.6 One-LoopRenormalization...... 73 5.7 AsymptoticFreedom ...... 75

6 Use of the background ﬁeld method 77 6.1 Backgroundﬁeldmethodforscalarﬁelds ...... 77 6.2 Background ﬁeld method for Yang-Mills theories ...... 79 6.3 Calculation of the one-loop β-function for Yang-Mills ...... 81

7 Lattice ﬁeld theory – Lattice gauge Theory 84 7.1 Lattices ...... 84 7.2 Scalarﬁeldsonalattice ...... 85 7.3 Latticegaugetheory ...... 87 7.4 Theintegrationmeasure forthegaugegaugeﬁeld ...... 89 7.5 Wilsonloops...... 90 7.6 Lattice gauge dynamics for U(1) and SU(N)...... 92

2 8 Global Internal Symmetries 94 8.1 Symmetriesinquantummechanics ...... 95 8.2 Symmetriesinquantumﬁeldtheory...... 96 8.3 Continuous internal symmetry with two scalar ﬁelds ...... 96 8.4 Spontaneous symmetry breaking in massless free ﬁeld theory ...... 100 8.5 Whychooseavacuum? ...... 103 8.6 Generalized spontaneous symmetry breaking (scalars) ...... 104 8.7 Thenon-linearsigmamodel ...... 107 8.8 The general G/H nonlinearsigmamodel ...... 108 8.9 Algebraofchargesandcurrents ...... 109 8.10 ProofofGoldstone’sTheorem ...... 111 8.11 The linear sigma model with fermions ...... 112

9 The Higgs Mechanism 115 9.1 TheAbelianHiggsmechanism...... 115 9.2 The Non-Abelian Higgs Mechanism ...... 117

10 Topological defects, solitons, magnetic monopoles 118 10.1 Topological defects - domain walls ...... 118

3 1 Classical Yang-Mills Theory

Free ﬁeld theories were described by quadratic Lagrangians, so that the equations of motion are linear. Interacting ﬁeld theories will be described by Lagrangians which are polynomial in the ﬁelds of degree higher than two, or which are non-polynomial in the ﬁelds. We shall insist on Poincar´einvariance, though it will sometimes be interesting to consider ﬁeld theories with higher symmetry (such as conformal ﬁeld theories) or with less or diﬀerent symmetries (such as Galilean or Schr¨odinger invariance, but we shall not do so here.) Sometimes, the eﬀects of a certain sector of the theory may be summarized in the form of external sources, thereby simplifying the practical problems involved. When we do so, the external sources will be assumed to be subject to deﬁnite transformation properties under Poincar´etransformations.

1.1 Interacting Lagrangians We begin by presenting a brief list of some of the most important fundamental interacting ﬁeld theories for elementary particles whose spin is less or equal to 1. These interacting ﬁeld theories are described by local Poincar´einvariant Lagrangians.

1. Scalar ﬁeld theory The simplest interacting Lagrangian is for a single real scalar ﬁeld φ, and it involves a kinetic term plus an interaction potential V (φ),

1 = ∂ φ∂µφ V (φ) (1.1) L 2 µ −

2. The linear σ-model This is a theory of N real scalar ﬁelds φa subject to quartic self-couplings,

2 1 N λ N = ∂ φa∂µφa φaφa v2 (1.2) L 2 µ − 4 − a=1 a=1 ! X X The denomination ”linear sigma model” is somewhat misleading as clearly neither the Lagrangian nor the ﬁeld equations are linear. The term arose historically from the fact that this theory has an SO(N) internal symmetry which is realized linearly on the ﬁelds φa. This realization, along with its non-linear σ-model counterpart, will be explained in detail later. Instead of the quartic interaction potential, one could have a more general non-derivative interaction potential with or without signiﬁcant internal symmetries.

4 3. Electrodynamics This theory governs most of the world that we see, namely of electrons interacting with electro-magnetic ﬁelds. It involves a single gauge ﬁeld Aµ, which accounts for the electro-magnetic ﬁeld, and a single Dirac fermion ψ of electric charge e, which accounts for the electron and the positron. Its Lagrangian is given by, 1 = F + ψ¯(iγµD m)ψ D ψ = ∂ ψ eA ψ (1.3) L −4 µν µ − µ µ − µ Electrically charge scalar ﬁelds may be included as well, giving rise to scalar electro- dynamics.

4. Yang-Mills theory Yang-Mills theory is the subject of the present section, and will be the most important interacting ﬁeld theory for the understanding of the standard model of strong and electro-weak interactions, as well as for virtually any topic in modern quantum ﬁeld theory. Here, we shall just present its Lagrangian, and defer to later the proper motivation and construction of the theory. Yang-Mills theory is associated with a Lie algebra with structure constants f abc for a, b, c =1, 2, , dim( ). The Lagrangian for pureG Yang-Mills theory is given by, ··· G 1 = F a F aµν F a = ∂ Aa ∂ Aa + g f abcAb Ac (1.4) L −4 µν µν µ ν − ν µ µ ν a c X X It is possible to couple these various theories to one another, as we shall see later.

1.2 Abelian gauge invariance To motivate Yang-Mills theory, which is also referred to as non-Abelian gauge theory, we return brieﬂy to the theory of electrodynamics which we know well and which is based on gauge symmetry. Consider a charged complex ﬁeld ψ(x), which transforms non-trivially under U(1) gauge transformations, denoted here by g(x),

ψ(x) ψ′(x)= g(x) ψ(x) (1.5) −→ The transformation g(x) satisﬁes the unitarity condition g(x)†g(x) = 1 at every point x in space-time, and may alternatively be represented by a real-valued phase ﬁeld θ(x),

g(x)= eiθ(x) (1.6)

These U(1) gauge transformations are internal symmetries which commute with the Lorentz group, as is clear from the fact that the gauge transformation ﬁelds g(x) and θ(x) are scalars

5 under Lorentz transformations. As a result, any local bilinear of the type ψ¯(x)γψ(x) is invariant under U(1) gauge transformations,

ψγψ¯ (x) ψ¯′γψ′(x)= ψγψ¯ (x) (1.7) −→ where γ stands for any one of the 16 γ-matrices,

γ = I, γµ, γµν , γµν γ5, γµγ5, γ5 (1.8) C1b1 { } The derivative of the ﬁeld does not transform homogeneously, but instead we have,

∂ ψ(x) g(x) ∂ ψ(x)+ ∂ g(x) ψ(x) (1.9) µ −→ µ µ ¯ µ so that the kinetic term iψγ ∂µψ is not invariant under U(1) gauge transformations. In QED, the ordinary derivative which appears in the kinetic term is upgraded to a covariant derivative which involves the gauge ﬁeld Aµ, and transforms under gauge trans- formations in exactly the same way the original ﬁeld does,

Dµψ(x) = ∂µψ(x)+ iAµψ(x) (1.10) ′ ′ Dµψ (x) = g(x) Dµψ(x)

This requirement on the covariant derivative then implies a deﬁnite transformation law for the gauge ﬁeld Aµ, which is easily obtained from the above formulas, and we ﬁnd,

A′ = A + i g−1∂ g = A ∂ θ (1.11) C1b2 µ µ µ µ − µ { }

µ We are now guaranteed that iψγ¯ Dµψ is invariant, and this kinetic term (up to a redeﬁni- tion of the ﬁeld A eA , to be discussed in more detail later) is indeed the key building µ → µ block of QED, and gives the only interaction in the theory.

By the Schwarz identity [∂µ, ∂ν ] = 0, ordinary derivatives commute with one another. Covariant derivatives, however, do not generally commute, and instead yield the ﬁeld strength Fµν of the gauge ﬁeld Aµ,

[Dµ,Dν]= iFµν (1.12)

This equation may be used to deﬁne the ﬁeld strength Fµν. By construction, and using the fact that the U(1) gauge transformations form an Abelian group, the ﬁeld strength Fµν is gauge invariant.

6 1.3 Non-Abelian gauge invariance Instead of considering a single Dirac fermion ﬁeld ψ, we shall now consider a multiplet ψ(x) of N Dirac fermion ﬁelds whose components are ψa(x) for a = 1, , N. It will be convenient to think of the multiplet ψ(x) as a column vector, ···

ψ1(x) ψ (x) ψ(x)= 2 (1.13) psi:mult { } ··· ψ (x) N A natural kinetic term K for these ﬁelds is given by adding the kinetic ﬁelds of each component, and we have,L

N = iψγ¯ µ∂ ψ = iψ¯ γµ∂ ψ (1.14) LK µ a µ a a=1 X Note that the multiplet ﬁeld ψ¯ is deﬁned by ψ¯ = ψ†γ0 where ψ† now includes the transpo- sition of the column matrix (1.13).

Besides Poincar´einvariance, the kinetic term K is invariant under constant, i.e. space- time independent, unitary transformations g underL which the multiplet ψ transforms by, ψ(x) ψ′(x)= g ψ(x) −→ N ψ (x) ψ′ (x)= g b ψ (x) (1.15) a −→ a a b Xb=1 Here g is an arbitrary N N matrix satisfying g†g = I, with components g b. The set of all × a unitary N N matrices is denoted by U(N) and forms a group under matrix multiplication. This group× is non-Abelian as soon as N 2, and reduces to the Abelian U(1) group when N = 1. These groups are all Lie groups,≥ i.e. parametric groups with analytic dependence on the parameters. They are also compact in the since the equation g†g = I that deﬁnes the set U(N) guarantees that U(N) is closed and bounded.1 We now wish to construct a Lagrangian for the multiplet ﬁeld ψ which is invariant under local U(N) gauge transformations, where the transformations are upgraded to being arbitrary x-dependent functions. Speciﬁcally, the transformation laws are as follows, ψ(x) ψ′(x)= g(x)ψ(x) −→ N ψ (x) ψ′ (x)= g b(x)ψ (x) (1.16) C1c1 a −→ a a b { } Xb=1 1The other compact Lie groups that we shall need belong to the the so-called classical series and consists of the inﬁnite series SO(N) and Sp(2N) (see my classical mechanics course).

7 where g(x) U(N) at every point x in space-time. As in the case of Abelian gauge symmetries,∈ the non-Abelian transformations g(x) are Lorentz scalars and commute with the Lorentz group. Therefore, local fermion bilinears of the type ψ¯(x)γψ(x) with γ as in (1.8) are invariant under U(N) gauge transformations,

ψ¯(x)γψ(x) ψ¯′(x)γψ′(x)= ψ¯(x)γψ(x) (1.17) −→

A covariant derivative Dµ is deﬁned so that the transformation law of Dµψ coincides with the transformation law of ψ. This requires introducing a non-Abelian gauge ﬁeld Aµ(x), which is an N N matrix valued ﬁeld, so that × Dµψ = ∂µψ + iAµψ (1.18) ′ ′ Dµψ (x) = g(x) Dµψ(x)

To work out the proper transformation law of Aµ, we need to take careful account of the novelty that the ﬁelds are now all non-commuting matrices,

′ (∂µ + iAµ) g(x)ψ(x)= g(x)(∂µ + iAµ)ψ(x) (1.19)

A term g(x)∂µψ(x) is common on both left and right sides of the equation and may be omitted. After this simpliﬁcation, all dependence on the derivatives of ψ has cancelled, and we are left with the equation,

′ ∂µg(x)+ iAµg(x) ψ(x)= g(x) iAµψ(x) (1.20) The equation holds for all ψ(x), and this ﬁeld may therefore be deleted from the equation,

′ ∂µg(x)+ iAµg(x)= g(x) iAµ (1.21)

′ Expressing the transformed ﬁeld Aµ in terms of the original ﬁeld Aµ and the gauge trans- formation g, we ﬁnd, we have,

A′ (x)= g(x)A (x)g−1(x)+ i∂ g(x) g−1(x) (1.22) C1c2 µ µ µ { }

In the special case where N = 1, the gauge ﬁeld Aµ has just one component, and the gauge transformation g commutes with Aµ, so that in this case we simply recover (1.11).

We proceed to compute the ﬁeld strength Fµν of the non-Abelian gauge ﬁeld Aµ by acting with the commutator of the covariant derivatives,

[Dµ,Dν]ψ = ∂µ + iAµ, ∂ν + iAν ψ = iFµν ψ (1.23) h i where Fµν takes the following form, F ∂ A ∂ A + i[A , A ] (1.24) µν ≡ µ ν − ν µ µ ν 8 The transformation properties of the ﬁeld strength may be read oﬀ from those of the covariant derivatives, namely, ′ ′ ′ [Dµ,Dν]ψ = g[Dµ,Dν]ψ (1.25) ′ which implies Fµν g = gFµν and hence also, F F ′ (x)= g(x)F (x) g−1(x) (1.26) µν −→ µν µν Notice that while for QED, where N = 1, the relation between ﬁeld strength and ﬁeld was linear, here it is non-linear. Thus, the requirement of non-Abelian gauge invariance produces a novel interaction. another way to say this is that in QED the photon does not carry electric charge and thus will not have a basic interaction with itself, while in Yang-Mills theory the gauge particle will carry charge, and thus will couple to itself.

1.3.1 The reality condition for physical Yang-Mills ﬁelds Next, we address the nature of the gauge ﬁeld A . The gauge ﬁeld A is an N N matrix µ µ × with complex entries. Any such matrix-valued ﬁeld may be decomposed with the help of (1) (2) two Hermitian matrix-valued ﬁelds Aµ , Aµ , where (1) (2) Aµ = Aµ + iAµ (1.27) Under U(N) gauge transformations of (1.22), the ﬁelds transform as follows, ′ (1) ′ (2) (1) (2) † † Aµ + iAµ = g Aµ + iAµ g + i(∂µg) g (1.28) † where we have suppressed the dependence on x, and made use of the fact that g g = I. Now † the matrix i(∂µg) g is Hermitian by construction, as may be seen by taking the derivative of the relation g g† = I, † † (∂µg)g + g(∂µg ) = 0 (1.29) † and then using this result to compute the Hermitian conjugate of i(∂µg) g , † † i (∂ g) g† = i g(∂ g†) = i(∂ g) g† (1.30) µ − µ µ (1) (1) As a result, the transformation rules for Aµ , Aµ are as follows, ′ (1) (1) † † Aµ = gAµ g + i(∂µg) g ′ (2) (2) † Aµ = gAµ g (1.31) (2) Now Aµ must vanish for a variety of reasons. First of all, incorporating it in the Dirac Lagrangian, its contribution will fail to be CPT invariant, and in fact will give an imaginary part to the Lagrangian. Second, we know that a quantum theory of a vector ﬁeld requires gauge invariance to eliminate the negative norm states produced by quantizing the vector (2) ﬁeld. But Aµ is not a gauge ﬁeld, and the associated negative norm states created by (2) this ﬁeld cannot be eliminated. Therefore, and henceforth, we set Aµ = 0, so that Aµ is Hermitian.

9 1.3.2 Lagrangian of U(N) Yang-Mills theory We are now in a position to write down a fully Lorentz and gauge invariant kinetic term for the multiplet ﬁeld ψ, with the help of the U(N) gauge ﬁeld Aµ,

¯ µ iψγ Dµψ (1.32) provided ψ transforms under the rules of (1.16) and Aµ transforms under the rules of (1.22). In addition the following pure gauge part is Lorentz-invariant and is invariant also under U(N) gauge transformations,

1 µν 1 µν 2 tr Fµν F 2 tr(Fµν)tr(F ) (1.33) −4g2 − 4g1 Combining these two ingredients, we may now write down a basic Lagrangian for Yang- Mills theory with gauge group U(N) and gauge ﬁeld Aµ coupled to a multiplet of N Dirac fermions ψ. We do this by assembling all the invariants that we have obtained so far, including the kinetic term for the fermions, and the F 2 term for the gauge ﬁeld,

1 µν 1 µν ¯ µ = 2 tr Fµν F 2 tr(Fµν)tr(F )+ ψ iγ Dµ m ψ (1.34) L −4g2 − 4g1 − Here, we have included a Dirac mass term for the fermions, which is both Lorentz and gauge invariant. (Terms like ψγ¯ µψ are gauge invariant, but of course not, by themselves, 2 2 Lorentz invariant.) The mass parameter m and the gauge couplings g1 and g2 must be real in order to have CPT invariance, but are otherwise arbitrary.

1.4 General gauge groups and general representations Although the Yang-Mills theory for the gauge group U(N) developed in the previous section is perfectly consistent, it is both too restrictive and too general for physically signiﬁcant applications. For example, in the theory of strong interactions, all quarks couple to the strong interaction Yang-Mills ﬁeld, but they do not do so as proposed in this earlier model. From the point of view of the strong interactions he quarks are organized as follows,

uc, dc,cc, sc, tc, bc (1.35) where the index c runs over the three colors c =1, 2, 3 or more prosaically red white blue. If we had followed the model of the preceding paragraph, we would have organized the quarks into a big multiplet with N =3 6 = 18 Dirac fermion ﬁelds and coupled an U(18) Yang-Mills ﬁeld to these fermions. But× this is not how Nature is structured. Instead, it is only a SU(3) Yang-Mills theory that couples in an identical manner to all six ﬂavors of quarks, and acts only on the color indices a =1, 2, 3.

10 Therefore, when we have a multiplet of N fermions, it is not necessary to have full U(N) gauge invariance. All that is required is that we have a gauge group G with the fermion multiplet transforming under a N-dimensional unitary representation of G. In other words, if the group composition in G is denoted by ⋆, the a unitary representation of dimension N is a map , T : G U(N) such that (g ⋆ g )= (g ) (g ) (1.36) T → T 1 2 T 1 T 2 for any g1, g2 G. If the representation is faithful, then G must be a subgroup of U(N), but this is not∈ necessary. For example, ifT we included also leptons in the fermion multiplet, then the strong interaction SU(3) would leave those invariance. To proceed, we need to know a little bit more about Lie groups. For all practical purposes, we shall need only compact Lie groups, which are such that every one of its ﬁnite dimensional representations is equivalent to a unitary representation. That’s a good setting, because we will always be dealing with a ﬁnite number of ﬁelds, so the representation must be ﬁnite dimensional, and also it must be unitary in order to ﬁt into U(N). An invariant subgroup H of a group G is a subset of G such that g ⋆H⋆ g−1 = H for any g G. The group G itself, and the identity group I are always invariant subgroups ∈ { } under the above deﬁnition. If G = U(N1) U(N2), then H = U(N1) is a non-trivial invariant subgroup of G, and so is H = U(N ).× Also, for N 2 we have the decomposition 2 ≥ U(N) = SU(N) U(1) so that G has non-trivial invariant subgroups U(1) and SU(N). A group G is said× to be simple if it has no invariant subgroups, other than itself and the identity subgroup I . The groups SU(N) are simple for N 2, but the Abelian group { } ≥ U(1) is not simple. Simple compact Lie groups are the building blocks of all compact Lie groups. In fact, Elie Cartan classiﬁed all simple Lie groups (actually more precisely all semi-simple simple Lie groups) in the 1920’s. The Cartan classiﬁcation includes, the classical series SU(N), N 2,SO(N), N 5, Sp(2N) ≥ ≥ the exceptional series E8, E7, E6, F4, G2 (1.37)

Note that we have the identiﬁcations Sp(2) = SU(2), as well as SO(3) = SU(2)/Z2, SO(4) = SU(2) SU(2)/Z2, SO(5) = Sp(4)/Z2 and SO(6) = SU(4)/Z2, which all have to do with spinors.× The groups in the exceptional series will not concern us here. The main theorem we shall use here is that any compact Lie group G is the product of a number r of simple compact Lie groups G , ,G with a number s of U(1)-factors, 1 ··· r G = G G U(1) U(1) (1.38) 1 ×···× r × 1 ×···× s up to possible factors of discrete groups, which we shall ignore here. The semi-simple factors are taken from the Cartan classiﬁcation. As you already know, the fundamental gauge group of the Standard Model of Particle Physics takes the form, SU(3) SU(2) U(1) (1.39) c × L × Y where the subscripts refer respectively to “color”, “left”, and “hyper-charge”.

11 1.5 Yang-Mills theory for a simple gauge group Let G be simple compact Lie group, and let be an N-dimensional representation of G. Since any ﬁnite-dimensional representation ofT a compact group is equivalent to a unitary representation, is unitary without loss of generality. Elements of the group G will be T denoted by g, and the corresponding N-dimensional representation matrix of G at the element g will be denoted by (g). The action of a gauge transformation g(x) valued in G T on an N-component Dirac spinor ψ is now speciﬁed by the representation under which ψ transforms, T

ψ(x) ψ′(x)= (g(x))ψ(x) (1.40) −→ T The covariant derivative D = DT depends on the representation and is deﬁned by, µ µ T T T Dµ ψ = ∂µψ + iAµ ψ (1.41)

T The matrix ﬁeld µ depends not only on the gauge group G, but also on the speciﬁc A T representation under which the spinor ψ transforms, and the superscript on Aµ is there to indicateT this dependence. In particular, it will be our task to disentangleT the dependences of AT on G and on . To do so, we use the transformation law of the µ T covariant derivative, which by deﬁnition is given as follows,

(DT )′ψ′(x)= (g(x)) DT ψ(x) (1.42) µ T µ By an argument analogous to the one we used for the case of G = U(N), this implies the following transformation law for the gauge ﬁeld for an simple arbitrary gauge group G,

(AT )′ = (g)AT (g)−1 + i ∂ (g) (g)−1 (1.43) µ T µ T µT T To analyze this equation, we need a little Lie group and Lie algebra theory.

1.5.1 A little Lie group and Lie algebra theory It is convenient to restrict to inﬁnitesimal gauge transformations and to deal with the Lie algebra of the Lie group G. Thus, we expand the Lie group element g(x) near the identity I G, soG that in the representation we have, ∈ T dim G (g(x)) = I + i ua(x)T a + (u2) (1.44) T N O a=1 X where ua are real-valued functions which parametrize the group elements near the identity. The generators T a are representation matrices of the Lie algebra in the representation . Since G is a Lie group, the T a span its associated Lie algebra G. Using the fact that forT G 12 −1 −1 any pair of elements g1, g2 G the group commutator (g1) (g2) (g1 ) (g1 ) must be of the form (g) for some g∈ G, one deduces the existenceT ofT the structureT T relations of , T ∈ G dim G [T a, T b]= i f abc T c (1.45) struct { } c=1 X Here, f abc are the structure constants of , which are real and anti-symmetric in its ﬁrst G two indices. In view of the Jacobi identity of its representation matrices, [[T a, T b], T c]+ [[T b, T c], T a] + [[T c, T a], T b] = 0, the structure constants satisfy the Lie identity, dim G f abαf αcd + f bcαf αad + f caαf αbd = 0 (1.46) fjac { } α=1 X This identity is crucial in Lie algebra theory. Supposing a set of relations (1.45) were given for some arbitrary assignment of constants f abc. The relations (1.45) actually correspond to a Lie algebra if and only if the constants f abc satisfy (1.46). Since the representation is unitary, it follows that the generators T a are Hermitian, T (T a)† = T a (1.47) The normalization of the generators T a depends on the normalization of the coeﬃcient functions ua, which is arbitrary. A convenient normalization may be imposed as follows. Assuming that the representation is non-trivial and unitary, the dim G dim G matrix a b T × tr(T T ) is real symmetric, and positive deﬁnite. Therefore, we may choose a basis for ua, and thus a basis for T a, in which tr(T aT b) is diagonal, and its positive eigenvalues may be scaled to a common value, so that we ﬁx the following normalization,2 tr(T aT b)= I( )δab (1.48) T The coeﬃcient I(T ) is referred to as the quadratic index of the representation . Group theory actually allows one to calculate the index for all representations, but we shallT post- pone this problem to later. With this normalization, the structure constants f abc obey one further property: f abc is totally anti-symmetric in the three indices a, b, c. The structure constants themselves are representation matrices of , as may be seen by expressing the generators as follows, G ( a)αβ = if aαβ (1.49) calF F − { } Note the factor of i needed to convert the real anti-symmetric matrices f to Hermitian generators . The size of this matrix is dim G dim G so that the dimension of the F × corresponding representation is dim G. This representation exists for any Lie group (or Lie algebra) and is referred to as the adjoint representation. It is always real, and always has the same dimension as the Lie algebra itself.

2Note that if G had been non-compact, then some of the generators T a will be non-hermitian and tr(T aT b) will not be a positive deﬁnite matrix, so that one cannot choose the above normalization.

13 1.5.2 Returning to gauge ﬁelds

T Restricting the transformation law of Aµ to inﬁnitesimal gauge transformations gives the 3 following relation to ﬁrst order in the gauge transformation parameters ua,

(AT )′ = AT + iua T a, AT ∂ u T a (1.50) µ µ µ − µ a a X T a The only part of Aµ that transforms as a gauge ﬁeld is a linear combination of T generators, which is the part that must live in the Lie algebra (by an argument analogous to the G (2) one we used earlier to rule out the anti-Hermitian part iAµ when we discussed the U(N) T case). Any other contributions to Aµ must vanish. Therefore, we must have,

T a a Aµ (x)= Aµ(x) T (1.51) a X and the gauge transformation rule reduces to,

(Aa )′ = Aa ∂ ua f abcubAc (1.52) µ µ − µ − µ Xb,c Note that the gauge transformation rule on the gauge ﬁeld depends only on the gauge Lie algebra and is independent of the representation of the fermions. In fact, the gauge G T ﬁeld always transforms under the adjoint representation of , as may be seen by just the contribution from the non-derivative part. G It is often convenient to recast transformations in inﬁnitesimal form, so that we have,

δψ = ψ′ ψ = i ua T a ψ − a X δAa = (Aa )′ Aa = (DGu)a (1.53) µ µ − µ − µ G where Dµ is the covariant derivative acting on the adjoint representation,

a G a a abc b c b b (Dµ u) = ∂µu f Aµu = ∂µu + i Aµ u (1.54) − F ! Xb,c Xb where b are the Hermitian matrices of the adjoint representation given in (1.49). F Finally, we compute the ﬁeld strength. We may do this using the covariant derivative DT in any representation of the gauge group. Thus, we have, µ T T T T Dµ ,Dν = iFµν (1.55)

3 dim G Henceforth, we shall denote the summation a=1 simply by a. P P 14 where

T a a Fµν = Fµν T (1.56) a X a and the components Fµν are given by,

F a = ∂ Aa ∂ Aa f abcAb Ac (1.57) µν µ ν − ν µ − µ ν Xb,c 1.6 Gauge invariant Lagrangians for simple gauge groups In this subsection, we shall construct general gauge invariant Lagrangians for a simple gauge group G, with Lie algebra . We assume that the theory has a Dirac spinor ﬁeld G ψ transforming under a unitary representation ψ of , and a scalar ﬁeld φ transforming T G a a under a unitary representation φ of , with respective Lie algebra generators Tψ and Tφ . The corresponding covariant derivativesT G on these ﬁelds then take the following form,4

Tψ a a Dµψ = Dµ ψ = ∂µψ + i Aµ Tψ ψ a X Tφ a a Dµφ = Dµ φ = ∂µφ + i Aµ Tφ φ (1.58) a X a The basic invariant Lagrangian for the ﬁelds Aµ, ψ, and φ is given as follows, 1 = F a F aµν + ψ¯ iγµD ψ +(Dµφ)†D φ V (φ, ψ) (1.59) L −4g2 µν µ µ − a X where g is the gauge coupling and V (φ, ψ) is a potential which involves only the ﬁelds φ and ψ, but not their derivatives. To have gauge invariance, it suﬃces for the potential to be invariant under global (i.e. constant) transformations of G. The terms allowed in the potential depend to a large degree on the group G and the representations and . For Tψ Tφ any gauge group G, mass terms are allowed for the fermion ﬁeld and for the scalar ﬁeld, giving the allowed quadratic part of the potential,

† V2(φ, ψ)= mψψψ¯ + mφφ φ (1.60)

More generally for any gauge group, we can have arbitrary functions of these quadratic combinations, V (φ, ψ) = v(φ†φ, ψψ¯ ). Gauge invariant Yukawa interactions, of the type ψφψ¯ , are allowed when the tensor product of the corresponding representations ∗ Tψ ⊗Tφ ⊗Tψ

4Having speciﬁed the transformation law of the ﬁeld on which the covariant derivative acts, it is cus- tomary to no longer explicitly exhibit the representation dependence of the covariant derivative.

15 contains the singlet representation. Interactions of the quartic order and higher may be analyzed by the same criterion. The above Lagrangian exists for these ﬁelds in any space-time dimension d. The coupling of the gauge ﬁeld to the Dirac spinor ﬁeld ψ is parity conserving. The entire Lagrangian will be parity conserving provided the potential, including the Yukawa couplings, is also parity conserving. In generic odd dimension d, the above Lagrangian is the most general one, given the gauge group and the ﬁeld content. A nice example is the QCD Lagrangian for which G = SU(3) which is a simple group. There is no scalar ﬁeld, and we have the well-known quark contents,

uc, dc,cc, sc, tc, bc c =1, 2, 3 (1.61)

Each quark transforms under the deﬁning representation of SU(3) which has dimension a 3, sometimes denoted by 3. We may denote the six ﬂavors of quarks instead by ψf for f = u,d,c,s,t,b. The Lagrangian in four dimensions is very simple, 1 = F a F aµν + θ F a F˜aµν + ψ¯ (iγµD m ) ψ (1.62) LQCD −4g2 µν µν f µ − f f a a X X Xf where g is the QCD strong coupling constant, and mf are the quark masses. Note the addition of the F F˜ term, which exists in d = 4 but which dos not exist in other dimensions. The corresponding coupling θ is real, and periodic for reasons that will be explained later. The Lagrangian preserves parity, but the θF F˜ term violates CP. Note that we could have assembled the 18 Dirac fermion ﬁelds of the quarks, including all colors and ﬂavors, into a single 18-dimensional representation of SU(3). Of course, this representation is reducible into the direct sum of the six quark ﬂavors, which is what allows us to have diﬀerent values for the quark masses.

1.6.1 Chiral gauge theories In even space-time dimension d, the above Lagrangian is not the most general one. The reason is that for even d, the Dirac spinor representation of the Lorentz group (or of the Euclidean rotation group for Euclidean signature) is reducible into the direct sum of two Weyl spinors, usually referred to as “left” and “right” Weyl spinors. We shall denote the corresponding projection operators onto left and right spinors by PL and PR respectively,

I = PL + PR PLPR = 0 (1.63)

Denoting the chirality matrix in arbitrary even dimension d by γd+1 (cfr. γ5 in d = 4), with the normalization (γd+1)2 = I, the projection operators are given as follows, 1 1 P = I + γd+1 P = I γd+1 (1.64) L 2 R 2 − 16 It is customary to use the following notations,

ψ = ψL + ψR ψL = PLψ ψR = PRψ (1.65)

Since gauge transformations are Lorentz scalars, the separation into left and right spinors does not interfere with gauge invariance. Therefore, the representations of under which G ψL and ψR transform do not have to be the same, and we shall denote them respectively a a by L and R with associated representation matrices TL and TR. This decoupling of left andT right chiralitiesT allows for further generalization of the Lagrangian in even space-time dimension d. We shall denote the corresponding covariant derivatives by,

a a DµψL = ∂µψL + i Aµ TL ψL a X a a DµψR = ∂µψR + i Aµ TR ψR (1.66) a X a The basic invariant Lagrangian for the ﬁelds Aµ, ψL, ψR, and φ is given as follows,

1 = F a F aµν + ψ¯ iγµD ψ + ψ¯ iγµD ψ +(Dµφ)†D φ V (φ, ψ , ψ ) (1.67) L −4g2 µν L µ L R µ R µ − L R a X Whether the Dirac mass term is allowed will now depend upon whether the tensor product ∗ contains the singlet representation or not. Similarly for other terms in the potential. TL ⊗TR A nice example of a chiral gauge theory is provided by the SU(2)L part of the weak interactions, in which we ignore color quantum numbers. The quarks may then be arranged into doublets, corresponding to the three generations,

u c t ψ = ψ = ψ = (1.68) 1 d 2 s 3 b which we may collectively denote by ψi with i =1, 2, 3. The SU(2)L of the weak interactions couples only to left-chirality quarks, which transform as doublets under SU(2)L, but not to the right chirality quarks, which therefore transform as singlets under SU(2)L. The theory also has a scalar ﬁeld φ (the Higgs ﬁeld) which transforms also as a doublet under SU(2)L. The Lagrangian is given as follows,

1 3 = F a F aµν + ψ¯ iγµD ψ + ψ¯ iγµ∂ ψ +(Dµφ)†D φ V (1.69) L −4g2 µν iL µ iL iR µ iR µ − a i=1 X X

Dirac mass terms, given by ψ¯iLψjR + ψ¯Rj ψLi, are not gauge invariant, and therefore not allowed. This is where the Higgs ﬁeld comes to the rescue. It allows for a Yukawa potential

17 and a quartic potential,

3 2 † 2 † 2 ¯ V (φ, ψiL, ψiR)= mφφ φ + λ (φ φ) + (ψiLφ)MijψjR + c.c. (1.70) i,j=1 X

The combination ψ¯iLφ represents the contraction of the SU(2)L doublets of the Higgs ﬁeld φ and the left spinor ψiL and the resulting combination is gauge invariant. Furthermore, 2 2 λ is the Higgs self-coupling, and mφ is the Higgs mass term. In the Standard Model, we 2 2 will have λ > 0 and mφ < 0, which will lead to the Higgs eﬀect and to giving masses to the gauge ﬁelds and to the quarks, as we shall discuss later. The matrix M is the so-called Cabbibo-Kobayashi-Maskawa matrix, which is responsible for mixing the couplings and masses of weak doublets (once the Higgs eﬀect will have taken place).

1.7 Bianchi identities and gauge ﬁeld equations

T The covariant derivative Dµ = Dµ is a diﬀerential operator on the space of ﬁelds and therefore must satisfy the Jacobi identity,

[Dµ, [Dν,Dρ]] + [Dν, [Dρ,Dµ]] + [Dρ, [Dµ,Dν]] = 0 (1.71)

Using [Dρ,Dσ]= iFρσ, the equation is equivalent to,

[Dµ, Fνρ] + [Dν, Fρµ] + [Dρ, Fµν ] = 0 (1.72)

These commutators reduce to just evaluating the covariant derivative, and give rise to the Bianchi identities for the ﬁeld strength,

DµFνρ + DνFρµ + DρFµν = 0 (1.73)

In 4-dimension, this equality is equivalent to a more familiar form of the Bianchi identities,

1 ǫµνρσD F = D F˜µν =0 F˜µν = εµνρσF (1.74) ν ρσ ν 2 ρσ where εµνρσ is the totally antisymmetric tensor in four dimensions. The Bianchi identities µν generalize two of the four Maxwell equations of QED, ∂ν F˜ = 0.

1.7.1 The action in terms of a current The action for the Yang-Mills ﬁeld is given by the integral over the Lagrangian (density). For any simple gauge group G, we may express the gauge ﬁeld and the ﬁeld strength in the

18 adjoint representation where the representation matrices a are normalized by the index of the adjoint representation, F

tr a b = I( )δab (1.75) F F G In any space-time dimension d, the action for the Yang-Mills ﬁeld is then given by,

1 1 S[A, J]= ddx tr(F F µν) tr(J µA ) (1.76) I( ) −4g2 µν − µ G Z where g is the coupling constant. The current also transforms under the adjoint represen- µ µa a tation, J = a J and for example for a Dirac fermion ψ which transforms under a representation of GF with representation matrices T a, takes the form,5 PT J aµ = ψγ¯ µT aψ (1.77)

One may express the same action in terms of component ﬁelds by,

1 S[A, J]= ddx F a F aµν J aµAa (1.78) −4g2 µν − µ a Z X In the special case where the space-time dimension if four, namely d = 4, one may add µν a further topological term, namely θtrFµν F˜ . This is possible only in four dimensions because only then does the ε tensor exist. The term is topological, because one can easily prove the following identity,

µν µ tr FµνF˜ = ∂µK (1.79) where Kµ is the so-called Chern-Simons term,

4i Kµ = εµνρσtr 2A ∂ A + A A A (1.80) ν σ ρ 3 ν ρ σ So, trF F˜ is a total derivative on space-time – but of a gauge non-invariant object Kµ. As a result, this term will have vanishing variations at any ﬁnite point in space-time, and will therefore not contribute to the ﬁeld equations. It will have, however, important quantum eﬀects due to tunneling eﬀects generated by instanton conﬁgurations.

5For a scalar ﬁeld φ, a term of the type AAφφ is present in the action, and produces a more complicated form for the current which depends on A.

19 1.7.2 Deriving the ﬁeld equations from the action

µ Assuming that the current J is independent of Aµ, the variation of the action with respect to Aµ takes the following form,

1 1 δS[A, J]= ddx tr(δF F µν) tr(J µδA ) (1.81) I( ) −2g2 µν − µ G Z To derive the ﬁeld equations, we use the following straightforward identities,

δDµ = i δAµ δF = D δA D δA (1.82) µν µ ν − ν µ

The gauge transformation rule for δAµ is homogeneous even though the transformation rule for Aµ is inhomogeneous. More generally and more mathematically familiar from GR, the diﬀerence between two connections acting on the same ﬁelds transforms as a tensor. µν Thus the variation of the trF F term will now involve tr((DµδAν)F ). We would like to integrate by parts and, to do so, we need a new formula,

tr (D δA )F µν = ∂ tr δA F µν tr δA D F µν (1.83) µ ν µ ν − ν µ To prove this formula, it suﬃces to write out the ingredients,

tr (D δA )F µν = tr ∂ δA + i[A , δA ] F µν µ ν { µ ν µ ν } = ∂ tr(δA F µν) tr δA (∂ F µν + i[A , F µν]) (1.84) µ ν − ν µ µ which gives the result announced above. Omitting the total derivative term in the variation,

1 1 δS[A, J]= ddx tr(δA D F µν) tr(J µδA ) (1.85) I( ) g2 ν µ − µ G Z and requiring the action to be stationary gives the ﬁeld equations,

µν 2 ν DµF = g J (1.86)

Note that consistency of the ﬁeld equations requires the current to be covariantly conserved, since we have, 1 i g2D J ν = D D F µν = [D ,D ]F µν = [F , F µν] = 0 (1.87) ν ν µ 2 ν µ −2 µν For the Dirac fermi ﬁeld ψ, the current J aµ = ψγ¯ µT aψ is indeed covariantly conserved, as ay be established by using the Dirac equation for ψ in the presence of the gauge ﬁeld.

20 1.8 Gauge-invariant Lagrangians for general gauge groups Finally, we put together all that we have learned to construct Lagrangians for general compact gauge groups. Recall that, up to ﬁnite discrete subgroups whose eﬀects we shall ignore for the moment, a general compact gauge group may be decomposed into simple factors Gi and U(1) factors, with associated gauge ﬁelds listed underneath,

G = G G G U(1) U(1) U(1) 1 × 2 ×···× r × 1 × 2 ×···× s A A A B B B (1.88) 1µ 2µ ··· rµ 1µ 2µ ··· sµ

We have fermion multiplets of left and right Weyl spinors ψL and ψR which transform a under representations L and R of G with respective representation matrices (TiL, TjL) a T T and (TiR, TjR), for i =1, ,r and j =1, ,s. We also have a scalar φ which transforms under a representation ···of G with representation··· matrices S . The Lagrangian is then S i given as follows,

r 1 s 1 = tr(F µνF ) H Hµν L − 4g2I( ) i iµν − 4e2 jµν j i=1 i i j=1 j X G X +ψ¯ iγµD ψ + ψ¯ iγµD ψ +(D φ)†(Dµφ) V (φ, ψ , ψ ) (1.89) L µ L R µ R µ − L R where

r dim Gi s a a DµψL = ∂µψL + i AiµTiLψL + i BiµTjLψL i=1 a=1 j=1 X X X r dim Gi s a a DµψR = ∂µψR + i AiµTiRψR + i BiµTjRψR i=1 a=1 j=1 X X X r dim Gi s a a Dµφ = ∂µφ + i AiµSi φ + i BiµSjφ (1.90) i=1 a=1 j=1 X X X

The potential V depends only on φ and ψL, ψR, and does not involve derivatives of these ﬁelds. It must be gauge invariant, a condition that reduces to requiring that V be invariant under global G transformations. The best example is of course the Standard Model, which has r = 2 and s = 1, and gauge group

G = SU(3) SU(2) U(1) (1.91) c × L × Y We shall give it in detail when we discuss the Higgs mechanism.

21 2 Functional Integrals For Gauge Fields

The functional methods appropriate for gauge ﬁelds present a number of complications, especially for non-Abelian gauge theories. In this chapter, we begin with a rapid and some- what sketchy discussion of the Abelian case and then move on to a systematic functional integral quantization of the Abelian and non-Abelian cases.

2.1 Functional integral for Abelian gauge ﬁelds The starting point will be taken to be the Hamiltonian, supplemented by Gauss’s law. It is convenient to describe the system in temporal gauge A0 = 0, where we have, 1 1 H = dd−1x E2 + B2 A J (2.1) 2 2 − · Z where J is a conserved external current and we the following expressions for electric and magnetic ﬁelds, ∂A E = B = A (2.2) ∂t ∇ × Conﬁguration space is described by all A(x), hence a “position basis” of Fock space consists of all states of the form A(x) for all A(x) (2.3) F ≡ | i The Hilbert space of physical statesn (recall the diﬀerenceo between the Fock space and the physical Hilbert space, as discussed for free spin 1 ﬁelds) is obtained by enforcing on the states the constraint of Gauss’s law, G(A)= E J 0 (2.4) ∇· − which commutes with the Hamiltonian so that [H,G(A)] = 0. Thus, we have

= ψ such that G(A) ψ =0 (2.5) Hphys | i∈F | i Of course, we are only interestedn in the physical states. In particular,o we are interested only in time-evolution on the subspace of physical states. Therefore, we introduce the projection operator onto as follows, Hphys = ψ ψ (2.6) P | ih | |ψi∈HXphys and the projected evolution operator may be decomposed as follows, N e−itH = lim e−itH/N (2.7) Hphys N→∞ P

The repeated insertion of the projection operator in the above formula is of course redun- dant since G commutes with H, but this redundant form will ultimately prove to be the

22 most useful one. We always assume that the full Hamiltonian, using the dynamics of J is used, so that it is time independent. Otherwise T -order ! The projection operator onto physical states may also be constructed as a functional integral, though admittedly, this construction is a bit formal,

1 = Λ exp i dd−1xΛ(x)( E J 0)(x) (2.8) P Λ D ∇· − D Z Z Clearly is the identityR on , and zero anywhere else. Summing over only physical P Hphys states is equivalent to summing over all states A(x) and using the projector to project the sum onto physical states only. | i P 1 e−itH = A E Λ (2.9) Hphys Λ D D D D Z Z Z t 1 1 Rexp i dt dd−1x E ∂ A E2 B2 + Λ( E J 0)+ A J · 0 − 2 − 2 ∇· − · Z0 Z where it is understood that the usual boundary conditions on the functional integral vari- ables are implemented at times 0 and t. Now re-baptize Λ as A0, and express the result in terms of a generating functional for the source J µ,

1 1 Z[J] A exp i ddx F F µν J Aµ (2.10) ≡ Λ D µ −4 µν − µ D Z Z However, this is reallyR only a formal expression. In particular, the volume Λ of the D space of all gauge transformation is wildly inﬁnite and must be dealt with before one can R truly make sense of this expression. Recall that the expressions that came in always assumed the form Z[J µ]/Z[0] so that an overall factor of the volume of the gauge transformations cancels out. Thus in the expression for Z[J µ] itself, we may ab initio ﬁx a gauge, so that the volume of the gauge orbit is eliminated. We shall give a full discussion of this phenomenon when dealing with the non-Abelian case in the next subsection; here we take an informal approach. One of the most convenient choices is given by a linear gauge condition,

1 A δ(∂µA c) exp i ddx F F µν J µA (2.11) D µ µ − −4 µν − µ x Z Y Z Since no particular slice is preferred, it is natural to integrate with respect to c, say with a Gaussian damping factor,

d 2 d µ 2 c e−iλ/2 R d xc (x) δ(∂µA (x) c(x)) = e−iλ/2 R d x(∂ Aµ) (2.12) D µ − x Z Y 23 and we recover our good old friend:

1 1 Z[J]= A exp i ddx F F µν λ(∂ A)2 J µA (2.13) D µ −4 µν − 2 · − µ Z Z The J-dependence of Z[J] is independent of λ, so we can formally take λ 0 and regain → our original gauge invariant theory. It is straightforward to evaluate the free ﬁeld result,

0 T A (x)A (y) 0 = G (x y) (2.14) h | µ ν | i µν − where the Green function G is te inverse of the diﬀerential operator η ✷ (1 λ)∂ ∂ , µν − − µ ν and may be calculated by taking the Fourier transform,

d4k 1 λ k k e−ik(x−y) G (x y) = i η + − µ ν (2.15) µν − (2π)d µν λ k2 + iǫ k2 + iǫ Z As a result, the ratio,

Z[J] 1 = exp ddx ddyJ µ(x)G (x y)J ν(y) (2.16) Z[0] 2 µν − Z Z is independent of λ provided J µ is conserved. In particular when any gauge independent quantity is evaluated, the dependence on λ will disappear. When λ one recovers Landau gauge, which is manifestly transverse; when λ 1, one recovers→ ∞ another very → convenient Feynman gauge.

2.2 Functional Integral for Non-Abelian Gauge Fields We consider a Yang-Mills theory based on a compact gauge group G with structure con- stants f abc, a, b, c = 1, , dimG. As discussed previously, if G is a direct product of r simple and s U(1) factors,··· then there will be r + s independent coupling constants. Here, we shall deal with only one simple, compact, non-Abelian group G. The basic ﬁeld in the a theory is the Yang-Mills ﬁeld Aµ. The starting point is the classical action,

1 S[A; J] = ddxF a F aµν ddxJ aµAa −4g2 µν − µ Z Z F a = ∂ Aa ∂ Aa f abcAb Ac (2.17) µν µ ν − ν µ − µ ν where summation over repeated gauge indices a is assumed. µ For this action to be gauge invariant, the current must be covariantly conserved DµJ = µ 0, which may render J dependent on the ﬁeld Aµ itself. This complicates the discussion, as compared to the Abelian case. Actually, we shall make one of two possible assumptions.

24 1. The current J µ arises from the coupling of other ﬁelds (such as fermions or scalars) in a complete theory that is fully gauge invariant.

2. The current J µ is an external source, but will be used to produce correlation functions µν of gauge invariant operators, such as trFµνF .

In both cases, the full theory will really be gauge invariant.

2.3 Canonical Quantization

Canonical quantization is best carried out in temporal gauge with A0 = 0. To choose this gauge, the equation ∂0U = igUA0 must be solved, which may always be done in terms of the time-ordered exponential,

t ′ ′ U(t, x)= T exp ig dt A0(t , x) U(t0, x) (2.18) t0 n Z o In ﬂat space-time with no boundary conditions in the time direction, this solution always exists. If time is to be period (as in the case of problems in statistical mechanics at ﬁnite temperature) then one cannot completely set A0 = 0. It is customary to introduce Yang-Mills electric and magnetic ﬁelds by 1 Ba ǫ F a i ≡ 2 ijk jk 1 1 Ea F a = ∂ Aa (2.19) i ≡ g2 0i g2 0 i

a a and Ei is the canonical momentum conjugate to Ai . The Hamiltonian and Gauss’ law are given by,

g2 1 H[E, A; J] = dd−1x EaEa + BaBa J aAa 2 i i 2g2 i i − i i Z Ga(E, A) = ∂ Ea f abcAbEc (2.20) i i − i i The canonical commutation relations are

[Ea(x), Ab(y)]δ(x0 y0)= iδabδ δ(d)(x y) (2.21) i j − − ij − The operator Ga commutes with the Hamiltonian (for a covariantly conserved J µ). The action of Ga(E, A) = Ga(x) on the ﬁelds is by gauge transformation, as may be seen by taking the relevant commutator,

[Ga(x), Ab(y)]δ(x0 y0)= ∂ δ(4)(x y)δab f abcAc(x)δ(d)(x y) (2.22) i − i − − i − 25 To clarify we integrating Ga against an inﬁnitesimal gauge transformation ωa,

d−1 a Gω = d xG (x)ωa(x) (2.23) Z in terms of which the above gauge transformation formula takes the form,

b b [Gω, Ai (y)] = (Diω) (y) (2.24) Gauge transformations form a Lie algebra as usual,

[Ga(x),Gb(y)]δ(x0 y0)= if abcGc(x)δ(d)(x y) (2.25) − − Thus, the operator Ga generates inﬁnitesimal gauge transformations. As in the case of the Abelian gauge ﬁeld, Gauss’s law cannot be imposed as an operator equation because it would contradict the canonical commutation relations between Ai and Ei. Thus, it is a constraint that should be imposed as an initial condition. In quantum ﬁeld theory, it must be imposed on the physical states. The Fock and physical Hilbert spaces are deﬁned in turn by

Aa(x) F ≡ | i i n o ψ such that Ga(E, A) ψ =0 (2.26) Hphys ≡ | i∈F | i n o It is interesting to note the physical signiﬁcance of Gauss’s law.

2.4 Functional Integral Quantization This proceeds very similarly to the case of the quantization of the Abelian gauge ﬁeld. The starting points are the Hamiltonian in A0 = 0 gauge and the constraint of Gauss’s law which commuted with the Hamiltonian. We construct the evolution operator projected onto the physical Hilbert space by repeated (and redundant) insertions of the projection operator. Recall the deﬁnition and functional integral representation of this projector,

ψ ψ P ≡ | ih | |ψi∈HXphys 1 = Λa(x) exp i dd−1xΛa(x) D Ea J a) (2.27) Λa D i i − 0 D Z n Z o As in the case of the AbelianR gauge ﬁelds, we construct the evolution operator projected onto the physical Hilbert space by repeated insertions of , P N −itH −i t H e = lim e N (2.28) phys N→∞ P

26 Its functional integral representation is obtained in analogy with the Abelian case,

−itH 1 a a a e = Λ Ai Ei phys Λa D D D D Z Z Z t g2 1 R ′ d−1 a a a a a a exp i dt d x Ei ∂0Ai Ei Ei 2 Bi Bi × 0 − 2 − 2g n Z Z +AaJ a +Λa(D Ea J a) i i i i − 0 a o together with the customary boundary conditions on the ﬁeld Ai and free boundary con- a ditions on the canonically conjugate momentum Ei . Next, we recall that Gauss’s law resulted from the variation of the non-dynamical ﬁeld a A0, which had been set to 0 in this gauge. But here, Λ plays again the role of a non- dynamical Lagrange multiplier and we rebaptize it as follows, Λa(x) Aa(x) (2.29) → 0 a The integration over the non-dynamical canonical momentum Ei may be carried out in an algebraic way by completing the square of the Gaussian, g2 1 Ea∂ Aa EaEa BaBa + AaJ a D AaEa AaJ a (2.30) i 0 i − 2 i i − 2g2 i i i i − i 0 i − 0 0 1 1 1 = (g2Ea ∂ Aa + D A )2 + (∂ Aa D A )2 BaBa Aa J aµ −2g2 i − 0 i i 0 2g2 0 i − i 0 − 2g2 i i − µ a a The integration measure over Ei is invariant under (functional) shifts in Ei , and we use D a this property to realize that the remaining Gaussian integral over Ei is independent of a a Aµ and Jµ, and therefor produces a constant. The remainder is easily recognized as the Lagrangian density with source that was used as a starting point. As a concrete formula, we record the matrix element of the evolution operator taken between (non-physical) states A and A in the Fock space, | 1i | 2i t −itH a ′ d−1 1 a aµν a aµ A2 e A1 = Aµ exp i dt d x 2 Fµν F AµJ (2.31) h | | i N D 0 −4g − Z Z Z with the following boundary conditions,

A(0, x) = A1(x)

A(t, x) = A2(x) (2.32) Time ordered correlation functions are deduced as usual and may be summarized by the following generating functional, 1 eiW [J] = Aa eiS[A,J] S[A, J]= ddx F a F aµν J aµAa (2.33) D µ −4g2 µν − µ Z Z Of course, this integral is formal, since gauge invariance leads to the integration of equivalent a copies of Aµ. Therefore, we need to gauge ﬁx, a process outlined in the next subsection.

27 2.5 The redundancy of integrating over gauge copies We wish to make sense out of the functional integral of any gauge invariant combination of operators which we shall collectively denote by . For example, the following operators are gauge invariant, O (x)= F a F aµν (x) O+ µν (x)= F a F˜aµν (x) (2.34) O− µν An inﬁnite family of operators may be formed out of a time ordered product of a string O of and operators, O+ O− = (x ) (x ) (y ) (y ) (2.35) O O+ 1 ···O+ m O− 1 ···O+ n We have the general formula, Aa eiS[A] 0 T 0 = D µ O (2.36) h | O| i Aa eiS[A] R D µ where the action is evaluated at vanishingR current, S[A] = S[A, 0], and is therefore gauge invariant. The key characteristic of this formula is that is a gauge invariant operator. a O The measure Aµ is also gauge invariant. The way to see this is to notice that a measure results fromD a metric on the space of all gauge ﬁelds. Once this metric has been found and properly deﬁned, the measure will follow. The metric may be taken to be given by the following norm,

δAa 2 d4x δAa δAaµ (2.37) k µk ≡ µ Z a The variation δAµ of a gauge ﬁeld transforms homogeneously and thus the metric is invari- ant. Therefore, we can construct an invariant measure. Thus, we realize now that all pieces of the functional integral for 0 0 are invariant under local gauge transformations. But this renders the numerator andh |O| thei denominator inﬁnite, since in each the integration extends over all gauge copies of each gauge ﬁeld. However, since the integrations are over gauge invariant integrands, the full integrations will factor out a volume factor of the space of all gauge transformations times the integral over the quotient of all gauge ﬁelds by all gauge transformations. Concretely, = all Aa (x) A { µ } = all Λa(x) (2.38) G { } Thus we have symbolically,

Aa eiS[A] = Vol( ) Aa eiS[A] (2.39) D µ O G D µ O ZA ZA/G 28 and in particular, the volume of the space of all gauge transformations cancels out in the formula for 0 0 , h |O| i a iS[A] a iS[A] A e Aµ e 0 0 = A D µ O = A/G D O (2.40) h |O| i Aa eiS[A] Aa eiS[A] R A D µ R A/G D µ The second formula does not haveR the inﬁnite redundancyR integrating over all gauge trans- formations of any gauge ﬁeld. In the case of the Abelian gauge theories, the problem was manifested through the µν fact that the longitudinal part of the gauge ﬁeld does not couple to the Fµν F term in the action, a fact that renders the kinetic operator non-invertible. A gauge ﬁxing term µ 2 λ(∂µA ) was added; its eﬀect on the physical dynamics of the theory is none since the current is conserved. For the non-Abelian theory, things are not as simple. First of all there a aµν are now interaction terms in the Fµν F term, but also, the current is only covariantly conserved. The methods used for the Abelian case do not generalize to the non-Abelian case and we have to invoke more powerful methods.

2.6 The Faddeev-Popov gauge ﬁxing procedure Geometrically, the problem may be understood by considering the space of all gauge ﬁelds and studying the action on this space of the gauge transformations , as depicted in A G Fig 1. The meaning of the ﬁgures is as follows (a) The space of all gauge ﬁelds is schematically represented as a square. The • A action of the group of gauge transformations on is indicated by the thin slanted lines with arrows. (Mathematically, is the spaceA of all connections in a ﬁber bundle A with structure group G over R4.) (b) There is no canonical way of representing the coset / . Therefore, one must • choose a gauge slice, represented by the fat line . To avoidA G degeneracies, the gauge S slice should be taken to be transverse to the orbits of the gauge group . (Mathe- matically, the gauge slice is a section of the ﬁber bundle.) G (c) Transversality guarantees – at least locally – that every point A′ in gauge ﬁeld • space is the gauge transformation of a point A on the gauge slice . A S (d) It is a tricky problem whether – globally – every gauge orbit intersects the gauge • slice just once or more than once. In physics, this problem is usually referred to as the Gribov ambiguity. (Mathematically, it can be proven that interesting slices produce multiple intersections.) Fortunately, in perturbation theory, the ﬁelds remain small and the Gribov copies will be inﬁnitely far away in ﬁeld space and may thus be ignored. Non-perturbatively, one would have to deal with this issue; fortunately, the best method is lattice gauge theory, where one never is forced to gauge ﬁx.

29 G G

S

(a) A (b) A

GG

A’

A S

S

(c) A (d) A

Figure 1: Each square represents the space of all gauge ﬁeld; each slanted thin line represents an orbit of ; each fat line representsA a gauge slice . Every gauge ﬁeld A′ may G S be brought back to the slice by a gauge transformation, shown in (c). A gauge slice may S intersect a given orbit several times, shown in (d) – this is the Gribov ambiguity. fig:14 { }

30 The choice of gauge slice is quite arbitrary. To remain within the context of QFT’s governed by local Poincar´einvariant dynamics, we shall most frequently choose a gauge compatible with these assumptions. Speciﬁcally, we choose the gauge slice to be given by a local equation a(A)(x) = 0, where (A) involves A and derivatives ofS A in a local manner. Possible choicesF are F

a(A)= ∂ Aaµ a = Aa (2.41) F µ F 0 But many other choices could be used as well. To reduce the functional integral to an integral along the gauge slice, we need to factorize a the measure Aµ into a measure along the gauge slice times a measure along the orbits of the gauge transformationsD . There is a nice gauge-invariantS measure g on resulting from the following gauge invariantG metric on , D G G δg 2 ddx tr g−1δg g−1δg (2.42) k k ≡ − Z If the orbits were everywhere orthogonal to , then the factorization would be easy G S to carry out. In general it is not possible to make such a choice for and a non-trivial Jacobian factor will emerge. To compute it, we make use of the Faddeev-PopovS trick.

Denote a gauge transformation g of a gauge ﬁeld Aµ by

g −1 −1 Aµ = gAµg + i(∂µg)g (2.43)

Notice that the action of gauge transformations on Aµ satisﬁes the group composition law of gauge transformations,

h g −1 −1 −1 −1 (Aµ) = g hAµh + i(∂µh)h g + i(∂µg)g −1 −1 gh = (gh)Aµ(gh) + i∂µ(gh)(gh) = Aµ (2.44) which will be useful later on. Let a(A) denote the loval gauge slice function, the gauge slice being determined by a(A) =F 0. The quantity a(Ag) will vanish when Ag belongs to , and will be non-zero F F S otherwise. We deﬁne the quantity [A] by J 1= [A] g δ a(Ag)(x) (2.45) J D F x Z Y The functional δ-function looks a bit menacing, but should be understood as a Fourier integral,

δ a(Ag)(x) Λa exp i ddxΛa(x) a(Ag)(x) (2.46) F ≡ D F x Y Z n Z o 31 g h gh Using the composition law (Aµ) = Aµ and the gauge invariance of the measure g, it follows that is gauge invariant. This is shown as follows, D J [Ah]−1 = g δ a((Ah)g)(x) J D F x Z Y = g δ a(Agh)(x) D F x Z Y ′ = (g′h−1) δ a(Ag )(x) D F x Z Y = [A]−1 (2.47) J The deﬁnition of the Jacobian readily allows for a factorization of the measure of J A into a measure over the splice times a measure over the gauge orbits . This is done as follows, S G

A eiS[A] = A g a(Ag)(x) [A] eiS[A] D µ O D µ D F J O A A G x Z Z Z Y = g A a(AU )(x) [A] eiS[A] D D µ F J O G A x Z Z Y = g A a(A)(x) [A] eiS[A] (2.48) D D µ F J O G A x Z Z Y In passing from the ﬁrst to the second line, we have simply inverted the orders of integration over and ; from the second to the third, we have used the gauge invariance of the A G measure Aµ, of and S[A]; from the third to the fourth, we have used the fact that the functionalD δ-functionO restricts the integration over all of to one just over the slice . A S Notice that the integration over all gauge transformation function U is independent of A and is an irrelevant constant. D R The gauge ﬁxed formula is independent of the gauge function . F Therefore, we end up with the following ﬁnal formula for the gauge-ﬁxed functional integral for Abelian or non-Abelian gauge ﬁelds,

a iS[A] A Aµ x (A)(x) [A] e 0 0 = D F J O (2.49) h |O| i R AQ a(A)(x) [A] eiS[A] A D µ x F J It remains to compute and toR deal withQ the functional δ-function. J 2.7 Calculation of the Faddeev-Popov determinant At ﬁrst sight, the Jacobian would appear to be given by a very complicated functional integral over U. The key observationJ that makes a simple evaluation of possible is that J 32 a(A) only vanishes (i.e. its functional δ-function makes a non-vanishing contribution) on Fthe gauge slice. To compute the Jacobian, it suﬃces to linearlize the action of the gauge transformations around the gauge slice. Therefore, we have

g = I + iωaT a + (ω2) O Ag = A D ω + (ω2) µ µ − µ O D ωa = ∂ ωa f abcAb ωc (2.50) µ µ − µ and the gauge function also linearizes,

a(Ag) = a(A D ω)(x)+ (ω2) F F µ − µ O δ a(A)(x) = a(A)(x) ddy F D ωb(y)+ (ω2) F − δAb (y) µ O Z µ = a(A)(x)+ ddy ab(x, y)ωb(y)+ (ω2) (2.51) F M O Z where the operator is deﬁned by M δ a(A)(x) ab(x, y) Dy F (2.52) M ≡ µ δAb (y) µ Now the calculation of becomes feasible. J [A]−1 = U δ a(AU )(x) J D F G x Z Y = ωa δ a(A)(x)+ ddy ab(x, y)ωb(y) D F M x Z Y Z = (Det )−1 (2.53) M Inverting this relation, we have δ (A)(x) [A] = Det = Det Dy F (2.54) J M µ δA (y) µ Given a gauge slice function , this is now a quite explicit formula. In QFT, however, we were always dealing with localF actions, local operators and these properties are obscured in the above determinant formula.

2.8 Faddeev-Popov ghosts We encountered a determinant expression in the numerator once before when delaing with intgerations over fermions. This suggests that the Faddeev-Popov determinant may also

33 be cast in the form of a functional intgeration over two complex conjugate Grassmann odd ﬁelds, which are usually referred to as the Faddeev-Popov ghost ﬁelds and are denoted by ca(x) andc ¯a(x). The expression for the determinant is then given by a functional integral over the ghost ﬁelds,

Det = ca c¯a exp iS [A; c, c¯] M D D ghost Z n o S [A; c, c¯] ddx d4y c¯a(x) ab(x, y)cb(y) (2.55) ghost ≡ M Z Z ab If the function is a local function of Aµ, then the support of (x, y) is at x = y and the above doubleF integral collapses to a single local integral involvingM the ghost ﬁelds.

2.9 Axial and covariant gauges First of all, there is a class of axial gauges where no ghosts are ever needed ! The gauge a function must be linear in Aµ and involve no derivatives on Aµ. They involve a constant vector nµ, and are therefore never Lorentz covariant, a(A)(x)= Aa (x)nµ, F µ a δF (A)(x) ab (d) µ b = δ δ (x y)n δAµ(y) −

y δF (A)(x) y (d) µ Dµ = ∂µδ (x y)n = (x, y) (2.56) δAµ(y) − M

Therefore, in axial gauges, the operator is independent of A and thus a constant which may be ignored. Thus, no ghost are requiredM as the Faddeev-Popov formula would just inform us that they decouple. Actually, since axial gauges are not Lorentz covariant, they are much less useful in practice than it might appear. It turns out that Lorentz covariant gauges are much more useful. The simplest such gauges are speciﬁed by a function f a(x) as follows,

a(A)(x) ∂ Aaµ(x) f a(x) (2.57) F ≡ µ − This choice gives rise to the following functional integral,

Z = A δ(∂ Aaµ f a) Det eiS[A] (2.58) O D µ µ − M O A x Z Y Since this functional integral is independent of the choice of f a, we may actually integrate over all f a with any weight factor we wish. A convenient choice is to take the weight factor

34 to be Gaussian in f a,

a i d a a aµ a iS[A] f exp 2 d xf f A Aµ x δ(∂µA f ) Det e Z = D − D − M O (2.59) O n o R R f a Rexp iQ ddxf af a D − 2 R n R o The denominator is just again a Aµ-independent constant which may be omitted. The f a-intgeral in the numerator may be carried out since the functional δ-function instructs a aµ us to set f = ∂µA , so that

a a ZO = Aµ Dc c¯ exp iS[A]+ iSgf [A]+ iSghost[A; c, c¯] (2.60) A D D O Z Z Z n o where the gauge ﬁxing terms in the action are given by 1 S [A]= ddx (∂ Aaµ)2 (2.61) gf −2ξ µ Z n o We obtain a more explicit form for the ghost action by computing , M ν ab ab y δ∂x Aν(x) (x, y) = Dµ M δAµ(y) = (Dy)ab∂µδ(4)(x y) (2.62) µ x − Inserting this operator into the expression for the ghost action, we obtain,

S [A; c, c¯] ddx ddy c¯a(x) ab(x, y)cb(y) ghost ≡ M Z Z = ddx ddyc¯(x)Dy ∂µδ(x y)c(y) µ x − Z Z d a a = d x∂µc¯ (x)(Dµc) (2.63) Z Thus, the total action for these gauges is given as follows, 1 1 S [A; c, c¯]= ddx F a F aµν (∂ Aaµ)2 + ∂ c¯a(Dµc)a (2.64) tot −4 µν − 2ξ µ µ Z and we have the following general formula for vacuum expectation values of gauge invariant operators,

A c c¯ eiStot[A;c,c¯] 0 0 = D µ D D O (2.65) h |O| i A c c¯ eiStot[A;c,c¯] R D µR D D In the section on perturbation theory,R we shallR show how to use this formula in practice.

35 3 Perturbative Non-Abelian Gauge Theories

In this section, the Feynman rules will be derived for a gauge theory based on a simple group G, governed by a single gauge coupling constant g. To carry out perturbation theory, we rescale the gauge ﬁeld.

A gA (3.1) µ → µ The pure gauge action, and the gauge ﬁxing terms and Faddeev-Popov ghost contribution may be expressed in terms of the fundamental ﬁelds as follows, 1 1 = (∂ Aa ∂ Aa )∂µAνa + g(∂ Aa ∂ Aa )f abcAµbAνc Lgauge −2 µ ν − ν µ 2 µ ν − ν µ 1 g2f abcf adeAb Ac AµdAνe −4 µ ν λ = ∂ Aµa∂ Aνa + ∂ c¯a∂µca gf abc∂ c¯aAµbcc (3.2) Lghost − 2 µ ν µ − µ Fermions ψ and (complex) bosons φ, minimally coupled to the gauge ﬁeld in representations Tψ and Tφ respectively, may also be included. The corresponding actions are = ψ¯i i∂/δ j gA/a(T a) j m δ j ψ (3.3) Lψ i − ψ i − ψ i j = ∂ φ∗i∂µφ + igAa ∂µφ∗i(T a) jφ φ∗i(T a) j∂µφ Lφ µ i µ φ i j − φ i j g2Aa Aµbφ∗i(T a) j(T b) kφ V (φ) − µ φ i φ j k − abc a a Here, f are the structure constants of the Lie algebra of G and Tφ and Tψ are the representation matrices corresponding to the representations Tφ and Tψ of G under which φ and ψ transform. They obey the structure relations,

a b abc c [Tφ , Tφ] = if Tφ a b abc c [Tψ , Tψ] = if Tψ (3.4) Finally, V (φ) is a potential which we shall assume to be bounded from below. In this section, we shall specialize to developing perturbation theory in a so-called unbroken phase, characterized by the fact that the minimum of the potential we choose to expand about is symmetric under the group G. (The spontaneously broken, or Higgs phase will be discussed in later sections.) By a suitable shift in φi by constants, this minimum may be chosen to be at φi = 0, so that ∂V ∂V (0) = ∗i (0) = 0 (3.5) ∂φi ∂φ Furthermore, by adding an unobservable constant shift to the Lagrangian, the potential at the minimum may be chosen to vanish, V (0) = 0. Invariance of the minimum under G

36 ﬁnally requires that the double derivative of the potential at the minimum be proportional to the identity,

2 ∂ V 2 j ∗i (0) = mφ δi (3.6) ∂φ ∂φj

The parameter mφ is the mass of the scalar in this unbroken phase.

3.1 Feynman rules for Non-Abelian gauge theories Propagators are given by,

i k k 1 gauge = − δ η ξ µ ν ξ =1 k2 + iǫ ab µν − k2 + iǫ − λ i ghost = δ k2 + iǫ ab i fermion = δ k/ m + iǫ ij − ψ αβ i scalar = δ (3.7) k2 m2 + iǫ ij − φ The vertices are given by,

gauge cubic = gf abc(2π)4δ(p + q + r) η (p q) + η (q r) + η (r p) µν − ρ νρ − µ ρµ − ν h i gaugequartic = ig2(2π)4δ(p + q + r + s) f eabf ecd(η η η η ) − µρ νσ − µσ νρ h+f eacf edb(η η η η ) µσ ρν − µν ρσ +f eadf ebc(η η η η ) µν νρ − µρ σν i ghost = g(2π)4δ(p q + k)f abcp − − µ

fermion = ig(2π)4δ(p q k)(γ ) (T a) − − − µ αβ ψ ij

scalar cubic = g(2π)4δ(p q k)(T a) (p + q ) − − φ ij µ µ

scalar quartic = ig2(2π)4δ(p + q + r + s)η T a, T b (3.8) − µν { φ φ}ij

37 k µ ν gauge propagator a b

k ghost propagator a b

k fermion propagator j i

k complex scalar propagator j i

Figure 2: Feynman rules : propagators fig:C3 { }

38 µ a

p gauge cubic q

r ρ b ν c

µ a σ d p s gauge quartic q r ρ ν b c

µ a k ghost cubic

q p

µ a

k fermion - complex scalar cubic

p' p

µ a ν b

k k' complex scalar quartic p p' i j

Figure 3: Feynman rules : vertices fig:C4 { }

39 3.2 Renormalization, gauge invariance and BRST symmetry The point of departure for the quantization and renormalization of non-Abelian gauge theories is a gauge invariant Lagrangian, such as in the pure Yang-Mills case (recall that we have rescaled A gA ), µ → µ 1 = F a F µνa (3.9) Lgauge −4 µν To renormalize this theory, one might be tempted to guess that only gauge-invariant coun- terterms will be required. The existence of gauge invariant regulators (such as dimensional regularization) would seem to further justify this guess. Perturbative quantization, how- ever, forces us add a gauge ﬁxing term, which in turn requires a Faddeev-Popov ghost, λ = ∂ Aµa∂ Aνa + ∂ c¯aDµca (3.10) Lghost − 2 µ ν µ Neither of these contributions is gauge invariant. As soon as some terms in the Lagrangian fail to be invariant, it is no longer viable to use gauge invariance to restrict the form of the counterterms. Luckily, the gauge and ghost Lagrangians actually exhibit a more subtle symmetry, referred to as BRST symmetry (after Becchi, Rouet, Stora; Tuytin 1975). This symmetry will eﬀectively play the role of gauge invariance in the gauge ﬁxed theory. To begin, the ghost ﬁelds are decomposed into real and imaginary parts, 1 1 ca = (ρa + iσa)c ¯a = (ρa iσa) (3.11) √2 √2 − Both ρa and σa are real odd Grassmann-valued Lorentz scalars. In terms of these real ﬁelds, the Lagrangian becomes λ = ∂ Aµa∂ Aνa + i∂ ρaDµσa (3.12) Lghost − 2 µ ν µ We now introduce a real odd Grassmann-valued constant ω. The product ωσa is then a real even Grassmann-valued Lorentz scalar function, and it is this combination that will be used to make inﬁnitesimal gauge transformations on the non-ghost ﬁelds in the theory,

δAa = (D (ωσ))a µ − µ a a δψ = ig ωσ Tψ ψ a a δφ = ig ωσ Tφ φ (3.13) One refers to these as ﬁeld-dependent gauge transformations . Clearly, the gauge part of the Lagrangian as well as the fermion and boson parts and are invariant. Lgauge Lψ Lφ 40 Remarkably, local transformation laws for the ghost ﬁelds may be found so that also the ghost part of the Lagrangian is invariant, Lghost δρa = iλ ω ∂µAa − µ 1 δσa = gωf abcσbσc (3.14) −2

To establish invariance of ghost, we ﬁrst establish invariance of Dµσ, under the transfor- mation laws given above, L δ (D σ)a = ∂ δσa gf abc(δAb σc + Ab δσc) µ µ − µ µ 1 = gωf abc∂ σbσc + gf abcAb gωf cdeσdσe − µ 2 µ +gf abc ∂ ωσb gf bdeAd ωσe σc (3.15) µ − µ b The derivative terms in ∂µσ manifestly cancel, and the remaining terms may be regrouped as follows, 1 δ (D σ)a = g2Ab ωσdσe f abcf cde + f acdf cbe f acef cbd = 0 (3.16) µ 2 µ − This expression vanishes by the Jacobi identity satisﬁed by the structure constants. Using the invariance of Dµσ, is now straightforward to complete the proof of invariance of ghost, and we have L µ a νa µ a a δ ghost = λ∂ δAµ(∂νA )+ i∂ δρ (Dµσ) L − µ a νa µ νa a = λ∂ (Dµωσ) (∂ν A )+ λ∂ (ω∂νA )(Dµσ) µ νa a = ∂ λ(∂ν A )(Dµσ) (3.17) Since the Lagrangian transforms with a total derivative, the BRST transformation is actu- ally a symmetry.

3.3 Slavnov-Taylor identities Assuming that BRST symmetry is preserved under renormalization, will imply that the F 2 terms in the Lagrangian continue to appear in a gauge invariant combination, and that the only ghost contributions of the Lagrangian are of the form given in ghost. Thus, the form of the bare Lagrangian will be L 1 λ = (∂ Aa ∂ Aa )∂µAνa 0 ∂ Aµa∂ Aνa Lb −2 µ 0ν − ν 0µ 0 − 2 µ 0 ν 0 1 1 + g (∂ Aa ∂ Aa )f abcAµbAνc g2f abcf adeAb Ac AµdAνe 2 0 µ 0ν − ν 0µ 0 0 − 4 0 0µ 0ν 0 0 +∂ c¯a∂µca g f abc∂ c¯aAµbcc µ 0 0 − 0 µ 0 0 0 +ψ¯ i∂/ g A/aT a m ψ (3.18) 0 − 0 0 ψ − 0 0 41 Here, we have included the contribution of a fermion ﬁeld, but omitted scalars. Multiplica- tive renormalization of all the ﬁelds yields the following relation between the bare ﬁelds and the renormalized ﬁelds,

1 1 1 a 2 a a ˜ 2 a 2 A0µ = Z3 Aµ c0 = Z3 c ψ0 = Z2 ψ (3.19)

The vertices are also given independent Z-factors, so that the starting point for the bare Lagrangian, expressed in renormalized ﬁelds, is given by 1 1 = Z (∂ Aa ∂ Aa )∂µAνa λZ ∂ Aµa∂ Aνa Lb −2 3 µ ν − ν µ − 2 λ µ ν 1 1 + gZˆ (∂ Aa ∂ Aa )f abcAµbAνc g2Z f abcf adeAb Ac AµdAνe 2 1 µ ν − ν µ − 4 4 µ ν +Z˜ ∂ c¯a∂µca gZ˜ f abc∂ c¯aAµbcc 3 µ − 1 µ +Z ψ¯ i∂/ψ gZ ψ¯ A/aT aψ mZ ψψ¯ (3.20) 2 − 1 ψ − m The identiﬁcation between the two formulations of the bare Lagrangian give the following relations,

ˆ −3/2 2 2 −2 g0 = gZ1 Z3 g0 = g Z4Z3 ˜ ˜−1 −1/2 −1 −1 g0 = gZ1 Z3 Z3 g0 = gZ1Z2 Z3 (3.21)

Elimination of g and g0 gives the Slavnov-Taylor identities,

ˆ2 ˜ ˜ ˆ −1 ˆ −1 −1 Z1 = Z3Z4 Z1Z3 = Z1Z3 Z1Z3 = Z1Z2 (3.22)

A gauge/BRST-invariant regularization/renormalization will preserve these identities.

42 4 Anomalies

In quantum ﬁeld theory, an anomaly stands for the phenomenon where a symmetry of the classical theory is inconsistent with quantization. Anomalies were encountered early on in the development of quantum ﬁeld theory, though they were not initially recognized as such. In modern quantum ﬁeld theory, anomalies play a dominant role. Mathematically, anomalies originate from summations and integrations that fail to be absolutely conver- gent, but are conditionally convergent and therefore depend on the way they are deﬁned. From this point of view, the ﬁrst encounters of anomalies in math may be traced back to Eisenstein in the 19-th century. In modern mathematics, anomalies are intimately related to problems of cohomology and Atiyah-Singer index theory. In this section, we shall discuss two concrete examples of anomalies in Poincar´einvariant quantum ﬁeld theories, one for scaling symmetry, and the other for chiral symmetry and chiral gauge symmetry.

4.1 Massless 4d QED, Gell-Mann–Low Renormalization

Recall the form of the vacuum polarization tensor Πµν in QED, deﬁned with electron mass m, electron charge e, and Pauli-Villars regulator M. Current conservation requires it to be transverse, so that its general tensorial form may be reduced to,

Π (k, m, M) = (k2η k k )Π(k, m, M) (4.1) µν µν − µ ν The one-loop contribution is well-known, and given by,

e2 1 m2 α(1 α)k2 Π(k, m, M) = 1 dαα(1 α) ln − − + (e4) (4.2) − 2π2 − M 2 O Z0 When the electron mass is non-zero, it is natural (though not necessary) to renormalize at 2 vanishing photon momentum, and to require ΠR(0, m) = 1, so that we have up to order e ,

Π (k, m)=1+Π(k, m, M) Π(0, m, M) (4.3) R −

The function ΠR then takes the form, e2 1 m2 α(1 α)k2 Π (k, m) = 1 dαα(1 α) ln − − + (e4) (4.4) R − 2π2 − m2 O Z0 However, as we let m tend to zero, the quantity Π(0, m, M) diverges, and this renormaliza- tion scheme leads to a infrared divergent result. This led Gell-Mann and Low to renormalize “oﬀ-shell” at a mass-scale µ that has no direct relation with the electron mass (or with photon mass which we have assume to be zero here anyway),

ΠR(k,m,µ) = 1 (4.5) k2=−µ2

43 so that,

e2 1 m2 α(1 α)k2 Π (k,m,µ)=1 dαα(1 α) ln − − + (e4) (4.6) R − 2π2 − m2 + α(1 α)µ2 O Z0 − Now, we may safely let m 0, → e2 k2 Π (k, 0,µ)=1 ln − + (e4) (4.7) R − 12π2 µ2 O The cost of renormalizing the massless theory is the introduction of a new scale µ, which was not originally present in the theory.

4.2 Scale Invariance of Massless QED But now think for a moment about what this means in terms of scaling symmetry. When m = 0, classical electrodynamics is scale invariant, 1 ξ = F F µν + (∂ Aµ)2 + ψ¯(i∂/ eA/)ψ (4.8) L −4 µν 2 µ − A scale transformation xµ x′µ = λxµ acts on a general ﬁeld φ (x) of dimension ∆ by, → ∆

′ ′ −∆ Aµ : ∆=1 φ∆(x) φ∆(x )= λ φ∆(x) (4.9) → ψ : ∆= 3 2 The classical theory is invariant since the eﬀects of scale transformations on the various terms in the Lagrangian are as follows,

F F µν(x) λ−4 F F µν(x) µν → µν (∂ Aµ)2 λ−4 (∂ Aµ)2 µ → µ ψ¯(i∂/ eA/)ψ λ−4 ψ¯(i∂/ eA/)ψ (4.10) − → − where we have assumed that e and ξ do not transform under scale transformations. Thus λ−4 , and the action S = d4x is invariant. Actually, you can show that is invariantL → L under all conformal transformationsL of xµ; inﬁnitesimally δxµ = δλxµ for a scaleL R transformation; δxµ = x2δcµ xµ(δc x) for a special conformal transformation. Thus, classical massless QED is scale− invariant· ; nothing in the theory sets a scale. A fermion mass term would, however, spoil scale invariance since ψψ¯ λ−3ψψ¯ , assum- ing that m is being kept ﬁxed. Similarly, a photon mass would also spoil→ scale invariance. We shall not include such terms here and keep these ﬁelds strictly massless, so that we have exact scale invariance of the action in the classical theory.

44 Now, think afresh of the one-loop renormalization problem: when m = 0, one-loop renormalization forced us to introduce a scale µ in order to carry out renormalization. One cannot quantize QED without introducing a scale, and thereby breaking scale invariance. This eﬀect is sometimes referred to as dimensional transmutation. What we have just identiﬁed is a ﬁeld theory, massless QED, whose classical scale symmetry has been broken by the process of renormalization, i.e. by quantum eﬀects. Whenever a symmetry of the classical Lagrangian is destroyed by quantum eﬀects caused by UV divergences, this classical symmetry is said to be anomalous. In this case, it is the scaling anomaly. In fact, you can think of

4.3 Chiral Symmetry The next example is chiral symmetry. The anomaly that will appear for this symmetry is referred to as the chiral anomaly or also the Adler-Bell-Jackiw (ABJ) anomaly, and was historically in 1969 the ﬁrst instance where one started thinking of anomalies as classical symmetries being destroyed by quantization. We consider again massless QED,

1 = F F µν + ψ¯(i∂/ eA/)ψ (4.11) L −4 µν − Setting m = 0 has enlarged the symmetry of the classical Lagrangian to include conformal invariance. But setting m = 0 also produces another symmetry enhancement: chiral symmetry. To see this, decompose ψ into left and right spinors, in a chiral basis where the chirality matrix γ5 is diagonal,

γ5ψ =+ψ ψ = ψ + ψ L L (4.12) L R γ5ψ = ψ R − R The Lagrangian then becomes,

1 = F F µν + ψ¯ (i∂/ eA/)ψ + ψ¯ (i∂/ eA/)ψ (4.13) L −4 µν L − L R − R

Clearly, ψL and ψR may be rotated by two independent constant phases,

iθL ψL e ψL U(1)L U(1)R → iθR (4.14) × ψR e ψR → and the Lagrangian is invariant. Any symmetry that transforms left and right spinors diﬀerently is referred to as a chiral symmetry. Now, we re-arrange these transformations

45 as follows,

iθ+ ψL e ψL U(1)V : → 2θ+ = θL + θR ψ eiθ+ ψ R → R iθ− ψL e ψL U(1)A : → −θ− 2θ− = θL θR (4.15) ψR e ψR − → or, in 4-component spinor notation

iθ+ U(1)V : ψ e ψ → − (4.16) U(1) : ψ eiγ5θ ψ A → As symmetries, they have associated conserved currents. the vector current jµ ψγ¯ µψ • ≡ the axial vector current jµ ψγ¯ µγ ψ • 5 ≡ 5 Conservation of these currents may be checked directly by using the Dirac ﬁeld equation,

µ µ iγ ∂µψ = eγ Aµψ i∂ ψγ¯ µ = eψγ¯ µA (4.17) − µ µ One then has, ∂ jµ = ∂ ψγ¯ µψ + ψγ¯ µ∂ ψ = ieψγ¯ µA ψ ieψγ¯ µA ψ =0 µ µ µ µ − µ ∂ jµ = ∂ ψγ¯ µγ ψ ψγ¯ γµ∂ ψ = ieψγ¯ µγ A ψ + ieψγ¯ γµA ψ = 0 (4.18) µ 5 µ 5 − 5 µ 5 µ 5 µ 4.4 The axial anomaly in 2-dimensional QED It is very instructive to investigate the fate of chiral symmetry in the much simpler 2- dimensional version of QED, given by the same Lagrangian, 1 = F F µν + ψ¯(i∂/ eA/)ψ (4.19) L −4 µν − Contrarily to 4-dim QED, this theory is not scale invariant, even for vanishing fermion mass, since the dimensions of the various ﬁelds and couplings are given as follows, [e] = 1, [A] = 0, [ψ]=1/2. But it is chiral invariant, just as 4-dim QED was, and we have two classically conserved currents, µ ¯ µ µ ¯ µ j = ψγ ψ j5 = ψγ γ5ψ (4.20) A considerable simpliﬁcation arises from the fact that the Dirac matrices are very simple, 0 1 0 1 1 0 γ0 = σ1 = γ1 = iσ2 = γ5 = σ3 = (4.21) 1 0 1 0 0 1 − − 46 As a result of this simplicity, the axial and vector currents are related to one another. This µ fact may be established by computing the products γ γ5 explicitly,

0 1 3 2 1 γ γ5 = σ σ = iσ = γ =+γ1 − − (4.22) γ1γ = iσ2σ3 = σ1 = γ0 = γ 5 − − − 0 Introducing the rank 2 antisymmetric tensor ǫµν = ǫνµ, normalized to ǫ01 = 1, the above relations may be recast in the following form, −

γµγ = ǫµν γ = jµ = ǫµν j (4.23) vectoraxial2d 5 ν ⇒ 5 ν { Consider the calculation of the vacuum polarization to one-loop, but this time in 2-dim QED. The vacuum polarization diagram is just the correlator of the two vector currents in the free theory, Πµν (k)= e2 0 T jµ(k)jν( k) 0 , and is given by, h | − | i d2p i i Πµν (k)= e2 tr γµ γν (4.24) − (2π)2 k/ + p/ p/ Z A priori, the integral is logarithmically divergent, and must be regularized. We choose to regularize in a manner that preserves gauge invariance and thus conservation of the current jµ. As a result, we have Πµν (k)=(k2ηµν kµkν)Π(k2). Gauge invariant regularization may be achieved using a single Pauli-Villars− regulator with mass M, for example. Just as in 4-dim QED, current conservation lowers the degree of divergence and in 2-dim, vacuum polarization is actually ﬁnite. 1 d2p α(1 α) α(1 α) Π(k2)=4e2 dα − − (4.25) (2π)2 (p2 + α(1 α)k2)2 − (p2 + α(1 α)k2 M 2)2 Z0 Z − − − Carrying out the p-integration, ie2 1 α(1 α) α(1 α) Π(k2)= dα − − (4.26) π α(1 α)k2 − α(1 α)k2 + M 2 Z0 − − − − Clearly, as M , the limit is ﬁnite and the contribution from the Pauli-Villars regulator actually cancels.→∞ One ﬁnds the ﬁnite result, ie2 Π(k2)= (4.27) −πk2 Notice that, even though the ﬁnal result for vacuum polarization is ﬁnite, the Feynman integral was superﬁcially logarithmically divergent and requires regularization. It is during this regularization process that we ensure gauge invariance by insisting on the conservation of the vector current jµ. Later, we shall see that one might have chosen NOT to conserve the vector current, yielding a diﬀerent result.

47 µν µ ν Next, we study the correlator of the axial vector current, Π5 (k)= e 0 j5 (k)j ( k) 0 . µν µν h | − | i Generally, Π5 and Π would be completely independent quantities. In 2-dim, however, the axial current is related to the vector current by (4.23), and we have

eΠµν (k)= ǫµρΠ ν(k)=(ǫµν k2 ǫµρk kν )Π(k2) (4.28) 5 ρ − ρ µν Conservation of the vector current requires that kνΠ5 (k) = 0, which is manifest in view of the transversality of Πµν (k). The divergence of the axial vector current may also be calculated. The contribution from the correlator with a single vector current insertion is ie2 k eΠµν (k)= k ǫµν k2Π(k2)= k ǫµν (4.29) kpi0 µ 5 µ − π µ { } which is manifestly non-vanishing. As a result, the axial vector is not conserved and chiral symmetry in 2-dim QED is not a symmetry at the quantum level : chiral symmetry suﬀers an anomaly. Translating this into an operator statement: e ∂ jµ = F ǫµν (4.30) anomeq µ 5 −2π µν { } Remarkably, this result is exact (Adler-Bardeen theorem). Higher point correlators are convergent and classical conservation valid.

4.5 The axial anomaly and regularization The above calculation of the axial anomaly was based on our explicit knowledge of the vector current two-point function (or vacuum polarization) to one-loop order. Actually, the anomaly equation (4.30) is an exact quantum ﬁeld theory operator equation, which in particular is valid to all orders in perturbation theory. To establish this stronger result, there must be a more direct way to proceed than from the vacuum polarization result. First, consider the issue of conservation of the axial vector current when only the fermion is quantized and the gauge ﬁeld is treated as a classical background. The correlator µ 0 j5 0 A is then given by the sum of all 1-loop diagrams shown in ﬁgure xx. Consider ﬁrst theh | graphs| i with n 2 external A-ﬁelds. Their contribution to 0 jµ 0 is given by ≥ h | 5 | iA d2p 1 1 1 Πµ;ν1···νn (k; k , ,k )= en tr γµγ γν1 γνn (4.31) 5 1 ··· n − (2π)2 5 p/ + ℓ/ ··· p/ + ℓ/ p/ + ℓ/ Z 1 n n+1 where we use the notation l l = k , As we assumed n 2, this integral is UV i+1 − i i ≥ convergent, and may be computed without regularization. In particular, the conservation of the axial vector current may be studied by using the following simple identity, (and the fact that l l = k by overall momentum conservation k + k + + k = 0), n+1 − 1 − 1 ··· n

1 1 ν1 1 1 k/γ5 γ = γ5 + γ5 (4.32) p/ + ℓ/n+1 p/ + ℓ/1 − p/ + ℓ/1 p/ + ℓ/n+1

48 Hence the divergence of the axial vector current correlator becomes, d2p 1 1 k Πµ;ν1···νn (k; k , ,k ) = en tr γ γν1 γνn (4.33) µ 5 1 ··· n (2π)2 5 p/ + ℓ/ ··· p/ + ℓ/ Z 1 n d2p 1 1 1 en tr γ γν2 γνn γν1 − (2π)2 5 p/ + ℓ/ ··· p/ + ℓ/ p/ + ℓ/ Z 2 n n+1 When n 3, each term separately is convergent. Using boson symmetry in the external ≥ A-ﬁeld (i.e. the symmetry under the permutations on i of (ki, νi)), combined with shift symmetry in the integration variable p, all contributions are found to cancel one another. When n = 2, each term separately is logarithmically divergent, and shift symmetry still µ applies. Thus, we have shown that ∂µj5 receives contributions only from terms ﬁrst order in A, which are precisely the ones that we have computed already in the preceding subsection. To see how the contribution that is ﬁrst order in A arises, we need to deal with a contribution that is superﬁcially logarithmically divergent, and requires regularization. One possible regularization scheme is by Pauli-Villars, regulators. The logarithmically divergent integral requires just a single regulator with mass M, and we have, d2p 1 1 1 1 Πµν(k, M)= e tr γµγ γν γν (4.34) 5 − (2π)2 5 p/ + k/ p/ − p/ + k/ M p/ M Z − − The conservation of the vector current, which is a necessary condition to have U(1)V gauge invariance, is readily veriﬁed, using the following identity, 1 1 1 1 1 1 1 1 k/ k/ = + (4.35) p/ + k/ p/ − p/ + k/ M p/ M p/ − p/ + k/ − p/ + k/ M p/ M − − − − and the fact that one may freely shift p in a logarithmically divergent integral. The diver- gence of the axial vector current is investigated using a similar identity, 1 1 1 1 trk/γ γν γν 5 p/ + k/ p/ − p/ + k/ M p/ M − − 1 1 1 1 = trγ γν + 5 p/ + k/ − p/ − p/ + k/ M p/ M − − 1 1 +2Mtr γ γν (4.36) 5 p/ + k/ M p/ M − − The ﬁrst line of the integrand on the rhs may be shifted in the integral and cancels. The second part remains, and we have d2p 1 1 k Πµν (k, M)= 2eM tr γ γν (4.37) kpi1 µ 5 − (2π)2 5 p/ + k/ M p/ M { } Z − − Of course, this integral is superﬁcially UV divergent, because we have used only a single Pauli-Villars regulator. To obtain a manifestly ﬁnite result, at least two PV regulators

49 should be used; it is understood that this has been done. To evaluate (4.37), introduce a single Feynman parameter, α and shift the internal momentum p accordingly,

1 d2p trγ (p/ + (1 α)k/ + M)γν (p/ αk/ + M) k Πµν (k, M)= 2eM dα 5 − − (4.38) kpi2 µ 5 − (2π)2 (p2 M 2 + α(1 α)k2)2 { } Z0 Z − − µ1 µn µ ν µν The trace trγ5γ γ vanishes whenever n is odd, while for n = 2, we have trγ5γ γ =2ǫ . In the numerator· of (4.38), the contributions quadratic in p, those quadratic in k and those quadratic in M cancel because they involve traces with odd n; those linear in p cancel because of p p symmetry of the denominator. This leaves only terms linear in M and → − ν µν linear in k. Finally, using trγ5k/γ =2ǫ kµ and the familiar integral

d2p 1 i 1 = (4.39) (2π)2 (p2 M 2 + α(1 α))2 4π M 2 α(1 α)k2 Z − − − − we have ie 1 M 2 k Πµν (k, M)= ǫµν k dα (4.40) µ 5 − π µ M 2 α(1 α)k2 Z0 − − As M is viewed as a Pauli-Villars regulator, we consider the limit M , which is smooth, →∞ and we ﬁnd,

µν ie µν lim kµΠ5 (k, M)= ǫ kµ (4.41) M→∞ − π which is in precise agreement with (4.29). Clearly, the anomaly contribution arises here solely from the Pauli-Villars regulator. Note that if we restored factors of ~, the right hand side would have exactly one factor of ~, since its eﬀect is purely quantum mechanical and has no classical counterpart.

4.6 The axial anomaly and massive fermions If a fermion has a mass, then chiral symmetry is violated by the mass term. This may be seen directly from the Dirac equation for a massive fermion in QED,

µ µ iγ ∂µψ = mψ + eγ Aµψ i∂ ψγ¯ µ = mψ¯ + eψA¯ γµ (4.42) − µ µ Chiral transformations may still be deﬁned on the fermion ﬁelds and the axial vector current may still be deﬁned by the same expression. For massive fermions, however, chiral transformations are not symmetries any more and the axial vector current fails to be µ conserved. Its divergence is easy to compute classically from the Dirac equation, ∂µj5 = ¯ µ 2imψγ5ψ. Hence the Pauli-Villars regulators violate the conservation of j5 , and this eﬀect

50 survives as M . In fact, one readily establishes the axial anomaly equation for a fermion of mass→m, ∞ e ∂ jµ =2imψγ¯ ψ ǫµν F (4.43) µ 5 5 − 2π µν In fact, it is impossible to ﬁnd a regulator of UV divergences which is gauge invariant and preserves chiral symmetry. Dimensional regularization makes 2 d, but now we have a problem deﬁning γ5 = γ0γ1. →

4.7 Solution to the massless Schwinger Model Two-dimensional QED is a theory of electrically charged fermions, interacting via a static Coulomb potential, with no dynamical photons. The potential is just the Green function 2 G(x, y) for the 1-dim Laplace operator, satisfying ∂xG(x, y)= δ(x, y), and given by 1 G(x, y)= x y (4.44) 2| − | The force due to this potential is constant in space. It would take an inﬁnite amount of energy to separate two charges by an inﬁnite distance, which means that the charges will be conﬁned. Interestingly, this simple model of QED in 2-dim is a model for conﬁnement ! Remarkably, 2-dim QED was completely solved by Schwinger in 1958. To achieve this solution, one proceeds from the equations for the conservation of the vector current, the anomaly of the axial vector current, and Maxwell’s equations,

µ ∂µj = 0 e ∂ jµ = ǫµν ∂ j = ǫµν F µ 5 µ ν −2π µν µν ν ∂µF = ej (4.45)

Of course, it is crucial to the general solution that these equations hold exactly. Notice that this is a system of linear equations, a fact that is at the root for the existence of an exact solution. We recast the axial anomaly equation in the following form, e ∂ j ∂ j = F (4.46) µ ν − ν µ −π µν µ Finally, taking the divergence in ∂µ, using ∂µj = 0 and Maxwell’s equations, we obtain,

e e2 ✷jν = ∂ F µν = jν (4.47) −2π µ − π which is the ﬁeld equation for a free massive boson, with mass m2 = e2/π. Actually, this boson is a single real scalar φ, which may be obtained by solving explicitly the conservation

51 µ µ µν equation ∂µj = 0 in terms of φ by j = ǫ ∂νφ. Notice that the gauge ﬁeld may also be expressed in terms of this scalar, (or the current), by solving the chiral anomaly equation, π π A = j + ∂ θ = ǫ ∂ν φ + ∂ θ (4.48) µ − e µ µ − e µν µ where θ is any gauge transformation. This is just the London equation of superconductivity, expressing the fact that the photon ﬁeld is massive now. This last equation is at the origin of the boson-fermion correspondence in 2-dim. Remarkably, the physics of the (massless) Schwinger model is precisely what we would expect from a theory describing conﬁned fermions. The salient features are as follows,

The asymptotic spectrum contains neither free quarks, nor free gauge particles; • The spectrum contains only fermion bound states, which are gauge neutral; • The spectrum has a mass gap, even though the fermions were massless. • It is also interesting to include a mass to the fermions, but this model is much more complicated, and no longer exactly solvable. Its bosonized counterpart is the massive Sine- Gordon theory.

4.8 The axial anomaly as an operator equation How can we understand that the anomaly equation is an operator equation ? Both the vector and the axial vector currents are composite operators, and they need to be properly deﬁned starting from the canonical ﬁelds in the theory. Generally in the deﬁnition of composite operators, one splits the points at which they are inserted, and then takes the limit. For the axial vector current, a ﬁrst attempt at doing so would be,

µ ¯ µ ¯ µ j5 (x) = lim ψ(y)γ γ5ψ(x) 0 ψ(y)γ γ5ψ(x) 0 (4.49) y→x −h | | i The problem is, however, that as the operators are now inserted at distinct points, the operator deﬁned in this way is no longer gauge invariant, since under a gauge transformation we would have,

ψ(x) eieθ(x)ψ(x) → ψ(y) eieθ(y)ψ(y) (4.50) → To obtain a gauge-invariant operator, we need to modify the regularization used here. One way to do this is by inserting the line integral of the gauge ﬁeld between the points x and

52 y, deﬁned by,

y µ U(y, x) = exp ie Aµ(z)dz (4.51) Zx which transforms as follows under the gauge transformation θ,

U(y, x) exp ie θ(y) ıe θ(x) (4.52) → { − } and deﬁne the gauge-invariant version of the axial current as follows,

µ ¯ µ ¯ µ j5 (x) = lim ψ(y)U(y, x)γ γ5ψ(x) 0 ψ(y)U(y, x)γ γ5ψ(x) 0 (4.53) y→x −h | | i Now, however, the equation for the divergence of the current is modiﬁed due to the insertion of U(y, x) in conjunction with the divergence of the correlator 0 ψ¯(y)γµγ ψ(x) 0 as y x. h | 5 | i → This eﬀect may be calculated explicitly, and precisely reproduces the anomaly equation, yet once again.

4.9 The non-Abelian axial anomaly in 2 dimensions So far, we have dealt only with Abelian symmetries and their possible anomalies. In fact, it is not very diﬃcult to generalize our calculations to the case of non-Abelian symmetries. To begin, let us consider a multiplet of N massless Dirac fermions in 1+1 dimensions. The free Lagrangian is,

N N = ψ¯aiγµ∂ ψa = ψ†ai∂ ψa + ψ†ai∂ ψa L0 µ L + L R − R a=1 a=1 X X n o The manifest classical symmetry of this Lagrangian (but, as we shall see later, not in fact the largest symmetry of ) is U(N) U(N) , under which ψ and ψ each transforms L0 L × R L R with its U(N). This set of fermions may be thought of – in a 4 dimensional analogy – as N ﬂavors of massless quarks, and the SU(N) SU(N) part of the symmetry group would L × R be chiral symmetry. We shall discuss the role of the extra U(1) components later. A simple interaction to introduce is by minimally coupling these fermions to an SU(N)V gauge ﬁeld, where SU(N) is the diagonal subgroup of SU(N) SU(N) . This coupling is explicitly V L × R written as follows (Henceforth we consider a matrix notation for the N components of ψa and group them into a column matrix which we again denote by ψ; also, we absorb the gauge coupling constant into the gauge ﬁeld.)

= ψγ¯ µ(i∂ A )ψ = ψ† (i∂ A )ψ + ψ† (i∂ A )ψ LA µ − µ L + − + L R − − − R N a a a Here, we have also used a matrix notation for Aµ = a=1 AµT and T are the (Hermitian) representation matrices of the Lie algebra of SU(N). Recall that they satisfy the following P 53 structure relations [T a, T b]= if abcT c where the structure constants f abc are real and completely anti-symmetric in all indices.

The SU(N)V gauge current is just the vector current; classical SU(N)L SU(N)R invariance also produces an axial vector current ×

jµa = ψγ¯ µT aψ µa ¯ µ a j5 = ψγ γ5T ψ (4.54)

These currents transform in a non-trivial way under SU(N)V . Thus, in the presence of an external gauge ﬁeld Aµ as we have here, there is no sense to having a conserved current, since this type of equation would not be invariant under SU(N)v. Instead, it is easy to see (using the Dirac equation), that the above currents are covariantly conserved :

(D jµ)a = ∂ jµa f abcAb jµc =0 µ µ − µ (D jµ)a = ∂ jµa f abcAb jµc = 0 (4.55) µ 5 µ 5 − µ 5

The SU(N)V gauge symmetry can be maintained at the quantum level. In fact there is again a multitude of UV regulators that preserve the symmetry such as dimensional regularization, Pauli-Villars regulators, or heat-kernel methods. (You may wish to work out an example or establish the Ward identities !) We shall thus assume that all Green functions µ are regularized and renormalized in an SU(N)V -invariant way, and we have Dµj = 0 as an exact operator equation which may be used in all Green functions. What happens to the (covariant) conservation of the axial vector current upon quantiza- µa tion ? We may ﬁrst look at the contribution that comes from ∂µj5 . The lowest non-trivial ∗ µa νb Green function is 0 T j5 j 0 and the Feynman integral corresponding to it is exactly h | | i a b 1 ab the same as in the Abelian case (with the additional factor of trT T = 2 δ ). Thus, to ﬁrst order in the gauge ﬁeld, we have

e2 ∂ jaµ = ǫµν (∂ Aa ∂ Aa )+ (A2) µ 5 4π µ ν − ν µ O

This equation is of course not SU(N)v gauge invariant (though it is invariant under global SU(N)v !), and we now have to ﬁnd out how to make it gauge invariant. Since the axial vector current transforms as a tensor in the adjoint representation of SU(N)v, we know µ that there must be a gauge invariant formulation of the covariant derivative of j5 . This must of course agree order by order in powers of the gauge ﬁeld, and the lowest non-trivial aµ order is uniquely ﬁxed by the above equation for ∂µj5 . Thus – without doing any further Feynman diagram calculations – we ﬁnd that we must have, e2 (D jµ)a = ǫµν F a (4.56) µ 5 4π µν 54 where F is the ﬁeld strength of the SU(N)V gauge ﬁeld Aµ F a = ∂ Aa ∂ Aa f abcAb Ac (4.57) µν µ ν − ν µ − µ ν The above anomaly equation can again be shown to be exact, i.e. there are no contributions from higher graphs with more insertions of the gauge ﬁeld. To conclude this subsection we make a comment about an often repeated confusion. In the case of the Schwinger model, we had U(1)V gauge invariance and global U(1)A axial symmetry. Both these transformations form groups, as just indicated by the notation. In fact any combination of the two also forms a group. The reason for this is that all these symmetries just act by commuting phase rotations. In the non-Abelian case, this ceases to be the case. The invariance groups are SU(N) SU(N) , and the subgroups are SU(N) , L × R L SU(N)R and the diagonal (vector) subgroup SU(N)V . As soon as N 2 however, the set a ≥ of axial generators γ5T does not form a closed Lie algebra, as can be seen from the fact that their commutator belongs to SU(N)V .

4.10 The Polyakov - Wiegmann calculation of the eﬀective action In the preceding discussion, we have found the non-Abelian generalization of the axial anomaly equations that we studied in the Schwinger model for the Abelian case. In the Schwinger model, the anomaly equations were suﬃcient to compute the entire eﬀective action for fermions in the presence of a gauge ﬁeld. In this subsection, we follow a calcu- lation of Polyakov and Wiegmann and show that also here the complete eﬀective action for the fermions in the presence of an SU(N)V gauge ﬁeld can be computed. The eﬀective action for the fermions in the presence of a background gauge ﬁeld in SU(N)V is deﬁned as follows,

−iW [Aµ] 2 µ µ e = DψDψ¯ exp i dx ψiγ¯ Dµψ = Det(γ Dµ) (4.58) Z n Z o The fundamental object is the vacuum expectation value of the gauge current, in the presence of the gauge ﬁeld Aµ, which is given by

µa ¯ µ a δW [Aµ] j (x)= 0 ψγ T ψ(x) 0= a (4.59) h | | δAµ(x) In view of the two-dimensional identity that we also used in the Abelian case, µa µν a j5 = ǫ jν (4.60) we may eliminate the axial vector current in favor of the vector current, for which we will now have the following two identities, which are exact, µ µ ∂µj + i[Aµ, j ] = 0 1 ǫµν (∂ j + i[A , j ]) = ǫµν F µ ν µ ν 4π µν 55 These equations are much more complicated than in the Abelian case since they are no longer linear. Nevertheless, they can be solved exactly and in closed form. To see this, we work in light-cone coordinates x± =(x0 x1)/√2 for the two-dimensional space-time under consideration. Next, we parametrize the± gauge ﬁeld by two SU(N)-valued functions, g, h, in the following way,

A = ig−1∂ g A = ih−1∂ h + − + − − −

Notice that this looks pure gauge, but is not as long as h = g. By making an SU(N)V 6 −1 gauge transformation by h, we choose a gauge in which A− = 0, and only A+ = ig ∂+g is non-vanishing. Under these conditions, the equations for the vector current become,−

+ − + ∂+j + ∂−j + i[A+, j ] = 0 1 ∂ j+ ∂ j− + i[A , j+] = ∂ A + − − + −4π − + The diﬀerence of these equations is given by, 1 ∂ j− = ∂ A (4.61) − 8π − +

− + and is easily solved by j = A+/(8π). The remaining equation is one for j , and takes the following form in this gauge, i ∂ j+ + [g−1∂ g, j+]= ∂ g−1∂ g (4.62) + + 8π − + Multiplication to the left by g and to the right by g−1 allows us to recast the equation in the following form, i ∂ g j+g−1 = ∂ ∂ gg−1 (4.63) + 8π + − This equation in turn is also easily integrated, and we ﬁnd, i i j+ = g−1∂ g j− = g−1∂ g (4.64) 8π − −8π + which gives the complete solution for the vector current in this gauge. Restoring both h and g by using SU(N)V gauge covariance of the vector current, we obtain, i j+ = ( h−1∂ h + g−1∂ g) 8π − − − i j− = (+h−1∂ h g−1∂ g) (4.65) 8π + − +

56 4.11 The Wess-Zumino-Witten term Can we ﬁnd the eﬀective action from the expression for these currents ? Again, we shall restrict to the gauge choice A− = 0, and attempt to integrate up the current equations. 1 δW [A ]= d2x tr(j+δA )= d2x tr g−1∂ g δ g−1∂ g µ + 8π − + Z Z The variation of the derivative is easily found

−1 −1 −1 −1 δ(g ∂+g)= ∂+(g δg) + [g ∂+g,g δg]

Putting the above equations together, we have 1 δW [A ]= d2x tr (g−1δg)∂ (g−1∂ g) µ 8π − + Z Can this variation come from a local action ? Well, there are not that many possibilities. For example, consider the variation of the standard kinetic term for a scalar ﬁeld

δtr g−1∂ gg−1∂ g = tr∂ (g−1δg)g−1∂ g tr∂ (g−1δg)g−1∂ g + − − + − − − + With the help of this equality, one may split up the change in the eﬀective action in terms of the kinetic term (symmetric in the interchange of + and -) and another part which is anti-symmetric under the interchange of + and -. We obtain, 1 W [A ]= d2x tr g−1∂ gg−1∂ g) + δS [g] µ 16π − + WZ Z Here SWZ[g] is given by its variation, 1 δS [g] = d2xtr g−1δg∂ (g−1∂ g) g−1δg∂ (g−1∂ g) WZ 16π − + − + − Z 1 = d2x ǫµν tr g−1δg∂ (g−1∂ g) (4.66) 32π µ ν Z Using symmetry of the double derivative contribution, this expression can be simpliﬁed, and we obtain 1 δS [g]= d2x ǫµν tr (g−1δg)(g−1∂ g)(g−1∂ g) WZ 32π µ ν Z There is in fact no way that this equation can be integrated up to a local and manifestly SU(N) SU(N) invariant action. Instead, we shall have the following construction. As L × R we know the variation of SWZ, we may always write down the full action as an integral

57 over an interpolating ﬁeld. We introduce a continuous function γ(τ, x) and 0 τ 1 and such that, ≤ ≤ γ(0, x)=1 γ(1, x)= g(x)

The action SWZ is now obtained by integrating the equation for δSWZ along the interpo- lating ﬁeld, 1 1 ∂γ S [g] S [1] = dτ d2xǫµν tr γ−1 γ−1∂ γγ−1∂ γ (4.67) WZ − WZ 32π ∂τ µ ν Z0 Z This term is called the Wess-Zumino-Witten term. If we view the two-dimensional space-time as a sphere S2, then, we may think of the domain of integration of the above three dimensional integral as that of a disk D which is just a half three sphere, whose only boundary is the two-sphere of space-time. We may then write the entire Wess-Zumino integrand in a simple geometrical fashion 1 S [g]= d3x ǫαβγ tr γ−1∂ γγ−1∂ γγ−1∂ γ WZ 96π α β γ ZD where ǫαβγ is the three dimensional completely anti-symmetric tensor. We shall see many properties of the WZW action a little later. Here, the main point is that using the anomaly equations in the non-Abelian fermion case, we have again been able to solve for the eﬀective action completely.

4.12 The Abelian axial anomaly in 4 dimensions In four-dimensional QED the 2-point function of the vector and axial vector currents ρ µ 0 T j5 (z)j (x) 0 vanishes by Lorentz invariance and parity conservation of QED. How- h | | i ρ µ ν ever, there are now triangle diagrams associated with the correlator 0 T j5 (z)j (x)j (y) 0 , which do not vanish, as their existence is allowed by Lorentz invarianch |e and parity in QED.| i The triangle graph again fails to be absolutely convergent in four dimensions, as its integral behaves superﬁcially as d4k/k3. Regularizing, for example with Pauli-Villars regulators, makes the triangles ﬁnite,∈ uniquely deﬁned, and gauge invariant. We ﬁrst deﬁne the Fourier transform of the triangle graph by, i(2π)4δ(p q q )M ρµν (p, q , q ) − 1 − 2 5 1 2 = d4z d4x d4y e−ipz+iq1x+iq2y 0 T jρ(z)jµ(x)jν (y) 0 (4.68) h | 5 | i Z Z Z Formally, the Fourier transform is given by the Feynman rules for the triangle graph, 4 ρµν d k µ i ν i ρ i ν i µ i ρ i i 5 (p, q1, q2)= 4 tr γ γ γ γ5 + γ γ γ γ5 (4.69) M − (2π) k/ k/ + q/2 k/ q/1 k/ k/ + q/1 k/ q/2 Z − − But this integral is only conditionally convergent, and will be uniquely determined by Pauli- Villars regularization, which by construction preserves the vector current and thus gauge

58 invariance. The regularized interal takes the form, d4k N i i i i ρµν (p, q , q ) = C tr γµ γν γργ M5 1 2 − (2π)4 n k/ M k/ + q/ M 5 k/ q/ M n=0 n 2 n 1 n Z X − − − − ν i µ i ρ i +γ γ γ γ5 (4.70) k/ Mn k/ + q/1 Mn k/ q/2 Mn − − − − The contribution n = 0 is just from the original massless fermion, for which we have,

M0 =0 C0 = 1 (4.71) To eliminate the linear superﬁcial divergence, we need, N

Cn = 0 (4.72) n=0 X after which the integral is logarithmically superﬁcially divergent. We shall use N = 3, with suitably chosen M and C to render the integrals convergent, and allow p ρµν to be split n n ρM5 into a diﬀerence of convergent integrals, as we did in two dimensions. Using p = q1 + q2, we may write the contribution on the ﬁrst line above with the help of the following identity, p γργ =(k/ + q/ M )γ + γ (k/ q/ M )+2M γ (4.73) ρ 5 2 − n 5 5 − 1 − n n 5 with a similar identity for the second line in which µ and ν are interchanged at the same time that q1 and q2 are. The contributions from the ﬁrst two terms on the right of the above identity then cancel with similar contributions from the second line, after suitably shifting the integration momentum in these convergent integrals, and only the contributions from the regulator terms remain, so that we have (after simplifying signs and factors of i), d4k N 1 1 1 p ρµν = 2C M tr γµ γν γ ρM5 − (2π)4 n n k/ M k/ + q/ M 5 k/ q/ M n=0 n 2 n 1 n Z X − − − − ν 1 µ 1 1 +γ γ γ5 (4.74) k/ Mn k/ + q/1 Mn k/ q/2 Mn − − − − Rationalizing and using Feynman parameters converts the ﬁrst term in the integrand to,

1 1−α tr γµ(k/ + M )γν(k/ + q/ + M )γ (k/ q/ + M ) n 2 n 5 − 1 n 2 dα dβ 3 (4.75) 0 0 2 2 2 2 2 Z Z (k + αq2 βq1) + αq + βq (αq2 βq1) M − 2 1 − − − n Shifting the integration momentum k k αq + βq , the integrand becomes, → − 2 1 tr γµ(k/ αq/ + βq/ + M )γν(k/ + q/ αq/ + βq/ + M )γ (k/ q/ αq/ + βq/ + M ) − 2 1 n 2 − 2 1 n 5 − 1 − 2 1 n 3 k2 + αq2 + βq2 (αq βq )2 M 2 2 1 − 2 − 1 − n In view of the γ5 factor, the only terms which contribute in the numerator are linear in Mn. Since the denominator is even in k, all terms odd in k in the numerator lead to a vanishing

59 contribution to the integral over k. Finally, all terms quadratic in k in the numerator also vanish in view of the γ5 factor. The resulting k-integral is then ﬁnite by power counting, so that only a single Pauli-Villars regular is needed so that we may set N = 1, M1 = M and C1 = 1. Since the integrals are convergent, we shall take M at the end of the calculation,− so that the momentum dependence of the denominator→∞may be omitted. We then have,

1 1−α d4k tr(γ γµγνq/ q/ ) p ρµν = 4M 2 dα dβ 5 1 2 (4.76) ρM5 − (2π)4 (k2 M 2)3 Z0 Z0 Z − The integral over α, β is readily carried out giving a factor of 1/2. The integral over k is given as follows,

d4k M 2 i = (4.77) (2π)4 (k2 M 2)3 −32π2 Z − and the trace over spinor space is evaluated using,

γ = iγ0γ1γ2γ3 tr(γ γµγνγργσ)= 4iεµνρσ (4.78) 5 5 − to give

1 p ρµν = εµνρσqρqσ (4.79) ρM5 4π2 1 2

Upon Fourier transforming back to position space, we convert the momentum factor q1 and q2 into derivatives on the gauge ﬁeld, and therefore into ﬁeld strengths times an extra factor of 1/2 to account for anti-symmetrization. Keeping careful track of factors of i and minus signs, we obtain the ﬁnal anomaly equation,

e2 1 ∂ jµ = F F˜µν F˜µν εµνρσF (4.80) µ 5 16π2 µν ≡ 2 ρσ The Adler-Bardeen theorem states that this result is exact at 1-loop, and may be viewed as an equation valid at the operator level for both ψ and Aµ. It may again be derived alternatively by point-splitting the fermion operators when deﬁning the axial vector current, and retaining the eﬀects of the singularity as the two fermions come close together.

4.13 Anomalies in gauge currents Consider a classical Lagrangian for a massless Dirac fermion ψ coupled to a vector gauge ﬁeld Aµ with ﬁeld strength Fµν as in QED, but also coupled to an axial vector gauge ﬁeld

60 B with ﬁeld strength G = ∂ B ∂ B , µ µν µ ν − ν µ 1 1 = F µνF GµνG + ψiγ¯ µD ψ L −4 µν − 4 µν µ Dµψ = ∂µψ + ieAµψ + igBµγ5ψ (4.81)

Does this theory deﬁne a consistent quantum theory ? This question could be asked in two space-time dimensions, or in four space-time dimensions. We shall investigate the question in the latter case. Note ﬁrst that all interactions proceed through the gauge ﬁelds coupling to the fermions. When there four or more gauge currents on a fermion loop, both vector and axial vector gauge invariance can be maintained. Furthermore, vector current conservation can be enforced in all triangle and vacuum polarization graphs. Thus, it is only for the triangle graphs involving at least one axial vector that the anomaly eﬀect shows up, and produces a coupling of the non-conserved axial vector current to a gauge ﬁeld. As a result, the Bµ gauge ﬁeld cannot be quantized in a completely gauge invariant way, and therefore its negative norm states cannot be consistently decoupled. This eﬀect was discovered by Gross and Jackiw and independently by Bouchiat, Iliopoulos, and Meyer in 1973, and is of fundamental importance in the quantization of gauge theories with chiral fermions (including in the Standard Model).

Figure 4: Gross-Jackiw and Bouchiat-Iliopoulos-Meyer double triangle graphs fig:C5 { } 61 In the Gross-Jackiw two-point function, gauge invariance is maintained on the A- propagators, but the anomaly then implies that gauge-invariance fails on the B-propagators, so that the negative norm states in the B-ﬁeld no longer decouple. In the Bouchiat- Iliopoulos-Meyer four-point function, negative norm states are allowed to propagate in the B-propagator. The upshot is that the axial anomaly can destroy gauge invariance in any current coupled to a dynamical gauge ﬁeld, and consistent unitary theories must be free of such anomalies. Two comments are in order. 1. Note that there is an easy modiﬁcation of the A B model in which both A and B − gauge invariance are preserved: introduce two Dirac fermions with the same electric charge, but opposite chiral charge. The triangle graphs contributing to the anomaly then add up to zero. 2. Since the Dirac fermion is massless in this theory, its left and right chirality contribu- tions separate, and we may choose to set Aµ = Bµ and gauge only the left chirality. But this cannot be done because the left current always contains the axial vector and one cannot make the left current by itself conserved.

4.14 Non-Abelian axial anomalies We begin by considering a Dirac fermion ψ in a representation of the gauge group G a T with representation matrices T of the Lie algebra coupled to a vector gauge ﬁeld Aµ, = ψγ¯ µ(i∂ gA )ψ (4.82) Lψ µ − µ and deﬁne the vector and axial vector currents as usual, jµa = ψγ¯ µT aψ jµ = jµaT a µa ¯ µ a µ µa a j5 = ψγ γ5T ψ j5 = j5 T (4.83) Assuming covariant conservation of the vector current, and therefore exact gauge invariance, the lowest order contribution to the anomaly in the axial vector current is given by the triangle graphs for the Abelian anomaly supplemented by the insertion of the matrices T a, i(2π)4δ(p q q )M ρµν cab(p, q , q ) − 1 − 2 5 1 2 = d4z d4x d4y e−ipz+iq1x+iq2y 0 T jρc(z)jµa(x)jνb(y) 0 (4.84) h | 5 | i Z Z Z Formally, the Fourier transform is given by the Feynman rules for the triangle graph, 4 ρµν cab d k µ a i ν b i ρ c i i 5 (p, q1, q2) = 4 tr γ T γ T γ γ5T M − (2π) k/ k/ + q/2 k/ q/1 Z − ν b i µ a i ρ c i +γ T γ T γ γ5T (4.85) k/ k/ + q/1 k/ q/2 − where the trace now also includes the trace over the representation space of . Recall that T the Abelian anomaly was already symmetric under Bose exchange of (µ, q1)and (ν, q2). The

62 insertion of the additional representation matrices produces a contribution antisymmetric in a, b which is ﬁnite and does not contribute to the anomaly, as well as a contribution which is symmetric in a, b and which does contribute to the anomaly. As a result, there is a simple relation between the Abelian and non-Abelian anomalies to this order, given by,

p ρµν cab(p, q , q )= dcab( ) p ρµν (p, q , q ) (4.86) ρM5 1 2 T ρM5 1 2 where the symbol dcab is deﬁned as follows,

1 dcab( )= tr T c T a, T b (4.87) T 2 { } Without proof, we state here the full axial vector anomaly equation,

g2 D jµ = F F˜µν (4.88) µ 5 16π2 µν

We shall give some of the properties of the d-symbol in the next section.

4.15 Non-Abelian chiral anomaly Suppose we had only a Weyl fermion ψ of left chirality in a representation of the gauge L T group G. By charge conjugation for four space-time dimensions, one can actually express a right chiral fermion ﬁeld ψR in a representation R of G in terms of a left chiral fermion c T ∗ ﬁeld ψR transforming in the complex conjugate representation L = R . So, using only left chirality fermions represents no loss of generality. The LagrangianT isT then,

= ψ¯ γµ(i∂ gA )ψ (4.89) Lψ L µ − µ L µa ¯ µ a There is now only the left current, jL = ψLγ T ψL, and the triangle graph of three left currents must be Bose symmetric under the permutation of its legs. This ﬁxes the graph uniquely. Note that when no vector current is available, there is no sense in which we can preserve the conservation of the vector current. All the anomaly eﬀects here must be in the left current alone. The anomaly equation is then given by,

g2 i (D jµ)c = εµνρσ∂ tr T c A ∂ A gA A A (4.90) µ L 24π2 µ ν ρ σ − 2 ν ρ σ Note that the right side is again proportional to dcab( ). The 1/3 in the pre-factor arises T from the Bose-symmetrization of the three left currents. Finally, the right side manifestly fails to be gauge-invariant, which is the manifestation of the anomaly in this context.

63 4.16 Properties of the d-symbols Of particular interest is to know for which representations, and which groups G the d- symbol vanishes. Recall that we have restricted from the beginning to be a unitary representation of a compact group G. When the representation is real,T the representation T matrices T a must be purely imaginary (since we have extracted a factor of i to make them Hermitian), and therefore they must be anti-symmetric, (T a)t = T a. As a result, the d-symbol vanishes, since we have, − t 2dabc( ) = tr T aT bT c + T aT cT b = tr T aT bT c + T aT cT b T = tr T cT bT a + T bT cT a = 2dabc( ) (4.91) − − T Thus, dabc( ) can be non-zero only for complex representation. The groups that have com- plex representationsT are U(1), SU(N), and possibly the spinor representations of SO(N). All other representations are real or pseudo-real and have vanishing d-symbol. In particular, one very important case is when G contains a U(1)-factor, G = U(1) H and H is semi-simple. The d-symbol evaluated on the index c corresponding to the U×(1)- generator while the other two indices a, b correspond to generators of H gives, dcab = tr(T c) tr(T aT b) (4.92) × But tr(T aT b) = 0 for generators of a semi-simple algebra, so the only way this anomaly 6 can be absent is that, tr(T c) = 0 (4.93) This is the case for the SU(2) U(1) anomaly, where the U(1) charges in the Standard L × Y Y Model are indeed traceless and sum up to zero within the multiplet of left fermions in each family. On left fermions, U(1)Y is proportional to electric charge. The multiplets are, ν uα e (4.94) e− dα where α is the color index α = 1, 2, 3. While the electric charge +e/3 of u and e/3 of d − only add up to +e/3, there are three colors of each, adding the charges up to +e, which is precisely cancelled by the negative charge of the electron. Having dispensed with the U(1) factors in G, we next consider the case where G is a semi-simple Lie algebra, given by the product of r simple factors, G = G G G (4.95) 1 × 2 ×···× r a a The Lie algebra generators Ti in factor Gi must be traceless trGi (Ti ) = 0 for all values of i = 1, ,r, since otherwise the algebra Gi would contain a commuting U(1) factor and this is··· in contradiction with the assumption of being simple.

64 When all three Lie algebra generators belong to mutually distinct simple factors, the • vanishing of the trace in each factor guarantees that d vanishes; When two Lie algebra generators belong to the same simple factor, and the third • to a distinct factor, then the traceless property in the last factor guarantees that the d-symbol vanishes. This applies for example for the SU(2)L generators to have vanishing anomaly with SU(3)c on left fermions, since the generators of SU(2)L are manifestly traceless. This leaves only the case where all three generators belong to the same simple factor, • which we address next.

When G is simple, the tensor dabc( ) has the following properties (including a summary T of the properties already stated above for general Lie algebras)

1. dabc( ) is invariant under the action of G in the adjoint representation; T 2. dabc( ∗)= dabc( ); T − T 3. dabc( )= dabc( )+ dabc( ); T1 ⊕ T2 T1 T2 4. dabc( )= dabc( ) dim( )+ dabc( ) dim( ); T1 ⊗ T2 T1 T2 T2 T1 5. dabc( ) is unique, up to overall normalization, so that we may deﬁne a numerical valueT d( ) by dabc( ) = d( )dabc where dabc is the value of the d-symbol in some ﬁxed faithfulT representationT T of G; 6. dabc( ) = 0 in all representations of SO(N), N = 6; all representations of Sp(2N); T 6 and all representations of E6, E7, E8,G2, F4; 7. dabc( ) = 0 in the deﬁning representation of SU(N) for N 3, and in the spinor T 6 ≥ representation of SO(6) = SU(4)/Z2, which is the deﬁning representation of SU(4). Properties 6 and 7 follow from the structure of the De Rham cohomology group H5(G, R).

65 5 One-Loop renormalization of Yang-Mills theory

In the ﬁgures 5, 6, 8 and 9 below, a complete list is given of all one-loop contributions to Yang-Mills theory, with fermions, but without scalars, that can contribute to renormaliza- tion at one-loop order. Contributions to the 1-point function are given by ﬁgure 5. They must vanish by translation and Lorentz invariance, as may indeed be veriﬁed explicitly by making use of the Feynman rules.

(1.a) (1.b) (1.c)

Figure 5: One-loop Yang-Mills : the 1-point functions fig:C5 { }

5.1 Calculation of Two-Point Functions Contributions to the 2-point function are given by ﬁgure 6. Only diagram (2.b) in ﬁgure 6 manifestly vanishes in dimensional regularization. The fermion self-energy graph (2.e) may be obtained from the same graph in QED by tagging on the group theory factors.

(2.a) (2.b) (2.c)

(2.d) (2.e) (2.f)

Figure 6: One-loop Yang-Mills : the 2-point functions fig:C6 { } The contributions (2.a), (2.b), (2.c) and (2.d) are vacuum polarization eﬀects for the Yang-Mills particle, and will yield the renormalization factor Z3. Their contribution is expected to be transverse in view of gauge invariance. Graphs (2.e) and (2.f) will yield the renormalization constants Z2 and Z˜3 respectively.

5.1.1 Vacuum polarization by gluons The novel diagrams are redrawn in ﬁgure 7 and given detailed labeling conventions.

66 κ k c κ -k c ' p a µ k+p -p b ν -p b ν p a µ

k -k-p d λ k+p d λ'

(2.a) (2.c)

Figure 7: Detailed labeling of vacuum polarization by gluons and ghosts graphs fig:C9 { } The gluon contribution is given by graph (2.a), as follows,

1 ddk i i Γab (p) = g2µ2ǫf adcf bcd − − (5.1) (a)µν 2 (2π)d k2 (k + p)2 Z ′ ′ κ κ λ λ ′ k k ′ (k + p) (k + p) ηκκ ξ ηλλ ξ × − k2 − (k + p)2 [η (2p + k) + η ( 2k p) + η (k p) ] × µλ κ κλ − − µ κµ − λ [η ′ (2p + k) ′ + η ′ ′ ( 2k p) + η ′ (k p) ′ ] × νλ κ κ λ − − ν κ ν − λ It is useful to arrange this calculation according to the powers of ξ, 1 ddk 1 N P Q Γab (p)= g2µ2ǫf acdf bcd M ξ µν ξ µν + ξ2 µν (a)µν 2 (2π)d k2(k + p)2 µν − k2 − (k + p)2 k2(k + p)2 Z with the following expressions,

M = (d 6)p p + (4d 6)p k + (4d 6)k k + 2k2 +2kp +5p2 η µν − µ ν − {µ ν} − µ ν { } µν N = k2p p +( 6k2 6kp)p k +( k2 6kp + p2)k k +(k2 +2kp)2η µν µ ν − − {µ ν} − − µ ν µν P = (2k2 p2)p p 2pkp k + (2p2 k2)k k +(k2 p2)2η µν − µ ν − {µ ν} − µ ν − µν Q = (p2η p p )(p2η p p )kσkρ (5.2) µν µσ − µ σ νρ − ν ρ

We see that the integral involving Qµν is UV convergent and will not participate in renor- malization. Notice that it is also manifestly transverse.

We shall compute in detail only the contribution involving Mµν . First, the term in Mµν 2 2 2 which is proportional to ηµν may be re-expressed as follows, 2k +2kp +5p = k +(k + p)2 +4p2; the terms in k2 and (k + p)2 cancel in dimensional regularization. Second, we introduce a single Feynman parameter, so that ddk M (k,p) 1 ddk M (k αp,p) = µν = dα µν − (5.3) Iµν (2π)d k2(k + p)2 (2π)d (k2 + α(1 α)p2)2 Z Z0 Z − 67 Terms odd in Mµν (k αp,p) which are odd in k do not contribute to the integral, and we are left with − M (k αp,p) (d 6 α(1 α)(4d 6))p p +4η p2 + (4d 6)k k (5.4) µν − ∼ − − − − µ ν µν − µ ν 2 The integration is now Lorentz invariant and we may replace kµkν by ηµν k /d. The re- maining k-integrations may be carried out by ﬁrst analytically continuing to Euclidean time k0 ik0 and then using the Euclidean integral formulas, → E ddk 1 Γ(n d/2) 1 = − (5.5) (2π)d (k2 + M 2)n (4π)d/2Γ(n) (M 2)n−d/2 Z As a result, we have Γ(2 d/2) 1 d/2−2 = i − dα α(1 α)p2 (5.6) Iµν (4π)d/2 − − Z0 4d 6 (d 6)p p α(1 α)(4d 6)p p +4p2η + − α(1 α)p2η × − µ ν − − − µ ν µν 2 d − µν − The α-integration may be carried out using the Euler B-function, and we obtain, Γ(2 d/2) Γ(d/2 1)2 = i − ( p2)d/2−2 (d 6)p p +4p2η − (5.7) Iµν (4π)d/2 − − µ ν µν Γ(d 2) − n 4d 6 Γ(d/2)2 + (4d 6)p p + − p2η − − µ ν 2 d µν Γ(d) − o Picking up only the pole part in ε where d = 4 2ε, and using Γ(2 d/2) = Γ(ε) 1/ε, − − ∼ we get i 19 22 = ( p2)−ε p2η p p (5.8) Iµν (4π)2 ε − 6 µν − 6 µ ν The entire diagram is evaluated using analogous calculations, and we ﬁnd, ig2( p2/µ2)−ε 19 22 Γab (p) = − f acdf bcd p2η p p + ξ(p2η p p ) (5.9) (a)µν 32π2 ε 6 µν − 6 µ ν µν − µ ν Notice that the gluon exchange graph by itself fails to be transverse, for any choice of gauge ξ.

5.1.2 Vacuum polarization by ghosts The ghost contribution is given as follows, ddk i i Γab (p) = g2µ2ǫf adcf bcd k (k + p) (c)µν − (2π)d k2 (k + p)2 µ ν Z 1 ddk η k2/d α(1 α)p p = g2µ2ǫf acdf bcd dα µν − − µ ν (5.10) − (2π)d (k2 + α(1 α)p2)2 Z0 Z − 68 Retaining only the pole parts in ε, we ﬁnd, ig2( p2/µ2)−ε 1 2 Γab (p)= − f acdf bcd p2η + p p (5.11) (c)µν 32π2ε 6 µν 6 µ ν Putting together the gluon and ghost contributions to vacuum polarization, we obtain, ig2( p2/µ2)−ε 13 Γab (p)= − f acdf bcd p2η p p (1 ξ) (5.12) (a,b,c)µν 32π2ε µν − µ ν 3 − − 5.1.3 Vacuum polarization by fermions The vacuum polarization contribution due to fermions is given by graph (2.d) of ﬁgure 6, and is proportional to the value in QED. We express this as follows,

ab 2 a b Γ(d)µν (p) = g tr(iTψ )(iTψ)Γµν(p) ddk i i Γ (p) = (5.13) µν − (2π)d k/ m k/ + p/ m Z − − The pole part of this contribution is given by ig2( p2/µ2)−ε 8 Γab (p)= − − tr(T aT b) p2η p p (5.14) (d)µν 32π2ε 3 ψ ψ µν − µ ν 5.2 Indices and Casimir Operators Some group theory is helpful in organizing the ﬁnal presentation of the renormalization eﬀects, to one-loop order and beyond. First, the generators of a simple Lie algebra G in a unitary representation will be denoted by T a = T a. They satisfy the Lie algebra structure T T relations, [T a, T b]= if abcT c, and will be normalized as follows. We begin by denoting the dimension of by dim( ). For any simple Lie algebra, G, and any representation , the T T T trace tr(T aT b) is proportional to δab, and the sum T aT a is proportional to the identity matrix IT in the representation space of . The corresponding coeﬃcient are referred to as the quadratic index I( ), and the quadraticT Casimir C( ), T T tr(I ) = dim( ) T T tr(T aT b) = I( ) δab T T aT a = C( ) I (5.15) T T It suﬃces to normalize either I( ) or C( ) is a single representation of G to ﬁx the normalizations of the structure constantsT f abcTand I( ) and C( ) for all . For Lie algebras SU(N), SO(N) and Sp(2N), it is customary to useT the deﬁningT representationsT = D T of dimensions N, N, and 2N respectively, and we require I(D)=1/2. For exceptional groups, one uses one of the fundamental representations to ﬁx the normalization.

69 The index and the Casimir are related to one another, as may be derived by evaluating tr(T aT a) respectively from the equations deﬁning I( ) and C( ), and we obtain, T T I( ) dim(G)= C( ) dim( ) (5.16) T T T where dim(G) denotes the dimension of the Lie algebra, which is, of course, the dimension of the adjoint representation. Under direct sums and tensor products, the dimension, index and Casimir behave as follows, dim( ) = dim( ) + dim( ) T1 ⊕ T2 T1 T2 dim( ) = dim( ) dim( ) T1 ⊗ T2 T1 × T2 I( ) = I( )+ I( ) T1 ⊕ T2 T1 T2 I( ) = I( ) dim( )+ I( ) dim( ) (5.17) T1 ⊗ T2 T1 T2 T2 T1 from which we deduce that dim( )C( ) + dim( )C( ) C( ) = T1 T1 T2 T2 T1 ⊕ T2 dim( ) + dim( ) T1 T2 C( ) = C( )+ C( ) (5.18) T1 ⊗ T2 T1 T2 Using these rules, we may evaluate some of these quantities for G = SU(N), the deﬁning representation D, and the adjoint representation G, related by D D∗ = G 1, ⊗ ⊕ 1 N 2 1 dim(D)= N I(D)= C(D)= − 2 2N dim(G)= N 2 1 I(G)= N C(G)= N (5.19) − The representation matrices of the adjoint representation are just the structure constants (T a) c = ( i f a) c. Here, we have been careful to include a factor of i in order to make G b − b − hermitian, as required by our earlier conventions. As a result, we have TG tr(T aT b )=( i f a) d( i f b) c = f acdf bcd = I(G)δab (5.20) G G − c − d We recognize the combination that entered the one-loop vacuum polarization corrections by gluons and ghosts.

5.3 Summary of two-point functions We collect the pole contributions to the renormalization factors for 2-point functions, g2 Z = 1+ (2 2ξ)C(ψ) + (g4) 2 32π2ε {− − } O g2 13 8 Z = 1+ (1 ξ) C(G) T (ψ) + (g4) 3 32π2ε 3 − − − 3 O g2 1 Z˜ = 1+ 1+ ξ C(G) + (g4) (5.21) 3 32π2ε 2 O 70 Even with the help of the Slavnov-Taylor identities, we do not have suﬃcient information, so far, to compute the relation between g and g0.

5.4 Three-Point Functions The following renormalization factors are determined by these graphs,

g2 1 Z = 1+ 2+ ξ C(G) (2 2ξ)C(ψ) + (g4) 1 32π2ε − 2 − − O g2 4 3 8 Zˆ = 1+ + ξ C(G) T (ψ) + (g4) 1 32π2ε 3 2 − 3 O g2 Z˜ = 1+ (1 ξ)C(G) + (g4) (5.22) 1 32π2ε {− − } O Some explanation of the group theoretic factors is in order, for graphs (3.e), (3.f), (3.g) and (3.h). In graphs (3.e) and (3.g), the group theoretic factor is

1 f abcT bT c = f abc[T b, T c] 2 i = f abcf bcdT d 2 i = C(G)T a (5.23) 2 where = for (3.e) and = ψ for (3.g). In graphs (3.f) and (3.h), the group theoretic factor isT G T

T bT aT b = T bT bT a + T b[T a, T b] Xb Xb Xb = C( )T a + i f abcT bT c T Xb,c 1 = C( )T a I( )T a (5.24) T − 2 T where = G for (3.f) and = ψ for (3.h). T T

5.5 Four-Point Functions The graphs contributing to the 4-point functions are given in ﬁgure 9. The graphs (4.a-e) contribute to the Yang-Mills 4-point function and are expected to received contributions that need to be renormalized. These are summarized by the renormalization factor Z4,

71 (3.a) (3.b) (3.c)

(3.d) (3.e) (3.f)

(3.g) (3.h)

Figure 8: One-loop Yang-Mills : the 3-point functions fig:C7 { }

72 which is g2 2 8 Z =1+ +2ξ C(G) I(ψ) + (g4) (5.25) 4 32π2ε −3 − 3 O On the other hand, the graphs (4.f-i), as well as (4.j-k) are superﬁcially log divergent, yet would require counterterms not already represented in the Lagrangian. These contributions are actually UV convergent, essentially because on at least one ghost vertex, the vertex mo- mentum factor is acting on the external leg and does not contribute to the loop momentum counting. Finally, the graphs (4.l-o) are UV convergent because their superﬁcial degree of divergence is -2. Thus, Z4, listed above, is in fact the only required renormalization for 4-point functions.

5.6 One-Loop Renormalization The above results for the renormalization factors verify the Slavnov-Taylor identities, as may be seen by explicit calculation. There are 3 identities to be checked, which we shall express in terms of 3 combinations predicted to be 1 by the Slavnov-Taylor identities,

Zˆ2 Z−1 Z−1 = 1+ (g4) 1 3 4 O Z˜ Z˜−1 Zˆ−1 Z = 1+ (g4) 1 3 1 3 O Z Z−1 Zˆ−1 Z = 1+ (g4) (5.26) 1 2 1 3 O

The quantity Zλ is not renormalized at all.

Next, we derive the relation between the bare coupling g0 and the renormalized coupling g. The starting point is the relation

ε ˆ −3/2 g0 = gµ Z1 Z3 (5.27) Using the expressions found above to one-loop order, we get

g2 b g = gµε 1+ 1 + (g4) 0 32π2 ε O 11 4 b = C(G)+ I(ψ) (5.28) 1 − 3 3 Inverting and squaring both sides, up to this order in g gives an equivalent relation,

−2ε −2ε 1 µ µ 2 2 = 2 2 b1 + (g ) (5.29) g0 g − 16π ε O

The proper interpretation of this relation is as follows. If the bare coupling g0 and the regulator ε are kept ﬁxed, then the relation gives the dependence of the renormalized

73 (4.a) (4.b) (4.c)

(4.f) (4.d) (4.e)

(4.g) (4.h) (4.i)

(4.j) (4.k) (4.l)

(4.m) (4.n) (4.o)

Figure 9: One-loop Yang-Mills : the 4-point functions fig:C8 { }

74 coupling g on the renormalization scale µ. Therefore, it is appropriate to denote the renormalized coupling by g(µ) instead of by g. The renormalized coupling should have a ﬁnite limit as ε 0, keeping µ ﬁxed. This requires a special choice for the dependence of → the bare coupling g0 on the regulator ε, 1 b =Λ−2ε 1 + c (5.30) g2 −16π2ε 0 Here, the pole in ε is required by the requirement that g(µ) should have a ﬁnite limit as ε 0, but the constant c is arbitrary. A scale Λ is required on dimensional grounds. Notice → that the choice of g0 is independent of µ and does not involve g(µ). Using this choice and multiplying through by µ2ε gives

1 b µ 2ε µ 2ε = 1 1 + c (5.31) g(µ)2 16π2 ε − Λ Λ This expression for g(µ) now has a ﬁnite limit as ε 0, but it involves the unknown constant c. Taking the diﬀerence for two diﬀerent values→ of µ eliminates this unknown and we get,

2ε ′ 2ε 1 1 b1 µ µ 2 ′ 2 = lim 2 (5.32) g(µ) − g(µ ) ε→0 16π ε ( Λ − Λ )! The dependence on Λ drops out, and we ﬁnd, 1 1 b µ = 1 ln + (g2) g(µ)2 − g(µ′)2 8π2 µ′ O 11 4 b = C(G)+ I(ψ) (5.33) 1 − 3 3 As a check, the coupling constant renormalization of QED may be recovered from this relation by setting the gauge group to be G = U(1), and g = e, so that C(G) = 0 and 2 I(ψ)= j qj , for Dirac fermions j with charge qje. We indeed recover,

P 1 1 1 µ 2 = q2 ln + (e2) (5.34) e(µ)2 − e(µ′)2 12π2 j µ′ O j X This agrees precisely with the formulas derived in the section on QED.

5.7 Asymptotic Freedom For a genuinely non-Abelian Yang-Mills theory, with simple gauge algebra G, the Casimir C(G) is positive, so that, unlike in QED, the coeﬃcient b1 may become negative. When

75 b1 < 0, the above equation implies that

µ>µ′ g(µ) < g(µ′) (5.35) ⇒ This means that as the energy scale at which the coupling g is probed is increased, the coupling decreases. This eﬀect is the opposite of electric screening which took place in QED. In particular, it implies that as the energy is taken to be asymptotically high, the coupling will vanish, hence the name asymptotic freedom. Notice that the perturbative calculations in an asymptotically free theory should become more reliable as the energy is increased. Conversely, as the energy is lowered, the coupling increases. Of course, at some point, the coupling will then fail to be in the perturbative regime and higher order corrections, as well as truly non-perturbative eﬀects will have to be included. Nonetheless, the trend is indicative that the theory may develop a genuinely large coupling as the energy scale is lowered. For the strong interactions, such increase in coupling is certainly a necessary condition to have quark conﬁnement. The phenomenon of growing coupling with lower energy scales is often referred to as infrared slavery. To get a quantitative insight into the theories which are asymptotically free, we shall examine more closely the case of Yang-Mills theory for the gauge group G = SU(N) with Nf Dirac fermions in the deﬁning representation of SU(N), which is of dimension N. Group theory informs us that, 1 C(G)= N I(ψ)= N (5.36) 2 f so that 11 2 b = N + N (5.37) 1 − 3 3 f

Asymptotic freedom is realized as long as b1 < 0, or

11 N 2 N > 0 (5.38) − f in other words, when there are not too many fermions. Speciﬁcally for the color gauge group of QCD, we have N = 3, and the number of quark ﬂavors is Nf = 6, namely, the u, d, c, s, t, b. We see that 11 N 2 Nf = 21 > 0 and QCD with 6 ﬂavors of quarks is indeed asymptotically free. −

76 6 Use of the background ﬁeld method

Feynman diagrams and standard perturbation theory do not always provide the most ef- ﬁcient calculational methods. This is especially true for theories with complicated local invariances, such as gauge and reparametrization invariances. In this chapter, the back- ground ﬁeld method will be introduced and discussed. It will be utilized to provide a complete calculation of the renormalization eﬀects of the Yang-Mills coupling.

6.1 Background ﬁeld method for scalar ﬁelds The background ﬁeld method is introduced ﬁrst for a theory with a single scalar ﬁeld φ, in arbitrary dimension of space-time d. The classical action is denoted S0[φ], and the full quantum action, including all renormalization counterterms is denoted by S[φ]. The generating functional for connected correlators is given by,6 i i i exp W [J] Dφ exp S[φ] ddxJ(x)φ(x) (6.1) back1 − ~ ≡ ~ − ~ { } n o Z n Z o By the very construction of the renormalized action S[φ], W [J] is a ﬁnite functional of J. The Legendre transform is deﬁned by

Γ[φ¯] ddxJ(x)φ¯(x) W [J] (6.2) ≡ − Z where J and φ¯ are related by δW [J] δΓ[φ¯] φ¯(x)= J(x)= (6.3) δJ(x) δφ¯(x)

Recall that Γ[φ¯] is the generating functional for 1-particle irreducible correlators. To obtain a recursive formula for Γ, one proceeds to eliminate J from (6.1) using the above relations, i i δΓ[φ¯] exp Γ[φ¯] ddxJ(x)φ¯(x) = Dφ exp S[φ] ddxφ(x) (6.4) back2 ~ − ~ − δφ¯(x) { } Z Z Z Moving the J-dependence from the left to the right side, and expressing J in terms of Γ in the process, we obtain, i i δΓ[φ¯] exp Γ[φ¯]= Dφ exp S[φ] ddx (φ φ¯)(x) (6.5) back3 ~ ~ − − δφ¯(x) { } Z Z 6To clearly exhibit the order in the loop expansion of various contributions, we shall in this section exhibit Planck’s constant ~ explicitly.

77 The ﬁnal formula is obtained by shifting the integration ﬁeld φ φ¯ + √~ϕ, → i i δΓ[φ¯] exp Γ[φ¯]= Dϕ exp S[φ¯ + √~ϕ] √~ dxϕ(x) (6.6) ~ ~ − δφ¯(x) Z Z At ﬁrst sight, it may seem that little has been gained since both the left and right hand sides involve Γ. Actually, the power of this formula is revealed when it is expanded in powers of ~, since on the rhs, Γ enters to an order higher by √~ than it enters on the lhs. The ~-expansion is performed as follows,

S[φ] = S [φ]+ ~S [φ]+ ~2S [φ]+ (~3) 0 1 2 O Γ[φ¯] = Γ [φ¯]+ ~Γ [φ¯]+ ~2Γ [φ¯]+ (~3) (6.7) 0 1 2 O

The recursive equation to lowest order shows that Γ0[φ¯]= S0[φ¯]. The interpretation of the terms S [φ] for i 1 is in terms of counter-terms to loop order i, while that of Γ [ϕ] is that i ≥ i of i-loop corrections to the eﬀective action. The 1-loop contribution is readily evaluated,

2 ¯ i δ S [φ¯] exp iΓ [φ¯]= eiS1[φ] Dϕ exp ddxϕ(x) ddyϕ(y) 0 (6.8) 1 2 δφ¯(x)δφ¯(y) Z Z Z

For example, if S0 describes a ﬁeld φ with mass m and interaction potential V (φ), we have

1 1 S [φ]= ddx ∂ φ∂µφ m2φ2 V (φ) (6.9) 0 2 µ − 2 − Z As a result, the one-loop eﬀective action is given by,

¯ i exp iΓ [φ¯] = eiS1[φ] Dϕ exp ddxϕ ✷ + m2 + V ′′(φ¯) ϕ (6.10) 1 2 Z Z The Gaussian integral over ϕ may be carried out in terms of a functional determinant,

i Γ [φ¯] = ln Det ✷ + m2 + V ′′(φ¯) + S [φ¯] (6.11) 1 2 1 n o Upon expanding this formula in perturbation theory, the standard expressions may be recovered,

i Γ [φ¯]=Γ [0] + ln Det 1+ V ′′(φ¯)(✷ + m2)−1 + S [φ¯] (6.12) 1 1 2 1 h i which clearly reproduces the standard 1-loop expansion. Therefore, for scalar theories, the gain produced by the use of the background ﬁeld method is usually only minimal.

78 6.2 Background ﬁeld method for Yang-Mills theories For gauge theories, the background ﬁeld method presents important calculational advan- tages, which we illustrate by working out the coupling constant renormalization eﬀects in Yang-Mills theory. The starting point is the classical Yang-Mills action, 1 S [A]= trF F µν F = ∂ A ∂ A + i[A , A ] (6.13) 0 −2g2 µν µν µ ν − ν µ µ ν

We begin by expanding the quantum ﬁeld Aµ around a non-dynamical background A¯µ,

Aµ = A¯µ + g√~aµ F = F¯ + g√~(D¯ a D¯ a )+ ig2~[a , a ] (6.14) µν µν µ ν − ν µ µ ν where the background ﬁeld covariant derivative is deﬁned by

D¯ µaν = ∂µaν + i[A¯µ, aν] (6.15)

The expansion of the classical action is given by

1 1 1 µ ν S0[A] S0[A¯] = tr 2 F¯µν D¯ a (6.16) ~ − ~ − √~g Z + tr (D¯ a D¯ a )D¯ µaν iF¯µν[a , a ] − µ ν − ν µ − µ ν Z 1 + tr 2i√~gD¯ a [aµ, aν]+ ~g2[aµ, aν][a , a ] − µ ν 2 µ ν Z The action of gauge transformations on Aµ is unique, but its action on A¯µ and aµ is not. The most useful way of decomposing the gauge invariance of Aµ is to let the quantum ﬁeld aµ transform homogeneously,

A¯ A¯′ = UA¯ U −1 + i∂ U U µ → µ µ µ a a′ = Ua U −1 (6.17) µ → µ µ Notice that under this action of the gauge transformation each term in the expansions above transform homogeneously. As a result, the structure of the counter-terms is uniquely ﬁxed by this background ﬁeld gauge invariance. To one-loop order, we have

µν S1[A¯]= c trF¯µν F¯ (6.18) Z where c is to be determined by the renormalization conditions.

79 6.2.1 Background gauge ﬁxing and ghosts In analogy with the quantization of Yang-Mills theory in trivial background, here also the kinetic part for the aµ ﬁeld fails to be invertible and therefore gauge ﬁxing is required. It is convenient to work in background ﬁeld Lorentz covariant gauges, such as the one deﬁned by the following gauge function, δ (A)(x) (A)= D¯ aµ f F = D¯ δ(x y) (6.19) F µ − δaµ(y) µ − where all terms are valued in the adjoint representation of the gauge algebra. The external function f is to be integrated over. The Faddeev-Popov operator is deﬁned by (x, y)= D D¯ µδ(x y) (6.20) M µ − Notice that the ﬁrst covariant derivative is with respect to the full gauge ﬁeld Aµ, while the second one is only with respect to the background gauge ﬁeld A¯µ. The corresponding Faddeev-Popov ghost Lagrangian is thus given by

µ Sgh[A, c, c¯] = 2 tr Dµc¯ D¯ c Z µ µ = 2 tr D¯ µc¯ D¯ c + ig√~ [aµ, c¯] D¯ c (6.21) Z while the gauge ﬁxing Lagrangian that results from integrating out f gives rise to 1 2 S [A]= tr D¯ aµ (6.22) gf ξ µ Z The kinetic term for aµ may now be combined with the gauge ﬁxing term, using the following identities,

tr D¯ a D¯ µaν = tr a D¯ D¯ µaν ν µ − µ ν Z Z = tr a [D¯ µ, D¯ ν]a tr a D¯ µD¯ νa µ ν − µ ν Z Z = i tr F¯µν [a , a ] + tr(D¯ aµ)2 (6.23) − µ ν µ Z Z Finally, it is convenient to choose Feynman gauge, where ξ = 1, so that

1 1 1 µ ν S0[A] S0[A¯] = tr 2 F¯µν D¯ a (6.24) ~ − ~ − √~g Z h i + tr D¯ a D¯ µaν 2iF¯µν[a , a ] − µ ν − µ ν Z h 1 i + tr 2i√~gD¯ a [aµ, aν]+ ~g2[aµ, aν][a , a ] − µ ν 2 µ ν Z h i 80 Notice that the kinetic term for the aµ ﬁeld may be viewed as the kinetic term for 4 scalar bosons, indexed by the Lorentz label ν.

6.2.2 Coupling scalars and spinors

a a a Let Ts , Tf , and TG denote representation matrices in the representations of the gauge algebra for the scalars (s), spinors (f) and gauge ﬁelds (i.e. the adjoint representation ). We denote the associated background covariant derivatives, acting on Lorentz scalars, byG

D¯ = ∂ + iA¯a T a R = s, f, (6.25) Rµ µ µ R G The Lagrangians for scalars and fermions coupled to the gauge ﬁeld A are then given by 1 1 S [φ, A] = (D¯ µφ)tD¯ φ + ig√~aa φtT aD¯ µφ g2~aaµab φtT aT bφ s 2 s sµ µ s s − 2 µ s s Z S [ψ, ψ,¯ A] = ψ¯(iγµD¯ m)ψ g√~aa ψγ¯ µT aψ (6.26) f fµ − − µ f Z 6.3 Calculation of the one-loop β-function for Yang-Mills The 1-loop contributions are all given by Gaussian integrals, and thus may be calculated in terms of functional determinants, 1 iΓscalar = ln Det( D¯ µD¯ ) 1 −2 − s sµ fermion µ ¯ iΓ1 = + ln Det(iγ Dfµ) 1 iΓgauge = ln Det( D¯ µD¯ δ κ +2iF¯[ , ]) 1 −2 − G Gµ ν · · iΓghost = + ln Det( D¯ µD¯ ) (6.27) 1 − G Gµ The fermion determinant may be put into a form more akin of the gauge contribution by squaring the background Dirac operator, i (iγµD¯ )2 = D¯ D¯ µ [γµ,γν]F¯a T a (6.28) fµ − fµ f − 4 µν f and thus 1 i iΓfermion = ln Det D¯ D¯ µ [γµ,γν]F¯a T a) (6.29) 1 2 − fµ f − 4 µν f The interpretation of the various contributions to the eﬀective action are then as follows. The square of the background covariant derivative on scalar represents the coupling of a charged scalar particle to the Yang-Mills ﬁeld. The eﬀect on the coupling is diamagnetic. The term coupling F¯ to spin is paramagnetic.

81 In the renormalization of the coupling, we shall only be interested in the contributions proportional to S0[A]. Therefore, the contributions of the diamagnetic and paramagnetic pieces do not interfere, and we may expand the corrections as follows.

iΓfermion = 2 ln Det( D¯ µD¯ ) 1 − f fµ 1 ′ a a′ ′ µ ν a µ′ ν′ a′ ′ dx dx F¯ (x)F¯ ′ ′ (x ) ψ¯[γ ,γ ]T ψ(x)ψ¯[γ ,γ ]T ψ(x ) −64 µν µ ν h f f i Z Z iΓgauge = 2 ln Det( D¯ µD¯ ) 1 − − G Gµ 1 ′ a a′ ′ abc b c a′b′c′ b′ c′ ′ dx dx F¯ (x)F¯ ′ ′ (x ) f a a (x)f a a (x ) (6.30) −2 µν µ ν h µ ν µ ν i Z Z f The coeﬃcients in front of the double integrals arise as follows. Γ1 has a sign because it contains a closed fermion loop; it has a factor i2/2 from second order perturbation− theory; a factor of ( i/4)2 from the F¯-spin coupling in the action and an overall factor of 1/2 because − gauge 2 we took the square. Γ1 has a factor i /2 from second order perturbation theory; a factor of (2i)2 from the F¯-spin coupling in the action and a factor of (i/2)2 from the normalization of the structure constants f abc. The correlators are easily evaluated,

′ ′ ′ ψ¯[γµ,γν]T aψ(x)ψ¯[γµ ,γν ]T a ψ(x′) (6.31) h f f i d d ′ ′ ′ d k ′ d p i i = I(f)δaa tr[γµ,γν][γµ ,γν ] e−ik(x−x ) − (2π)d (2π)d p2 · (p + k)2 Z Z Using the γ-trace formula,

′ ′ ′ ′ ′ ′ tr[γµ,γν][γµ ,γν ]= 16(ηµµ ηνν ηµν ηνµ ) (6.32) − − evaluating the Feynman integral in dimensional regularization, and retaining only the di- vergent part, we have,

′ ′ ′ 2i ′ ′ ′ ′ ′ ψ¯[γµ,γν]T aψ(x)ψ¯[γµ ,γν ]T a ψ(x′) = I(f)δaa (ηµµ ηνν ηµν ηνµ )δ(x x′) (6.33) h f f i π2ǫ − −

The correlator of the aµ-ﬁelds is similarly evaluated,

′ ′ ′ ′ ′ i ′ ′ ′ ′ ′ f abcab ac (x)f a b c ab ac (x′) = I( )δaa (ηµµ ηνν ηµν ηνµ )δ(x x′) (6.34) h µ ν µ ν i −8π2ǫ G − −

Furthermore, the group theory dependence of the scalar determinants may be factored out as follows. For any representation r of , we have G 1 iI(r)Γ = ln Det( D¯ µD¯ ) (6.35) d −2 − r rµ 82 where the reduced eﬀective action Γd is independent of the representation r. Combining the above result, we have

scalar Γ1 = I(s)Γd (6.36) Γghost = 2I( )Γ 1 − G d 1 Γfermion = 4I(f)Γ I(f) F¯a F¯aµν 1 − d − 16π2ǫ µν Z 1 Γgauge = +4t ( )Γ + I( ) F¯a F¯aµν 1 2 G d 8π2ǫ G µν Z The normalization of the fermion vacuum polarization graph was computed long ago, and we have 1 Γfermion = δZ F¯a F¯aµν 1 4g2 3 µν Z g2 δZ = I(f) (6.37) 3 −6π2ǫ Therefore, i Γ = F¯a F¯aµν (6.38) d −192π2ǫ µν Z From this we calculate, 1 1 1 I I I 2 2 = 2 +2 ( ) 4 (f)+ (s) (diamagnetic) g − g0 48π ǫ G − 1 + 24I( )+12I(f) (paramagnetic) 48π2ǫ − G 1 = 22I( )+8I(f)+ I(s) (6.39) 48π2ǫ − G These results are exactly the ones found by direct, but painful, Feynman diagram calcula- tions.

83 7 Lattice ﬁeld theory – Lattice gauge Theory

Although most phenomena in electro-weak interactions at suﬃciently low energy, and most phenomena in the strong interactions at high energy are well-approximated by a pertur- bative evaluation in quantum ﬁeld theory, there is also a host of phenomena that require understanding the dynamics at strong coupling beyond perturbation theory. Conﬁnement in the strong interactions is one such non-perturbative eﬀect, while dynamical symmetry breaking, the triviality of λφ4, pair production in strong electric ﬁelds, and the dynamics of solitons and magnetic monopoles provide further examples. In very rare examples are exact solutions available. This happens almost exclusively in two space-time dimensions, where their exact solutions often provides invaluable guid- ance for understanding non-perturbative phenomena in higher dimensions. The Schwinger model, sine-Gordon theory (including solitons), the Gross-Neveu model of dynamical sym- metry breaking, and the ‘t Hooft model of two-dimensional QCD are all such examples that are either integrable, or very close to integrable. In supersymmetric Yang-Mills theories in four space-time dimensions, powerful methods which result from the holomorphic dependence on chiral superﬁelds, and various duality symmetries, may be used compute the leading parts of the low energy eﬀective action. of the theory in an exact manner. Modern methods of AdS/CFT allow one to access the ultra-strong coupling regime for ﬁeld theories with large numbers of ﬁelds (the so-called large N-limit). When integrability, supersymmetry, or any other special circumstances are not available, however, one resorts to lattice approximations of ﬁeld theory. In this section, we shall formulate ﬁeld theories, including Yang-Mills theories on lattices, and develop some basic analytical and numerical techniques to study their dynamics.

7.1 Lattices To formulate a QFT in d space-time dimensions on a lattice, we ﬁrst reformulate the QFT in Euclidean space-time by analytically continuing to imaginary time. This procedure has already been used when computing Feynman diagrams, and Euclidean correlators. Next, we need to choose a lattice to approximate Rd. There are of course zillions of possible choices. Any choice of lattice will break the translational and rotational symmetries of Rd. Some lattices, however, will preserve a discrete subgroup of translations and rotations, and it will almost always be advantageous to choose a lattice that has as large a discrete symmetry group as possible. The square lattice (a generic term used for the lattice in Rd generated by an orthonormal frame of vectors) is about as good as one can get for generic dimension d, and we shall restrict attention here to square lattices only. The square lattice Λ = Zd in Rd has a single length scale, the so-called lattice spacing a which sets the distance between any two nearest neighbors on the lattice. The lattice

84 spacing a should be thought of as a UV cutoﬀ on the theory, as distance scales smaller than a do not exist on such a lattice. The symmetries of this lattice are the discrete translations which form the group Zd, along with the symmetry group of reﬂections and rotations of the d-dimensional hypercube, Dd. Actually, it will often be useful to consider not the inﬁnite- size lattice, but rather a ﬁnite, but very large, chunk of the lattice, say of linear length N. Restricting the lattice in this way is important for example in numerical work since the number of variables a computer can deal with must generally be ﬁnite. Thus, the lattice may be wrapped by imposing, for example, periodic boundary conditions. The lattice then becomes Λ = (Z/NZ)d and now has an infrared cutoﬀ scale as well given by Na. Other boundary conditions can also be used, but preserve fewer symmetries.

• • • • • •

• • • 2• • • nx 1 nx • • • x• • •

• • • • • •

• • • • • •

Lattice points x are labelled by d integers x =(m1a, m2a, , mda). The d unit vectors µ ··· emanating at a point x will be denoted by nx. For the two-dimensional case, the lattice is µ shown in the above ﬁgure. By translation invariance on a square lattice the vectors nx are in fact independent of x, so we shall drop the subscript x, and simply use nµ.

7.2 Scalar ﬁelds on a lattice A scalar ﬁeld on the lattice is a function of the lattice points, also referred to as lattice ˆ sites, and will be denoted φ(x). A lattice derivative ∂µ acting on φ(x) in direction µ is deﬁned as follows, ∂ˆ φ(x)= φ(x + anµ) φ(x) (7.1) µ − It is now straightforward to write down the action for a canonical real ﬁeld φ, a2 S [φ]= ∂ˆ φ(x)∂ˆ φ(x)+ a4V (φ(x)) (7.2) Λ 2 µ µ x∈Λ µ ! X X 85 As the lattice spacing a tends to 0, and assuming that φ(x) varies slowly as a function of x, the classical lattice action SΛ[φ] tends to the continuum action,

1 S[φ]= ddx ∂ φ∂ φ + V (φ) (7.3) 2 µ µ Z using the following relations,

4 d 1 a d x ∂ˆµφ(x) ∂µφ(x) (7.4) d → a → xX∈Λ Z The lattice partition function is deﬁned by,

−SΛ[φ] ZΛ = dφ(x) e (7.5) x∈Λ ! Y Z and lattice correlators of φ are given by,

1 φ(x ) φ(x ) = dφ(x) φ(x ) φ(x ) e−SΛ[φ] (7.6) h 1 ··· p iΛ Z 1 ··· p Λ x∈Λ ! Y Z It will be understood that this prescription for the correlators is for the bare correlators. To renormalize the correlators, counter-terms will need to be added to the bare action. In continuum Lorentz-invariant QFT, these counter-terms are subject to the symmetries of the action and the measure. The same is true for the lattice formulation, but the amount of symmetry has now been reduced. There remains discrete translation invariance, which requires the counter-terms to have no explicit dependence on x. But there also remains discrete rotations and reﬂections, which are very helpful in constraining possible counter- terms. Consider for example kinetic terms. Since we no longer have the symmetry of continuous rotations in Rd, one may wonder how the counter-terms for the kinetic term are restricted. Take for example,

∂ˆµφ ∂ˆνφ (7.7)

When µ = ν, such terms are forbidden by reﬂection in the direction µ, leaving ν alone. In turn,6 all terms with µ = ν must be equal because of discrete rotations. But then we actually will have full rotation invariance in the continuum, so the restrictions by the discrete symmetries are very powerful. Although space-time symmetries of translations and rotations are broken by the lattice, internal continuous symmetries are naturally maintained. For example, take a theory with

86 N real scalar ﬁelds which we assemble into a column vector φ. It is straightforward to generalize the case with one scalar, and our action is now,

a2 S [φ]= ∂ˆ φt(x)∂ˆ φ(x)+ a4W (φtφ(x)) (7.8) Λ 2 µ µ x∈Λ µ ! X X for some potential of the square of the scalar ﬁeld W . It is manifest that the action is invariant under global SO(N) rotations g SO(N) of the the scalar ﬁeld multiplet, φ(x) φ′(x) = gφ(x). One may similarly∈ consider an SU(N)-invariant theory of N → complex scalar ﬁelds.

7.3 Lattice gauge theory Scalar ﬁelds live on lattice sites, but gauge ﬁelds live on links. It is not that hard to see why this is a good idea. Gauge transformations under the group G acting on a multiplet of N complex scalar ﬁelds φ(x) take the following form,

φ(x) φ′(x)= g(x)φ(x) g(x) SU(N) (7.9) → ∈ where φ transforms in an N-dimensional unitary representation of G, and g acts in this T same representation. Potential terms by themselves are functions of the ﬁeld φ only, but do not depend on its derivatives, and as a result invariance under global SU(N) rotations implies invariance under local SU(N) gauge transformations. This is just as it was in the case of the continuum theory.

• • • • • nµ − • • • • • nν nν − nµ • • x• • •

• • • • •

To make the kinetic terms invariant, we need a gauge ﬁeld that produces a gauge invariant out of scalar ﬁelds evaluated at diﬀerent lattices sites. Indeed, expanding the

87 lattice kinetic term gives, ∂ˆ φ†∂ˆ φ = (φ(x + nµ)† φ(x)†)(φ(x + nµ) φ(x)) µ µ − − = (φ(x + nµ)†φ(x + nµ)+ φ(x)†φ(x) φ(x + nµ)†φ(x) φ(x)†φ(x + nµ) (7.10) − − The two terms on the second line above are gauge invariant, since they only involve scalar ﬁelds evaluated at the same lattice sites. In fact, they are equivalent to scalar mass terms. The two terms on the last line are not gauge invariant, as the scalar ﬁelds are evaluated at diﬀerent lattice sites. What we need is a variable U(x, nµ) that lives on the link between x and x + nµ, and transforms inversely of the scalars in the representation at the lattice site x and † at the site x + nµ, T T φ(x) φ′(x)= g(x)φ(x) → U(x, nµ) U ′(x, nµ)= g(x)U(x, nµ)g(x + nµ)† (7.11) → It is natural to identify the hermitian conjugate as follows, U(x + nµ, nµ)= U(x, nµ)† (7.12) − We shall shortly identify U with the gauge ﬁeld in the continuum limit. We are now ready to write down a fully gauge invariant coupling of a scalar to the variable U(x, nµ),

d φ(x)†U(x, nµ)φ(x + nµ)+ φ(x + nµ)†U(x, nµ)†φ(x)+ a4W (φ†φ(x)) (7.13) x∈Λ µ=1 ! X X The mass term parts of the original version of the kinetic term have here been absorbed into the potential, without loss of generality. It remains to construct a gauge invariant action for the variable U itself. Clearly the trace of product of link variables U(x, nµ) around any closed loop on the lattice will be gauge invariant. The simplest, and most local choice is to take that loop to be a single plaquette, so that the pure gauge action is given by, c S [U]= tr U(x, nµ)U(x + nµ, nν)U(x + nµ + nν, nµ)U(x + nν, nν ) (7.14) Λ −g2 − − Xx∈Λ Xν6=µ with a normalization factor c that we shall determine from the a 0 limit of the action. Next, we need to make contact with the continuum action for the→ gauge ﬁeld and scalars as a 0. In the continuum, we know one object that transforms exactly as U(x, nµ) does, and it→ is given by the following formula,

x+nµ α P exp i dy Aα(y) (7.15) Zx integrated over yα(τ) along the straight line segment from x to x + nµ, parametrized by τ. Here the symbol P preceding the exponential stands for path-ordering. In the

88 approximation where the lattice spacing a is small (compared to the typical length scale of variations of the gauge ﬁeld Aµ, we may identify U with A as follows,

U(x, nµ) = exp iaA (x)+ (a2) (7.16) µ O n o Within the same approximation, the plaquette variable is obtained as follows,

U(x, nµ)U(x + nµ, nν)U(x + nµ + nν , nµ)U(x + nν, nν ) − − = exp i a2 F (x)+ (a3) µν O n 1 o = I + i a2 F (x) a4F (x)2 + (a5) (7.17) µν − 2 µν O

Taking the trace in group space cancels out the term linear in Fµν but retains the term quadratic in Fµν , so that we obtain,

tr U(x, nµ)U(x + nµ, nν)U(x + nµ + nν, nµ)U(x + nν, nν ) − − 1 = dim I( )a4F a (x)F a (x)+ (a5) (7.18) T − 2 T µν µν O Hence the normalization constant c should be chosen as follows, 1 c = (7.19) 2 I( ) T In the standard normalization of representation matrices of SU(N) or SO(N) in the deﬁn- 1 ing N-dimensional representations, we have I( ) = 2 , and the normalization constant reduces to c = 1. T

7.4 The integration measure for the gauge gauge ﬁeld The integration measure for the link variables U(x, nµ), which take values in G, must be invariant under the gauge transformations acting on U(x, nµ), namely a transformation g(x) acting to the left of U, and an independent transformation g(x + nµ) acting to its right. The problem is thus to ﬁnd an integration measure for U G that is invariant under both left and right transformations,7 ∈

U U ′ = g Ug−1 (7.20) → L R To construct an invariant measure, it suﬃces to have an invariant metric since, by standard Riemannian geometry, a measure gives rise to a unique volume form, and the symmetries

7For unitary representations, g−1 = g†, but we shall provide here a general discussion without making this assumption.

89 of the metric are preserved by the measure. The metric may be constructed from either left or right-invariant diﬀerential forms, deﬁned respectively by U −1 dU and dU U −1. By inspection, it is clear that U −1 dU is invariant under U U ′ = gU, while dU U −1 is invariant under U U ′ = Ug−1. The following metric on→G is invariant under both left and right transformations,→

1 ds2 = tr U −1dU U −1dU (7.21) −I( ) T The corresponding measure, denoted [dU] is the GL GR-invariant volume form on G, and is referred to as the Haar measure. It will often be× convenient to normalize the Haar measure so that the volume of G is one,

[dU] = 1 (7.22) ZG although this normalization is by no means standard. In the special case of G = U(1), the Haar measure is simply dθ, where θ is the angle parametrizing the circle S1 = U(1). For SU(2) the metric ds2 will be identical to the round metric on the sphere S3 = SU(2), and the Haar measure is the standard volume form on S3. For higher compact Lie groups, the Haar measure is more complicated, but one usually does not need its explicit form.

7.5 Wilson loops In a theory with only gauge ﬁelds, i.e. link variables U(x, nµ), all gauge-invariant observ- ables may be constructed as follows. Take a closed path C on the lattice, and take the trace of the product of all oriented link variables U(x, nµ) along the path C. This is an inﬁnite set of operators, which tends to be a bit unwieldy. On a square lattice, one may restrict the paths to rectangles of sides Lt, Lx N links. One may think of one of the directions being time-like while the other direction∈ is space-like. the corresponding observables are referred to as Wilson loops,

W (C)=tr U(x, nµ) (7.23) µ (x,nY)∈C The signiﬁcance of the Wilson loop in a theory with scalars and fermions which couple to the gauge ﬁeld may be seen as follows. We take scalars for simplicity. Consider a scalar ﬁeld φ with very large mass (compared to the scale of the dynamics of the gauge ﬁeld). If its mass is large, then it will remain virtually immobile on the lattice for the time span of

90 the gauge dynamics. This means that we can integrate out the φ-ﬁeld by performing just local Gaussian integrals, using the scalar correlator,

φi(x)φ†(y) = δ δi (7.24) h j iφ x,y j The only contributions that involve the gauge ﬁeld are obtained when links follow one another on the lattice, in the time-like direction nt,

φ†(x nt)U(x nt, nt)φ(x)φ†(x)U(x, nt)φ(x + nt, nt) h − − iφ(x) = φ†(x nt)U(x nt, nt)U(x, nt)φ(x + nt, nt) (7.25) − − where stands for integrating out φ at the site x only. h···iφ(x)

• • • • • •

• • • • • • Lx W (C)

• • • • • •

• • • • Lt • •

• • • • • •

Now consider the vacuum expectation value of the Wilson loop operator W (C) for a rectangular closed loop C of dimensions L , L N. In the limit where L 1 and x t ∈ t ≫ becomes much larger than the dynamical scale of system, the vacuum expectation value decays exponentially with Lt with a coeﬃcient set by the lowest energy state in the channel under consideration,

W (C) = exp E a L + (L0) (7.26) h i − g t O t The channel under consideration here is the presence of a very heavy scalar in the repre- sentation of the gauge group, and another very heavy scalar in the complex conjugate T ∗ representation , separated by the distance Lx in terms of lattice units a. Since both scalars are veryT heavy, the energy of the state corresponds simply to the static potential Vqq¯(r) at distance r = aLx between the two particles. Conﬁnement of either scalar or fermion quarks corresponds to this potential being linear with r for r large compared to

91 the dynamical scale of the system. The coeﬃcient of proportionality is referred to as the string tension σ, V (r)= σr + (r0) (7.27) qq¯ O in the limit of large r. With a linear potential, the force between the quarks is constant at large distances, and it would take an inﬁnite amount of energy to separate the two quarks to asymptotic states. Therefore, in the presence of conﬁnement, we expect the following behavior for the Wilson loop expectation value, W (C) = exp σ a2 L L + (L + L ) (7.28) h i − t x O t x Coulomb behavior would instead produce perimeter law fall-oﬀ.

7.6 Lattice gauge dynamics for U(1) and SU(N) Now consider the dynamics of the pure U(1) gauge theory, and compute the expectation value of a Wilson loop of area L L for L , L 1. To compute it, we need the integration t x t x ≫ over the links of the following combinations,

[dU]U m (U †)n = δmn (7.29) ZU(1) where we may parametrize U = eiθ for 0 θ 2π. We compute the expectation value for large values of the coupling g2. Every time≤ we≤ bring down a plaquette from the expansion of the lattice action we multiply by a power of 1/g2. Thus, the lowest order, and dominant contribution is obtained by bringing down the smallest number of plaquettes which gives a non-zero result. This is achieved by precisely ﬁlling up the entire Wislon loop by one plaquette per square.

• • • • • •

• • • • • • Lx W (C)

• • • • • •

• • • • Lt • •

• • • • • •

92 The value of the expectation value is then given by,

1 LxLy W (C) = (7.30) h i g2 so that we deduce the expression for the string tension as follows,

σ a2 = ln g2 (7.31)

The calculation in the non-Abelian case is basically identical, although for SU(N), we now use slightly more delicate identities,

[dU] = 1 (7.32) ZSU(N) and

j † j′ 1 j′ j [dU] U (U ) ′ = δ δ ′ (7.33) i i N i i ZSU(N) But the result is essentially the same. How can it be true that, at strong coupling g2, both the Abelian U(1) gauge theory and the non-Abelian SU(N) Yang-Mills theory obey the same conﬁnement law ? The explanation is that the U(1) gauge dynamics undergoes a phase transition from conﬁning to non-conﬁning when the coupling constant g2 is lowered as the lattice spacing a is being decreased in physical units.

93 8 Global Internal Symmetries

In Nature, diﬀerent quantum states may have similar physical properties and therefore may be grouped together into a multiplet of states which are transformed into one another under the action of a certain group. This procedure is familiar, of course, from space rotations or Poincar´etransformations. In both of these cases, the group transformations actually correspond to exact symmetries of space-rotations and Poincar´etransformations. But the symmetries need not always be exact to provide a useful framework for assembling diﬀerent states and ﬁelds into a single multiplet, and this is often the case with internal symmetries. Internal symmetries were considered by Heisenberg, who noticed that – from the point of view of the strong interactions – there is little distinction between a proton and a neutron. For one, their masses are surprisingly close. This led Heisenberg to assemble the proton and neutron states and ﬁelds into a doublet,

p m = 938.2 MeV p (8.1) n m = 939.5 MeV n

The group acting on this doublet is referred to as isospin SU(2)I . The strong interactions are insensitive to SU(2)I isospin rotations of the proton - neutron doublet. This observation leads one to postulate that the group SU(2)I is a global symmetry of the strong interactions, either exact, or approximate. From the point of view of electromagnetism, the proton and neutron are very diﬀerent, since the proton is electrically charged while the neutron is neutral, and therefore SU(2)I isospin is not a symmetry of the electromagnetic interactions. Global symmetries may be useful even if they are only approximate. In the case of SU(2)I , the symmetry is broken by mass and by electromagnetic interactions. It is also broken by the weak interactions. More modern examples are found by considering quarks instead of hadrons. The presently known quarks form the following array,

uc cc tc c = color r, w, b ∈{ } dc sc bc (8.2)

The strong interactions are sensitive only to the color of the quarks and not to their ﬂavor. Extending the notion of isospin to six quarks, the symmetry group would naturally be SU(6). Given that the masses of the quarks are approximately as follows,

mu = 5 MeV mc =1.5 GeV mt = 170 GeV

md = 7 MeV ms = 100 MeV mb =4.9 GeV (8.3) it is clear that SU(6) is very badly broken and little or no trace of it is left in physics. The u, d and s quarks may still be considered alike from the point of view of the strong

94 interactions as their masses are small compared to the typical strong interaction scale of the proton for example, and the associated SU(3) symmetry is useful. In quantum ﬁeld theory, elementary particles are described by local ﬁelds, whose dynam- ics is governed by a Lagrangian (or when there is no Lagrangian by the operator product algebra). Thus, to describe the above type of internal symmetries, we shall consider ﬁeld theories that are invariant under global symmetries. In classical ﬁeld theory, a symmetry is a transformation of the ﬁelds that maps any solution of the Euler-Lagrange equations into a solution of the same equations. Generally, the symmetry will map a solution to a diﬀerent solution. A speciﬁc solution may, however, be mapped into itself and be invariant under the transformation. Two distinct concepts of symmetry emerge from the above discussion; the symmetry of an equation versus the symmetry of a solution to the equation. At the much more elementary level of algebraic equations, for example, the simple equation x3 x = 0 has solutions x =0, 1, and has the following symmetry properties, − ± the symmetry of the equation is given by the transformation P : x x; • → − P maps the solution x = 0 into itself, so that x = 0 is invariant under P ; • P maps the solution x = +1 into the third solution x = 1 and vice-versa, so that • the solution x = +1 is not invariant under the symmetry −P of the equation.

The concept of symmetry for an object, such as a circle, a triangle, a square, a cube and so on, dates back to ancient Greek times, and probably to even earlier times. The concept of a symmetry of an equation, however, was considered only in relatively recent times by Lagrange and Galois in the early 19-th century.

8.1 Symmetries in quantum mechanics In quantum mechanics, for systems with a ﬁnite number of degrees of freedom, a symmetry is realized via a unitary operator U in Hilbert space. This holds for continuous as well as discrete symmetries, with the exception of time-reversal which is realized by an anti-unitary transformation. Internal symmetries commute with the Hamiltonian and obey,

U †HU = H (8.4)

Under an internal symmetry, the eigenstates of the Hamiltonian associated with a given eigenvalue E of H transform under a unitary representation (E). To see this, consider the U eigenspace of eigenstates E,α of the Hamiltonian H with given energy eigenvalue E, and α labeling the diﬀerent states| withi the same eigenvalue E, so that we have H E,α = E E,α . | i | i Next, calculate the action of the symmetry transformation U,

H U E,α = UH E,α = E U E,α (8.5) | i | i | i 95 It follows that U E,α has energy E, and must therefore be a linear combination of the states E, β , given| asi follows, | i U E,α = (E) E, β (8.6) | i U αβ| i Xβ Unitarity of U then guarantees unitarity of (E) which satisﬁes (E)† (E)= I, where I is the identity operator in the eigenspace of UH for eigenvalue E. ThusU internalU symmetries organize the quantum states at given energy into unitary representations of the symmetry group. This fact is often referred to as the symmetry is realized in the Wigner mode, or the symmetry is realized linearly.

8.2 Symmetries in quantum ﬁeld theory In quantum ﬁeld theory, there are essentially three possibilities. The starting point may be a classical Lagrangian assumed to have a symmetry in the classical sense. Upon quantization, it may be that this symmetry survives quantization, or alternatively that it suﬀers an anomaly and therefore ceases to be a symmetry of the system at the quantum level. U(1) gauge invariance in QED is an example of the ﬁrst case, while U(1)A symmetry in QED is an example of the second case. Recall that anomalies manifest themselves when regularization and renormalization eﬀects necessarily destroy the symmetry. When the symmetry survives quantization, there are two further possibilities, which are distinguished by the properties of the vacuum.

1. The symmetry leaves the vacuum invariant. This case is very much like the Wigner mode in quantum mechanics : the states in Hilbert space transform under unitary transformations. Global U(1) symmetry in QED is again an example of this mode; Poincar´einvariance is another, since all states are unitary multiplets of ISO(1,3).

2. The symmetry does not leave the vacuum invariant. This case has no counterpart in quantum mechanics (except when supersymmetry is considered). The symmetry is said to be spontaneously broken or in the Nambu-Goldstone mode.

The spontaneously broken case is similar to classical solutions of a given classical Euler- Lagrange equations never realizing the symmetries of the equation, though the diﬀerence is of course that there is no concept of vacuum in classical physics other than it being the solution with the largest amount of symmetry.

8.3 Continuous internal symmetry with two scalar ﬁelds A system with a single real ﬁeld and a polynomial potential does not have interesting continuous internal symmetries. The case with two real ﬁelds φ1,φ2, or equivalently a

96 complex ﬁeld φ =(φ1 + iφ2)/√2, subject to the discrete symmetries,

φ1 φ1 φ1 +φ1 R1 → − R2 → (8.7) φ +φ φ φ 2 → 2 2 → − 2 and renormalizability in four space-time dimensions gives,

1 1 1 1 λ2 λ2 λ2 = ∂ φ ∂µφ + ∂ φ ∂µφ m2φ2 m φ2 1 φ4 φ2φ2 2 φ4 (8.8) L 2 µ 1 1 2 µ 2 2 − 2 1 1 − 2 2 2 − 4! 1 − 12 1 2 − 4! 2

The symmetries R1, R2 are mutually commuting involutions, so that the symmetry group is Z Z where Z is the group Z = 1, 1 . 2 × 2 2 2 { − } For special arrangements of masses and couplings, however, the symmetry is actually Z Z 2 2 2 2 Z larger than 2 2. Speciﬁcally, if we require m1 = m2 and λ1 = λ2, we have a further 2 × 2 2 which acts by interchanging φ1 and φ2. Furthermore, when m1 = m2 and λ1 = λ = λ2, the Lagrangian has a continuous SO(2) U(1) global symmetry as well, ∼ ′ φ1 φ1 cos α sin α φ1 ′ = − (8.9) φ2 → φ sin α cos α φ2 2 where α is independent of x, but otherwise arbitrary. One way of seeing the presence of the symmetry more explicitly is by noticing that may be written in terms of a potential 2 2 L which only depends on the φ1 + φ2. More generally, any Lagrangian of the form, 1 1 = ∂ φ ∂µφ + ∂ φ ∂µφ V (φ ,φ ) (8.10) L 2 µ 1 1 2 µ 2 2 − 1 2 where the potential is invariant under SO(2), and is therefore of the form V (φ1,φ2) = 2 2 W (φ1 + φ2) for some function W . Alternatively, one may regroup the two real ﬁelds φ1,φ2 into a single complex ﬁeld φ = (φ1 + iφ2)/√2, in terms of which the U(1) acts by φ φ′ = eiαφ and the Lagrangian take the form, → = ∂ φ¯ ∂µφ W (2φφ¯ ) (8.11) L µ − Henceforth, the following SO(2)-invariant potential will be considered,

1 λ2 V (φ ,φ )= m2(φ2 + φ2)+ (φ2 + φ2)2 (8.12) 1 2 2 1 2 4! 1 2 For real λ2 and λ2 < 0, the potential is unbounded from below, which is physically unac- ceptable. Henceforth, we shall assume that λ2 and m2 are real, and that λ2 > 0, so that the potential is bounded from below for all values of m2, including m2 < 0. Next, the semi-classical vacuum conﬁguration will be determined and its symmetry properties will be examined in light of the preceding discussion.

97 8.3.1 The symmetric or unbroken phase m2 > 0 The form of the potential for m2 > 0 is represented in Fig 1 (a). The potential has a unique minimum at φ1 = φ2 = 0. This conﬁguration is the semi-classical approximation to the vacuum of the theory, and is invariant under SO(2) rotations. Thus, the system has unbroken symmetry. To study the spectrum, the ﬁeld is decomposed into creation and annihilation operators, and the Fock space construction is used. The starting point is the vacuum, corresponding here to the classical conﬁguration φi = 0. The ﬁeld is then decomposed as follows,

d d k 1 −ik·x † ik·x φi(x)= d ai(k)e + ai (k)e i =1, 2 (8.13) (2π) 2ωk Z where ω = √k2 + m2, k =(ω , k) and k x = k xµ, with the commutation relations, k µ k · µ a (k), a†(l) =2ω (2π)d−1δ δ(k l) (8.14) i j k ij − h i The generator of SO(2) rotations is deﬁned by, dd−1k Q = a† (k)a (k) a† (k)a (k) (8.15) (2π)d−1 2 1 − 1 2 Z and acts upon the creation and annihilation operators as follows,

[Q, ai(k)] = εijaj(k) † − † [Q, ai (k)] = +εijaj(k) (8.16) where ε12 = 1. The generator Q commutes with the Hamiltonian. The ground state is deﬁned by a (k) 0 = 0 for all k Rd, and for both i = 1, 2. The i | i ∈ equations deﬁning 0 are invariant under SO(2) and so is 0 , which satisﬁes Q 0 = 0. The excited states (to| i lowest order in the interaction) are constr| i ucted by applying| creationi operators to the vacuum, a† (k ) a† (k ) 0 (8.17) i1 1 ··· in n | i Since 0 is invariant under SO(2) it is manifest that the states transform under unitary representations| i of SO(2). This is the Wigner mode.

8.3.2 The spontaneously broken phase m2 < 0 The form of the potential for m2 < 0 is represented in Fig. 1 (b). The extrema of the potential are governed by the equations,

2 ∂V 2 λ 2 2 = m φi + (φ1 + φ2)φi =0 i =1, 2 (8.18) ∂φi 6

98 V V

φ2 0 φ2 0

φ1 φ1

2 2 Figure 10: Quartic Potentials for (a) m > 0 and (b) m < 0 fig:C1 { }

The ﬁrst solution φ1 = φ2 = 0 is isolated and corresponds to a local maximum. The remaining solutions satisfy the relation, m2 φ2 + φ2 = v2 v2 = 6 > 0 (8.19) 1 2 − λ2 and form a continuum of solutions, all of which have the same energy which is the minimum possible value for the potential. The minimum of the potential exhibits a continuous degeneracy. The set of all minimum conﬁgurations is invariant under SO(2), but no single value of φi in this set is invariant. This poses a challenge to the Fock space construction, which assumes uniqueness of the ground state at the outset. We begin by proceeding informally with the Fock space construction, despite this complication.

We shall make the assumption that one can pick a single value of φi and build a Fock space and thus a Hilbert space on this conﬁguration. (The validity of this assumption will be discussed later.) For example, the following choice may be made,

φ1 = v, φ2 = 0 (8.20) The study of the semi-classical spectrum reveals a characteristic feature that will be generic when the vacuum breaks a continuous symmetry. The spontaneously broken symmetry produces a massless boson, often referred to as a Nambu-Goldstone boson. To see this, at the semi-classical level, one proceeds to shifting the ﬁelds,

ϕ1 = φ1 + v ϕ2 = φ2 (8.21) and rewriting the Lagrangian in terms of the ϕi ﬁelds, we ﬁnd, 1 1 λ2 = ∂ ϕ ∂µϕ + ∂ ϕ ∂µϕ 4v2ϕ2 +4vϕ (ϕ2 + ϕ2)+(ϕ2 + ϕ2)2 (8.22) L 2 µ 1 1 2 µ 2 2 − 24 1 1 1 2 1 2 Manifestly, the ﬁeld ϕ2 is massless. A look at Fig 1 (b) readily reveals the origin of this phenomenon. As the semiclassical vacuum conﬁguration has φ1 = v but φ2 = 0, the

99 ﬂuctuations around this point purely in the direction of φ2 = ϕ2 encounter no potential barrier at all and proceed along the ﬂat direction of the potential, remaining at its minimum. This is the Nambu-Goldstone boson. The treatment so far has been semi-classical. In the full quantum ﬁeld theory, one is concerned with the full ground state of the Hamiltonian. In terms of the energy expectation value of any given state ψ>, | ψ H ψ E(ψ) h | | i (8.23) ≡ ψ ψ h | i the ground state ψ0 is deﬁned by the requirement E(ψ0) E(ψ) for all ψ . The following possibilities| i arise, ≤ | i ∈ H

1. U ψ = ψ Wigner mode | 0i | 0i 2. U ψ = ψ Goldstone mode | 0i 6 | 0i The analysis of the full quantum spectrum is more involved than the semi-classical approx- imation shows. We ﬁrst treat an exactly solvable example to illustrate the general ideas and to arrive at the important Coleman-Mermin-Wagner Theorem.

8.4 Spontaneous symmetry breaking in massless free ﬁeld theory Consider massless free ﬁeld theory of a single scalar ﬁeld φ(x) in ﬂat d dimensional space- time Rd, with Lagrangian, 1 = ∂µφ ∂ φ (8.24) L0 2 µ The Lagrangian is invariant under shifts of φ by a real constant c,

φ(x) φ′(x)= φ(x)+ c (8.25) → The corresponding Noether current and charge are

jµ = ∂µφ P = dd−1x j0 = dd−1x ∂0φ (8.26) Z Z Since the theory is massless, it is also invariant under scale and conformal transformations. As usual, we expect that the Hilbert space will arise from the Fock space construction on a unique Poincar´einvariant vacuum, say 0 . Is this vacuum invariant under the translations of φ ? Consider the 1-point function |0iφ(x) 0 . By Poincar´einvariance, this is an x- independent real constant φ . If 0 wereh invariant| | i under a translation φ φ + c, then φ 0 | i → 0 should also be invariant under this transformation. The transformation is by φ φ + c, 0 → 0 which leaves no ﬁnite φ0 invariant.

100 8.4.1 Adding a small explicit symmetry breaking term The nature of the vacuum is always an IR problem. It is natural to try and modify our theory in the IR by a term in the Lagrangian that breaks the symmetry φ φ+c explicitly. In this case, we can actually add a small mass term that will preserve the→ free ﬁeld nature of the problem and hence allow for an exact solution. The new Lagrangian will be (for Euclidean metric), 1 1 = ∂µφ∂ φ + m2(φ φ )2 (8.27) Lm,ϕ 2 µ 2 − 0 2 where φ0 is a real constant and m > 0. The original 0 may be recovered by letting 2 L m 0, in which limit the Lagrangian becomes independent of φ0. For given φ0, the → 2 2 minimum of the potential of m,ϕ is at φ0 for all m > 0. Thus, for any ﬁnite value m > 0, the one-point function is well-deﬁnedL and given by, 0 φ(x) 0 = φ (8.28) h | | i 0 The two-point function is given by, 0 φ(x)φ(y) 0 =(φ )2 + G (r) r2 =(x y)2 (8.29) h | | i 0 d − where the Green function is given by, ddk 1 G (r) = eik·(x−y) d (2π)d k2 + m2 Z ∞ d d k 2 2 = dτ eik·(x−y)e−τ(k +m ) (8.30) (2π)d Z0 Z Carrying out the Gaussian integration in k, we ﬁnd,

∞ d/2−1 1 −d/2 −τm2−r2/4τ 1 m Gd(r)= d/2 dτ τ e = d/2 Kd/2−1(mr) (8.31) (4π) 0 (2π) r Z Actually, G (r) always decays exponentially when r for ﬁxed m > 0. We are most d → ∞ interested, however, in the regime which is closest to the massless regime : this is when 0 < mr 1 instead. The leading behavior in this regime for the cases is given by ≪ 1 G (r) = e−mr 1 2m 1 G (r) ln(mr/2) 2 ∼ −2π Γ(d/2) G (r) d 3 (8.32) d ∼ 2(d 2)πd/2 rd−2 ≥ − We recognize the harmonic oscillator case with d = 1 and the Yukawa potential for d = 3. The cases d 3 actually admit ﬁnite limits when m 0 for ﬁxed r, and the resulting ≥ → 101 limits fall oﬀ to 0 as r . This behavior demonstrates that as d 3, the system gets locked into the vacuum→ expectation ∞ value 0 φ(x) 0 = φ and thus breaks≥ the translation h | | i 0 symmetry in the ﬁeld φ at all m and also in the limit m 0. → The cases d = 1 and d = 2 do not obey this pattern, as no ﬁnite limits exist. The case d = 1 is ordinary quantum mechanics. The ground state for m = 0 is indeed a state uniformly spread out in φ, which will produce divergent 0 φ 0 . The case d = 2 is also h | | i important. The failure of being able to take the limit m 0 is at the origin of the Coleman- Mermin-Wagner Theorem that no continuous symmetry→ can be spontaneously broken in 2 space-time dimensions.

8.4.2 Canonical analysis

To determine the vacuum of the theory, we construct the Hamiltonian H0 in terms of the ﬁeld φ and its canonical momentum π. Upon quantization, UV divergences may be renor- malized by normal ordering. Decomposing the ﬁeld in creation and annihilation operators,

φ(x) = a†(k)e−ik·x + a(k)e+ik·x k Z † H0 = k a (k)a(k) k | | Z 1 P = a(0) a†(0) (8.33) 2i − Here, a general notation is used for the summation over the momenta k. In d-dimensional ﬂat space, the summation corresponds to an integral over the continuous variable k,

dd−1k = d−1 (8.34) k (2π) 2 k Z Z | | but we shall also consider quantization on a torus, where the sum will be discrete. The Hamiltonian is positive and lowest energy is achieved when a state is annihilated by the operators a(k). But, we observe a variant of the usual (for example for massive ﬁelds) situation, since the operator a(0) corresponds to zero energy. Thus, minimum, i.e. zero energy, requires only that all the operators a(k) with k = 0 annihilate the minimum energy state. Its behavior under a(0) is undetermined by the6 minimum energy condition. † The translation charge P only depends upon a(0) a (0) and commutes with H0. Therefore, the zero energy states are distinguished by their− P -charge and may be labeled by this charge, p. The conjugate variable is Q =(a(0)+ a†(0))/2, and satisﬁes [Q, P ]= iV , where V is the volume of space.

102 8.5 Why choose a vacuum ? This question is justiﬁed because in ordinary quantum mechanics such degeneracies get lifted. For example, in the one-dimensional Hamiltonian, 1 1 H = p2 + mλ2(x2 v2)2 (8.35) 2m 2 − a state localized around x = v has a non-zero probability to tunnel to a state localized − around x = +v. In fact the transition probability is given by the imaginary time path integral, familiar from the WKB approximation,

x =+v x = v = x(t) e−S[x]/~ (8.36) h | − i D Z with the boundary conditions x( ) = v and x(+ )=+v. The Euclidean action is −∞ − ∞ given by,

∞ 1 1 S[x]= dt mx˙ 2 + mλ2(x2 v2)2 (8.37) 2 2 − Z−∞ Semi-classically, ~ is small and the integral will be dominated by extrema of S (satisfying the initial and ﬁnal boundary conditions). Solutions are given by the ﬁrst integral,

1 mλ2 mx˙ 2 (x2 v2)2 = 0 (8.38) 2 − 2 − whose solutions are

x (t)= v tanh λv(t t ) (8.39) c ± − o 3 The corresponding action is ﬁnite and given by S[xc]=4/3mλv ; the transition amplitude in this approximation is non-zero and given by,

3 ~ 4mλv x =+v x = v e−S[xc]/ = exp (8.40) h | − i ≈ 3~ These objects are called instantons. The true ground state is a superposition of the states localized at v and v. It would thus be absurd to choose one conﬁguration above the other. − What happens in QFT? Can one still tunnel from one conﬁguration to the other? Let’s investigate this with the help of the semi-classical approximation to the imaginary time path integral.

1 λ2 S[φ]= ddx ∂ φ∂µφ + (φ2 v2)2 (8.41) 2 µ 4 − Z 103 Assume there exists a solution φc(x), which is a minimum of the classical action. Consider σ its scaled version φ (x)= φc(σx),

1 λ2 S[φσ]= ddx σ2−d∂ φ ∂µφ + σ−d(φ2 v2)2 (8.42) 2 µ c c 4 c − Z Now σ = 1 better be a stationary point.

∂S[φσ] λ2 = ddx ∂ φ ∂µφ (2 d) d (φ2 v2)2 (8.43) ∂σ µ c c − − 4 c − σ=1 Z

But for d 2, both terms are negative, and cancellation would require, ≥ µ 2 2 ∂µφc∂ φc = 0 and φc = v (8.44)

Only for d = 1 can there be non-trivial solutions. This result is often referred to as Derrick’s theorem. Thus, for a scalar potential theory with d 2, it is impossible to tunnel from one localized state to the other via instantons. ≥ Even when vacuum degeneracy occurs for space-time dimension larger than 1, it may still be possible to have tunneling between the degenerate vacua. Famous examples of this phenomenon are precisely when instantons with ﬁnite action exist in space-time dimension larger than one, Non-linear σ-models in d = 2; • Higgs model with U(1) gauge symmetry in d = 2; • QCD in d = 4 with θ-vacua. • Finally, the Mermin-Wagner-Coleman Theorem states that in d = 2, a continuous (bosonic) symmetry can never be spontaneously broken. The reason for this resides in the fact that the massless Goldstone bosons that would arise through this breaking cannot deﬁne a properly interacting ﬁeld theory, since their infrared behavior is too singular.

8.6 Generalized spontaneous symmetry breaking (scalars)

Consider a ﬁeld theory of n real scalar ﬁelds φi, with i = 1, , n, arranged in a column vector φ. For simplicity, assume that the Lagrangian has a standard··· kinetic term as well as a potential V , 1 = ∂ φt∂µφ V (φ) (8.45) L 2 µ − It will be further assumed that V (φ) is bounded from below, and has its minima at ﬁnite values of φ. (Runaway potentials such as eφ generate special problems.)

104 It is assumed that is invariant under a continuous symmetry group G acting linearly • on φ; for this reason,L these models are often referred to as linear σ-models; Invariance of the kinetic term requires that G be a subgroup of SO(N); • The potential V (φ) must be invariant under the action of G; • The ground state of V (φ) is assumed to be degenerate, and all ground states are • mapped into one another under the action of G. When any given minimum φ0 is invariant under a subgroup H of G, which is strictly smaller than G, then the symmetry group G of the Lagrangian is spontaneously broken to the symmetry group H of the vacuum. Note that H may be the identity group.

8.6.1 Goldstone’s Theorem The Goldstone bosons associated with the spontaneous symmetry breaking from G H are described by the coset space G/H, and their number is dim G/H = dim G dim→H. − To prove the theorem, we use the fact that G acts transitively on G/H; i.e. without ﬁxed points. Every point in G/H may be obtained by applying some g G to any given reference point in G/H. Actually, this is true for the action of G on G (a∈ special case with H = 1). Indeed, take a reference point g0 in G then to reach a point g G, it suﬃces to −1 ∈ multiply to the left by g g0 . Thus, it suﬃces to analyze the problem around any point on G/H, and then carry the result to all of G/H. Take the reference point in G/H to be φ0 which satisﬁes, ∂V = 0 (8.46) ∂φi 0 φi=φi

Now the set of all solutions transform under G. Decompose the Lie algebra of G into its subalgebra and the orthogonal complement under the Cartan-Killing form,G : H M ρ(T a) φ0 =0 T a = ∈ H (8.47) G H⊕M ρ(T a) φ0 =0 T a 6 ∈M The Goldstone boson ﬁelds now correspond to the directions of the potential from φ0 where φ0 is charged, but the potential is still at its minimum value. These are precisely the directions in G/H represented here by its tangent space . M 8.6.2 Example 1

The N scalar ﬁelds φi, for i =1, , N, are grouped into a column vector φ, subject to the following Lagrangian, ··· 1 = ∂ φt∂µφ W (φtφ) (8.48) L 2 µ − 105 invariant under SO(N). Assume the minimum of v occurs at φtφ = v2 > 0. The symmetry of the Lagrangian is G = SO(N), but the symmetry of any vacuum conﬁguration is only H = SO(N 1) so that G/H = SN−1 and there are N 1 Goldstone Bosons. − −

8.6.3 Example 2

2 The N scalar ﬁelds Mij for i, j =1, , N may be regrouped into a real square matrix M valued ﬁeld subject to the Lagrangian,···

1 1 λ2 = tr ∂ M T ∂µM m2trM T M tr(M T M)2 (8.49) L 2 µ − 2 − 4! This Lagrangian is invariant under the following G = O(N) O(N) transformations, L × R M g MgT g O(N) , g O(N) (8.50) → L R L ∈ L R ∈ R Any stationary point M of the potential obeys the equation

6m2 M T M v2I M T =0 v2 = (8.51) − N − λ2 When m2 > 0, the solution M = 0 is unique and the full G symmetry leaves the vacuum invariant. The interesting case is when m2 < 0, which we now analyze systematically. The eigenvalues of M T M must either vanish or equal v2. The general solution to this equation may be organized according to the number n of vanishing eigenvalues, with T 2 0 n N, and M M = v In. For each n, the conﬁguration deﬁnes a stationary point of≤ the potential,≤ but only one case n = N is an absolute minimum of the potential, which corresponds to the vacuum for the theory. This may be veriﬁed by substituting the solutions for M T M into the potential and evaluating the potential. The general solution for M to the equation M T M = v2I is given by M = vg with g SO(N), which under an N ∈ O(N) O(N) transformation is equivalent to M = vI . The residual symmetry of L × R N M = vIN is calculated as follows,

g (vI ) gT =(vI ) g = g O(N) (8.52) L N R N ⇒ L R ∈ D Hence the dynamics of this model shows the following pattern, 1 O(N) O(N) O(N) + N(N 1) Goldstonebosons (8.53) L × R → D 2 − The Goldstone boson ﬁelds parametrize the coset G/H = (O(N) O(N) )/O(N) , L × R D following the general pattern announced earlier. Other examples of ﬁeld theories may be analyzed with the same semi-classical methods.

106 8.7 The non-linear sigma model Often one is interested in describing only the dynamics of the Goldstone bosons at low energy/momentum. All other excitations are then massive and decouple. In this limit, the remaining Lagrangian consists of the kinetic term on the scalar ﬁelds φa, subject to the constraint that the potential on these ﬁelds equals its minimum value, 1 = ∂ φt∂µφ V (φ)= V (φ0) (8.54) L 2 µ The Lagrangian is still invariant under G, and so is the condition for the minimum. But G is spontaneously broken to H, and the model only describes the Goldstone bosons. The true dynamical ﬁelds are no longer the ﬁelds φ, since they are related to one another by the constraint V (φ)= V (φ0). Introducing true dynamical ﬁelds may be done by solving the constraint. It is best to illustrate the procedure with an example. Consider a vector φ of n real scalar ﬁelds φ , i =1, , N with a quartic potential, i ··· 1 λ 6m2 V (φ) = m2(φtφ)+ (φtφ)2 v2 = > 0 (8.55) 2 4! − λ

0 Let the vacuum expectation value of φi be φi = vδi,N , and denote the vector of the remain- 0 ing N 1 ﬁelds by πi = φi for i = 1, , N 1. The constraint V (φ) = V (φ ) is given t − 2 ··· − 2 t by φ φ = v . Eliminating the only non-Goldstone ﬁeld φN using φN = √v π π. The resulting Lagrangian is now a nonlinear sigma model, with Lagrangian −

1 1 (πt∂ π)(πt∂µπ) = ∂ πt∂µπ + µ (8.56) LGB 2 µ 2 v2 πtπ − A number of remarks:

1. G/H = SN−1; indeed there are N 1 independent ﬁelds π. − 2. H = SO(N 1) is realized linearly on π by rotations. − 3. The Lagrangian is now non-polynomial in π. 4. This implies that it is non-renormalizable (in d = 4). This is no surprise since we discarded higher energy parts of the theory. 5. Before eliminating φ , the Lagrangian was invariant under SO(N). What has N L happened with this symmetry? Has it completely disappeared or is its presence just more diﬃcult to detect ?

We shall now address the last point. Since SO(N 1) leaves φN invariant, the only concern is with the transformations in SO(N) modulo those− in SO(N 1) which leave φ invariant. − N 107 Using SO(N 1) symmetry, these are equivalent to rotations g between π1 and φN , which leave π , i =2−, 3, , N 1 invariant, i ··· − π cos α sin α π g 1 = 1 (8.57) φ sin α cos α φ N − N

Since φN has been eliminated from the Lagrangian GB, only the action of g on π1 is of interest in examining the symmetries of , L LGB g(π )= π cos α + √v2 πtπ sin α (8.58) 1 1 − As a result, the Lagrangian is still invariant g( ) = . But, the novelty is that the LGB LGB symmetry is realized in a nonlinear fashion; hence the name.

8.8 The general G/H nonlinear sigma model The wonderful thing about Goldstone boson dynamics is that, given the symmetries G and H, the corresponding Lagrangian is unique up to an overall constant. The basic as- LGB sumption producing this uniqueness is that GB has a total of two derivatives. Thus, within this approximation, the dynamics of GoldstoneL bosons is independent of the precise mi- croscopic theory and mechanism that produced the symmetry breaking and the Goldstone bosons. In other words, given G/H, the Goldstone boson dynamics is universal. The construction is as follows. The group G is realized in terms of matrices for which we still use the notation g. The Lie algebra admits the following orthogonal decomposition, G = . The (independent) Goldstone ﬁelds in the theory are taken to be φi with Gi =1H⊕M, , dim = dim G dim H. Instead of working with the independent ﬁelds φ , it ··· M − i is much more convenient to parametrize the Goldstone ﬁelds by the group elements g G, ∈ g(φi). Of course, g contains too many ﬁelds, which will have to be projected out. To do this, one proceeds geometrically. −1 The combinations g ∂µg take values in . In fact, they are left-invariant 1-forms on G, since under left-multiplication of the ﬁeldGg by a constant element γ G we have, ∈ ω (g) g−1∂ g ω (γg)= g−1γ−1∂ γg = ω (g) (8.59) µ ≡ µ µ µ µ On the other hand, under right-multiplication by a function h H, ω (g) transforms as a ∈ µ connection or gauge ﬁeld,

−1 −1 −1 −1 ωµ(gh)= h g ∂µ(gh)= h ωµ(g)h + h ∂µh (8.60)

Therefore, the ortogonal projection of ω (g) onto transforms homogeneously under H, µ M

M −1 ωµ (g) g ∂µg (8.61) ≡ M

108 and the following universal sigma model Lagrangian is f 2 = tr ωM(g)ωMµ(g) (8.62) LGB 2 µ is invariant under left multiplication of the ﬁeld g by constant γ G and under right multiplication of g by any function in h. Therefore, precisely describes∈ the dynamics LGB of Goldstone bosons in the right coset G/H. The parameter f is a constant with the dimensions of mass in d = 4 but dimension 0 in the important case of d = 2.

8.9 Algebra of charges and currents

A local classical Lagrangian is invariant under a continuous symmetry δaφi provided the Lagrangian changes by a total divergence of a local quantity Xµ, i.e. δ = ∂ Xµ. There aL µ a exists an associated conserved current constructed as follows, ∂ jµ = L δ φ Xµ ∂ jµ = 0 (8.63) a ∂ φ a i − a µ a i µ i X and the associated Noether charge is time-independent,

d−1 0 ˙ Qa(t)= d x ja(t, x) Qa = 0 (8.64) Z The commutator with the charge reproduces the ﬁeld transformation,

δaφi(x) = [Qa, φi(x)] (8.65) When the symmetry has no anomalies, all these results generalize to the full QFT. The symmetry charges form a representation of a Lie algebra acting on the ﬁeld operators G and on the Hilbert space. We have the following structure relations where fabc are the structure constants of , G

[Qa, Qb] = i fabcQc µ µ [Qa, jb (x)] = i fabcjc (x) [jo(t, x), φ (t, y)] = δ(d−1)(x y)(T ) jφ (t, y) (8.66) a i − a i j One may ask what the algebra of the currents is. The form of the time components of the currents was conjectured by Gell-Mann; that of a time and a space component of the current was worked out by Schwinger,

0 0 (d−1) 0 [ja(t, x), jb (t, y)] = i fabcδ (x, y) jc (t, x) [j0(t, x), ji(t, y)] = i f δ(d−1)(x y) ji(t, x)+ Sij ∂jδ(d−1)(x y) (8.67) a b abc − c ab − ij Sab is a set of constants, referred to as the Schwinger terms.

109 8.9.1 Example 1 : Free massless fermions

We consider N fermion ﬁelds ψi for i =1, , N arranged into a column matrix ψ. For a massless Dirac fermion ψ in even space-time··· dimension, we may decompose ψ into left and right fermions ψL and ψR, and we have the Lagrangian,

= ψiγ¯ µ∂ ψ = ψ¯ iγµ∂ ψ + ψ¯ iγµ∂ ψ (8.68) L0 µ L µ L R µ R is invariant under 1. U SU(N) ψ ψ′ = U ψ ψ ψ L ∈ L iL → L L L R → R 2. U SU(N) ψ ψ = U ψ ψ ψ R ∈ R R → R R R L → L 3. eiθL U(1) ψ ψ′ = eiθL ψ ψ ψ ∈ L L → L L R → R 4. eiθR U(1) ψ ψ′ = eiθR ψ ψ ψ ∈ R R → R R L → L The corresponding currents are

SU(N) jµa ψ¯ T aγµψ L L ≡ L L SU(N) jµa ψ¯ T aγµψ R R ≡ R R U(1) jµ ψ¯ γµψ L L ≡ L L µ ¯ µ U(1)R jR = ψRγ ψR (8.69)

8.9.2 Example 2 : Free massive fermions A non-zero mass term will always spoil the full chiral symmetries. For example, if equal mass m = 0 is given to all components of ψ, 6 = mψψ¯ (8.70) Lm L − then is invariant only under SU(n) U(1) , deﬁned by U = U and θ = θ . Lm V × V L R L R 8.9.3 Example 3 : Real scalars

Next, we consider a theory with N real scalar ﬁelds φi grouped into a column matrix φ, governed by the following Lagrangian, 1 = ∂ φt∂µφ W (φtφ) (8.71) L 2 µ − the Lagrangian is invariant under SO(N) rotations, with inﬁnitesimal generators T a,

a a t a δaφ(x) = T φ(x) (T ) = T µ t a − ja (x) = ∂µφ T φ(x) (8.72)

110 8.10 Proof of Goldstone’s Theorem We denote the ﬁelds collectively as φ (x), with i =1, , N, and the symmetry acts by, i ···

δaφi(x) = [Qa,φi(x)] (8.73)

The transformation properties of correlation functions are now readily investigated. We consider a correlator of n ﬁelds,

δ 0 Tφ (x ) φ (x ) 0 ah | i1 1 ··· in n | i = 0 T δ φ (x ) φ (x ) 0 + + 0 Tφ (x ) δ φ (x ) 0 h | a i1 1 ··· in n | i ··· h | i1 1 ··· a in n | i = 0 T [Q ,φ (x )] φ (x ) 0 + + 0 Tφ (x ) [Q ,φ (x )] 0 h | a i1 1 ··· in n | i ··· h | i1 1 ··· a in n | i = 0 T Q φ (x ) φ (x ) 0 0 Tφ (x ) φ (x )Q 0 (8.74) h | a i1 1 ··· in n | i−h | i1 1 ··· in n a| i

The above results are generally valid, whether the symmetry generated by Qa is realized linearly or is spontaneously broken. We shall now analyze these two cases separately.

1. If Q 0 = 0 expresses the fact that the vacuum is invariant under the symmetry a| i generated by Qa. The symmetry is unbroken, and we have,

δ 0 Tφ (x ) φ (x ) 0 = 0 (8.75) ah | i1 1 ··· in n | i

Therefore, the correlator is invariant under the symmetry transformation Qa.

2. If Q 0 = 0, there will exist at least one correlator that is not invariant (otherwise a| i 6 one can show that Qa = 0).

So now, let Q 0 = 0, and a| i 6 δ 0 Tφ (x ) φ (x ) 0 = 0 (8.76) ah | i1 1 ··· in n | i 6 µ Consider the associated correlator with the current ja inserted, and take its divergence,

∂z 0 T jµ(z)φ (x ) φ (x ) 0 (8.77) µh | a i1 1 ··· in n | i Since the current is conserved, the only non-zero contributions come from the T product.

This may be seen by working out the eﬀect on a single φik (xk) as follows,

∂z T jµ(z)φ (x ) = ∂z θ(zo xo )jµ(z)φ (x ) θ(xo zo)φ (x )jµ(z) µ a ik k µ − k a ik k − k − i k a = δ(zo xo )[jo(z), φ (x )] = δ(4)(z x )(T )j φ (t x ) − k a ik k − k a ik j i k = δ(4)(z x )[Q ,φ (x )] (8.78) − k a ik k 111 Thus

z µ ∂µ 0 T ja (z)φi1 (x1) φin (xn) 0 h | n ··· | i = δ(4)(z x ) 0 Tφ (x ) [Q ,φ (x )] φ (x ) 0 (8.79) − k h | i1 1 ··· a ik k ··· in n | i Xk=1 Fourier transform in z

µ iqµ 0 T ja (q)φi1 (x1) φin (xn) 0 h n| ··· | i = e−iq·xk 0 Tφ (x ) [Q ,φ (x )] φ (x ) 0 (8.80) h | i1 1 ··· a ik k ··· in n | i Xk=1 The right-hand side admits a ﬁnite, and non-vanishing, limit at q 0: → µ lim iqµ 0 T ja (q)φi1 φin 0 = δa 0 Tφi1 φin 0 = 0 (8.81) q→0 h | ··· | i h | ··· | i 6

How can the left-hand side be = 0? Only if the correlators 6 0 T jµ(q)φ φ 0 (8.82) h | a i1 ··· in | i has a pole in q as q 0: → qµ 0 T jµ(q)φ φ 0 = i δ 0 Tφ (x ) φ (x ) 0 + (q0) (8.83) h | a i1 ··· in | i − q2 ah | i1 1 ··· in n | i O

This necessarily signals the presence of a massless bosonic particle with the same internal µ quantum numbers as the current ja whose couplings necessarily involve a ﬁrst derivative.

8.11 The linear sigma model with fermions This model, invented by Gell-Mann and Levy (1960) provides a simple and beautiful il- lustration of the spontaneous symmetry breaking mechanisms, both for bosons and for fermions. The model served as a prototype for the later construction of the Weinberg- Salam model (1967).

p ψ = (π+, π0, π−, σ) (8.84) n + 0 − Under strong isospin, SU(2)I , the (p, n), (π , π , π ) and (σ) multiplets transform respec- tively as a doublet, a triplet and a singlet. The Lagrangian is the sum of the bosonic Lσ 112 and fermionic parts, Lψ 1 λ2 2 ∂ σ∂µσ + ∂ ~π ∂µ~π (σ2 + ~π2) f 2 Lσ ≡ 2 µ µ · − 4! − π = ψ¯ iγµ∂ + g(σ + iπaτ aγ ) ψ (8.85) Lψ µ 5 It is convenient to recast this in a more compact and transparent way, by deﬁning the 2 2 × matrix-valued ﬁelds Σ, σ + iπ3 iπ1 + π2 Σ= σI + iπaτ a = (8.86) 2 iπ1 π2 σ iπ3 − − † 2 2 2 2 Clearly, we have Σ Σ=(σ + ~π )I2, det(Σ) = σ + ~π , and

2 1 λ 2 = tr ∂ Σ†∂µΣ tr Σ†Σ f 2I Lσ 4 µ − 48 − π 2 = ψ¯ iγµ∂ ψ + ψ¯ iγµ∂ ψ + gψ¯ Σ†ψ + gψ¯ Σψ (8.87) Lψ L µ L R µ R R L L R Both Lagrangians are manifestly invariant under chiral symmetry:

Σ g Σg+ g SU(2) g SU(2) → L R L ∈ L R ∈ R ψ g ψ L → L L ψ g ψ (8.88) R → R R

The diagonal U(1)V of fermion number (or baryon number for a model of nucleons) acts on both ψL and ψR but does not act on Σ. The associated currents may be arranged in vector and axial vector currents: 1 τ jµ = (jµ + jµ )= ǫ π ∂µπ + ψγ¯ µ a ψ a 2 La Ra abc b c 2 1 τ jµ = (jµ jµ )= σ∂µπ π ∂µσ + ψγ¯ µγ a ψ (8.89) 5a 2 La − Ra a − a 5 2 µ Notice that σ is a singlet under SU(2)V , so ja cannot involve σ. Both currents are classically µa µa conserved ∂µj = ∂µj5 = 0, although, depending upon the theory, there may be anomalies in the axial currents.

8.11.1 Unbroken phase

2 When fπ < 0, the symmetry is unbroken, and the system is in the unbroken phase, where we have 0 σ 0 = 0 πa 0 = 0. The structure of the spectrum is as follows, h | | i h | | i the σ-ﬁeld and the πa are massive with the same mass square given by λ f 2; • − 12 π the ψ-ﬁeld is massless. • 113 8.11.2 Spontaneously broken phase

2 When fπ > 0, the symmetry is spontaneously broken, and the system is in the sponta- a neously broken phase, where we have 0 σ 0 = fπ = 0 and 0 π 0 = 0. The structure of the spectrum is as follows, h | | i 6 h | | i σ-ﬁeld is massive; • πa are massless Goldstone bosons of SU(2) SU(2) SU(2) ; • L × R → V ψ are massive: M = gf . • ψ π 8.11.3 Small explicit breaking Finally, it is useful to exhibit the eﬀects of small explicit symmetry breaking as well. For example, one may introduce a term that conserves SU(2)V but breaks the axial symmetry, = cσ c constant (8.90) Lb − which is not invariant under SU(2)L SU(2)R. Thus, the vector current is still conserved, but not the axial vector current is not.× Its violation may be evaluated classically, µa µ ∂µj =0 ∂µj5a = cπa (8.91) Before this linear σ-model Lagrangian had been proposed, theories were developed using the dynamics only of symmetry currents. The above results in the linear model were then advocated as basic independent assumptions. These hypotheses went under the names of the conserved vector current hypothesis (CVC) and the partially conserved axial vector current (PCAC) respectively, and were proposed as such by Feynman and Gell-Mann. Computing the mass of the πa ﬁeld proceeds as follows. One starts with the axial vector current, τ jµ = σ∂µπ π ∂µσ + ψγ¯ µγ a ψ (8.92) 5a a − a 5 2 Its matrix element between the vacuum and the one-π state is obatined using the approx- imation that 0 σ∂µπ πb(p) = 0 σ 0 0 ∂µπ πb(p) , so that h | a| i h | | ih | a| i 0 jµ (x) πb(p) = ipµf δ e−ipx (8.93) h | 5a | i π ab Taking the divergence of this equation, 0 ∂µjµ (x) π (p) = δ p2f e−ipx = 0 cπ (x) π (p) = c δ e−ipx (8.94) h | 5a | b i ab π h | a | b i ab As a result, we ﬁnd 2 mπfπ = c (8.95) This expression gives an explicit relation between the mass of the π pseudo-Goldstone bosons and the amount c of explicit symmetry breaking. Clearly, the use of approximate symmetries is still very helpful in solving certain quantum ﬁeld theory problems.

114 9 The Higgs Mechanism

In an Abelian gauge theory, massive gauge particles and ﬁelds may be achieved by simply 2 µ adding a mass term m AµA to the Lagrangian. This addition is not gauge invariant. Yet, the longitudinal part of the gauge particle decouples from any conserved external current and the on-shell theory is eﬀectively gauge invariant. 2 a µa In a non-Abelian gauge theory, the addition of a mass term m AµA has a more dramatic eﬀect and the resulting gauge theory fails to be unitary. Therefore, the problem of producing massive gauge ﬁelds and particles in a manner consistent with unitarity is an important and incompletely settled problem. The Higgs mechanism is one such way to render gauge particles massive. It requires the addition of elementary scalar ﬁelds to the theory. It was invented by Brout, Englert (1960) and Higgs (1960). As we shall discuss later, it is also possible to produce massive gauge particles with scalar ﬁelds that are not elementary, but rather arise as composites out of a more basic interaction. This mechanism has a very long history, starting with Anderson (1958), Jackiw and Johnson (1973), Cornwall and Norton (1973), and Weinberg (1976). Susskind revived the mechanism in a simpler and more general setting (1978) and the mechanism is usually referred to as ”Technicolor” now.

9.1 The Abelian Higgs mechanism The simplest model originates in the theory of superconductivity. The Landau-Ginzburg model uses a U(1) gauge ﬁeld Aµ and a complex scalar ﬁeld φ. The gauge ﬁeld may be thought of as the photon ﬁeld while the scalar arises as the order parameter associated with the condensation of Cooper pairs (which are electrically charged). A simple relativistic Lagrangian that encompasses these properties is as follows,

2 1 λ 2 = F F µν + i∂ φ eA φ 2 φ 2 v2 (9.1) L −4 µν | µ − µ | − 4 | | − We assume that λ2, v2 > 0. When e = 0, the model is easily analyzed, since it separates into a free Maxwell theory and a complex scalar theory which exhibits spontaneous symmetry breaking. The particle contents is thus

1. Two massless photon states; 2. One massless Goldstone scalar, corresponding to the phase of φ; 3. One massive scalar, corresponding to the modulus of φ.

The scalar ﬁelds do mutually interact. When e = 0, the gauge and scalar ﬁelds interact, and we want to determine the masses of the various6 particles. To isolate the Goldstone ﬁeld, we assume that the scalar takes

115 on a vacuum expectation value in a deﬁnite direction of the minimum of the potential, 0 φ 0 = v, and we introduce associated real ﬁelds ϕ ,ϕ as follows. h | | i 1 2 1 φ = v + (ϕ1 + iϕ2) 0 ϕ1,2 0 = 0 (9.2) √2 h | | i In terms of these ﬁelds, the Lagrangian becomes 1 1 1 = F F µν + (∂µϕ eA ϕ )2 + (∂µϕ + ev√2A + eA ϕ )2 L −4 µν 2 1 − µ 2 2 2 µ µ 1 λ2 1 1 (√2vϕ + ϕ2 + ϕ2)2 (9.3) − 4 1 2 1 2 2 The spectrum of masses is governed by the quadratic approximation to the full action, which is readily identiﬁed, 1 1 1 λ2v2 = F F µν + ∂µϕ ∂ ϕ + (∂µϕ + ev√2A )2 ϕ2 (9.4) Lquad −4 µν 2 1 µ 1 2 2 µ − 2 1 ′ Eﬀectively, the ﬁeld ϕ2 becomes unphysical, as any ϕ2 bat be changed into any other ϕ2 by a gauge transformation. One particular gauge choice is ϕ2 = 0, in which case, we have 1 1 λ2v2 = F F µν + ∂µϕ ∂ ϕ + e2v2A Aµ ϕ2 (9.5) Lquad −4 µν 2 1 µ 1 µ − 2 1

The gauge ﬁeld has mass √2ev, the scalar ϕ1 remains a massive scalar and the Goldstone ﬁeld ϕ2 has disappeared completely. Actually, to make the gauge ﬁeld Aµ massive, a third degree of polarization was required and this is provided precisely by the ﬁeld ϕ2. One says that the Goldstone boson has been eaten by the gauge ﬁeld. Thus, for e = 0, the particle contents is given by, 6

1. three massive photon states with mass mγ = √2ev; 2. no Goldstone bosons; 3. one massive scalar. To avoid the impression that the above conclusions might be an artifact of the quadratic approximation, we shall treat the problem now more generally. To this end, we use the following decomposition φ(x)= ρ(x)eiθ(x) 0 ρ 0 = v (9.6) h | | i The Lagrangian in terms of these ﬁelds is 1 λ2 = F F µν + ∂µρ∂ ρ +(∂ θ eA )2 (ρ2 v2)2 (9.7) L −4 µν µ µ − µ − 4 − It is clear now that, to all orders in perturbation theory, the phase θ(x) is a gauge artifact and may be rotated away freely.

116 9.2 The Non-Abelian Higgs Mechanism Instead of analyzing this problem in general, we ﬁrst concentrate on a representative speciﬁc example. Let

117 10 Topological defects, solitons, magnetic monopoles

Spontaneous symmetry breaking (either of a continuous or of a discrete symmetry) gener- ically implies the existence of topological defects. The minimum energy conﬁguration is a vacuum state in which the symmetry breaking expectation value is pointed in the same direction throughout space. However, excited states appear when the symmetry breaking expectation value points in one direction in one part of space, but in a diﬀerent direction in a diﬀerent part of space. In between the two phase regions, the scalar ﬁeld must vary away from its minimum energy values, and this type variation costs total energy. The conﬁguration is generally referred to as a domain wall. Higher topological defects include vortices, Skyrmions, and magnetic monopoles.

10.1 Topological defects - domain walls In any dimension of space-time d 3, we consider the two-dimensional subspace-time con- ≥ sisting of the time direction, and the direction transverse or perpendicular to the separating line between the two adjacent phases. For simplicity, we examine the theory corresponding to broken φ4-theory, with eﬀective two-dimensional Lagragian given by,

1 λ2 = ∂ φ∂µφ (φ2 v2)2 (10.1) L 2 µ − 2 − Look for static domain-wall solutions that are time-independent and interpolate between the ground state conﬁgurations φ v and φ +v. (cf. instanton problem in one → − → space-time dimension.) The Euler-Lagrange equation admits an integral of motion,

∂2φ 2λ2φ(φ2 v2) = 0 x − − (∂ φ)2 λ2(φ2 v2)2 =const. (10.2) x − − Solutions for which φ v as x correspond to the above constant being exactly zero. When this is the→ case, ± the integral→ ±∞ of motion is easily solved by elementary methods, and we ﬁnd a one-parameter family of solutions,

dφ(x) = λdx φ (x)= v tanh vλ(x x ) (10.3) φ2 v2 ± 0 ± − 0 − where x0 is an arbitrary integration constant which corresponds to the central location of the solution. The energy of the static solution is ﬁnite, corresponds to the mass density M of the domain wall, and is given by,

∞ 1 λ2 v3 M = dx (∂ φ )2 + (φ2 v2)2 (10.4) 2 x 0 4 0 − ≈ λ2 Z−∞ 118 Since the ﬁeld equations are Poincar´einvariant, it is clear that a boosted solution moving with velocity u is also a solution to the ﬁeld equations. It is given by, 1 φu(x, t)= φ0 γ(x ut) γ = (10.5) − √1 u2 − The energy density of the moving domain wall is then given by the relativistic relation,

E = γMc2 (10.6)

This behavior is of course analogous to that of elementary particles. For weak coupling λ2v4−d 1, these solutions correspond to semi-classical particles with very large masses. ≪ 10.1.1 Topological defects of higher order More generally, topological defects are classiﬁed by the vanishing of a scalar ﬁeld along some sub-space-time. For example, considering the situation in four space-time dimensions, d = 4, we have the following classes of topological defects,

φ(x)=0 at a point particle ∼ φ(x) = 0 along a line vortex ∼ φ(x) = 0 along plane domain wall ∼ By ignoring one of the space-like dimensions, we decrease the dimension from d = 4 to d = 3. The domain wall of d = 4 becomes a vortex in d = 3, the vortex of d = 4 appears as a point-like particle in d = 3.

119