Quantum Field Theory

Home , Field equation, Ground state, Quantum field theory

A Physicist’s Primer

Gordon Walter Semenoff Copyright c 2019 Gordon Walter Semenoff

PUBLISHEDBY PUBLISHER

BOOK-WEBSITE.COM

Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this ﬁle except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “ASIS” BASIS, WITHOUTWARRANTIESORCONDITIONSOFANYKIND, either express or implied. See the License for the speciﬁc language governing permissions and limitations under the License.

First printing, May 2019 Contents

I Many Particle Physics as a Quantum Field Theory

1 Many particle physics ...... 9 1.1 A preview of this chapter9 1.2 Introduction9 1.3 Non-relativistic particles 10 1.3.1 Identical particles...... 13 1.3.2 Spin...... 16 1.4 Second Quantization 17 1.5 The Heisenberg picture 20 1.6 Summary of this chapter 23

2 Degenerate Fermi and Bose Gases ...... 25 2.1 A preview of this chapter 25 2.2 The limit of weak interactions 25 2.3 Degenerate Fermi gas and the Fermi surface 28 2.3.1 The ground state |O > ...... 28 2.3.2 Particle and holes...... 29 2.3.3 The grand canonical ensemble...... 31 2.4 Bosons 33 2.5 Summary of this chapter 38 3 Classical ﬁeld theory and the action principle ...... 41 3.1 The Action Principle 41 3.1.1 The Action...... 42 3.1.2 The action principle and the Euler-Lagrange equations...... 43 3.1.3 Canonical momenta, Poisson brackets and Commutation relations...... 47 3.2 Noether’s theorem 48 3.2.1 Examples of symmetries...... 49 3.2.2 Proof of Noether’s Theorem...... 50 3.3 Phase symmetry and the conservation of particle number 51 3.4 Translation invariance 53 3.5 Galilean symmetry 54 3.6 Scale invariance 57 3.6.1 Improving the energy-momentum tensor...... 58 3.6.2 The consequences of scale invariance...... 59 3.7 Special Schrödinger symmetry 60 3.8 The Schrödinger algebra 61 3.9 Summary of this chapter 63

II Relativistic Symmetry and Quantum Field Theory

4 Space-time symmetry and relativistic ﬁeld theory ...... 69 4.1 Quantum mechanics and special relativity 69 4.2 Coordinates 73 4.3 Scalars, vectors, tensors 75 4.4 The metric 76 4.5 Symmetry of space-time 77 4.6 The symmetries of Minkowski space 77

5 The Dirac Equation ...... 79 5.1 Solving the Dirac equation 81 5.2 Lorentz Invariance of the Dirac equation 84 5.3 Phase symmetry and the conservation of electric current 86 5.4 The Energy-Momentum Tensor of the Dirac Field 87 5.5 Summary of this chapter 90

6 Photons ...... 93 6.1 Relativistic Classical Electrodynamics 93 6.2 Covariant quantization of the photon 94 6.2.1 Field equations and commutation relations...... 94 6.2.2 Massive photon (Optional reading)...... 102 6.3 Space-time symmetries of the photon 103 6.4 Quantum Electrodynamics 104 6.5 Summary of this chapter 106

III Functional methods and quantum electrodynamics

7 Functional Methods and Correlation Functions ...... 111 7.1 Functional derivative 111 7.2 Functional integral 113 7.3 Photon Correlation functions 114 7.3.1 Generating functional for correlation functions of free photons...... 117 7.3.2 Photon Generating functional as a functional integral...... 119 7.4 Functional differentiation and integration for Fermions 121 7.5 Generating functionals for non-relativistic Fermions 125 7.5.1 Interacting non-relativistic Fermions...... 127 7.6 The Dirac field 128 7.6.1 Two-point function for the Dirac field...... 128 7.6.2 Generating functional for the Dirac field...... 130 7.6.3 Functional integral for the Dirac field...... 131 7.7 Summary of this chapter 131

8 Quantum Electrodynamics ...... 135 8.1 Quantum Electrodynamics 135 8.2 The generating functional in perturbation theory 139 8.3 Wick’s Theorem 140 8.4 Feynman diagrams 141 8.5 Connected Correlations and Goldstone’s theorem 145 8.5.1 Connected correlation functions...... 145 8.5.2 Goldstone’s Theorem...... 147 8.6 Fourier transform 147 8.7 Furry’s theorem 150 8.8 One-particle irreducible correlation functions 151 8.9 Some calculations 152 8.9.1 The photon two-point function...... 152 8.9.2 The Dirac field two-point function...... 155 8.9.3 Traces of gamma matrices...... 158 8.9.4 Feynman Parameter Formula...... 158 8.9.5 Dimensional regularization integral...... 159 8.10 Quantum corrections of the Coulomb potential 160 8.11 Renormalization 162 8.11.1 The Ward-Takahashi identities...... 163 8.12 Summary of this Chapter 164 9 Formal developments ...... 167 9.1 In-fields, the Haag expansion and the S-matrix 167 9.2 Spectral Representation 168 9.2.1 Gauge invariant scalar operators...... 169 9.2.2 The Dirac field...... 170 9.3 S-matrix and Reduction formula 173 9.3.1 Some intuition about asymptotic behaviour...... 173 9.3.2 In and out-fields in QED...... 174 9.3.3 Proof of the LSZ identities...... 178 9.3.4 An Example: Electron-electron scattering...... 180 9.4 More generating functionals 182 9.4.1 Connected correlation functions...... 182 9.4.2 One-particle irreducible correlation functions...... 184 9.4.3 Charge conjugation symmetry ...... 186 Many Particle Physics as a I Quantum Field Theory

1 Many particle physics ...... 9 1.1 A preview of this chapter 1.2 Introduction 1.3 Non-relativistic particles 1.4 Second Quantization 1.5 The Heisenberg picture 1.6 Summary of this chapter

2 Degenerate Fermi and Bose Gases .. 25 2.1 A preview of this chapter 2.2 The limit of weak interactions 2.3 Degenerate Fermi gas and the Fermi surface 2.4 Bosons 2.5 Summary of this chapter

3 Classical ﬁeld theory and the action principle ...... 41 3.1 The Action Principle 3.2 Noether’s theorem 3.3 Phase symmetry and the conservation of particle number 3.4 Translation invariance 3.5 Galilean symmetry 3.6 Scale invariance 3.7 Special Schrödinger symmetry 3.8 The Schrödinger algebra 3.9 Summary of this chapter

1. Many particle physics

1.1 A preview of this chapter In this chapter, we will formulate our first example of a quantum field theory. We begin with the study of a quantum mechanical system with non-relativistic, identical, interacting particles. We will define the problem at hand as that of needing to find a solution of the Schrödinger equation which describes that system, subject to the appropriate boundary conditions. We will discuss the two cases of particle exchange statistics, Fermions and Bosons. We shall then find a way to rewrite the quantum many-particle problem as a quantum field theory. For this, we introduce field operators and the problem is posed as a field equation and commutation relations that the field operators of the quantum field theory should satisfy.

1.2 Introduction In this chapter we will attempt to develop intuition for the answer to the question “what is a quantum field theory”. We will do this by studying a system with many particles. For now, we will assume that the particles are non-relativistic. The generalization to relativistic particles and relativistic quantum field theory will be discussed in later chapters. We will assume that the problem in front of us is quantum mechanical, that is, that we want to find a solution of the Schrödinger equation for the system as a whole and then use that solution, the wave function, to answer questions about the physical state of the system. In order to describe the quantum mechanical problem for a large number of particles in an elegant way, we will develop a procedure which is called “second quantization”. In non-relativistic quantum mechanics, when the total number of particles is finite, second quantization gives an alternative, but at the same time completely equivalent formulation of the problem of solving the Schrödinger equation. This formulation is convenient for some applications, such as perturbation theory which is widely used to study many-particle systems and it can be relevant to many interesting physical scenarios. Metals, superconductors, superfluids, trapped cold atoms and nuclear matter are important examples. The formalism is particularly useful in that it allows us to take the “thermodynamic limit” which is an idealization of such a system that considers 10 Chapter 1. Many particle physics

the limit as both the volume of the system and the total number of particles in the system go to infinity, with the density – the number of particles per unit of volume – kept finite. The system can simplify somewhat in that limit. Moreover, it can be a good approximation to real systems, where the number of particles in a macroscopic system is typically very large, of order Avogadro’s number,6.02 × 1023 and the size of the system is macroscopic, many orders of magnitude greater than the natural sizes of the components of the system, like the Compton wave-lengths of the particles for example. Later, in subsequent chapters, we will generalize the second quantized system that we find in order to make it relativistic, that is, so that it can describe particles with velocities approaching the velocity of light. In this generalization, the analog of second quantization is essential. Relativistic quantum mechanics is necessarily a many-particle theory and the number of particles is always infinite, so there is no convenient description of it using a many-particle Schrödinger equation. In both the relativistic and non-relativistic cases, the second quantized theory is a quantum field theory, that is, a quantum mechanical theory where fields are the dynamical variables. In classical field theory, a field is simply a function of space and time coordinates whose value at a given time and point in space has a physical interpretation. A familiar example of a classical field theory is classical electrodynamics where the electric field and magnetic field are the classical fields. We can think of classical electrodynamics as a mechanical theory where the dynamical variables are these classical fields and the mechanical problem is to determine the time evolution of the dynamical variables, in this case, to determine the electric and magnetic fields as functions of the space and time coordinates. This is done by solving Maxwell’s equations. In a quantum field theory, instead of being ordinary functions, like the electric and magnetic fields which are studied in classical electrodynamics, the fields in a quantum field theory are space and time-dependent operators which act on vectors in a Hilbert space, the space of possible quantum states of the quantum field theory. In such a theory, the physical entities, those attributes which can be measured by doing experiments, for example, are the expectation values and correlations of various operators. We will eventually get a much more precise picture of how this works.

1.3 Non-relativistic particles We will begin by studying the non-relativistic quantum mechanics of a system of identical particles. Particles are identical if all of their physical properties, such as their mass, electric charge, spin, et cetera, are identical. Generally we are interested in describing the behaviour of a large number of such particles. This can have many applications in physics, to any system where many identical degrees of freedom are involved, from the study of the collective properties of the electrons in a metal to the molecules of a gas or a liquid, to the behaviour of a superﬂuid or a superconductor. Our central goal here is not a comprehensive overview of such applications, which is in itself a fascinating subject, but, rather, our aim is to gain intuition about quantum ﬁeld theory. We will begin by assuming that we can study a many-particle system by studying its Schrödigner equation. The Schrödinger equation contains the Hamiltonian, which is generally the energy of the system as a function of the dynamical variables. We will assume that the dynamical variables are the momenta and the positions of each of the particles. The kinetic energy of an assembly of particles is given by the sum over their individual kinetic energies

N ~p2 total kinetic energy = ∑ i i=1 2m

where ~pi is the momentum of the i’th particle and each particle has mass m. Particles also have a potential energy by virtue of their mutual interactions. We will assume that the potential energy is a function of the positions of the particles, V(~x1,...,~xN). If the particles 1.3 Non-relativistic particles 11 are identical, this potential energy should be a symmetric function of the positions, in that, if we interchange any two of the positions, the value of the potential is left unchanged. We will also generally assume that the total potential energy is due to two-body interactions, that is, that it can be written as a sum N total potential energy = V(~x1,...,~xN) = ∑ V(~xi,~x j) i< j=1 where V(~xi,~x j) is the energy that is stored in the interaction between particle i and partical j. If the particles are identical

V(~xi,~x j) = V(~x j,~xi) for each pair (i j). We will assume that this is always the case. For the most part, we shall assume that they are functions of relative positions of the particles so that

V(~xi,~x j) = V(~xi −~x j) The Hamiltonian for such a system is given by the sum of the kinetic energy and the interaction energy. It has the form

N 2 N ~pi H(~p1,...,~pN,~x1,...,~xN) = ∑ + ∑ V(~xi −~x j) (1.1) i=1 2m i< j=1

Here,~xi is the position and ~pi is momentum of the i’th particle and the index i runs over the labels of the particles, i = 1,2,...,N. In the quantum mechanics of non-relativistic particles, the positions and momenta are operators. We will temporarily denote operators with a hat, so that they are {~ˆx1,...,~xˆN ~pˆ1,...,~pˆN}. The precise property that deﬁnes them as operators are the commutation relations

h a bi a a xî ,xˆj = 0 , xî , pˆ jb = ih¯δi jδ b , pîa, pˆ jb = 0 (1.2)

There the labels i, j take values in the set {1,2,...,N} and they label the distinct particles. The indices a,b take the values {1,2,3} and they label the three Cartesian components of the position or momentum vector of each particle. The right-hand-side of the non-zero commutation relations contains Planck’s constant, h¯. It is necessary to find a workable “representation” of the commutation relations (1.2) between a the position and momentum operators. A common way to do this is to think of the operators xî and a a pî as operating on functions of all of the coordinates, ϕ(~x1,...,~xN), with the operation of xî being a simply the multiplication of the function by the variable xi a a xî ϕ(~x1,...,~xN) = xi ϕ(~x1,...,~xN) a a and the operation ofp î as proportional to the partial derivative by xi , ∂ pîaϕ(~x1,...,~xN) = −ih¯ a ϕ(~x1,...,~xN) ∂xi It is easy to see that this definition reproduces the commutation relation for position and momentum, a a a xî , pˆ jb ϕ(~x1,...,~xN) = xî pˆ jb ϕ(x1,...,xN) − pˆ jbxî ϕ(x1,...,xN)

a h¯ ∂ h¯ ∂ a = xi b ϕ(x1,...,xN) − b [xi ϕ(x1,...,xN)] i ∂x j i ∂x j a = [ih¯δi jδ b]ϕ(x1,...,xN) 12 Chapter 1. Many particle physics

The Hamiltonian in equation (1.1) is a function of positions and momenta. If positions and momenta become operators, the Hamiltonian also becomes an operator,1

2 N ~p N N h2 N ˆ ~ ~ ~ ~ î ~ ~ ¯ ~ 2 ~ ~ H ≡ H(pˆ1,..., pˆN,xˆ1,...,xˆN) = ∑ + ∑ V(xî −xˆj) = − ∑ ∇i + ∑ V(xî −xˆj) (1.3) i=1 2m i< j=1 i=1 2m i< j=1

The Schrödinger equation determines how the quantum state evolves from an initial time to a later time ∂ ih¯ ψ(~x ,...,~x ,t) = Hˆ ψ(~x ,...,~x ,t) ∂t 1 N 1 N " N 2 N # h¯ ~ 2 ~ ~ = − ∑ ∇i + ∑ V(xˆi − xˆj) ψ(~x1,...,~xN,t) (1.4) i=1 2m i< j=1

Here, ψ(~x1,...,~xN,t) is the wave-function which should be interpreted as the probability amplitude that that particles occupy positions ~x1,...,~xN at time t. It should be normalized so that the total probability is unity, Z 3 3 2 d x1 ...d xN |ψ(~x1,...,~xN,t)| = 1

We can present the Schrödinger equation (1.4) as a time-independent equation by making the ansatz

−iEt/h¯ ψ(~x1,...,~xN,t) = e ψE (~x1,...,~xN)

Then (1.4) implies that

" N 2 # −h¯ ~ 2 EψE (~x1,...,~xN) = ∑ ∇i + ∑V(~xi −~x j) ψE (~x1,...,~xN) (1.5) i=1 2m i< j

The solution of this equation with boundary conditions should give us the wave-functions and the energies, E of stationary states. Here, ψE (~x1,...,~xN) is called an “eigenstate” or “eigenvector” of the Hamiltonian and E is the “eigenvalue” which is associated with it. Wave-functions with different energy eigenvalues are orthogonal, Z 3 3 † d x1 ...d xNψE (~x1,...,~xN)ψE0 (~x1,...,~xN) = δEE0

1 ~ We shall often use the notation where ∇ = (∇1,∇2,∇2) or ∇a with a = 1,2,3 is the gradient operator with components

∂ ∂ ∂ ~∇ ≡ , , ∂x1 ∂x2 ∂x3 a Given that xi , a = 1,2,3, i = 1,2,...,N is the a’th component of the Cartesian coordinates of i’th particle, ∂ ∇ia ≡ a , p ja = −ih¯∇ ja ∂xi We also denote Laplace’s operator as !2 !2 !2 ~ 2 ∂ ∂ ∂ 2 2~ 2 ∇i ≡ 1 + 2 + 3 , ~pi = −h¯ ∇i ∂xi ∂xi ∂xi 1.3 Non-relativistic particles 13

Generally, equation (1.5) is difﬁcult to solve when the interaction potential is non-trivial. In fact, there are very few examples of interaction potentials where one can solve for the wave-functions or the energies exactly. One of them is the case of free particles, when the potential is zero. In that case, the explicit wave-function can be found, it is simply constructed from plane-waves,

1 N ~ i∑i=1 ki·~xi ψE (~x1,...,~xN) = 3 e (2π) 2 and the energy eigenvalue is

N h¯ 2~k2 E = ∑ i i=1 2m

If the initial state, say at time t = 0 is given by a function ψ0(~x1,...,~xN), the wave-function at any time is given by N 3 3 Z d k jd y j 2~ 2 ~ −ih¯ k j t/2mh¯+ik·(~x j−~y j) ψ(~x1,...,~xN,t) = ∏ 3 e ψ0(~y1,...,~yN) j=1 (2π)

In fact, in this simple case, the integrations over~k j can be done to get

N " 3 # Z d y j 2 ψ(~x ,...,~x ,t) = eim|~x j−~y j| /2ht¯ ψ (~y ,...,~y ) 1 N ∏ 2 3 0 1 N j=1 (2πh¯ t/im) 2 This formal expression is a solution of the initial value problem for the quantum state of N free particles.

1.3.1 Identical particles There is one important aspect of the problem which we have ignored until now and which must be discussed here. We have constructed the Hamiltonian so that the particles are identical. They have the same masses and the interaction between any pair of particles is governed by the same two-body potential as the interaction between any other pair of particles. The Hamiltonian is unchanged if we trade the labels on the indices of the particles. That is, if we make the substitution

{~x1,~x2,...,~xN;~p1,~p1,...,~pN} → {~xP(1),~xP(2),...,~xP(N);~pP(1),~pP(2),...,~pP(N)} where the permutation {1,2,...,N} → {P(1),P(2),...,P(N)} is a re-ordering of the integers {1,2,...,N}. There are N! different possible permutations, including the identity. We require that the permutation act on the indices of both the particles and the momenta. This guarantees that the commutation relations (1.2) as well as the Hamiltonian (1.3) are left unchanged by the transformation. This permutation symmetry of the Hamiltonian has an important consequence. Consider the Schrödinger equation (1.5) and let us assume that we manage to solve the equation to ﬁnd an allowed value of the energy, E, and the wave-function which corresponds to it, ψE (~x1,...,~xN). The permutation symmetry then tells us that ψE (~xP(1),...,~xP(N)) also obeys the same equation, (1.5), for any of the N! distinct permutations. What is more, the normalization of the wave-functions are identical Z 3 3 † d x1 ...d xNψE (~xP(1),...,~xP(N))ψE (~xP(1),...,~xP(N)) Z 3 3 † = d xP−1(1) ...d xP−1(N)ψE (~x1,...,~xN)ψE (~x1,...,~xN) Z 3 3 † = d x1 ...d xNψE (~x1,...,~xN)ψE (~x1,...,~xN) 14 Chapter 1. Many particle physics where, P−1(i) is the integer that P maps onto the integer i and we have used the fact that 3 3 3 3 d xP−1(1) ...d xP−1(N) is an inconsequential re-ordering of d x1 ...d xN. Then, there are two possibilities. The ﬁrst possibility is that, using permutations, we have found some new quantum states which are not equivalent to the one that we began with. That is, for some permutation, P, the wave function ψE (~x1,...,~xN) and the wave function ψE (~xP(1),...,~xP(N)) are truly distinct wave functions representing distinct quantum states. In order to describe distinct quantum states, the state vectors must be linearly independent. The test for linear independence is to ask whether the equation whether the equation

c1ψE (~x1,...,~xN)) + c2ψE (~xP(1),...,~xP(N)) = 0 (1.6) has a solution where c1 and c2 are not zero. If such a solution exists, they are linearly dependent. If both c1 and c2 must be zero, they are linearly independent. If the states are truly distinct quantum states, the only solution of equation (1.6) has both c1 and c2 equal to zero. Then ψE (~x1,...,~xN) and ψE (~xP(1),...,~xP(N)) are two different quantum states with the same energy eigenvalue E. Let us examine this possibility. If we consider all permutations and find the linearly independent states which are generated, we find a degenerate set of state vectors which are transformed into each other by permutations. The degeneracy would be a prediction of our quantum mechanical model. It is up to us to compare what we find with the real physical system which we are describing in order to see if the degeneracies which would result are indeed there. When the degeneracy is two-fold or greater, the particles which are being described are said to obey “parastatistics”. In parastatistical systems, the degeneracies can depend on the total number of particles. Nature does not seem to make use of parastatistics.2 For any three-dimensional many-particle system, and for any permutation P, the wave function ψE (~x1,...,~xN) and the wave function ψE (~xP(1),...,~xP(N)) are linearly dependent and represent the same quantum state. Such particles are said to be “indistinguishable”. This indistinguishability is extremely important to us. It is responsible for the stability of atoms, for example, via the Pauli exclusion principle applied to identical electrons. Nature would be very different if electrons were distinguishable particles. When particles are indistinguishable, the equation

c1ψE (~x1,...,~xN) + c2ψE (~xP(1),...,~xP(N)) = 0 has a solution where both c1 and c2 are non-zero for any permutation P. Then, the wave-functions must be proportional to each other,

ψE (~xP(1),...,~xP(N)) = c[P] ψE (~x1,...,~xN) where c[P] = − c2[P] . If the wave function is normalized, then |c[P]| = 1 and, considering a c1[P] permutation which, for example, exchanges the positions of just two particles, where doing the permutation twice returns the wave-function to its original form. Then c2[P] = 1 and we would conclude that c[P] = 1 or c[P] = −1. Then, considering the fact that any permutation can be built up out of successive interchanges of pairs of particles, we can see that, for any permutation, there are two possibilities, the ﬁrst is where the wave-function is a completely symmetric function of its arguments, 3

c[P] = 1 , ∀P

2There are some examples of unusual statistics when the effective dimension of a quantum system is one or two, where permutations have a topological interpretation and the wave-function can have a richer structure. Particles which obey such statistics are called “anyons”. 3We shall use the mathematical symbol ∀ as shorthand for “for all”. 1.3 Non-relativistic particles 15 and

ψE (~x1,...,~xN)) = ψE (~xP(1),...,~xP(N)) for any permutation, P. The particles are called Bosons, or are said to obey “Bose-Einstein statistics”. The second possibility is where the wave-function is a totally anti-symmetric function of the positions arguments,

c[P] = (−1)deg[P] and

deg[P] ψE (~x1,...,~xN) = (−1) ψE (~xP(1),...,~xP(N)) for a permutation P and where the degree deg[P] is the number of interchanges of pairs that are needed to implement the permutation. Particles which obey statistics of this sort are called Fermions, or are said to obey “Fermi-Dirac statistics”. In the quantum many-body systems that are found in nature, particles that have identical properties are identical particles and they are either Fermions of Bosons. Given a solution of the Schrödinger equation we can construct a wave-function for Bosons by symmetrizing over the positions of the particles, so that

ψb(~x1,...,~xN,t) = cb ∑ψ(~xP(1),...,~xP(N),t) P

On the other hand, if the particles that the wave-function is intended to describe are Fermions, then we should anti-symmetrize over the positions of the particles,

deg(P) ψ f (~x1,...,~xN,t) = c f ∑(−1) ψ(~xP(1),...,~xP(N),t) P

Here, the summations are over all N! possible permutations, including the trivial one. In each of these expressions, the constants cb and c f should be adjusted to correctly normalize the resulting wave-function. When the wave-function is either completely symmetric or anti-symmetric, the probability density

† ψ (~x1,...,~xN,t)ψ(~x1,...,~xN,t) is a completely symmetric function of its arguments, (~x1,...,~xN). Since the particles are iden- † 3 3 3 tical, the quantity ψ (~x1,...,~xN,t)ψ(~x1,...,~xN,t)d x1,d x2,...d xN should be interpreted as the probability at time t for ﬁnding the system with particles occupying the inﬁnitesimal volumes 3 3 3 d x1,d x2,...d xN which are each centered on the points~x1,...,~xN, respectively, with no reference to which particles occupy which volumes. It should be normalized so that

Z 3 3 † d x1 ...d xNψ (~x1,...,~xN,t)ψ(~x1,...,~xN,t) = 1

This has the interpretation that the total probability for ﬁnding the N particles somewhere is equal to one. 16 Chapter 1. Many particle physics

1.3.2 Spin There is one elaboration which we should discuss before proceeding to develop our current discussion further. That is the issue of spin. If we want to describe realistic many-particle systems of atoms or electrons, the particles in question generally have spin and their wave-functions must carry an index to label their spin state. To describe these, we add an index to the total wave-function for each particle, so that the wave-function is

ψσ1σ2...,σN (~x1,~x2,...,~xN,t)

For spin J, the indices σi each run over 2J + 1 values σi = −J,...,J which correspond to the spin states of a single particle. The wave-function of a system of identical particles must then be either symmetric or anti-symmetric under simultaneous permutations of the spin and position variables of the particles. Generally, Bosons have integer spins and Fermions have half-odd integer spin. In summary, for Bosons, J is an integer and

ψσ1...,σN (~x1,...,~xN,t) = ψσP(1)...,σP(N) (~xP(1),...,~xP(N),t) , ∀P For Fermions, J is a half-odd-integer and

deg[P] ψσ1...,σN (~x1,...,~xN,t) = (−1) ψσP(1)...,σP(N) (~xP(1),...,~xP(N),t) , ∀P where, when we implement the permutation, we permute both the spin and the position labels. The Hamiltonian can also have spin-dependent interactions. In that case, the potential energy is generally a hermitian matrix which operates on spin indices. For two-body interactions, the σiσ j two-body gets spin indices as Vρiρ j (~xi −~x j) and its operation on the wave-function is the mapping

J ρiρ j ψσ1...σi...σ j...σN (~x1,...,~xN,t) → ∑ Vσiσ j (~xi −~x j)ψσ1...ρi...ρ j...σN (~x1,...,~xN,t) ρiρ j=−J

We will see shortly that this sort of interaction is very easy to implement in second quantization. Before we continue let us consider a two examples. First, there is a spin-orbit interaction. For such an interaction, we need to understand how to measure the “spin” that is contained in a many-particle 1 wave-function. For this, we assume that the particles have spin J = 2 and we introduce the Pauli matrices, ~σ, as

0 1 0 −i 1 0 σ 1 = , σ 2 = , σ 3 = , (1.7) 1 0 i 0 0 −1

Then, the expectation value of the spin is simple the expectation value of the spin matrix, deﬁned as 1 2~σ for each particle D E ~Σ = Z N 1 dx ...dx σ1...,ρi...σN †(~x ,...,~x ,t)~ τi (~x ,...,~x ,t) (1.8) 1 N ∑ ψ 1 N σρi ψσ1...τi...σN 1 N i=1 2 Here and in the following, we are using the Einstein summation condition for repeated up and down indices. In each term on the right-hand-side, each of the indices σ1,σ2,...,σi−1,σi+1,...,σN and τi and ρi are all summed from −J to J. We have omitted the summation symbols

J J J J J J ∑ ... ∑ ∑ ... ∑ ∑ ∑ σ1=−J σi−1=−J σi+1=−J σN =−J τi=−J ρi=−J 1.4 Second Quantization 17

A typical spin-dependent interaction is the spin-spin interaction which we could add to the spin-independent interaction to get

σ σ σ 1 σ V i j (~x −~x ) = δ σi δ j v (~x −~y) + ~σ σi ·~σ j v (~x −~y) (1.9) ρiρ j i j ρi ρ j 0 4 ρi ρ j ss We leave writing down a spin-orbit interaction as an exercise.

1.4 Second Quantization Second quantization is a technique which summarizes the many-particle quantum mechanical problem contained in (1.4), together with either Bose or Fermi statistics in an elegant way. To implement second quantization, we begin by constructing an abstract basis for the states of the N-particle system. We define the Schrödinger field operator, ψ(~x) which depends on one position variable,~x. In spite of the use of the symbol ψ, this operator should not be confused with a wave-function, it is an operator whose important property is that it obeys the commutation relations which will be listed in equations (1.10) or (1.11) below. There is one such operator for each different kind of identical particle, for example in a gas of electrons where the electron can exist in two spin 1 1 states, the field operator would have the spin index, ψσ (~x) with σ = − 2 , 2 labelling the spin. We shall also need the Hermitian conjugate of the field operator, ψ†σ (~x). This should be regarded as the hermitian conjugate of the operator ψσ (~x) in the sense that

† †σ † † †σ (ψσ (~x)|state >) =< state|ψ (~x) , (ψσ (~x)[operator]) = [operator] ψ (~x) In the case of particles with Bose-Einstein statistics, these operators satisfy the commutation relations

†ρ ρ †σ †ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y), ψσ (~x),ψρ (~y) = 0, ψ (~x),ψ (~y) = 0 (1.10) where, as usual, the square bracket denotes a commutator ([A,B] = AB − BA). In the case of particles with Fermi-Dirac statistics, the commutators should be replaced by anti-commutators so that the operators satisfy the anti-commutation relations

†ρ ρ †σ †ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y), ψσ (~x),ψρ (~y) = 0, ψ (~x),ψ (~y) = 0 (1.11) We use the curly brackets to denote an anti-commutator, ({A,B} = AB + BA). †σ The operators ψσ (~x) and ψ (~x) can be thought of as annihilation and creation operators for a particle at point~x and in spin state σ. To see this, consider the following construction. We begin with a speciﬁc quantum state which we shall call the “empty vacuum” |0 >. It is the state where there are no particles at all. Its mathematical deﬁnition is that it is the state which is annihilated by the operators ψσ (~x) for all values of the position~x and spin label σ,

ψσ (~x)|0 >= 0 ∀~x,σ (1.12) The adjoint of the above statement is that the Hermitian conjugate and the dual state to the vacuum also have the property

< 0|ψ†σ (~x) = 0 ∀~x,σ (1.13)

Then, we create particles which occupy the distinct points ~x1,...,~xn and are in spin states †σ σ1,...,σN by repeatedly operating ψ i (~xi) on the vacuum state 1 σ1...σN †σ1 †σN |~x1,...,~xN,> = √ ψ (~x1)...ψ (~xN)|0 > (1.14) N! 18 Chapter 1. Many particle physics

σ † σ ...σ Since the operators ψ i (~xi) either commute or anti-commute with each other, |~x1,...,~xN > 1 N is automatically either totally symmetric or anti-symmetric under permutations of the position coordinates and spins and it is therefore appropriate for either Bosons or Fermions, respectively. Similarly,

1 <~x1,...,~xN|σ ...σ = √ < 0|ψσ (~x) ...ψσ (~x1) (1.15) 1 N N! N 1 The inner product is

ρ1...ρN <~x1,...,~xN|σ1...σN |~y1,...,~yN,> 1 ρ ρ = (− )D(P) (~x −~y ) P(1) ... (~x −~y ) P(N) ∑ 1 δ 1 P(1) δσ1 δ N P(N) δσN N! P where D(P) = 0 for Bosons and D(P) = deg[P] for Fermions. σ ...σ In second quantization, the vectors |~x1,...,~xN,> 1 N are used to construct a state of the quantum system in the following way. A candidate for the wave-function of the system is a function of the N positions and the time, ψσ1...σN (~x1,...,~xN,t). We consider a state vector in the Hilbert space of the N-particle system, ψσ1...σN (~x1,...,~xN,t) and we form the quantity Z 3 3 σ1...σN |ψ(t) >= d x1 ...d xNψσ1...σN (~x1,...,~xN,t)|~x1,...,~xN,> (1.16)

There is a one-to-one correspondence between the state vectors |ψ(t) > and the functions ψσ1...σN (~x1,...,~xN,t).

If we have a function, ψσ1...σN (~x1,...,~xN,t), we simply form the corresponding |ψ(t) > by forming the integrals in equation (1.16). If, on the other hand, we are given |ψ(t) >, we can ﬁnd the function which corresponds to it by taking the inner product,

<~x1,...,~xN|σ1...σN |ψ(t) >= ψσ1...σN (~x1,...,~xN,t) (1.17)

This gives us two languages in which we can discuss the same quantity. Now, let us assume that ψσ1...σN (~x1,...,~xN,t) is the wave-function. That is, it satisﬁes the Schödinger equation (1.4). Second quantization will then give us the wave-function described as the state Z 3 3 σ1...σN |ψ(t) >= d x1 ...d xNψσ1...σN (~x1,...,~xN,t)|~x1,...,~xN,> (1.18)

Unit normalization of the wave-function ψσ1...σN (~x1,...,~xN,t) results in unit normalization of the state |Ψ(t) >, Z 3 †σ1...σN < Ψ(t)|Ψ(t) >= d x1 ...d~xNψ (~x1,...,~xN,t)ψσ1...σN (~x1,...,~xN,t)

= 1

We can ask the question as to what is the equation which |Ψ(t) > must satisfy that is equivalent to the fact that ψσ1...σN (~x1,...,~xN,t) satisﬁes the Schrödinger equation. To answer this question, we consider the operator

Z 2 Z 3 h¯ ~ †σ ~ 1 3 3 †σ †ρ σ˜ ρ˜ H = d x ∇ψ (~x) · ∇ψ (~x) + d xd yψ (~x)ψ (~y)V (~x −~y)ψ ˜ (~y)ψ ˜ (~x) 2m σ 2 σρ ρ σ (1.19) 1.4 Second Quantization 19

This operator is the Hamiltonian in the second quantized language. It is easy to see that ψσ1...σN (~x1,...,~xN,t) obeys the Schrödinger equation (1.4) when |Ψ(t) > satisﬁes the equation ∂ ih¯ |Ψ(t) >= H |Ψ(t) > (1.20) ∂t Furthermore, one can construct an initial state |Ψ(0) > using the initial many-particle wave-function

ψσ1...σN (~x1,...,~xN,t = 0). The state at later times is then uniquely determined by (1.20) which has the formal solution

|Ψ(t) >= e−iHt/h¯ |Ψ(0) > and the wave-function at any time that can be extracted from it by taking the inner product,

ψσ1...σN (~x1,...,~xN,t) =<~x1,...,~xN|σ1...σN |Ψ(t) > and it must coincide with the solution of the many-body Schrödinger equation (1.4). Thus, the mathematical problem of solving the second-quantized operator equation (1.20) is identical in all respects to the mathematical problem of solving the many-particle Schrödinger equation (1.4), they are solved when we ﬁnd the wave-function ψσ1...σN (~x1,...,~xN,t) or equivalently the state |Ψ(t) >. We thus have two equivalent formulations of the same theory. σ ...σ The state |~x1 ...~xN > 1 N that we have constructed should be thought of as the quantum mechanical state where the N particles can be found occupying the positions~x1,...,~xN and the spin σ ...σ states σ1 ...σN. To see this, we note that |~x1 ...~xN > 1 N is an eigenstate of the density operator, which we form from the product of a creation and annihilation operator,

σ† ρ(~x) = ψ (~x)ψσ (~x)

σ ...σ (where we are using the summation convention for the spin index σ). Operating on |~x1,...,~xN > 1 N , we discover that these states are eigenstates of the density with eigenvalues given by a sum of delta-functions

N ! σ1...σN σ1...σN ρ(~x) |~x1 ...~xn > = ∑ δ(~x −~xi) |~x1 ...~xn > (1.21) i=1 which is just what we would expect for a group of N particles localized at positions~x1,...,~xN. This formula holds for both Bosons and Fermions. The second quantized Schrödinger equation (1.20) does not contain the explicit information that there are N particles. The number of particles can be measured by the number operator which is an integral over space of the density operator, Z 3 †σ N = d xψ (~x)ψσ (~x)

The states, |Ψ(t) >, which we have constructed are eigenstates of the particle number operator,

N |Ψ(t) > = N |Ψ(t) >

Furthermore, the Hamiltonian commutes with the number operator,

[N ,H ] = 0 (1.22)

This can easily be checked explicitly using the algebra for the operators ψ(~x) and ψ†(~x). The result is that the number operator N and the Hamiltonian H can have simultaneous eigenvalues and that the total number of particles will be preserved by the time evolution of the system. 20 Chapter 1. Many particle physics

1.5 The Heisenberg picture Let us pause to review what we have done so far. We have found two different descriptions of the many-particle quantum system. The ﬁrst one is conventional quantum mechanics where we should solve the partial differential equation to ﬁnd the wave-function of the many-particle system. We have summarized the relevant equations in the inset below.

Many-particle quantum mechanics The wave-function must obey the Schrödinger equation:

N 2 ∂ −h¯ ~ 2 ih¯ ψσ1...σN (~x1,...,~xN,t) = ∑ ∇i ψσ1...σN (~x1,...,~xN,t) ∂t i=1 2m ρiρ j + ∑Vσiσ j (~xi −~x j)ψσ1...... ρi...ρ j...σN (~x1,...,~xN,t) i< j

The wave-function should be normalized, Z 3 3 †σ1...σN d x1 ...d xNψ (~x1,...,~xN,t)ψσ1...σN (~x1,...,~xN,t) = 1

Bosons have symmetric wave-functions

ψσ1...σN (~x1,...,~xN,t) = ψσP(1)...σP(N) (~xP(1),...,~xP(N),t)

Fermions have anti-symmetric wave-functions

deg(P) ψσ1...σN (~x1,...,~xN,t) = (−1) ψσP(1)...σP(N) (~xP(1),...,~xP(N),t)

for any permutation P.

The other is the second quantized picture, where we are given Hamiltonian and number operators containing fields and the operator nature of the fields is defined by their commutation relations. These and the equation of motion for the quantum state are summarized in the inset below. We have seen in the above development how the mathematical problem which is defined in the inset above and the inset below are equivalent.

Second quantization in the Schrödinger Picture The Schrödinger equation is

∂ ih¯ |Ψ(t) >= H |Ψ(t) > , N |Ψ(t) >= N |Ψ(t) > ∂t The Number and Hamiltonian operators are

Z J 3 †σ N = d x ∑ ψ (~x)ψσ (~x) σ=−J Z 2 Z 3 h¯ ~ †σ ~ 1 3 3 †σ 0 †σ ρρ0 H = d x ∇ψ (~x) · ∇ψ (~x) + d xd yψ (~x)ψ (~y)V 0 (~x −~y)ψ (~y)ψ 0 (~x) 2m σ 2 σσ ρ ρ 1.5 The Heisenberg picture 21

and [N ,H] = 0. For Bosons, the commutation relations are

†ρ ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y) †σ †ρ ψσ (~x),ψρ (~y) = 0 , ψ (~x),ψ (~y) = 0

For Fermions, the anti-commutation relations are

†ρ ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y) †σ †ρ ψσ (~x),ψρ (~y) = 0 , ψ (~x),ψ (~y) = 0

The latter set of equations, those in the inset above, are essentially the definition of a quantum †σ field theory. The quantum fields are the field operators ψσ (~x) and ψ (~x). Note that they do not depend on time. Instead, the state |Ψ(t) > is time dependent. The reason for this is that, like our many-particle problem (1.4), we have formulated the problem in the Schrödinger picture of quantum mechanics where operators are time independent and the states carry the time dependence. The Heisenberg picture is an alternative and equivalent formulation of quantum mechanics. It is related to the Schrödingier picture that we have developed so far by a time dependent unitary transformation of the operators and state vectors. The unitary transformation begins with the observation that, if we know the state of the system at an initial time, say at t = 0, we can find a formal solution of the equation of motion for the state vector,

|Ψ(t)i = e−iHt/h¯ |Ψ(0)i which uses a unitary operator that is obtained by exponentiating the Hamiltonian, exp(−iHt/h¯). We can thus set the state vector to its initial condition (assuming t = 0 is where we must impose an initial condition) by a unitary transformation. Going to the Heisenberg picture simply does this unitary transformation to all of the operators to get an equivalent description of the theory where the operators are time-dependent and the states are independent of time. The unitary transformation of the operators is

0 iHt/h¯ −iHt/h¯ †σ 0 iHt/h¯ †σ −iHt/h¯ ψσ (~x,t) = e ψσ (~x)e , ψ (~x,t) = e ψ (~x)e (1.23)

In the Heisenberg picture, the states are time independent. For a given physical situation the quantum state is simply given by the initial state of the system. The operators, on the other hand, become time dependent and it is their time dependence which carries the information of the time evolution of the quantum system. In quantum ﬁeld theory, particularly the relativistic quantum ﬁeld theory which we shall study later on, the equations of motion are more commonly presented in the Heisenberg picture. †σ Unlike the time-independent operators ψσ (~x) and ψ (~x) which we introduced in order to 0 †σ 0 construct second quantization, the Heisenberg picture operators ψσ (~x,t) and ψ (~x,t) now depend on time and their time dependence contains dynamical information, which is determined by equations (1.23). This information can also be given as a differential equation, the Heisenberg equation of motion, which can be obtained by taking a time derivative of equations (1.23).

∂ ∂ ih¯ ψ(~x,t) = [ψ(~x,t),H] , ih¯ ψ†(~x,t) = ψ†(~x,t),H (1.24) ∂t ∂t These are the usual algebraic operator equations which are meant to be solved to find the time dependence of the operators in the Heisenberg picture. In (1.24), and elsewhere when it is clear from 22 Chapter 1. Many particle physics the context, we will drop the prime on the Heisenberg picture fields. They are still distinguished from the Schrödinger picture fields in that they are time dependent and the Schrödinger picture fields are not. The Heisenberg picture field operators have an equal-time commutator algebra, which can be obtained from (1.10) or (1.11) by multiplying from the left and right by e−iHt/h¯ and eiHt/h¯ , respectively. This leads to the canonical equal-time commutation relations for Bosons,

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y) (1.25) †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 (1.26) or the canonical equal-time anti-commutation relations for Fermions,

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), (1.27) †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 (1.28) At this point the reader should take careful note of the fact that this algebra holds only when the times in both of the operators are the same. The time-derivative of the time-dependent field ψσ (~x,t) can be computed from the Heisenberg equation of motion (1.24) using the equal time commutation relations. It is given by an equation which looks like a non-linear generalization of the Schrödinger equation 2 Z ∂ h¯ ~ 2 3 ρρ0 †σ 0 ih¯ + ∇ ψ (~x,t) = d yV 0 (|~x −~y|)ψ (~y,t)ψ 0 (~y,t)ψ (~x,t) (1.29) ∂t 2m σ σσ ρ ρ Again, the non-relativistic quantum mechanics problem is presented as a quantum field theory. The †σ operators ψσ (~x,t) and ψ (~x,t) are the quantized fields. They satisfy the equal time commutation relations in (1.25) and (1.26) for Bosons or the anti-commutation relations (1.27) and (1.28) for Fermions. These define their algebraic properties as quantum mechanical operators. Their time evolution is determined by solving the non-linear field equation (1.29). That field equation has been presented in a standard form, with the “Schrödinger wave operator”

∂ h¯ 2 ih¯ + ~∇ 2 ∂t 2m operating on the field on the right-hand-side and with an additional non-linear interaction term. We should note the similarity of the field equation with the non-linear Schrödinger equation for a single particle. However, as we have said before, ψσ (~x,t) is not a wave-function of a single particle, it is an operator which obeys the equal time (anti-)commutation relations. We thus have our third presentation of the many-particle problem, the field equation of a quantum field theory plus the equal-time commutation or anti-commutation relations which define the quantum fields as operators. We would fix the total number of particles by requiring that states are eigenvectors of the number operator N with eigenvalue N. This is compatible with the field equation when d dt N = 0, which is the case for the example that we are considering. The Heisenberg formulation of the many-particle problem is summarized in the inset below.

Second quantization in the Heisenberg picture The ﬁeld equation is

2 Z ∂ h¯ ~ 2 3 ρρ0 †σ 0 ih¯ + ∇ ψ (~x,t) = d yV 0 (~x −~y)ψ (~y,t)ψ 0 (~y,t)ψ (~x,t) ∂t 2m σ σσ ρ ρ 1.6 Summary of this chapter 23

Equal-time commutation relations for Bosons are

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0

Equal-time anti-commutation relations for Fermions are

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0

d Since dt N = 0 and we could ﬁx the total particle number by requiring that states are eigenstates of N with eigenvalue N.

1.6 Summary of this chapter In a quantum mechanical system of N identical non-relativistic particles with spin, the wave-

function, ψσ1...σN (~x1,...,~xN,t), is a function of time, t, the positions,~x1,...,~xN, and it depends on the spin states σ1 ...σN of the particles. If a particle has spin J, its index σ runs over 2J + 1 values. It must solve the Schrödinger equation,

N 2 ∂ −h¯ ~ 2 ih¯ ψσ1...σN (~x1,...,~xN,t) = ∑ ∇i ψσ1...σN (~x1,...,~xN,t) ∂t i=1 2m ρiρ j + ∑Vσiσ j (~xi −~x j)ψσ1...ρi...ρ j...σN (~x1,...,~xN,t) i< j with appropriate boundary conditions. Here, we have assumed a two-body interaction (that is, the interaction of any two particles does not depend on the positions and spins of the other particles). If the identical particles are Bosons, the wave-function is completely symmetric under the simultaneous permutations of the labels of spins and positions, σ1~x1 ...σN~xN → σP(1)~xP(1) ...σP(N)~xP(N). If the identical particles are Fermions, it is completely anti-symmetric. Wave-functions should be normalized, Z 3 3 †σ1...σN d x1 ...d xN ψ (~x1,...,~xN,t)ψσ1...σN (~x1,...,~xN,t) = 1

The equivalent second quantized theory in the Schrödinger picture has the state vector |Ψ(t) > obeying the Schrödinger equation and being an eigenstate of the number operator with eigenvalue N, ∂ ih¯ |Ψ(t) >= H |Ψ(t) > , N |Ψ(t) >= N |Ψ(t) > ∂t where the number and Hamiltonian operators are Z 3 †σ N = d xψ (~x)ψσ (~x) Z 2 Z 3 h¯ ~ †σ ~ 1 3 3 †σ †σ 0 ρρ0 H = d x ∇ψ (~x) · ∇ψ (~x) + d xd yψ (~x)ψ (~y)V 0 (~x −~y)ψ 0 (~y)ψ (~x) 2m σ 2 σσ ρ ρ The quantized ﬁelds obey the commutation relations for Bosons:

†ρ ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y) †σ †ρ ψσ (~x),ψρ (~y) = 0 , ψ (~x),ψ (~y) = 0 24 Chapter 1. Many particle physics or the the anti-commutation relations for Fermions:

†ρ ρ ψσ (~x),ψ (~y) = δσ δ(~x −~y) †σ †ρ ψσ (~x),ψρ (~y) = 0 , ψ (~x),ψ (~y) = 0

We use the notation [A,B] ≡ AB − BA for a communator and {A,B} ≡ AB + BA are for an anti- commutator.

The equivalent second quantized theory in the Heisenberg picture is deﬁned by the ﬁeld equation:

2 Z ∂ h¯ ~ 2 3 ρρ0 †σ 0 ih¯ + ∇ ψ (~x,t) = d yV 0 (~x −~y)ψ (~y,t)ψ 0 (~y,t)ψ (~x,t) ∂t 2m σ σσ ρ ρ and, if the particles are Bosons, the equal-time commutation relations for the time-dependent ﬁelds,

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 or, if the particles are Fermions, the equal-time anti-commutation relations for the time-dependent ﬁelds,

†ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0

The wave equation and equal time commutation or anti-commutation relations above are a definition of the many-particle problem which is closest in spirit to a quantum field theory. In this formalism, it will turn out that the number operator, constructed from the time-dependent fields, Z 3 †σ N = d xψ (~x,t)ψσ (~x,t) (1.30) is independent of the time. If an initial state is an eigenstate of N with eigenvalue N, it will remain so at later times. Moreover, we can set the time that is inside the integral in (1.30) to whatever value we choose. We can thus show that it obeys the algebra

ρ† ρ† [N ,ψσ (~x,t)] = −ψσ (~x,t) , N ,ψ (~x,t) = ψ (~x,t) which tells us that if |ϕ > is an eigenstate of N with eigenvalue N,

N |ϕ >= N|ϕ >

ρ† then ψσ (~x,t) and ψ (~x,t) are a lowering and raising operators for particle number,

ρ† ρ† N ψ (~x,t)|ϕ >= (N + 1)ψ (~x,t)|ϕ > N ψσ (~x,t)|ϕ >= (N − 1)ψσ (~x,t)|ϕ >

N is a positive semi-deﬁnite operator and that its lowest eigenvalue is the empty vacuum, which obeys ψ(~x,t)|0 >= 0 for all values of ~x,t. Then N |0 >= 0 and all of the eigenvalues of N are non-negative integers. 2. Degenerate Fermi and Bose Gases

2.1 A preview of this chapter In this chapter, we will study the Heisenberg representation quantum ﬁeld theories of non-relativistic many-particle systems what we developed in the previous chapter in the limit where the volume is very large, the density is ﬁnite and the inter-particle interactions is weak. This will introduce the idea of Fermi energy and Fermi surface for Fermions, the concept of particles and holes, and it will allow us to study some of the properties of a weakly interacting Fermi gas. We will also introduce the concept of a Bose condensate for a many-boson system and study the low energy excitations of a wealky interacting system of Bosons.

2.2 The limit of weak interactions As an example of our use of a quantum field theory to describe a quantum mechanical system with many identical particles, let us the special case where the particles interact with each other so weakly that, to a first approximation, we can ignore the interactions. In the Hamiltonians which we discussed in the previous chapter, this happens when we can ignore the terms containing the interaction potential V(~x −~y). We will assume that the particles are either Bosons or Fermions. The beginning of our development applies equally well to both cases. However, as we shall see later, there are dramatic differences in the low energy states of a system of Fermions or a system of Bosons. We will work in the Heisenberg picture of quantum mechanics. In that picture, the quantum field is a time- (as well as space-) dependent operator which obeys the wave equation

∂ h¯ 2 ih¯ + ~∇2 ψ (~x,t) = 0 (2.1) ∂t 2m σ It also obeys equal time commutation relations (for Bosons)

†ρ ρ 3 ψσ (~x,t),ψ (~y,t) = δσ δ (~x −~y) (2.2) †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 (2.3) 26 Chapter 2. Degenerate Fermi and Bose Gases or the equal tme anti-commutation relations (for Fermions)

†ρ ρ 3 ψσ (~x,t),ψ (~y,t) = δσ δ (~x −~y) (2.4) †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) } = 0 (2.5)

We will take equations (2.1), (6.42) and (6.43) or equations (2.1), (2.4) and (2.5) as the definition of the quantum field theory and, in the following, we will proceed to find a solution of it. Here, 1 we have retained the spin index, σ,ρ,... to denote the spin state. If the spin is 2 , as it is for an 1 1 electron, this index runs over the two values, − 2 , 2 denoting the two spin states. For a spin J atom, this index will run over 2J + 1 values which label the 2J + 1 different spin states. To simplify the notation, we will generally consider spinless Bosons, so in formulae which are specific to Bosons, we will drop this index from the field operators. It is straightforward to generalize to Bosons with spin if it is needed. The field equation (2.1) is a linear partial differential equation which we can easily solve using a Fourier transform. Here, we are assuming that the three dimensional space is open infinite Euclidean space, called R3, and that the fields have boundary conditions such that their Fourier transform exists. A general solution of the wave equation is

3 2~ 2 Z d k i~k·~x−i h¯ k t/h 2m ¯ ~ ψσ (x,t) = 3 e ασ (k) (2.6) (2π) 2 If this were a wave equation for a classical field, and if (2.6) were a classical solution of that classical field equation, (2.6) is a complete solution in the sense that the function of wave-numbers, ~ ασ (k), can be completely determined by initial data. To do this, we take the Fourier transform of the field at an initial time. For example, if we know that at an initial time, say t = 0, the field ~ is given by the function ψinσ (~x), then we would determine ασ (k) by taking an inverse Fourier transform of equation (2.6) with respect to the space variables, and with the time set to t = 0 to get

Z 3 ~ d x −i~k·~x ασ (k) = 3 e ψσ (~x,0) (2π) 2 ~ We would use this formula to determine the function of wave-vectors, ασ (k). Plugging the result into equation (2.6) then determines the solution of the classical differential equation. However, here, the differential equation is one which must be obeyed by the field operators, so ~ we have a slightly different sense as to how equation (2.6) is a solution of the problem. Now, ασ (k) is a wave-number-dependent operator. As well as solving the wave equation, which equation (2.6) ~ accomplishes, we must determine the properties of the operators, ασ (k), so that the field ψσ (~x,t) in equation (2.6) satisfies the commutation relations (6.42) and (6.43) or the anti-commutation relations (2.4) and (2.5) . ~ †σ ~ It will indeed satisfy those relations if ασ (k) and α (k) satisfy

h ~ †ρ i ρ 3 ~ ασ (k),α (~p) = δσ δ (k −~p) (2.7) h ~ i h †σ ~ †ρ i ασ (k),αρ (~p) = 0 , α (k),α (~p) = 0 (2.8) for Bosons or

n ~ †ρ o ρ 3 ~ ασ (k),α (~p) = δσ δ (k −~p) (2.9) n ~ o n †σ ~ †ρ o ασ (k),αρ (~p) = 0 , α (k),α (~p) = 0 (2.10) 2.2 The limit of weak interactions 27 for Fermions. To see this explicitly for the case of Bosons, consider

3 3 Z d k Z d p ~ h¯~k2 h¯~p2 h i †ρ ik·~x−i 2m t −i~p·~y+i 2m t ~ †ρ ψσ (~x,t),ψ (~y,t) = 3 3 e e ασ (k),α (~p) (2.11) (2π) 2 (2π) 2

Z 3 Z 3 ~ 2 2 d k d p ~ h¯k h¯~p ρ ik·~x−i 2m t −i~p·~y+i 2m t 3 ~ = 3 3 e e δσ δ (k −~p) (2.12) (2π) 2 (2π) 2 3 Z d k ~ = δ ρ eik·(~x−~y) = δ ρ δ 3(~x −~y) (2.13) σ (2π)3 σ where we have used the fact that the Fourier transform of the Dirac delta function is given by the formula 3 Z d k ~ eik·(~x−~y) = δ 3(~x −~y) (2π)3 Then, in order to complete the solution of the problem, we must find the Hilbert space on ~ †σ ~ which ασ (k) and α (k) operate. This is straightforward, and similar to what we have done in the previous section for the Schrödinger picture field operators (in fact these are the Fourier transform of the field operators at time t = 0 where they should be identical to those operates). We begin with the state where there are no particles at all, the empty vacuum |0 >. Here, we will define it as the ~ ~ state which is annihilated by all of the annihilation operators ασ (k) for all values k, σ, ~ ~ ασ (k)|0 >= 0 for all k,σ Similarly

< 0|α†σ (~k) = 0 for all~k,σ

Then, we construct the multi-particle states by repeatedly operating on the vacuum state with the creation operator α†σ (~k), 1 σ1σ2...σN †σ1 †σ2 †σN |~k1,~k2,...,~kN > = √ α (~k1)α (~k2)...α (~kN)|0i (2.14) N! Because the creation operators either commute with each other for Bosons or they anti-commute with each other for Fermions, the states in equation (2.14) are either totally symmetric or totally antisymmetric, respectively, under permuting indices of the momenta and the spins. For a permutation P of the numbers {1,2,...,N}, and for Bosons,

~ ~ ~ σP(1)σP(2)...σP(N) ~ ~ ~ σ1σ2...σN |kP(1),kP(2),...,kP(N) > = |k1,k2,...,kN > and for Fermions

~ ~ ~ σP(1)σP(2)...σP(N) deg[P] ~ ~ ~ σ1σ2...σN |kP(1),kP(2),...,kP(N) > = (−1) |k1,k2,...,kN > where deg[P] is the degree of the permutation, the number of interchanges of neighbouring integers which must be made in order to implement the permutation. The dual states are

~ ~ ~ 1 ~ ~ ~ < k1,k2,...,kN|σ σ ...σ = √ < 0|ασ (kN)...ασ (k2)ασ (k1) 1 2 N N! N 2 1 and

ρ ρ ...ρ 0 ~ ~ ~ 0 1 2 N < k1,k2,...,kN|σ1σ2...σN |~p1,~p2,...,~pN > = 28 Chapter 2. Degenerate Fermi and Bose Gases

δ 0 ρ ρ = NN (− )σ[P] (~k −~p )... (~k −~p ) P(1) ... P(N) ∑ 1 δ 1 P(1) δ N P(N) δσ1 δσN N! P

where σ[P] = 0 for Bosons and σ[P] = deg[P]. When we plug the solution (2.6) into the Hamilto- nian, we get

Z 2 2~ 2 3 h¯ k †σ ~ ~ H0 = d k ∑ α (k)ασ (k) (2.15) σ=1 2m

Here we have used the subscript on H0 to denote the free ﬁeld Hamiltonian. Also, the number operator is

Z 2 3 †σ ~ ~ N = d k ∑ α (k)ασ (k) (2.16) σ=1 The quantum states that we have constructed are eigenstates of both the Hamiltonian and the number operator,

N h¯ 2~k2 ~ ~ ~ σ1σ2...σN i ~ ~ ~ σ1σ2...σN H0|k1,k2,...,kN > = ∑ |k1,k2,...,kN > i=1 2m

σ1σ2...σN σ1σ2...σN N |~k1,~k2,...,~kN > = N |~k1,~k2,...,~kN >

Note that the energy of a basis state is given by the sum of the energies of the particles in the state. This is a result of the fact that the particles are not interacting, so their total energy is just the sum of their individual energies. What is more, there individual energies are entirely due to their kinetic energy, which for a non relativistic particle is ~p2/2m where, where the momentum is given in terms of the wavenumber by ~p = h¯~k. The general solution of the time-dependent Schrödinger equation for the quantum state of the system is

2~ 2 N h¯ ki Z −i ∑i=1 t/h¯ 3 3 2m ~ ~ ~ ~ σ1...σN |Ψ(t)i = d k1 ...d kN e φσ1...σN (k1,...,kN)|k1,...,kN > (2.17)

~ ~ The function φσ1...σN (k1,...,kN) is totally symmetric for Bosons and totally antisymmetric for Fermions. It is the initial value of the many-body wave-function of the system. That is, the function ~ ~ φσ1...σN (k1,...,kN) must be determined by initial data. The appropriate initial data would be the ~ ~ quantum state at an initial time, say t = 0, from which one could obtain φσ1...σN (k1,...,kN) by doing ~ ~ a Fourier transform of the initial wave-function ψσ1...σN (k1,...,kN,t = 0). The time-dependent state vector in equation (2.17) is a complete solution of the problem in the Schrödinger picture. For illustrative purposes, we can also consider the equivalent in the Heisenberg picture. There, it is the time dependent operator in equation (2.6).

2.3 Degenerate Fermi gas and the Fermi surface 2.3.1 The ground state |O > There is a profound difference between the ground state of a system of Bosons and a system of Fermions. When the particles are Bosons, the creation operators commute with each other and, h i2 for example, α†(~k) simply creates two particles in the state with wave-number~k. The lowest energy state simply has all of the Bosons in the single-particle state which has the lowest energy, N |O >= α†(0) |0 >. 2.3 Degenerate Fermi gas and the Fermi surface 29

On the other hand, when the particles are Fermions, their many-particle wave-function must be anti-symmetric. This is reﬂected in second quantization by the fact that the operators which create the Fermions anti-commute with each other. Then the algebra of creation operators in imply 0 0 h i2 α†σ (~k)α†σ (~k0) = −α†σ (~k0)α†σ (~k) and α†σ (~k) |0 >= 0. We cannot create a state where two Fermions have the same spin σ and the same wave-vector~k. This is the manifestation of the Pauli principle in this second quantized framework: two Fermions cannot occupy the same quantum state. This means that, in the ground state, the Fermions must have distinct wave-vectors and spins. Then the lowest energy state of a system of free Fermionic particles must be gotten by populating 2~ 2 N h¯ ki the spin states and all of the the wave-vector states that are closest to zero, so that E = ∑i=1 2m is minimal. These wave-vectors occur in the interior of a sphere with radius kF , called the Fermi wave-vector,

  J 1 † |O >=  ∏ ∏ α σ (~k)|0 > (2.18) c =−J |~k|≤kF σ

where the constant c should be chosen so that the state is normalized.

There is one particle in each state with wavenumber of magnitude less than or equal to kF . Here hk¯ F is called the Fermi momentum and kF is the Fermi wavenumber. It is the upper bound of wave-numbers that Fermions can have in the ground state. The highest energy states that are populated have the energy

h¯ 2k2 ε = F (2.19) F 2m

is called the Fermi energy. The boundary of the set of occupied states in wave-vector space, those states with |~k| = kF and which have the Fermi energy is called the Fermi surface.

2.3.2 Particle and holes

In all of the discussion so far, we have been assuming that the particles which we are studying occupy infinite three dimensional space. Moreover, their wave-vectors also occupy infinite three dimensional space and their spectrum is continuous. This makes the expression (2.18) that we used to define the Fermion ground state problematic in that it is a product over a continuously infinite number of operators. Of course, we could always make sense of it by the common trick of assuming that the space were not quite infinite, but a large but finite box where the fields has, for example, periodic boundary conditions. Then, the wave-vectors would be discrete and equation (2.18) would be an ordinary product over a very large number of discrete factors. Instead of this, we will find a simpler, albeit more formal way to deal with the definition of the ground state which remains in infinite volume and avoids infinite products altogether. In a construction such as this, the normalization constant c in equation (2.18) would be equal to one. We shall re-define what we mean by annihilation and creation operator for a particle as follows: as

~ ~ †σ ~ †σ ~ ~ ασ (k) = aσ (k) , α (k) = a (k) , |k| > kF (2.20) σ ~ †σ ~ † ~ ~ ~ β (k) = a (k) , βσ (k) = aσ (k) , |k| ≤ kF (2.21) 30 Chapter 2. Degenerate Fermi and Bose Gases

We now have two sets of creation and annihilation operators with the anti-commutator algebra

†ρ ρ ~ ασ (k),α (q) = δσ δ(k −~q) (2.22) †σ †ρ ασ (k),αρ (q) = 0 , α (k),α (q) = 0 (2.23) n σ † o σ ~ β (k),βρ (q) = δ ρ δ(k −~q) (2.24)

σ ρ n † † o {β (k),β (q)} = 0 , βσ (k),αρ (q) = 0 (2.25) ρ ρ† ρ {ασ (k),β (q)} = 0 , α (k),β (q) = 0 (2.26) n † o n ρ† † o ασ (k),βρ (q) = 0 , α (k),βρ (q) = 0 (2.27)

The reader should note well that we have not introduced any new concept here. We have simply re-labeled some of the same creation and annihilation operators that we had previously deﬁned. The reason for this re-labeling was so that, using the deﬁnition in equation (2.18), we can see that the Fermion ground state is annihilated by the new annihilation operators,

~ ρ ~ ~ ασ (k)|O >= 0 , β (k)|O >= 0 , for all k,σ,ρ (2.28)

Then, there are apparently two types of excitations of this ground state. One is obtained by creating a particle in a wave-vector state that is outside of the Fermi surface. This is done with the creation †σ operator α (~k) (which has |~k| > kF by deﬁnition (2.21)). Such an excitation is called a “particle”. The other excitation is gotten by annihilating a particle which is already contained in |O > and † ~ whose wave-vector is inside the Fermi surface. This is done by operating with βρ (k) (which has |~k| < kF by deﬁnition (2.21)). Such an excitation is called a “hole”. We will also assume that the ground state is normalized so that

< O|O >= 1 (2.29)

The ﬁeld operator is given

3 2~ 2 Z d k i~k·~x−i h¯ k t/h 2m ¯ ~ ψσ (x,t) = e ασ (k) + ~ 3 |k|>kF (2π) 2 3 2~ 2 Z d k −i~k·~x−i h¯ k t/h¯ + e 2m β †(~k) (2.30) ~ 3 σ |k|

Z 3 3 †σ d k (2J + 1)kF ρ =< O|ψ (~x,t)ψσ (~x,t)|O >= (2J + 1) 3 · 1 = 2 k

We can solve this equation to determine Fermi wave-number kF and the Fermi energy εF in terms of the density,

1 2 6π2ρ 3 h¯ 2 6π2ρ 3 k = , ε = F 2J + 1 F 2m 2J + 1 2.3 Degenerate Fermi gas and the Fermi surface 31

The ground state energy U = uV is the expectation value of the Hamiltonian, and the internal energy density u is

5 h¯ 2 (2J + 1)h¯ 2 6π2ρ 3 u = < O|~∇ψ†σ (~x,t) ·~∇ψ (~x,t)|O >= (2.31) 2m σ 20π2m 2J + 1 In fact, we can plug the expression for the ﬁeld operator in equation (2.30) into the particle number and the Hamiltonian to ﬁnd the expressions Z Z 3 †σ ~ ~ 3 † ~ σ ~ N = ρV + d kα (k)ασ (k) − d kβσ (k)β (k) k>kF k

Z 2~ 2 Z 2~ 2 3 h¯ k †σ ~ ~ 3 h¯ k † ~ σ ~ H = uV + d k α (k)ασ (k) − d k βσ (k)β (k) k>kF 2m knegative energy. We could ﬁx up the latter fact by adding a a constant to the single-particle energy so that it is deﬁned with respect to h¯ 2~k2 h¯ 2~k2 the Fermi energy, 2m → 2m − εF . This would translate into a replacement of the Hamiltonian in the above formula by ! Z h2~k2 Z h2~k2 3 ¯ †σ ~ ~ 3 ¯ † ~ σ ~ H − εF N = uV − εF ρ + d k − εF α (k)ασ (k) + d k − εF βσ (k)β (k) k>kF 2m k

2.3.3 The grand canonical ensemble In many practical circumstances in the quantum field theory of Fermions, rather than fixing the total number of particles, as we have been doing so far, it is useful to consider an open system where particles can enter and leave the system. In this case, it is advantageous to study the system with a modified Hamiltonian

H0 = H − µN (2.32)

so that the expectation value of H0 is the appropriate “free energy” that is needed in order to study an open system. In that case, the parameter µ is the chemical potential. In principle, it can be adjusted in order that the system has a given density. If we use H0 to generate the time evolution of the ﬁelds, the ﬁeld equation becomes

∂ h¯ 2 ih¯ + ~∇ 2 + µ ψ (~x,t) = 0 (2.33) ∂t 2m σ The chemical potential has the a thermodynamic deﬁnition,

∂u µ = ∂ρ V where u is the internal energy which we found in equation (2.35) in the previous section. Using equation (2.35), we ﬁnd that, for our example of non-interacting Fermions at zero temperature,

h¯ 2k2 µ = F = ε 2m F 32 Chapter 2. Degenerate Fermi and Bose Gases that is, the chemical potential is equal to the Fermi energy. The chemical potential has the statistical mechanics interpretation as the energy that is gained by adding a single particle to the system. In this example, this is clearly equal to the Fermi energy, since if we add one particle, we must add it with an energy greater than or equal to the Fermi energy. The solution of the ﬁeld equation (2.33) is

3 2~ 2 3 2~ 2 Z d k i~k·~x−i h¯ k −ε t/h Z d k −i~k·~x−i h¯ k −ε t/h 2m F ¯ ~ 2m F ¯ † ~ ψσ (x,t) = e ασ (k) + e β (k) ~ 3 ~ 3 σ |k|>kF (2π) 2 |k|

The grand canonical potential Φ = φV is given by the expectation value of the Hamiltonian H0 in equation (2.32), which we can ﬁnd as

5 (2J + 1)h¯ 2 6π2ρ 3 φ = u − µρ = − (2.35) 30π2m 2J + 1

Notice that, if we use the thermodynamic deﬁnition of the pressure of the Fermi gas is given by

5 ∂ (2J + 1)h¯ 2 6π2ρ 3 P = − (uV) = (2.36) ∂V N 30π2m 2J + 1 we see that, the grand canonical potential for a fee Fermi gas is given by

Φ = −PV and the grand canonical potential density is given by the negative of the pressure, φ = −P. As we observed at the end of the last section, when we plug the solution (2.34) into the Hamiltonian, we get

Z 2 2~ 2 ! 0 3 h¯ k †σ H = d k − εF α (k)ασ (k) ~ ∑ |k|≥kF σ=1 2m Z 2 2~ 2 3 h¯ k † σ + d k − εF β (~k)β (~k) − PV (2.37) ~ ∑ σ |k|≤kF σ=1 2m

2 2 h¯ ~k − The absolute value 2m εF is a positive number, so both particles and holes have positive energies, when the energy is deﬁned in this way. The states of the quantum theory are found by beginning with the ground state, |O >, and † ~ †σ ~ operating with the creation operators ασ (k) and β (k). A basis for the Fock space is

| >, †σ (k)| >, †(k)| >, †σ1 (k ) †σ2 (k )| >, † (k ) † (k )| >, O α O βσ O α 1 α 2 O βσ1 1 βσ2 2 O

†σ1 (k ) † (k )| >, †σ1 (k ) †σ2 (k ) †σ3 (k )| >,... α 1 βσ2 2 O α 1 α 2 α 3 O

The quantum state for a particle is

† ασ (k)|O > 2.4 Bosons 33

It has has energy and particle number which we can obtain by operating the Hamiltonian and particle number operators (with ground state energy and particle number subtracted) 2~ 2 ! 0 † h¯ k † (H + PV)α (k)|O >= − εF α (k)|O > (2.38) σ 2m σ † † ~ (N −Vρ)ασ (k)|O >= (+1) ασ (k)|O > , |k| > kF (2.39) where we remember that −PV is the ground state energy and Vρ is the ground state particle number. 2 2 ~ h¯ ~k ~ We see that a particle in momentum state k has positive energy, 2m − εF > 0 when |k| > kF , and its particle number is one. The quantum state for a hole is β †σ (k)|O > It has energy and particle number

h2~k2 0 †σ ~ ¯ †σ ~ (H + PV)β (k)|O >= − εF β (k)|O > (2.40) 2m †σ †σ (N −Vρ)β (~k)|O >= (−1) β (~k)|O > , |~k| ≤ kF (2.41)

2 2 h¯ ~k − Its energy is also positive, given by 2m εF and its particle number is negative one.

2.4 Bosons Now, if instead of Fermions, we examine the states of Bosons, we ﬁnd that the many-particle state at ﬁrst sight has a much simpler structure. Arbitrarily many particles can occupy the lowest energy state, so the ground state of a system of free Bosons is given by 1 N |O >= √ α†(~k = 0) |0 > (2.42) N!

This state is an eigenstate of the Hamiltonian, H0 in (2.15), with eigenvalue equal to zero and it is an eigenstate of the particle number operator, N in (2.16), with eigenvalue equal to N. This state, where the Bosons have a macroscopic occupation of a single eigenstate of the Hamiltonian, usually the ground state, is called a “Bose-Einstein condensate”. The macroscopic occupation of a single quantum state gives the Bose-Einstein condensate profound properties. It is a superfluid, which is a fluid with vanishing viscosity. It can flow past barriers without friction or dissipation. There is a beautiful argument due to Landau which relates superfluidity to the spectrum of small oscillations of the fluid, the so-called quasi-particles. Let us briefly review Landau’s argument which is based on Galilean relativity. First of all, let us review some facts about Galilean relativity. (There will be a more detailed presentation of Galilean relativity in the next chapter.). In Newtonian mechanics, the momentum and the energy of a mass M, moving with velocity~v are given by 1 ~P = M~v , E = Mv2 , (2.43) 2 respectively. According to Galilean relativity, if we view the same particle from a different reference frame, one which is moving with velocity ~V with respect to the first frame, the momentum and energy will be ~P0 = ~P − M~V (2.44) 1 1 E0 = M(~v −~V)2 = E −~P ·~V + MV 2 (2.45) 2 2 34 Chapter 2. Degenerate Fermi and Bose Gases

Equations (2.44 and (2.45) tell us how to transforms the momentum and the energy when we view the system in a reference frame which is moving with velocity~v. Now, let us consider a fluid flowing through a capillary with uniform velocity~v with respect to the walls of the capillary. We begin by viewing the fluid in its own rest frame. In its rest frame, it has vanishing velocity and momentum and it has energy E0, the ground state energy of the static fluid. A superfluid will flow through a capillary without dissipation. Let us assume that our fluid is not a superfluid, that is, that the motion of the fluid is dissipative. Let us also assume that the process by which it dissipates is the production of ripples in the fluid, called quasi-particles. Let us assume that, in a small enough interval of time, only one quasi-particle is produced. The quasi-particle has momentum ~p and energy ω(p), which is a function of its momentum. After the quasi-particle is produced, in the rest frame of the fluid, the total momentum is that of the quasi-particle, ~p, and the total energy is that of the fluid at rest plus the quasiparticle energy, E0 + ω(p). By the transformation of Galilean relativity in equations (2.44) and (2.45), in the rest frame of the capillary whose velocity is ~V = −~v, the total momentum and energy are ~P0 = ~p − M~v (2.46) 1 E0 = E + ω(p) −~p ·~v + Mv2 (2.47) 0 2 We should compare this with the same motion where no quasi-particle is produced, and the momentum and energy would be

~P˜0 = −M~v (2.48) 1 E˜ 0 = E + Mv2 (2.49) 0 2 This process of producing a quasi-particle will proceed if it is energetically favourable. In the rest frame of the capillary. This is so if the energy of the state where the quasi-particle was produced is less than the energy of the state where it was not produced, E0 ≤ E˜ 0, that is if 1 1 E + ω(p) −~p ·~v + Mv2 ≤ E + Mv2 (2.50) 0 2 0 2 or if ω(p) ≤ ~p ·~v, at least for some values of ~p. This can happen when ω(p) v ≥ v ≡ minimum of (2.51) c p The last inequality tells us that dissipation is allowed only when the fluid velocity exceeds a 2 minimum critical velocity, vc. This critical velocity could vanish, for example, if ω(p) ∼ p , as it does for a normal fluid. On the other hand, if ω(p) ∼ p for small p = |~p|, the critical velocity could be non-zero and dissipation is not allowed for fluid flows with velocities smaller than the critical one. This is Landau’s criterion for a superfluid, that vc > 0. Let us now study the Bose-Einstein condensate in more detail. The ground state that we have written down in equation (2.42) has a fixed number of particles, N. What is more, all of the particles have vanishing kinetic energy and we have considered non-interacting particles so that they have no potential energy, so they are eigenstates of the free field theory Hamiltonian with vanishing energy. What is more, the energy does not depend on the total number of particles. In an open system, particles can wander in and out of the system without changing the energy. This means that the chemical potential is zero. We would thus expect an open system to be a superposition of states with different numbers of particles, rather than (2.42) we would have ∞ cN N |O >= ∑ √ α†(~k = 0) |0 > (2.52) N=0 N! 2.4 Bosons 35

In such a state, the ﬁeld operator has an expectation value

∞ N ∗ p h ~ † ~ i < O|ψ(~x,t)|O >= ∑ NcNcN+1 (N + 1) < 0|α(k = 0)α (k = 0)|0 > (2.53) N=0 where, also

∞ N 2 h ~ † ~ i ∑ |cN| < 0|α(k = 0)α (k = 0)|0 > = 1 N=0 needs to be defined using a regularization. . There is no way to determine the coefficients cN in the context of free field theory, they are simply arbitrary and the ground state of an open system of free Bosons at zero temperature is not unique. To fix this ambiguity and to make our considerations more realistic, but still solvable, we shall consider the system with a small, positive chemical potential and a weak, repulsive interaction, so that the Hamiltonian is

Z h¯ 2 λ H0 = d3x ~∇ψ†(x) ·~∇ψ(x) − µψ†(x)ψ(x) + ψ†(x)ψ†(x)ψ(x)ψ(x) (2.54) 2m 2

We are considering the case where both µ and λ are positive parameters. We have approximated the interactions by a delta-function two-body potential

V(~x −~y) = λδ(~x −~y) and we will take the limit of this theory where the interaction is weak, that is, where λ is sufficiently small that the interaction can be treated as a perturbation. Of course, λ is a constant with dimensions, so to say that λ is small means that it is smaller than other quantities with the same dimensions that we could make out of the other parameters of the theory .1 As we have stated above, once the system is open, the ground state is no longer an eigenstate of particle number but it can be a superposition of states with different particle numbers as in equation (2.52). We have also emphasized that the ground state cannot be determined by free field theory alone and interactions are needed. However, once interactions are present and they play an important role, equation (2.52) is no longer a good characterization of the ground state. On the other hand, our description of the many-particle theory of weakly interacting Bosons using quantum field theory is a useful starting point. To begin, we observe that a characteristic of a state which is a superposition of states with different particle numbers is the fact that the field operator has an expectation value,

< O|ψ(~x,t)|O >= η(~x,t) (2.55)

When this is the case, we can separate the ﬁeld operator into a classical and quantum part,

ψ(~x,t) = η(~x,t) + ψ˜ (~x,t) where

< O|ψ˜ (~x,t)|O >= 0

1 When written in terms of the s-wave scattering length, a, λ = 4πma , the criterion for weak coupling is that h¯ 2 1 aρ 3 << 1. 36 Chapter 2. Degenerate Fermi and Bose Gases

In the weak coupling limit, the classical part of the field operator satisfies the classical field equation,

∂ h¯ 2 ih¯ η(~x,t) = − ~∇2 − µ η(~x,t) + λ|η(~x,t)|2η(~x,t) ∂t 2m

The solutions of the above equation are

r µ η = 0 , η = eiθ λ where the phase θ is not ﬁxed by the equation. When both λ and µ are positive, the solution with q µ lower grand canonical potential is the second one, with η = λ . Here, we have made a choice of the phase. Then, at very weak coupling, the particle density and ground state energy density of the system are gotten by plugging this classical value of the ﬁeld into the number density and the energy density to get µ ρ = λ

λ λ P = −φ = µρ − ρ2 = ρ2 2 2 Let us assume that the ﬁeld operator at time zero is given by

3 Z d k ~ ψ˜ (~x,0) = eik·~xα(~k) (2π)3 with the commutation relations h i h i α(~k),α(~`) = 0 = α†(~k),α†(~`)

h i α(~k),α†(~`) = δ 3(~k −~`)

When we plug

r µ ψ(~x,0) = + ψ˜ (~x,0) λ √ into the Hamiltonian (2.54) and expand in powers of λ, we obtain

µ2 H = − V + . 2λ Z 2 √ 3 h¯ † † µ † † † d x ~∇ψ˜ ·~∇ψ˜ − µψ˜ ψ˜ + ψ˜ ψ˜ + ψ˜ ψ˜ + 4ψ˜ ψ˜ + O( λ) (2.56) 2m 2 ! λ Z h¯ 2~k2 = − ρ2V + d3k + µ α†(~k)α(~k)+ 2 2m Z µ h i √ + d3k α†(~k)α†(−~k) + α(−~k)α(~k) + O( λ) (2.57) 2 2.4 Bosons 37

The Hamiltonian no longer has the form of an energy times the number operator α†α. We need to do a change of variables in order to get it in this form. Consider the transformation

a(~k) coshϕ sinhϕ α(k) = (2.58) a†(−k) sinhϕ coshϕ α†(−k) α(~k) coshϕ −sinhϕ a(k) = (2.59) α†(−k) −sinhϕ coshϕ a†(−k) where ϕ is a function of |~k|. This is called a Bogoliubov transformation. Its speciﬁc form is designed to preserve the commutation relations, so that the new variables also obey h i h i a(~k),a(~`) = 0 = a†(~k),a†(~`)

h i a(~k),a†(~`) = δ 3(~k −~`) for which we need the property cosh2 ϕ − sinh2 ϕ = 1. We assume that ϕ is a real function of |~k|.

a(~k) coshϕ sinhϕ α(k) = (2.60) a†(−k) sinhϕ coshϕ α†(−k) α(~k) coshϕ −sinhϕ a(k) = (2.61) α†(−k) −sinhϕ coshϕ a†(−k)

When we substitute into the Hamiltonian, we obtain ! µ2 Z h¯ 2~k2 H = − V + d3k + µ cosh2 ϕa†(~k)a(~k) + sinh2 ϕa(−~k)a†(−~k) + 2λ 2m Z µ h i + d3k cosh2 ϕa†(~k)a†(−~k) + cosh2 ϕa(−~k)a(~k) 2 Z µ h i + d3k sinh2 ϕa(−~k)a(~k) + sinh2 ϕa†(~k)a†(−~k) + 2 ! Z h¯ 2~k2 + d3k + µ coshϕ sinhϕ a†(~k)a†(−~k) + a(−~k)a(~k) + 2m

Z µ h i + d3k coshϕ sinhϕ a†(−~k)a(−~k) + a(~k)a†(~k) + a†(~k)a(~k) + a(−~k)a†(−~k) (2.62) 2 The off-diagonal terms are proportional to ! µ h¯ 2~k2 cosh2 ϕ + sinhϕ2 + + µ coshϕ sinhϕ 2 2m

( ! ) 1 h¯ 2~k2 = .µ cosh2ϕ + + µ sinh2ϕ 2 2m and, we adjust ϕ so that this quantity vanishes, µ tanh2ϕ = − h¯ 2~k2 2m + µ 38 Chapter 2. Degenerate Fermi and Bose Gases

Then, when we plug this solution into the Hamiltonian, we ﬁnd that the Hamiltonian is !! µ2 Z d3k h¯ 2~k2 H = − V +V E(k) − + µ 2λ 2(2π)3 2m Z √ + d3kE(k)a†(~k)a(~k) + O( λ) (2.63)

where the new energies are v u !2 u h¯ 2~k2 E(k) = t + µ − µ2 (2.64) 2m

Here, we have been careful to keep track of terms which are produced by changing the order of a†(~k) and a(~k). The new excitation which is called a “quasi-particle” has a dispersion relation, for small |~k|, like a sound wave, E(k) ∼ vS|~k|. By Landau’s criterion, s E(k) λρh¯ 2 v = minimum of = = v c k m S

the critical velocity is just equal to the quasi-particle velocity. The weakly interacting Bose gas is a superﬂuid with critical velocity given by the expression above.

2.5 Summary of this chapter In the absence of interactions, the ﬁeld equation of a gas of non-relativistic particles is

∂ h¯ 2 ih¯ + ~∇2 + µ ψ (~x,t) = 0 ∂t 2m σ

For Fermions, µ = εF , the Fermi energy and the ﬁeld equation has the solution

3 2~ 2 Z d k i~k·~x−i h¯ k −ε t/h 2m F ¯ ~ ψσ (x,t) = e ασ (k)+ ~ 3 |k|>kF (2π) 2 3 2~ 2 Z d k −i~k·~x−i h¯ k −ε t/h¯ + e 2m F β †(~k) ~ 3 σ |k|

where the creation and annihilation operators for particles and holes satisfy the algebra

n ~ †ρ o ρ 3 ~ ασ (k),α (~p) = δσ δ (k −~p) n ~ o n †σ ~ †ρ o ασ (k),αρ (~p) = 0 , α (k),α (~p) = 0 n σ ~ † o σ 3 ~ β (k),βρ (~p) = δρ δ (k −~p) n σ ~ ρ o n † ~ † o β (k),β (~p) = 0 , βσ (k),βρ (~p) = 0 (2.65) n ~ ρ o n ~ † o ασ (k),β (~p) = 0 , ασ (k),βρ (~p) = 0 n †σ ~ ρ o n †σ ~ † o α (k),β (~p) = 0 , α (k),βρ (~p) = 0 2.5 Summary of this chapter 39

The ground state obeys

~ σ ~ ~ ασ (k)|O >= 0 , β (k)|O >= 0 , for all k , σ

The Hamiltonian and number operator are diagonal

Z 2 2~ 2 ! 0 3 h¯ k †σ H = d k − εF α (k)ασ (k) ~ ∑ |k|≥kF σ=1 2m Z 2 2~ 2 3 h¯ k † σ + d k − εF β (~k)β (~k) − PV ~ ∑ σ |k|≤kF σ=1 2m

Z Z N = d3k α†(k)α(k) − d3k β †(~k)β(~k) + ρV |~k|>kF |~k|≤kF

2 2 1 h¯ 2 3 2 3 where ρ is the density, εF = 2m 3π ρ , kF = 3π ρ and the equation of state of a cold Fermi gas is

2 2 2 (3π ) 3 h¯ 5 P = ρ 3 5 m

1 Here we have assumed that the spin J = 2 . The ground state of a Bose gas with a weak repulsive interaction is a Bose-Einstein condensate where the ﬁeld operator has a classical part,

ψ(~x,t) = η(~x,t) + ψ˜ (~x,t) , ψ†(~x,t) = η∗(~x,t) + ψ˜ †(~x,t) with

η(~x,t) =< O|ψ(~x,t)|O > , < O|ψ˜ (~x,t)|O >= 0

At sufﬁciently weak coupling, η(~x,t) satisﬁes the classical equation of motion,

∂ h¯ 2 ih¯ + ~∇2 + µ η(~x,t) = λη†(~x,t)η(~x,t)η(~x,t) (2.66) ∂t 2m which is called the Gross-Pitaevskii equation and, to the leading order in λ, the operator Ψ(~x,t) = ψ˜ (~x,t) satisﬁes the Bogoliubov-de Gennes equation ψ˜ †(~x,t)

h¯ 2 ~ 2 ∗ 2 ! ∂ − 2m ∇ − µ + 2η η η ih¯ Ψ(~x,t) = 2 Ψ(~x,t) + ... (2.67) ∂t ∗2 h¯ ~ 2 ∗ −η 2m ∇ + µ − 2η η where corrections are small when λ is small and one must ﬁnd a solution which obeys Ψ(~x,t) = 0 1 Ψ†(~x,t). A translation invariant solution of (2.66) is η = pµ/λ. The leading orders in 1 0 the density, the energy density and the pressure are obtained by plugging the classical solution ψ(~x,t) ∼ η into the number operator and Hamiltonian,

2 µ 4πh¯ 2 1 ρ = → µ = ρ 3 aρ 3 λ m 40 Chapter 2. Degenerate Fermi and Bose Gases

2 λ 2 2πh¯ 5 1 u = ρ = ρ 3 aρ 3 2 m

2 2πh¯ 5 1 P = ρ 3 aρ 3 m

4πh¯ 2a where we have used the expression λ = m with a the s-wave scattering length. These are the 1 ﬁrst terms in an expansion in the dimensionless number aρ 3 and corrections to these formula are suppressed by higher powers of this constant. The corrections to the internal energy are known to the next order, 2

2 3 2πh¯ 5 1 128 1 2 u = ρ 3 aρ 3 1 + √ aρ 3 + ... m 15 π Moreover, the solution of the Bogoliubov-de Gennes equation yields the quasi-particle spectrum

v 2 u 2 ! 2 u h¯ ~k2 h¯ p E(~k) = t + µ − µ2 ∼ 2πaρ |~k| + ... 2m 2m which, in agreement with Landau’s argument for the existence of a superﬂuid state, is linear in the wave-number for small wave-numbers.

2T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957). ;T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957). 3. Classical ﬁeld theory and the action principle

3.1 The Action Principle In the previous chapters, we have formulated the quantum mechanical many-particle system as a quantum field theory. The quantum field theory consisted of a field equation, which was a non-linear partial differential equation, in our example

∂ h¯ 2 ih¯ + ~∇ 2 + µ ψ (~x,t) = λψ†ρ (~x,t)ψ (~x,t)ψ (~x,t) (3.1) ∂t 2m σ ρ σ

which the field operators must satisfy together with some boundary conditions. 1 In addition to the field equation, the field operators were required to satisfy equal time commutation or anti-commutation relations which defined the nature of the operators themselves. In this chapter, we shall examine an alternative way of encoding the information that is contained in the field equation and commutation relations. We will begin with a classical field theory which is specified by writing down an action functional and then we will derive the field equation using the action principle. The action principle stipulates that the action functional is stationary when it is evaluated on those fields which are solutions of the equations of motion of the classical field theory, with appropriate boundary conditions. The classical action also contains information about the fields when they are viewed as generalized dynamical variables. From it, we can identify the generalized coordinates and their canonical momenta and as well as their Poisson brackets. This will give us a classical field theory which we could then quantize by the standard procedure of replacing the classical fields by quantum mechanical operators and Poisson brackets by commutators or anti-commutators. In this approach, the essential information that we need to define the quantum field theory, as it is defined in equations (??)-(??), is encoded in the classical action. One great benefit of being able to derive the field equations of the quantum field theory from an action principle is that the symmetries of the theory are symmetries of the action, as well as being symmetries of the equation of motion and commutation relations. Symmetries of the action

1 Here, and in the following, we will consider the special case of a contact interaction. Everything that we say can easily be generalized to more complicated interactions. 42 Chapter 3. Classical ﬁeld theory and the action principle

are often easier to identify than symmetries of the ﬁeld equations. In addition, the existence of the action and the action principle gives us a bridge between symmetries and conservation laws in the form of Noether’s theorem. This theorem states that, if a mechanical system has a continuous symmetry, and if its equation of motion is derived from an action, then the theory has a conserved charge that associated with that symmetry. In the following, after introducing the action and the action principle, we shall give two alternative proofs of Noether’s theorem. Finally, as we shall see in later chapters, the action is an important ingredient of the functional integral formulation of quantum ﬁeld theory.

3.1.1 The Action Consider the classical field theory, that is a dynamical theory of classical fields which we shall †σ denote by ψσ (~x,t) and ψ (~x,t). In spite of the notation, where take the widely used convention that uses the same symbols, these classical fields are not operators as they were in the quantum field theory that we have formulated in the preceding chapters. Here, they are simply functions, smooth differentiable mappings of space and time (t,~x) onto the complex numbers where ψσ (~x,t) and ψ†σ (~x,t) take their values. Generally, these functions that must obey some appropriate boundary conditions (for example, that they fall of fast enough at infinity that their Fourier transform exists) but they are not necessarily solutions of the equations of motion, for the moment they are just †σ arbitrary functions. The dagger symbol, which distinguishes ψ (~x,t) from ψσ (~x,t), and which we also take from the quantum field theory, in this case simply means complex conjugation of the complex-valued classical fields. Strictly speaking, what we have said in the paragraph above applies only to Bosons. If our theory, once it is eventually quantized, will describe Fermions rather than Bosons, they fields †σ ψσ (~x,t) and ψ (~x,t) are slightly more complicated objects, in that they anti-commute with each other.2 They are still not operators in the sense that the Fermion quantum fields are operators. Rather than the anti-commutators of Fermion field operators which we have studied in previous sections, they simply have anti-commutators where the right-hand-sides always vanish, which we can summarize as

†ρ 0 †ρ 0 ψσ (~x,t)ψ (~y,t ) + ψ (~y,t )ψσ (~x,t) = 0 0 0 ψσ (~x,t)ψρ (~y,t ) + ψρ (~y,t )ψσ (~x,t) = 0 ψ†σ (~x,t)ψ†ρ (~y,t0) + ψ†ρ (~y,t0)ψ†σ (~x,t) = 0

Clearly, the objects which obey the above rules are not ordinary functions. This means that, in our view, Fermions are never really described by classical fields. The closest that we can come are these anti-commuting functions. The detailed study of what these object are is an interesting one. However, we shall not require much of the details, other than a few simple rules which will allow us to use them for a few specific things. We will develop those rules as we need them. The anti-commutators given above are the first such rules. The second such rule concerns complex conjugation, where, when we conjugate a product of anti-commuting fields, we also reverse the order so that, for example,

†σ 0 † †ρ 0 ψ (~y,t )ψρ (~x,t) = ψ (~x,t)ψσ (~y,t )

In spite of the fact that they are not really classical in the case of Fermions, we will often use the term “classical ﬁelds” when referring to either Bosons and Fermions.

2 At this point, we do not expect that it is obvious to the reader why this is needed. For now, we can say that it will simplify our work with Fermions later on. 3.1 The Action Principle 43

The action is given by the integral over the space and time coordinates of a Lagrangian density,

Z Z †σ 3 ∂ S[ψσ ,ψ ] = dt d x L (ψ, ∂t ψ,∇ψ) (3.2)

The Lagrangian density is a function of the classical ﬁelds and their derivatives. In our case,

ih¯ ∂ ih¯ ∂ L (ψ, ∂ ψ,∇ψ) = ψ†σ (~x,t) ψ (~x,t) − ψ†σ (~x,t)ψ (~x,t) ∂t 2 ∂t σ 2 ∂t σ h¯ 2 − ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) + µψ†σ (~x,t)ψ (~x,t) 2m σ σ λ 2 − ψ†σ (~x,t)ψ (~x,t) (3.3) 2 σ

We will argue that, beginning with this classical field theory and applying rules for finding the equation of motion and then the rules of quantization in a straightforward way, we arrive at the non-relativistic quantum field theory that we have been discussing. †σ The action, S[ψσ ,ψ ] in equation (3.2) is the integral over time and space coordinates of the Lagrangian density, L , given in equation (3.3). The action is a functional. A functional is a mathematical object which maps functions onto numbers. Here, we imagine that we have classical †σ functions of the space and time coordinates, ψσ (~x,t) and ψ (~x,t) which we insert, together with their derivatives, into the expression (3.3) to form the Lagrangian density. We then insert the Lagrangian density into the integrand in equation (3.2) and perform the integral. The result is a number, the action has mapped the classical functions onto a number, in this case a real number.3

3.1.2 The action principle and the Euler-Lagrange equations

The action principal states that

The Action Principle: The action functional is stationary when it is evaluated on the field configurations which obey the classical equations of motion and the appropriate boundary conditions. Let us use this statement to find the equations of motion which correspond to a given action. For this, we need a way to decide when the action functional is stationary. Consider two classical †σ field configurations which differ by an infinitesimal function, the classical fields ψσ (~x,t),ψ (~x,t) †σ †σ †σ and the classical fields ψσ (~x,t)+δψσ (~x,t),ψ (~x,t)+δψ (~x,t) where δψσ (~x,t),δψ (~x,t) are functions of infinitesimal magnitude and arbitrary profile. The fields ψ†σ (~x,t) and δψ†σ (~x,t) are the complex conjugates of ψσ (~x,t) and δψσ (~x,t). It is still useful to treat them as independent †σ fields. The action evaluated on the first configuration is S[ψσ ,ψ ] and the action evaluated on †σ †σ the second configuration is S[ψσ + δψσ ,ψ + δψ ]. It is clear that these must differ by an infinitesimal amount. The action is stationary if the difference

†σ †σ †σ δS ≡ S[ψσ + δψσ ,ψ + δψ ] − S[ψσ ,ψ ]

3 For Fermions, the situation is a little more complicated as the functional of anti-commuting functions is not just a number but is itself an algebraic entity. Again, we will not take the time to deﬁne it here, as only some operational aspects of dealing with anti-commuting functions will be needed in the following. 44 Chapter 3. Classical ﬁeld theory and the action principle

†σ vanishes to linear order in the inﬁnitesimal functions δψσ and δψ . To linear order,

Z Z δS = dt d3x· " ∂L ∂ ∂L ∂L δψ (~x,t) + δ ψ (~x,t) + δ(∇ ψ (~x,t)) σ σ ∂ a σ ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ # ∂L ∂ ∂L ∂L +δψ†σ (~x,t) + δ ψ†σ (~x,t) + δ(∇ ψ†σ (~x,t)) †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) (3.4)

In the equation above, we have assumed that the Lagrangian density L (x) depends on the ﬁelds and their ﬁrst derivatives, that is, on the variables

∂ ∂ ψ (~x,t), ψ (~x,t),∇ ψ (~x,t),ψ†ρ (~x,t), ψ†ρ (~x,t),∇ ψ†ρ (~x,t) σ ∂t σ a σ ∂t a but otherwise it is quite general. The idea here is that, if we fix the space and time coordinates ∂ to a specific value, we must treat each of ψσ (~x,t), ∂t ψσ (~x,t) and ∇aψσ (~x,t) and their complex †σ ∂ †σ †σ conjugates ψ (~x,t), ∂t ψ (~x,t) and ∇aψ (~x,t) as independent variables. For fixed ~x and t, ∂ †σ and each value of σ, the partial derivatives by each of ψσ (~x,t), ∂t ψσ (~x,t), ∇aψσ (~x,t), ψ (~x,t), ∂ †σ †σ 4 ∂t ψ (~x,t) and ∇aψ (~x,t) are taken while holding all of the other variables fixed. The variation of the derivatives of the functions are defined as the derivatives of the variations, so that

∂ ∂ ∂ ∂ δ ψ (~x,t) ≡ δψ (~x,t) , δ ψ†ρ (~x,t) ≡ δψ†ρ (~x,t) ∂t σ ∂t σ ∂t ∂t

†ρ †ρ δ(∇aψσ (~x,t)) ≡ ∇a(δψσ (~x,t)) , δ(∇aψ (~x,t)) ≡ ∇a(δψ (~x,t))

Then, using the product rule

! ∂ ∂L δψ (~x,t) ∂t σ ∂ ∂( ∂t ψσ (~x,t)) ! ∂ ∂L ∂ ∂L = δψ (~x,t) + δψ (~x,t) ∂t σ ∂ σ ∂t ∂ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ(~x,t))

4 ∂ †σ ∂ †σ In the case of Fermions, for ﬁxed ~x and t, we must treat ψσ (~x,t), ∂t ψσ (~x,t), ∇aψσ (~x,t), ψ (~x,t), ∂t ψ (~x,t) †σ and ∇aψ (~x,t) as independent anti-commuting numbers. In addition, derivatives by anti-commuting numbers must also be anti-commuting entities. For example,

∂ ∂ ∂ ∂ ∂ ∂ 0 0 = 0 , = − , etc. ∂ψσ (~x,t) ∂ψσ (~x ,t ) ∂ψσ (~x,t) ∂ ∂ ∂ψσ (~x,t) ∂( ∂t ψρ (~x,t)) ∂( ∂t ψρ (~x,t)) Moreover, variables and derivatives by the variables also anti-commute with each other. For example,

∂ † σ † ∂ † ψρ (~x,t) f (ψ,ψ ) = δρ f (ψ,ψ ) − ψρ (~x,t) f (ψ,ψ ) ∂ψσ (~x,t) ∂ψσ (~x,t) ∂ ∂ ψ†ρ (~x,t) f (ψ,ψ†) = −ψ†ρ (~x,t) f (ψ,ψ†) ∂ψσ (~x,t) ∂ψσ (~x,t)

These rules should be sufficient for defining the variation of the action in the case of Fermions. 3.1 The Action Principle 45 we rewrite the expression for the variation of the action in equation (3.4) as Z Z δS = dt d3x· ( " # ∂L ∂ ∂L ∂L δψ (~x,t) − − ∇ σ ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ! ∂ ∂L ∂L + δψ (~x,t) + ∇ δψ (~x,t) σ ∂ a σ ∂t ∂(∇aψ (~x,t)) ∂( ∂t ψσ (~x,t)) σ " # ∂L ∂ ∂L ∂L +δψ†σ (~x,t) − − ∇ †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) ! ) ∂ ∂L ∂L + δψ†σ (~x,t) + ∇ δψ†σ (~x,t) (3.5) ∂ a †σ ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) We shall call the right-hand-side of equation (3.5) the variation of the action. Gauss’ theorem, applied in four-dimensional space-time, can be used to rewrite the last two terms in each line of (3.5) as surface integrals. Each of these terms are the four-dimensional volume integral of a total divergence, for example, from the first line of (3.5), ! ∂ ∂L ∂L δψ (~x,t) + ∇ δψ (~x,t) σ ∂ a σ ∂t ∂(∇aψ (~x,t)) ∂( ∂t ψσ (~x,t) σ is such a four-divergence. Gauss’ theorem allows us to rewrite its space-time volume integral as a surface integral at the boundaries of space and time. We shall assume that the boundary †σ †σ conditions for the functions ψσ (~x,t), ψ (~x,t)r, δψσ (~x,t) and δψ (~x,t) are such that the surface terms that are generated in this way all vanish. These are normally taken either as Dirichlet boundary conditions where the value of the field ψσ (~x,t) is fixed at large |~x| so that δψσ (~x,t) must vanish there or the Neumann boundary condition where δψσ (~x,t) is allowed to be nonzero but the xa ∂L component of the derivatives of the fields normal to the boundary, †σ , must go to zero at |~x| ∂∇aψ (~x,t) the boundary. There is also a boundary condition associated with the boundaries of the time integral,

†σ †σ δψ (~x,t)ψσ (~x,t) = 0 , ψ (~x,t)δψσ (~x,t) = 0 t=ti,t f t=ti,t f where ti and t f are initial and ﬁnal times (which we will usually take to be minus and plus inﬁnity, respectively). Then, assuming that these boundary conditions are obeyed, we can drop the total divergence terms from the variation of the action to get Z Z δS = dt d3x· ( " # ∂L ∂ ∂L ∂L δψ (~x,t) − − ∇ σ ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ " #) ∂L ∂ ∂L ∂L +δψ†σ (~x,t) − − ∇ (3.6) †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t))

†σ We are interested in specific elements of the set of all possible functions ψσ (~x,t) and ψ (~x,t) †σ where the action S[ψσ ,ψ ] is stationary. The action is stationary when the terms linear in the variations, which we have found in equation (3.6), vanish. This must be so for any profile of the †σ functions δψσ (~x,t) and δψ (~x,t). This requires that the coefficients of these functions under the 46 Chapter 3. Classical field theory and the action principle integrations in equation (3.6) must vanish. This gives us a set of differential equations which the classical field must obey, that is, the classical field equations (3.7) and (3.8) below. They are called the Euler-Lagrange equations

Euler-Lagrange Equations

∂L ∂ ∂L ∂L − − ∇ = 0 (3.7) ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ∂L ∂ ∂L ∂L − − ∇ = 0 (3.8) †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t))

Application of the Euler-Lagrange equations (3.7) and (3.8) to the action (3.2) yields equations for the classical fields which is identical to the one for the quantum fields in equation (3.1) plus an equation which is its complex conjugate. This gives the classical field equations. Modulo the ordering of operators in the interaction term, which is arbitrary in the classical equation (remembering minus signs for the case of Fermions where the classical functions anti-commute) but of course is important in the equation for quantum mechanical operators, the classical field equation is identical to the field equation of the quantum field theory. We will find ways to deal with the operator ordering ambiguity later, when we discuss specific computations. As well as the field equation, there are boundary conditions, which must be compatible with the boundary conditions which were used to eliminate boundary terms that were encountered when finding the linear variation of the action.

Euler-Lagrange equations Beginning with the action functional Z Z S = dt d3xL

where L is a function of the variables ∂ ∂ ψ (~x,t), ψ (~x,t),∇ ψ (~x,t),ψ†ρ (~x,t), ψ†ρ (~x,t),∇ ψ†ρ (~x,t) σ ∂t σ a σ ∂t a

the equations of motion resulting from the action principle are the Euler-Lagrange equations:

∂L ∂ ∂L ∂L − − ∇ = 0 ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ∂L ∂ ∂L ∂L − − ∇ = 0 †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) with the appropriate boundary conditions 3.1 The Action Principle 47

3.1.3 Canonical momenta, Poisson brackets and Commutation relations The other data that we need in order to define the quantum field theory are the canonical commutation relations. The form that these must take are also encoded in the action. In this non-relativistic field theory, the Lagrangian density is linear in the time derivative of the field. For this reason, it is easiest to think of the Lagrangian density as being a function on the phase space of the mechanical system, that is, it is a function of the generalized coordinates and momenta, rather than generalized coordinates and velocities. The analog in classical mechanics, where qi are the generalized coordinates and pi are the canonical momenta, and the set of all values of the generalized coordinates and momenta together comprise phase space, is the classical action on phase space Z d S = dtL(q(t), p(t)) , L = p (t) q (t) − H(q(t), p(t)) i dt i L is the Lagrangian and the phase space function H(q, p) is the Hamiltonian. The momenta and coordinates have the Poisson bracket qi, p j = δi j , qi,q j = 0 , pi, p j = 0

∂ which can be read from the ﬁrst, linear in time derivatives term in the Lagrangian, L = pi(t) δi j ∂t q j(t)+ .... The classical ﬁeld theory Lagrangian density (3.3) has a form analogous to this plus a total derivative, ∂ ∂ ih¯ L = ih¯ψ†σ (~x,t) ψ (~x,t) − H (ψ,ψ†) − ψ†σ (~x,t)ψ (~x,t) ∂t σ ∂t 2 σ

and the total time derivative can be removed by a canonical transformation.5 Then, we would identify the generalized coordinate as the field ψσ (~x,t) and the canonical momenta as being equal to its coefficient in the Lagrangian density, ih¯ψ†σ (~x,t). The Poisson brackets for the classical field theory are then ψ (~x,t),ihψ†ρ (~y,t) = δ ρ δ(~x −~y) , σ ¯ PB σ ψ (~x,t),ihψ (~y,t) = 0 , ψ†σ (~x,t),ihψ†ρ (~y,t) = 0 σ ¯ ρ PB ¯ PB and, when we quantize, we identify the commutator bracket with ih¯ times the Poisson bracket. This tells us that the commutator in the case of Bosons, or anti-commutator in the case of Fermions, in the field theory should be †ρ ρ 3 ψσ (~x,t),ψ (~y,t) = δσ δ (~x −~y) †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0 , ψ (~x,t),ψ (~y,t) = 0 This indeed matches the commutation or anti-commutation relation given in the field theory (??). In addition to the field equation and commutation relations, we learn that the Hamiltonian is given by

Z h¯ 2 H = d3x H (ψ,ψ†) ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) − µψ†σ (~x,t)ψ (~x,t) 2m σ σ λ 2 . + ψ†σ (~x,t)ψ (~x,t) (3.9) 2 σ which, modulo operator ordering, agrees with the expression for Hamiltonian which we derived earlier. 5Even with the total derivative, an equivalent result can be found by analyzing the Lagrangian system as a constrained ρ† system where, when the constraints are properly resolved, we would obtain the same bracket for ψσ (~x,t) and ψ (~x,t). One easy way to see this is to remember that a total time derivative term in a classical action can be removed by a canonical transformation and that the Poisson bracket is left unchanged by canonical transformations. 48 Chapter 3. Classical ﬁeld theory and the action principle

3.2 Noether’s theorem In the last section, we showed how the essential information which appears in the field equations and the commutation relations, if we take those as defining the quantum field theory, is also encoded in the classical action and the action principle. In this section we will show how, if the field equations can be derived from an action via the action principle, symmetries of the theory lead to conservation laws. By a conservation law, we mean an equation of continuity for a charge density and a current density ∂ R(~x,t) +~∇ · J~ (~x,t) = 0 ∂t where R(~x,t) is the charge density and J~ (~x,t) is the current density. The integral over space of the charge density, Z Q = d3xR(~x,t)

deﬁnes a charge. The time derivative of the total charge by the time is given by

d Z ∂ Z I Q = d3x R(~x,t) = − d3x~∇ · J~ (~x,t) = − d~a ·~∇ · J~ (~x,t) = 0 (3.10) dt ∂t where the latter integral is a surface integral of the normal component of the current density over the boundaries of the space. This formula is a statement of charge conservation. It says that the time rate of change of the total charge is equal to the total flux of the current through the boundaries of the system. We will normally use boundary conditions such that the current densities that we will consider go to zero sufficiently rapidly at spatial infinity that the surface integral vanishes and the total charge is therefore time independent. The symmetries which we shall study are those for which there exists the notion of an infinitesimal transformation. An example is translation invariance. A transformation is a change in the space coordinate,~x →~x˜ =~x +~c where~c is a vector whose components are constants. An infinitesimal transformation has a “parameter”,~c, a vector of infinitesimal magnitude. We define symmetry as follows. For our purposes, a symmetry is a particular transformation of the dynamical variables. A transformation of the dynamical variables is a replacement of the †ρ †ρ variables ψσ (~x,t) and ψ (~x,t) by new variables ψ˜σ (~x,t) and ψ˜ (~x,t) wherever they appear on the field equations or, equivalently, wherever they appear in the action. Generally, the new variables †ρ †ρ ψ˜σ (~x,t) and ψ˜ (~x,t) are functions of the old variables, ψσ (~x,t) and ψ (~x,t) as well as their derivatives, space-time coordinates and other parameters. For an infinitesimal symmetry, we will consider an infinitesimal change of variables

ψσ (~x,t) → ψ˜σ (~x,t) = ψσ (~x,t) + δψσ (~x,t) ψ†ρ (~x,t) → ψ˜ †ρ (~x,t) = ψ†ρ (~x,t) + δψ†ρ (~x,t) ,

†ρ where δψσ (~x,t) and δψ (~x,t) have inﬁnitesimal magnitude. For our present purposes, a symmetry is a particular inﬁnitesimal transformation of the dynamical variables such that, without use of the equations of motion, we can show that the linear variation of the Lagrangian density is by terms which can be assembled into partial derivatives by the space and time coordinates:

∂ δL = R(~x,t) +~∇ · J~(~x,t) (3.11) ∂t That is, we have identiﬁed a symmetry if, by examining the linear variation of the Lagrangian density, we can show that it can be written in the form (3.11) for some R and J~ which depend on the ﬁelds, their derivatives and perhaps the space and time coordinates. 3.2 Noether’s theorem 49

3.2.1 Examples of symmetries Consider the Lagrangian density in equation (3.3), which we copy here for the reader’s convenience,

∂ ih¯ ∂ ih¯ ∂ h¯ 2 L (ψ, ψ,∇ψ) = ψ†σ (~x,t) ψ (~x,t) − ψ†σ (~x,t)ψ (~x,t) − ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) ∂t 2 ∂t σ 2 ∂t σ 2m σ λ 2 +µψ†σ (~x,t)ψ (~x,t) − ψ†σ (~x,t)ψ (~x,t) σ 2 σ

Phase symmetry We can see by inspection that the Lagrangian density written above is unchanged if we make the substitution

−iθ ψσ (~x,t) → ψ˜σ (~x,t) = e ψσ (~x,t) ψ†σ (~x,t) → ψ˜ †σ (~x,t) = eiθ ψ†σ (~x,t)

The inﬁnitesimal transformation is

†ρ †ρ δψσ (~x,t) = −iψσ (~x,t) , δψ (~x,t) = iψ (~x,t)

where we have dropped the factor of θ on the right-hand-sides. Under this transformation,

δL = 0

so the above transformation is a symmetry. In this simple case, R = 0 and J~ = 0.

Space and time-translation invariance A second example is the case of a time translation and a space translation where

ψσ (~x,t) → ψ˜σ (~x,t) = ψσ (~x +~ε,t + ε) ψ†σ (~x,t) → ψ˜ †σ (~x,t) = ψ†σ (~x +~ε,t + ε)

The inﬁnitesimal transformations are obtained by Taylor expansion

∂ ψ˜ (~x,t) = ψ (~x +~ε,t + ε) = ψ˜ (~x,t) +~ε ·~∇ψ (~x,t) + ε ψ (~x,t) + ... σ σ σ σ ∂t σ ∂ ψ˜ †σ (~x,t) = ψ†σ (~x,t + ε) = ψ˜ †σ (~x,t) +~ε ·~∇ψ†σ (~x,t) + ε ψ†σ (~x,t) + ... ∂t

so that

∂ ∂ δψ (~x,t) = ~ε ·~∇ + ε ψ (~x,t) , δψ†ρ (~x,t) = ~ε ·~∇ + ε ψ†ρ (~x,t) σ ∂t σ ∂t

Inspection of the change in the Lagrangian density yields

∂ δL = ~ε ·~∇ + ε L ∂t

and the time translation is a symmetry. In this case, R = εL and Ja = εaL . 50 Chapter 3. Classical ﬁeld theory and the action principle

3.2.2 Proof of Noether’s Theorem We have defined a symmetry as a transformation of the fields under which the transformation of the Lagrangian density can be written in the form given in equation (3.11). This was assumed to be possible with use of algebra, but without the benefit of the Euler-Lagrange equations of motion. Now we shall assume that, in addition to this, the Euler-Lagrange equations are satisfied by the classical fields. We begin with the variation of the Lagrangian density which we found in equation (??) as6 ∂L ∂ ∂L ∂L δ ≡ δψ (~x,t) + δ ψ (~x,t) + δ(∇ ψ (~x,t)) L σ σ ∂ a σ ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ∂L ∂ ∂L ∂L +δψ†σ (~x,t) + δ ψ†σ (~x,t) + δ(∇ ψ†σ (~x,t)) †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) We can reorganize this expression to put it into the form where we can use the Euler-Lagrange equations. We get ( ) ∂L ∂ ∂L ∂L δ = δψ (~x,t) − − ∇ L σ ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ( ) ∂L ∂ ∂L ∂L + δψ†σ (~x,t) − − ∇ †σ ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) " !# ∂ ∂L †σ ∂L + δψσ (~x,t) + δψ (~x,t) ∂t ∂ ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L ∂L + (~x,t) + †σ (~x,t) ∇a δψσ δψ †σ ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) Now, we use the Euler-Lagrange equations to set the first two lines to zero. We obtain " !# ∂ ∂L †σ ∂L δL = δψσ (~x,t) + δψ (~x,t) ∂t ∂ ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L ∂L + (~x,t) + †σ (~x,t) ∇a δψσ δψ †σ (3.12) ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) We have shown that, when the equations of motions are used after the variation of the Lagrangian density, any variation of the Lagrangian density is given by total derivatives. Moreover, in equation (3.12) we know the total derivatives are for a given Lagrangian density. Then, we can equate the two different expressions that we have found for the variation of the Lagrangian density which we have found, the one in equation (3.12) and the one in equation (3.11), " # ∂ ∂ ∂L R +~∇ · J~ = δψ (~x,t) ∂t ∂t σ ∂ ∂( ∂t ψσ (~x,t)) ∂L ∂L + (~x,t) + †σ (~x,t) ∇a δψσ δψ †σ (3.13) ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) By combining the terms, we obtain the equation of continuity

∂ R(~x,t) +~∇ · J~ (~x,t) = 0 (3.14) ∂t 6We remind the reader that, in all cases, the variation of the derivative of a function is equal to the derivative of the ∂ ∂ variation, for example, δ ∂t ψ = ∂t (δψ), δ(∇aψ) = ∇a (δψ). 3.3 Phase symmetry and the conservation of particle number 51

where the charge and current densities are given by the expressions

∂L †ρ ∂L R(~x,t) = δψσ (~x,t) + δψ (~x,t) − R(~x,t) ∂ ∂ †ρ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L ∂L a(~x,t) = (~x,t) + †ρ (~x,t) − Ja(~x,t) J δψσ δψ †ρ ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) This is Noether’s theorem. In summary, Noether’s theorem tells us that, given a Lagrangian density L which is a function †σ of the variables ψσ (~x,t) and ψ (~x,t) and their ﬁrst derivatives by time and space coordinates,

Symmetry and Noether0s Theorem. †ρ †ρ †ρ Under ψσ (~x,t) → ψσ (~x,t) + δψσ (~x,t), ψ (~x,t) → ψ (~x,t) + δψ (~x,t), ∂ whenever, without equations of motion, δL = R(~x,t) +~∇ · J~(~x,t) (3.15a) ∂t ∂ R(~x,t) +~∇ · J~ (~x,t) = 0 where (3.15b) ∂t ∂L †ρ ∂L R(~x,t) = δψσ (~x,t) + δψ (~x,t) − R(~x,t) (3.15c) ∂ ∂ †ρ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L ∂L a(~x,t) = (~x,t) + †ρ (~x,t) − Ja(~x,t) J δψσ δψ †ρ (3.15d) ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t))

The charge density R(~x,t) defined in equation (3.15c) and the current density Ja(~x,t) defined in equation (3.15d) are the conserved Noether current (R(~x,t),Ja(~x,t)). In either classical or quantum field theory, a charge density and current density which obey a continuity equation such as the one above is called a conserved current. Here, we have assumed that the fields are classical. However, like the field equations, which we assume to still hold when the classical fields are replaced by quantum field theory operators, we will generally assume that the same conservation law holds at both the classical and quantum levels. Of course, the conservation law is a consequence of the field equations, so if the quantum fields obey the fields equations, the conservations laws that are constructed from them should also hold. However, of course, there are issues in the quantum field theory, such as operator ordering and the singularities which we shall encounter when we consider products of operators evaluated at the same space-time point which can conspire to ruin a conservation law. The conservation of currents in the quantum field theory should therefore always be checked with some care as there are known cases where it fails. The existence of the conserved current R(~x,t),J~ (~x,t) as a consequence of symmetry is the content of Noether’s theorem. We have derived it in the context of our non-relativistic quantum field theory. However, it, or straightforward generalizations of it, are valid for any field theory where the field equations can be obtained from an action by a variational principle.

3.3 Phase symmetry and the conservation of particle number Now, let us consider the Lagrangian density (3.3)and the inﬁnitesimal phase transformation †ρ †ρ δψ = iθψσ (~x,t) , δψ (~x,t) = −iθψ (~x,t) (3.16) which we have already identiﬁed as a symmetry, in this case, we found that δL = 0. The quantities R and J~ which we would use to construct the Noether current are both zero in this case, R = 0, J~ = 0. Then Noether’s theorem tells us that the charge density is

∂L †σ ∂L ρ(~x,t) = iθψσ (~x,t) − iθψ (~x,t) ∂ ∂ †σ ∂( ∂t ψ(~x,t)) ∂( ∂t ψ (~x,t) 52 Chapter 3. Classical ﬁeld theory and the action principle

†σ = −h¯θψ (~x,t)ψσ (~x,t) and the current density is ∂L ∂L ~ (~x,t) = i (~x,t) − i †σ (~x,t) J θψσ θψ †σ ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) ih¯ ←− −→ = −h¯θ ψ†σ (~x,t)( ∇ − ∇ )ψ (~x,t) 2m σ It is convenient to remove the factor of −h¯θ, it is a constant, and if a charge and current density obey the continuity equation, so do that charge and current density each multiplied by the same common constant. When we do this, ﬁnd the conserved current (for which we use the same notation)

i ∂L i †σ ∂L ρ(~x,t) = − ψσ (~x,t) + ψ (~x,t) (3.17a) h¯ ∂ h¯ ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) i ∂L i ∂L (~x,t) = − (~x,t) + †σ (~x,t) Ja ψσ a ψ a †σ (3.17b) h¯ ∂(∇ ψσ (~x,t)) h¯ ∂(∇ ψ (~x,t)) These are quite general, in that the above equation applies to any Lagrangian density which has a phase symmetry (and where the Lagrangian density depends on the ﬁelds and their ﬁrst derivatives only). By Noether’s theorem, the charge and current densities are guaranteed to obey the conservation law ∂ ρ(~x,t) +~∇ ·~J(~x,t) = 0 (3.18a) ∂t For the Lagrangian density (3.3) which we have been discussing the charge and current densities are

†σ ρ(~x,t) = ψ (~x,t)ψσ (~x,t) (3.19a) ih¯ ih¯ ~J(~x,t) = − ψ†σ (~x,t)∇ψ (~x,t) + ∇ψ†σ (~x,t)ψ (~x,t) (3.19b) 2m σ 2m σ The Noether charge is given by Z Z 3 3 †σ d xρ(~x,t) = d xψ (~x,t)ψσ (~x,t) ≡ N which is just the particle number which we have been using in previous chapters. Its derivative by time is Z Z I d 3 ∂ †σ 3 N = d x ψ (~x,t)ψ (~x,t) = − d x~∇ ·~J(~x,t) = − d~s ·~J(~x,t) dt ∂t σ where, in the last expression, we have used Gauss’ theorem to write the integral of the divergence of a vector field as the integral of the normal component of the vector on the boundary of the system. The last integral is interpreted as the negative of the total flux of particles leaving the system through the sphere at the infinite boundaries of space. It is equal to the time rate of change of the total particle number, as it should be for a conserved current. If that final surface integral vanishes the particle number is time-independent. The boundary condition that guarantees that this integral vanishes is the same boundary condition that would make the laplacian −~∇2 operating on the wave- function a hermitian differential operator. These are the boundary conditions which are normally imposed in a quantum mechanical system. Thus, we can expect that the boundary conditions result in conservation of the particle number. We have thus found that the time-independence of the particle number is a consequence of the symmetry of the theory under changes of phase of the field operator. 3.4 Translation invariance 53

3.4 Translation invariance The quantum ﬁeld theory that we have been discussion has a symmetry under constant translations of the space and time coordinates. Under a space and time translation,

†σ †σ †σ ψσ (~x,t) → ψ˜σ (~x,t) = ψσ (~x +~ε,t + ε) , ψ (~x,t) → ψ˜ (~x,t) = ψ (~x +~ε,t + ε) The inﬁnitesimal transformations are gotten by taking the leading order in a Taylor expansion in the parameters~ε and ε, ∂ ∂ δψ (~x,~t) = ~ε · ∇ + ε ψ (~x,t) , δψ†ρ (~x,~t) = ~ε · ∇ + ε ψ†ρ (~x,t) σ ∂t σ ∂t and we can use some simple algebra to show that the variation of the Lagrangian density is the combination of derivatives ∂ δL = ~∇ · (~εL ) + (εL ) ∂t This qualiﬁes the transformation as a symmetry where we identify R = εL and J~ =~εL . The Noether current then has charge density ih¯ †σ ~ ∂ ih¯ ~ ∂ †σ Tt = ψ (~x,t) ~ε · ∇ + ε ψ (~x,t − ~ε · ∇ + ε ψ (~x,t)ψ (~x,t)) − εL 2 ∂t σ 2 ∂t σ ih¯ ih¯ =~ε · ψ†σ (~x,t)~∇ψ (~x,t) − ~∇ψ†σ (~x,t)ψ (~x,t) 2 σ 2 σ 2 h¯ λ 2 + ε ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) − µψ†σ (~x,t)ψ (~x,t) + ψ†σ (~x,t)ψ (~x,t) 2m σ σ 2 σ (3.20) and the current density is

h¯ 2 ∂ ∂ T = − ∇ ψ†σ (~x,t) ~ε ·~∇ + ε ψ (~x,t) + ~ε ·~∇ + ε ψ†σ (~x,t)~∇ ψ (~x,t) a 2m a ∂t σ ∂t a σ

− εaL h¯ 2 ∂ ∂ =ε − ∇ ψ†σ (~x,t) ψ (~x,t) + ψ†σ (~x,t)∇ ψ(~x,t) 2m a ∂t σ ∂t a h¯ 2 + ε − ∇ ψ†σ (~x,t)∇ ψ (~x,t) − ∇ ψ†σ (~x,t)∇ ψ (~x,t) b 2m a b σ b a σ

−δabL } (3.21) From this Noether charge density (3.20) and current density (3.21), we identify the two-index object, called the energy-momentum tensor, which has components

∂ ∂L ∂ †σ ∂L Ttt = ψσ (~x,t) + ψ (~x,t) − L (3.22a) ∂t ∂ ∂t ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L †σ ∂L Ttb = ∇bψσ (~x,t) + ∇bψ (~x,t) (3.22b) ∂ ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂ ∂L ∂ ∂L = (~x,t) + †σ (~x,t) Tat ψσ a ψ a †σ (3.22c) ∂t ∂(∇ ψσ (~x,t)) ∂t ∇ ψ (~x,t)) ∂L ∂L = (~x,t) + †σ (~x,t) − Tab ∇bψσ a ∇bψ a †σ δabL (3.22d) ∂(∇ ψσ (~x,t)) ∂(∇ ψ (~x,t)) 54 Chapter 3. Classical ﬁeld theory and the action principle

We emphasize that the above equations are general and the continuity equations for components of the energy-momentum tensor

∂ T + ∇aT = 0 (3.23a) ∂t tt at ∂ T + ∇aT = 0. (3.23b) ∂t tb ab

hold, whatever the Lagrangian density is, as long as it is space and time translation invariant. Explicitly, for the quantum ﬁeld theory with the Lagrangian density (3.3), the energy-momentum tensor is given by

h¯ 2 T = ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) − µψ†σ (~x,t)ψ (~x,t). tt 2m σ σ λ 2 + ψ†σ (~x,t)ψ (~x,t) (3.24a) 2 σ ih¯ ih¯ T = − ∇ ψ†σ (~x,t)ψ (~x,t) + ψ†σ (~x,t)∇ ψ (~x,t) (3.24b) tb 2 b σ 2 b σ h¯ 2 ∂ ∂ .T = − ∇ ψ†σ (~x,t) ψ (~x,t) + ψ†σ (~x,t)∇ ψ(~x,t) (3.24c) at 2m a ∂t σ ∂t a h¯ 2 T = − ∇ ψ†σ (~x,t)∇ ψ (~x,t) + ∇ ψ†σ (~x,t)∇ ψ (~x,t) ab 2m a b σ b a σ 2 h¯ λ 2 + δ ~∇2 ψσ†(~x,t)ψ (~x,t) − ψσ†(~x,t)ψ (~x,t) (3.24d) ab 4m σ 2 σ

where, we we have used the ﬁeld equation to eliminate the time derivative terms in the Lagrangian density in order to obtain equation (3.24d) for Tab. When we eliminate these time derivatives using the equation of motion, the Lagrangian density is

2 h¯ 2 σ† λ σ† 2 L = − ~∇ ψ (~x,t)ψ (~x,t) + ψ (~x,t)ψ (~x,t) 4m σ 2 σ which is reflected in the last term in equation (3.24d). However, for economy of notation, we have not eliminated the time derivatives of the fields in Tat , as we shall not use its explicit form in what follows. If that form is needed, we would also have to use the field equation in the expression for Tat .

3.5 Galilean symmetry In classical mechanics, Newton’s second law of motion for a free particle,

d2 m ~x(t) = 0 (3.25) dt2 is invariant under replacing ~x by ~x +~vt where ~v is a constant (~x and t independent) vector. The symmetry of the equation of motion under this replacement is a form of non-relativistic relativity called Galilean symmetry. It tells us that the laws of physics hold equally well in either of two different reference frames, the original one, with space coordinates~x and time coordinate t˜ and a new one with space coordinates~x˜ and time coordinate t where the reference frames are related by the coordinate transformations ~x˜ =~x +~vt, t˜ = t 3.5 Galilean symmetry 55

Now, consider a system of particles which interact with each other in such a way that their equations of motion are 2 d ~ m 2~xi(t) = −∑∇iV(~xi −~x j) dt j where~xi with i = 1,..,N labels the positions of the particles. Here, we see that the N equations of motion are also invariant under Galilean transformation of the particle positions, the replacement of (~x1(t),~x2(t),...,~xN(t)) by

(~x˜1(t),~x˜2(t),...,~x˜N(t)) = (~x1(t) +~vt,~x2(t) +~vt,...,~xN(t) +~vt)

The classical mechanics of a free particle or an assembly of free particles interacting by a two-body potential are thus invariant under Galilean transformations. We expect that the quantum mechanics of systems such as these are also invariant. It is interesting to ﬁrst ask how this symmetry can be seen in Schrödinger’s equation for a single free particle, the quantum mechanical version of the system described by the classical equation (3.25). We expect that the quantum mechanical system is Galilean invariant if the classical one is. The single free particle Schrödinger equation is

∂ h¯ 2 ih¯ + ~∇2 ψ(~x,t) = 0 ∂t 2m We could try to boost this system by substituting~x →~x +~vt into the wave-function. This does not quite work. The wave equation which the “boosted” wavefunction ψ(~x +~vt,t) satisﬁes is a little different from the original Schrödinger equation,

∂ h¯ 2 ih¯ − ih¯~v ·~∇ + ~∇2 ψ(~x +~vt,t) = 0 ∂t 2m

It has an extra term “−ih¯~v ·~∇”. We next observe this extra term with an ~x-dependent change of phase of the wave-function. First of all,

∂ h¯ 2 m ih¯ + ~∇2 + ~v2 e−im~v·~x/h¯ ψ(~x +~vt,t) = 0 ∂t 2m 2

m 2 Now, there is a simpler extra term, “ 2~v ” which we can remove using another simple change of phase of the wave-function. We get

2 ∂ h¯ 2 −im~v·~x/h¯+i m~v2t/h¯ ih¯ + ~∇ e 2 ψ(~x +~vt,t) = 0 ∂t 2m

−im~v·~x/h¯+i m~v2t/h¯ which is the original Schrödinger equation. Our conclusion is that ψ(~x,t) and e 2 ψ(~x+ ~vt,t) satisfy the same Schrödinger equation. In as much as the Schrödinger equation describes the physics of the system, the upshot is that all physical processes will follow the same dynamical rules in the original reference frame and in the Galilean boosted reference frame if we map the wave-function from one frame to another by this transformation. This statement, as we have formulated it above, applies to single particle quantum mechanics. We can easily generalize it to the transformation of a many-particle wave-function

m 2 −im~v·(~x1+...+~xN )/h¯+i ~v t/h¯ ψ(~x1,...,~xN.t) → e 2 ψ(~x1 +~vt,...,~xN +~vt.t)

We can easily use this formula to understand how Galilean symmetry works in the quantum field theory formulation of the many-particle problem. However, there is a shortcut to doing this, 56 Chapter 3. Classical field theory and the action principle beginning with free particles. Here, we are actually interested in the classical field theory which yields that quantum field theory when one applies the rules of quantization, as we have discussed them earlier in this chapter. We know that the wave equation which the classical field satisfies, in the case where there are no interactions, is identical in form to the Schrödinger equation,

∂ h¯ 2 ih¯ + ~∇2 + µ ψ (~x,t) = 0 ∂t 2m σ

This tells us how to do the Galilean transformation of the classical ﬁeld:

−im~v·~x/h¯+i m~v2t/h¯ ψσ (~x,t) → ψ˜σ (~x,t) = e 2 ψσ (~x +~vt,t) (3.26)

We know by our discussion above that this must be a symmetry of the non-interacting theory. What remains to check is that it is also a symmetry of a theory with interactions. To check that this is indeed a symmetry of the classical ﬁeld theory with interactions, for example the theory with Lagrangian density given in equation (3.3), we consider the inﬁnitesimal transformation ~ δψσ (~x,t) = −im~v ·~x/h¯ +t~v · ∇ ψσ (~x,t) δψ†σ (~x,t) = im~v ·~x/h¯ +t~v ·~∇ ψ†σ (~x,t)

By plugging this transformation into the Lagrangian density above, we see that the variation of the Lagrangian density (without use of the equations of motion) is equal to

δL = ~∇ · (~vtL (~x,t))

The transformation in equations (??) and (??) is therefore a symmetry. The Noether charge and current densities corresponding to this symmetry are easy to ﬁnd in terms of components of the energy-momentum tensor and the number density and current,

Bb(~x,t) = tTtb(~x,t) + mxb ρ(~x,t) ,

Bba(~x,t) = tTab(~x,t) + mxb ja(~x,t)

Here, (ρ,~j) are the Noether current associated with phase symmetry that we found in equation (3.19a). It is straightforward to conﬁrm that the Galilean current in (??) and (??) is conserved. Noether’s theorem implies that in a Galilean invariant system

∂ b Ba(~x,t) + ∇ B (~x,t) = 0 ∂t ab Alternatively, if the system has space- and time-translation invariance and therefore a conserved energy-momentum tensor, we could form the Galilean Noether charge and current densities and, using the conservation law for the energy-momentum tensor (??) and (??) alone, we can write

∂ [ tT (~x,t) + x ρ(~x,t)] + ∇ [ tT (~x,t) + mx j (~x,t)] ∂t tb b a ab b a

= Ttb(~x,t) + m jb(~x,t) We obtain a condition that the energy-momentum tensor and particle current must obey in order to have Galilean invariance in the system,

Ttb(~x,t) = −m jb(~x,t) 3.6 Scale invariance 57

The conclusion is that the momentum density is equal to minus the particle mass times the particle current density in any translation and Galilean invariant system. In summary, the consequences of Galilean invariance are

Galilean invariance. im δaψ (~x,t) = t∇a − xa ψ (~x,t) (3.27a) σ h¯ σ †ρ im †ρ δaψ (~x,t) = t∇a + xa ψ (~x,t) (3.27b) h¯ b δaL = ∇ (tδbaL ) (3.27c)

Bb(~x,t) = tTtb(~x,t) + mxb ρ(~x,t) (3.27d)

Bba(~x,t) = tTab(~x,t) + mxb ja(~x,t) (3.27e) ∂ ∂ If T (~x,t) + ∇aT (~x,t) = 0, T (~x,t) + ∇aT (~x,t) = 0, (3.27f) ∂t tt at ∂t tb ab ∂ R (~x,t) +~∇ · J~ (~x,t) = 0 requires (3.27g) ∂t b b Tat (~x,t) + m ja(~x,t) = 0 (3.27h)

3.6 Scale invariance In some circumstances, the non-relativistic quantum ﬁeld theory that we have been discussing can have a symmetry under scaling of the space and time variables. If we examine the ﬁeld equation,

∂ h¯ 2 ih¯ ψ (~x,t) = − ~∇ 2 − µ ψ (~x,t) + λψ†ρ (~x,t)ψ (~x,t)ψ (~x,t) (3.28) ∂t σ 2m σ ρ σ

†σ and, if we assume that ψσ (~x,t) and ψ (~x,t) satisfy the ﬁeld equation, and, wherever ψσ (~x,t) and ψ†σ (~x,t) appear, we substitute7

†σ d †σ 2 ψ˜ (~x,t)) ≡ Λ 2 ψ (Λ~x,Λ t) (3.29) d 2 ψ˜σ (~x,t)) ≡ Λ 2 ψσ (Λ~x,Λ t)) (3.30)

†σ with Λ is a positive real number, we see that ψ˜σ (~x,t) and ψ˜ (~x,t) also satisfy the ﬁeld equation with some re-scaled parameters,

∂ h¯ 2 µ ih¯ ψ˜ (~x,t) = − ~∇ 2 − ψ˜ (~x,t) + λΛd−2ψ˜ †ρ (~x,t)ψ˜ (~x,t)ψ˜ (~x,t) (3.31) ∂t σ 2m Λ2 σ ρ σ where we have given the result for d dimensions. What is more, the factors in front of the ﬁelds †σ in equations (3.29) and (3.30) are determined by requiring that ψ˜σ (~x,t)) and ψ˜ (~x,t) satisfy the equal-time commutation or anti-commutation relations,

†ρ h d 2 d †σ 2 i d ρ d ρ d ψ˜σ (~x,t)),ψ˜ (~x,t) = Λ 2 ψσ (Λ~x,Λ t)),Λ 2 ψ (Λ~x,Λ t) = Λ δσ δ (Λ~x−Λ~y) = δσ δ (~x−~y)

where we have used the property of the Dirac delta function δ(Λx) = δ(x)/|Λ|. If we set µ → 0 and if d = 2 or λ → 0, we obtain a scale invariant quantum ﬁeld theory. The inﬁnitesimal symmetry

7Note that the space and time coordinates to not scale in the same way, in fact ~x → Λ~x and t → Λ2t. In general ~x → Λ~x and t → Λzt where z is called the dynamical critical exponent. For our free non-relativistic field theory, z = 2 whereas, for relativistic field theory that we will study in subsequent chapters, z = 1. 58 Chapter 3. Classical field theory and the action principle

transformations are ∂ d δψ (~x,t)) = 2t +~x ·~∇ + ψ (~x,t) (3.32) σ ∂t 2 σ ∂ d δψ†σ (~x,t)) = 2t +~x ·~∇ + ψ†σ (~x,t) (3.33) ∂t 2 The Noether current that is associated with this transformation is constructed from the energy- momentum tensor and the particle number current as

R(~x,t) = 2tTtt (~x,t) + xbTtb(~x,t) (3.34) d h¯ 2 Ja(~x,t) = 2tTat (~x,t) + x T (~x,t) − ∇aρ(~x,t) (3.35) b ab 2 2m When the system is scale invariant, this charge and current density obeys the equation of continuity,

∂ a R(~x,t) + ∇aJ (~x,t) = 0 ∂t This equation involves an identity between the momentum charge density and the particle number current,

d h¯ 2 2T (~x,t) + T (~x,t) − ~∇2ρ(~x,t) = 0 (3.36) tt aa 2 2m Any time and space-translation invariant theory with a conserved energy-momentum tensor and a number charge and current density where the energy-momentum tensor and the number current also obeys equation (3.36), then theory also has scale symmetry.

3.6.1 Improving the energy-momentum tensor It is sometimes convenient to consider an “improved” the energy-mometum tensor. Improvement is a procedure which adds a conserved, symmetric tensor to the energy momentum tensor in order to get a tensor with more favourable properties. In the present case, consider

d h¯ 2 T (~x,t) = T˜ (~x,t) + δ ~∇2 − ∇ ∇ ρ(~x,t) ab ab d − 1 4m ab a b

σ† where d is the dimension of space and, as usual, ρ(~x,t) = ψ (~x,t)ψσ (~x,t). Since

a h ~ 2 i ∇ δab∇ − ∇a∇b anything = 0

and therefore a a ∇ T˜ ab(~x,t) = ∇ Tab(~x,t) the added term does not affect continuity equations. Also, the quantity that has been added, is a total divergence of derivatives of the density

d h¯ 2 d h¯ 2 δ ~∇2 − ∇ ∇ ρ(~x,t) = ∇c (δ δ − δ δ )∇dρ(~x,t) d − 1 4m ab a b d − 1 4m ab cd ac bd

The derivatives ∇dρ(~x,t) will falls off rapidly at spatial inﬁnity, particularly when the density approaches a constant there. Then, using Gauss’ theorem, we see that Z Z 3 3 d x Tab(~x,t) = d x T˜ ab(~x,t) 3.6 Scale invariance 59

Finally, if the spatial part of the energy-momentum tensor is symmetric, Tab(~x,t) = Tab(~x,t) then so is T˜ ab(~x,t) = T˜ ab(~x,t). Finally, h¯ 2 T a = T˜ a + d ~∇2ρ(~x,t) a a 2m where we have written the expression for d space dimensions. Thus, by adjusting the constant c, we can adjust the trace of T˜ ab. The condition for scale invariance which we found in equation (3.36) was h¯ 2 2T + T a − d ~∇2ρ = 0 tt a 4m

In terms of T˜ ab the condition becomes

2Ttt + T˜ aa = 0

3.6.2 The consequences of scale invariance In summary, the consequences of scale invariance are

Scale invariance. ∂ d δψ (~x,t) = 2t +~x ·~∇ + ψ (~x,t) (3.37a) σ ∂t 2 σ ∂ d δψ†ρ (~x,t) = 2t +~x ·~∇ + ψ†ρ (~x,t) (3.37b) ∂t 2

∂ a δL = (2tL ) + ∇a (x L ) (3.37c) ∂t a S (~x,t) = 2tTtt (~x,t) + x Tta(~x,t) (3.37d) a Kb(~x,t) = 2tTtb(~x,t) + x T˜ ba(~x,t) d h¯ 2 − δ ~∇2 − ∇ ∇ xaρ(~x,t) (3.37e) ba b a d − 1 4m (3.37f) ∂ ∂ If T (~x,t) + ∇aT (~x,t) = 0, T (~x,t) + ∇aT (~x,t) = 0, (3.37g) ∂t tt at ∂t tb ab ∂ S (~x,t) +~∇ · K~ (~x,t) = 0 requires : (3.37h) ∂t ˜ a 2Ttt (~x,t) + Ta (~x,t) = 0 (3.37i)

˜ a The operator equation 2Ttt (~x,t) + Ta (~x,t) = 0 has interesting consequences. This identity must hold in any scale invariant ﬁeld theory. Its expectation value must also hold in any state of a scale invariant theory, even when the state itself is not scale invariant. In particular, the ground state |O > which we have discussed for a weakly interacting Fermi or Bose gas cannot be scale invariant since it contains a ﬁnite density of particles. However, if the theory happened to be scale invariant, we would have ˜ a 2 < O|Ttt (~x,t)|O > + < O|Ta (~x,t)|O >= 0

Generally, the expectation value of Ttt (~x,t) is the energy density. Moreover, the average of the expectation values of the diagonal components of T˜ ab(~x,t) is equal to the pressure. This tells us that, in any state of a scale invariant theory,

2u = dP (3.38) 60 Chapter 3. Classical ﬁeld theory and the action principle

Generally, the system that we are discussing is not scale invariant. In fact, since for the Lagrangian density (3.3), the improved energy-momentum tensor is

h¯ 2 T˜ = − ∇ ψ†σ (~x,t)∇ ψ (~x,t) + ∇ ψ†σ (~x,t)∇ ψ (~x,t) ab 2m a b σ b a σ d h¯ 2 1 + δ ~∇2 − ∇ ∇ ψσ†(~x,t)ψ (~x,t) d − 1 2m d ab a b σ λ 2 − δ ψσ†(~x,t)ψ (~x,t) (3.39) ab 2 σ where we have used the equation of motion to eliminate the time derivative terms. The trace condition for scale invariance is

λ 2 2T + T˜ = −2µψσ†(~x,t)ψ (~x,t) + (2 − d) ψσ†(~x,t)ψ (~x,t) (3.40) tt aa σ 2 σ which is non-zero. The system can only have scale invariance if the right-hand-side vanishes, as an operator. This can only happen if the chemical potential vanishes. This is not surprising, as the chemical potential has the dimensions of an energy and it should not be scale invariant. Also, outside of two dimensions, the interaction is not scale invariant. It turns out that the apparent scale invariance when µ = 0 and d = 2 is violated by a scale anomaly, so even in two dimensions, there is no scale invariance once the particles interact with each other with generic values of the coupling constant. There can be some special values of the coupling, “ﬁxed points” at which the theory is conjectured to be scale invariant. The Feschbach resonance, or unitary point of a cold atom gas is thought to be such a point.

3.7 Special Schrödinger symmetry In a translation, Galilean and scale invariant theory, there is always another symmetry, called the special Schrödinger symmetry. The special Schrödinger transformation of the ﬁelds is ∂ im~x2 d δψ (~x,t) = t2 +t~x ·~∇ − + t ψ (~x,t) σ ∂t h¯ 2 2 σ ∂ im~x2 d δψ†σ (~x,t) = t2 +t~x ·~∇ + + t ψ†σ (~x,t) ∂t h¯ 2 2 With this transformation, it can be shown that the Lagrangian density transforms by the total derivative terms,

∂ 2 δL = t L +~∇ · (t~xL ) ∂t The Noether charge and current densities are constructed in the standard manner. They are related to the energy-momentum tensor components and the particle and number current densities as

2 2 x R(~x,t) = t Ttt (~x,t) +tx T (~x,t) + m ρ(~x,t) b tb 2 2 2 2 x h¯ Ja(~x,t) = t Tat (~x,t) +tx T (~x,t) + m Ja(~x,t) − dt ∇aρ(~x,t) b ab 2 4m If the energy-momentum tensor is conserved, the above current is also conserved if the following expression vanishes

h¯ 2 2tT (~x,t) + xbT (~x,t) +tT (~x,t) + mxbJ (~x,t) −td ~∇2ρ(~x,t)) = 0 tt tb aa b 4m 3.8 The Schrödinger algebra 61

Scale and Galilean symmetry are enough to guarantee the above. We conclude that a translation invariant quantum ﬁeld theory which has a conserved energy-momentum tensor associated with the time and space-translation invariance, and which also also has a Galilean symmetry, so that

Ttb(~x,t) + mJb(~x,t) = 0

and a scale symmetry so that

h¯ 2 2T (~x,t) + T a(~x,t) − d ~∇2ρ(~x,t) = 0 tt a 4m

must also have a conserved current corresponding to the special Schrödinger symmetry. In summary, the transformations, Noether currents and the conditions for their conservation are

Special Schroedinger invariance. ∂ im~x2 d δψ (~x,t) = t2 +t~x ·~∇ − + t ψ (~x,t) (3.41a) σ ∂t h¯ 2 2 σ ∂ im~x2 d δψ†σ (~x,t) = t2 +t~x ·~∇ + + t ψ†σ (~x,t) (3.41b) ∂t h¯ 2 2

∂ 2 a δL = t L + ∇a (tx L ) (3.41c) ∂t 2 2 a x S˜(~x,t) = t Ttt (~x,t) +tx Tta(~x,t) + m ρ(~x,t) (3.41d) 2 x2 K˜ (~x,t) = t2T (~x,t) +txaT˜ (~x,t) + m J (~x,t) b bt ab 2 b d h¯ 2 − δ ~∇2 − ∇ ∇ xaρ(~x,t) (3.41e) ba b a d − 1 4m ∂ ∂ If T (~x,t) + ∇aT (~x,t) = 0, T (~x,t) + ∇aT (~x,t) = 0, (3.41f) ∂t tt at ∂t tb ab ∂ ~ S˜(~x,t) +~∇ · K˜(~x,t) = 0 requires (3.41g) ∂t ˜ a Tat (~x,t) + m ja(~x,t) = 0 and 2Ttt (~x,t) + Ta (~x,t) = 0 (3.41h)

3.8 The Schrödinger algebra

A translation, rotation and Galilean invariant quantum ﬁeld theory has conserved Noether charges the Hamiltonian, H, the linear momentum Pa, the angular momentum, Mab and a Noether charge Ba corresponding to Galilean boosts. In addition, Galilean symmetry makes use of the conserved number operator N . These charges are time independent and the expressions for them are 62 Chapter 3. Classical ﬁeld theory and the action principle somewhat simpler if we evaluate them at t = 0, where they are Z Z 3 3 H = d Ttt (~x,0) = d x H (~x,0) Z Z 3 3 Pa = − d x Tta(~x,0) = m d x ja(~x,0) Z Z 3 3 Mab = − d x Mab(~x,0) = − d x (xa Ttb(~x,0) − xb Tta(~x,0)) Z 3 = m d x (xa jb(~x,0) − xb ja(~x,0)) Z im Z Ba = d3x Ba(~x,0) = d3x xaρ(~x,0) h¯ Z N = d3x ρ(~x,t)

The densities in the above integrals are

2 h¯ †σ λ †σ 2 H (~x,t) = ~∇ψ (~x,t) ·~∇ψ (~x,t) + ψ (~x,t)ψ (~x,t) 2m σ 2 σ †σ ρ(~x,t) = ψ (~x,t)ψσ (~x,t) ih¯ ih¯ j (~x,t) = − ψ†σ (~x,t)∇ ψ (~x,t) + ∇ ψ†σ (~x,t)ψ (~x,t) a 2m a σ 2m a σ

2 a h¯ ∂ H (x,t),ψ (~y,t) = −∇ δ(~x −~y)∇aψ (~x,t) − ih¯ ψ (~x,t)δ(~x −~y) ρ x 2m ρ ∂t ρ 2 †ρ a h¯ †ρ ∂ †ρ H (x,t),ψ (~y,t) = −∇ δ(~x −~y)∇aψ (~x,t) − ih¯ ψ (~x,t)δ(~x −~y) x 2m ∂t †σ †σ [ρ(~x,t),ψσ (~y,t)] = −δ(~x −~y)ψσ (~x,t), ρ(~x,t),ψ (~y,t) = δ(~x −~y)ψ (~x,t) ih¯ ih¯ [j (~x,t),ψ (~y,t)] = δ(~x −~y)∇ ψ (~x,t) − ∇ (δ(~x −~y)ψ (~x,t)) a σ m a σ 2m a σ ih¯ ih¯ j (~x,t),ψ†σ (~y,t) = δ(~x −~y)∇ ψ†σ (~x,t) − ∇ δ(~x −~y)ψ†σ (~x,t) a m a 2m a

†σ We can use the equal-time commutation relations for the ﬁelds ψσ (~x,t) and ψ (~x,t) to get the commutators for the number density and current,

[ρ(~x,t),ρ(~y,t)] = 0 ih¯ [ρ(~x,t),j (~y,t)] = − ρ(~y,t)∇ δ(~x −~y) a 2m a ih¯ [j (~x,t),j (~y,t)] = − (j (~y,t)∇ + j (~x,t)∇ )δ(~x −~y) a b 2m a b b b These can be used to form the commutators of the Noether charges which are summarized as

[N ,H] = 0 [N ,Pa] = 0 [N ,Mab] = 0 [N ,Ba] = 0 (3.42)

[Pa,Pb] = 0 [Mab,H] = 0 [Pa,H] = 0. (3.43)

[Mab,Mcd] = δadMbc − δacMdb + δbcMda − δbdMac (3.44)

[Mab,Pc] = δacPb − δbcPa [Mab,Bc] = δacBb − δbcBa (3.45) N [B ,B ] = 0 [B ,H] = P [B ,P ] = −δ m (3.46) a b a a a b ab 2 3.9 Summary of this chapter 63

Note that the commutators of elements of the set {H,Pa,Mab,Ba,N } result in linear combinations of the elements themselves. Note that this statement requires that the number operator N is an element of the set. This property gives the linear vector space that is formed from all linear combinations of elements of the set {H,Pa,Mab,Ba,N } the structure of a Lie algebra. It is called the Galilean algebra. It contains a sub-algebra, the angular momenta Jab and their commutators with each other which is just the Lie algebra of rotations. In d-dimensional space, there are d(d − 1)/2 distinct Jab (three in d = 3). If we are interested in a theory that also has scale invariance, we can add two more charges, the dilatation, and the speial Schrödinger operator Z Z 3 m 3 a ∆ = d x D(~x,0) = d x x ja(~x,0) ih¯ Z Z ~x2 S = d3x D(~x,0) = m d3x ρ(~x,0) 2 These have commutators with the Galilean algebra elements

[∆,H] = 2H, [∆,Pa] = Pa, [∆,Ba] = −Ba, [∆,Mab] = 0 (3.47)

[S,H] = ∆, [S,Pa] = −Ba, [S,Ba] = 0, [S,Mab] = 0 (3.48) [S,∆] = 2S (3.49)

The Lie algebra which includes these charges has the basis {H,Pa,Mab,Ba,N ,∆,S} and it is called the Schrödinger algebra. We can use the equal-time commutation relation to ﬁnd 3 3 [∆,ψ (0,0)] = i ψ (0,0) ∆,ψ†σ (0,0) = i ψ†σ (0,0) σ 2 σ 2 An operator that has this property, that its commutator with ∆ is equal to itself times a constant is called a scaling operator and the constant is its scaling dimension. In this case, the scaling 3 dimension is equal to 2 . We can obtain other operators with higher scaling dimensions in two ways. One is by taking products of operators such as

†ρ1 †ρ2 ρ` ψσ1 (0,0)ψσ2 (0,0)....ψσk (0,0)ψ (0,0)ψ (0,0)....ψ (0,0)

3 is also a scaling operator with dimension (k + `) · 2 .

3.9 Summary of this chapter The action is the integral of the Lagrangian density over space and time

Z Z ∂ ∂ S[ψ ,ψ†σ ] = dt d3x L (ψ,ψ†, ψ, ψ†,∇ψ,∇ψ†) σ ∂t ∂t The Lagrangian density is a function of the classical ﬁelds and their derivatives. For Bosons, †σ ψσ (~x,t) and ψ (~x,t) are ordinary functions. For Fermions, they are anti-commuting functions. An example of a Lagrangian density for a non-relativistic many-particle system with spin-independent contact interactions is

ih¯ ∂ ih¯ ∂ h¯ 2 L = ψ†σ (~x,t) ψ (~x,t) − ψ†σ (~x,t)ψ (~x,t) − ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) 2 ∂t σ 2 ∂t σ 2m σ λ 2 +µψ†σ (~x,t)ψ (~x,t) − ψ†σ (~x,t)ψ (~x,t) σ 2 σ 64 Chapter 3. Classical ﬁeld theory and the action principle

The ﬁeld equations are the Euler-Lagrange equations

∂ ∂ ∂L ∂L − − ∇ = 0 L ∂ a ∂ψ (~x,t) ∂t ∂(∇aψ (~x,t)) σ ∂( ∂t ψσ (~x,t)) σ ∂ ∂ ∂L ∂L − − ∇ = 0 †σ L ∂ a †σ ∂ψ (~x,t) ∂t †σ ∂(∇aψ (~x,t)) ∂( ∂t ψ (~x,t)) with the appropriate boundary conditions. The field equation is ∂ h¯ 2 ih¯ ψ (~x,t) = − ~∇ 2 − µ ψ (~x,t) + λψ†ρ (~x,t)ψ (~x,t)ψ (~x,t) ∂t σ 2m σ ρ σ The Lagrangian density, when written in the form ∂ L = ih¯ψ†σ (~x,t) ψ (~x,t) − H (ψ,ψ†) ∂t σ †σ indicates that the canonical momentum conjugate to ψσ (~x,t) is ih¯ψ (~x,t), the Poisson bracket is ψ (~x,t),ihψ†σ (~y,t) = δ ρ δ(~x −~y) , σ ¯ PB σ from which we identify the commutation relations of the quantized fields, ψ (~x,t),ihψ†σ (~y,t) = δ ρ δ(~x −~y) → ψ (~x,t),ihψ†ρ (~y,t) = ihδ ρ δ(~x −~y) σ ¯ PB σ σ ¯ ¯ σ The fields must therefore obey the equal time commutation relations †ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 for Bosons or anti-commutation relations †ρ ρ ψσ (~x,t),ψ (~y,t) = δσ δ(~x −~y), †σ †ρ ψσ (~x,t),ψρ (~y,t) = 0, ψ (~x,t),ψ (~y,t) = 0 for Fermions. The classical field theory has a continuous symmetry if there exists an infinitesimal change of variables

ψσ (~x,t) → ψσ (~x,t) + δψσ (~x,t) ψ†ρ (~x,t) → ψ†ρ (~x,t) + δψ†ρ (~x,t) such that, without use of the equations of motion, the linear variation of the Lagrangian density can be assembled into partial derivatives, ∂ δL = R(~x,t) +~∇ · J~(~x,t) ∂t Then, Noether’s theorem states that the Noether current and charge densities

∂L †ρ ∂L R(~x,t) = δψσ (~x,t) + δψ (~x,t) − R(~x,t) ∂ ∂ †ρ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L ∂L (~x,t) = (~x,t) + †ρ (~x,t) − J (~x,t) Ja δψσ δψ †ρ a ∂(∇aψσ (~x,t)) ∂(∇aψ (~x,t)) 3.9 Summary of this chapter 65 obey the continuity equation

∂ R(~x,t) +~∇ · J~ (~x,t) = 0 ∂t which associates a conserved charge Z 3 QR = d xR(~x,t) with the symmetry in question. The energy-momentum tensor is constructed from the Noether currents corresponding to space and time translation symmetries. The components of the energy-momentum tensor are

Improved energy-momentum tensor

∂ ∂L ∂ †σ ∂L Ttt (~x,t) = ψσ (~x,t) + ψ (~x,t) − L ∂t ∂ ∂t ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂L †σ ∂L Ttb(~x,t) = ∇bψσ (~x,t) + ∇ψ (~x,t) ∂ ∂ †σ ∂( ∂t ψσ (~x,t)) ∂( ∂t ψ (~x,t)) ∂ ∂L ∂ ∂L (~x,t) = (~x,t) + †σ (~x,t) Tat ψσ a ψ a †σ ∂t ∂(∇ ψσ (~x,t)) ∂t ∂(∇ ψ (~x,t)) ∂L ∂L ˜ (~x,t) = (~x,t) + †σ (~x,t) − Tab ∇bψσ a ∇bψ a †σ δabL ∂(∇ ψσ (~x,t)) ∂(∇ ψ (~x,t)) d h¯ 2 + δ ~∇2 − ∇ ∇ ρ(~x,t) d − 1 4m ab a b

Explicit energy-momentum tensor

h¯ 2 T = ~∇ψ†σ (~x,t) ·~∇ψ (~x,t) − µψ†σ (~x,t)ψ (~x,t) tt 2m σ σ λ 2 + ψ†σ (~x,t)ψ (~x,t) 2 σ ih¯ T = − ∇ ψ†σ (~x,t)ψ (~x,t) − ψ†σ (~x,t)∇ ψ (~x,t) tb 2 b σ b σ h¯ 2 ∂ ∂ T = − ∇ ψ†σ (~x,t) ψ (~x,t) + ψ†σ (~x,t)∇ ψ(~x,t) at 2m a ∂t σ ∂t a h¯ 2 T˜ = − ∇ ψ†σ (~x,t)∇ ψ (~x,t) + ∇ ψ†σ (~x,t)∇ ψ (~x,t) ab 2m a b σ b a σ d h¯ 2 − ∇ ∇ ψσ†(~x,t)ψ (~x,t) d − 1 2m a b σ 2 1 h¯ λ 2 − δ ~∇2 ψσ†(~x,t)ψ (~x,t) + ψσ†(~x,t)ψ (~x,t) ab d − 1 4m σ 2 σ

Particle density and current

σ† ρ(~x,t) = ψ (~x,t)ψσ (~x,t) ih¯ ih¯ j (~x,t) = − ψσ†(~x,t)∇ ψ (~x,t) + ∇ ψσ†(~x,t)ψ (~x,t) a 2m a σ 2m a σ 66 Chapter 3. Classical ﬁeld theory and the action principle

Continuity equations

∂ T + ∇aT = 0 ∂t tt at ∂ T + ∇aT˜ = 0 ∂t tb ab ∂ ρ(~x,t) +~∇ ·~j(~x,t) = 0 ∂t

Relations required by symmetry

Rotation symmetry: T˜ ab(~x,t) = T˜ ba(~x,t) Galilean invariance: Tta(~x,t) = −mja(~x,t) ˜ a Scale invariance: 2Ttt (~x,t) + Ta (~x,t) = 0

Charge and current densities for space-time symmetries

(charge density, current density) symmetry

(Ttt ,Tat ) time translation d h¯ 2 −T ,−T˜ − (δ − ∇ ∇ )ρ space translation tb ab d − 1 4m ab a b xbTtc − xcTtb,xbT˜ ac − xcT˜ ab rotation d h¯ 2 d h¯ 2 x T − x T ,x T˜ − x T˜ + x δ ~∇2 − ∇ ∇ ρ − x δ ~∇2 − ∇ ∇ ρ rotation b tc c tb b ac c ab b d − 1 4m ac a c c d − 1 4m ab a b −tTtb − mxbρ,−tT˜ ab − mxbja Galilean boost d h¯ 2 2tT + xbT , 2tT + xbT˜ + δ ~∇2 − ∇ ∇ (xbρ) dilatation tt tb at ab d − 1 4m ab a b x2 x2 d h¯ 2 t2T +txbT + m ρ , t2T +txbT˜ + m j +t δ ~∇2 − ∇ ∇ (xbρ) special Schrodinger tt tb 2 at ab 2 a d − 1 4m ab a b Relativistic Symmetry and II Quantum Field Theory

4 Space-time symmetry and relativistic ﬁeld theory ...... 69 4.1 Quantum mechanics and special relativity 4.2 Coordinates 4.3 Scalars, vectors, tensors 4.4 The metric 4.5 Symmetry of space-time 4.6 The symmetries of Minkowski space

5 The Dirac Equation ...... 79 5.1 Solving the Dirac equation 5.2 Lorentz Invariance of the Dirac equation 5.3 Phase symmetry and the conservation of electric current 5.4 The Energy-Momentum Tensor of the Dirac Field 5.5 Summary of this chapter

6 Photons ...... 93 6.1 Relativistic Classical Electrodynamics 6.2 Covariant quantization of the photon 6.3 Space-time symmetries of the photon 6.4 Quantum Electrodynamics 6.5 Summary of this chapter

4. Space-time symmetry and relativistic ﬁeld theory

4.1 Quantum mechanics and special relativity

It is often said that quantum field theory is the natural marriage of Einstein’s special theory of relativity and the quantum theory. The point of this section will be to motivate this statement. We will begin with a single free quantum mechanical particle and ask what is wrong with simply assuming that its energy spectrum is 2 p 2 4 2 2 ~p given by the relativistic expression E(~p) = m c + c ~p , rather than the non-relativistic E(~p) = 2m . After 2 ~p2 all, for small momenta, the first expression is E(~p) ≈ mc + 2m + ... is very similar to the second expression with an additional constant in the energy. Let us assume that the particle travels on open, infinite three dimensional space. It is described by its position~x and momentum ~p which, for the quantum mechanical particle, are operators with the commutation relation a a [xˆ , pˆb] = ih¯δ b Momentum and energy are conserved and the energy and momentum are related by p E(~p) = m2c4 +~p2c2 (4.1)

where m is the rest mass of the particle and c is the speed of light. In the quantum mechanics of a single particle, we could consider a quantum state of the particle which is an eigenstate of its linear momentum,

pˆa |pi = pa |pi , i = 1,2,3 (4.2)

These eigenstates of momentum have a continuum normalization, so that

p|p0 = δ 3(~p −~p 0) .

Because the energy of the particle, given in equation (4.1) above, is a function only of the momentum momentum of the particle, an eigenstate of the momentum is also an eigenstate of the energy, that is p H |pi = m2c4 +~p2c2 |pi (4.3) 70 Chapter 4. Space-time symmetry and relativistic ﬁeld theory where H is the Hamiltonian. The Schrödinger equation which must be satisﬁed by the time-dependent state vector, |Ψ(t)i, is

This simple development would seem to be a complete solution of the quantum theory of a single relativistic particle. We can use it to answer questions about it. For example, let us consider the scenario where, at some initial time, say t = 0 the particle is localized at position~0. This state could be created by a measurement of the position of the particle. We construct an eigenstate of position, that is, one which obeys

xˆa|~x >= xa|x > (4.6) by superposing the complete set of momentum states as

Z |~x >= d3 p|~p > < ~p |~x > (4.7) where the overlap matrix is a plane wave

ei~p·~x/h¯ < ~p|~x >= 3 (4.8) (2πh¯) 2

Then, the wave function that evolves from an eigenstate of position with eigenvalue~0 at t = 0, at a time t later becomes √ Z −i m2c4+~p2c2t/h¯ 3 e |Ψ(t) >= d p|~p > 3 (4.9) (2πh¯) 2

We can now ask the question as to the amplitude for observing the particle at position~x after a time t has elapsed. The answer is simply the overlap of the position eigenstate |~xi with the above wave function evaluated at t. The result is √ Z e−i m2c4+~p2c2t/h¯−i~p·~x/h¯ <~x|Ψ(t) >=<~x|eiHt/h¯ |~0 >= d3 p (4.10) (2πh¯)3

Now, we find the difficulty.1 One of the postulates of the special theory of relativity states that the speed of light is a maximum speed. However, from equation (4.10), the probability amplitude is nonzero in the causally forbidden region, where |~x| > ct. There seems to be a nonzero amplitude for motion at speeds greater than that of light. A formal way to see that (4.10) is indeed nonzero in the forbidden region, is to consider t where it occurs in that equation as a complex variable. Then, (4.10) is analytic in the lower half of the complex t-plane. When t is real, the expression is a distribution which should be defined by its limit as complex t approaches the real axis from the lower half plane. Given that it is analytic in this domain, it cannot be zero in any region of the lower half plane plus the real axis except for discrete points, otherwise it would have to be zero everywhere. It is definitely not zero for all times, in fact when t = 0 it is a Dirac delta function. Thus, it cannot be zero in the entire region ct < |~x|.

1 This is in addition to the already obvious difﬁculty that the expression (4.10) is not Lorentz invariant. In fact, it transforms like the time derivative of a Lorentz invariant function. Let us overlook this issue for the time being. 4.1 Quantum mechanics and special relativity 71

To see this more explicitly, we can do the integral for the special case where m = 0. It becomes Z ∞ h i iHt/h¯ ~ 1 −ip[ct−|~x|]/h¯ −ip[ct+|~x|]/h¯ <~x|e |0 > = 2 pdp e − e 4π2h¯ i|~x| 0 ∂ 1 1 1 = lim − ε→0+ ∂(ct) 4π2i|~x| ct − |~x| − iε ct + |~x| − iε ∂ 1 P P = − − iπδ(ct − |~x|) + iπδ(ct + |~x|) ∂(ct) 4π2i|~x| ct − |~x| ct + |~x| ∂ 1 P = −iπδ((ct)2 −~x2)sign(t) + (4.11) ∂(ct) 2π2i (ct)2 − |~x|2

In the first line above, we have integrated the angles in spherical polar coordinates. In the second line above, −ip[ct−|~x|]/h¯ ∂ −ip[ct−|~x|]/h¯ we have used pe = i ∂(ct) e and we have defined the integral over the semi-infinite domain by introducing the positive infinitesimal parameter ε. In the third line, we have used the identity

1 P lim = + iπδ(x) ε→0+ x − iε x where P/x is the principal value distribution. Also, for the Dirac delta function 1 δ(t2 − a2) = (δ(t − a) + δ(t + a)) 2|a|

1 δ(t2 − a2)sign(t) = (δ(t − |a|) − δ(t + |a|)) 2|a| In equation (4.11), we see that the wave-function of a massless particle spreads in two ways. The ﬁrst is a wave which travels at the speed of light and is therefore conﬁned to the light cone - where |~x| = ct. The second is a principle value distribution which is non-zero everywhere, including in the forbidden region where |~x| > ct. This latter spreading of the wave packet violates causality. It tells us that, in our quantum mechanical system, the result of a measurement of the position of the particle at position~x after time t would indeed be possible. The particle could be observed as travelling faster than light. This would certainly seem to be incompatible with the principles of the special theory of relativity where objects are restricted to having sub-luminal speeds.

Figure 4.1: The wave packet is initially localized at~0 and as time evolves it spreads in such a way that there is a nonzero amplitude for detecting it in the vicinity of point~x. If it is detected at~x, since |ct| < |~x|, its classical velocity would be greater than that of light. 72 Chapter 4. Space-time symmetry and relativistic ﬁeld theory

Now that we have found a difﬁculty with causality, we need to ﬁnd a way to resolve it. We will resolve it by going beyond single-particle quantum mechanics to an extended theory where there is another process which competes with the one that we have described. The total amplitude will then be the sum of the amplitudes for the two processes and we will rely on destructive interference of the amplitudes to solve our problem, that is, to make the probability of detecting the particle identically zero in the entire forbidden region |~x| > ct.

Figure 4.2: We should add to the amplitude for the particle to travel from~0 to~x as in ﬁgure 4.1 the amplitude that a particle-anti-particle pair is created at ~x, the particle continues forward in time as it did in the ﬁrst process, the anti-particle propagates backward in time and annihilates the particle which was prepared in the state localized at~0.

To include the second process, we will begin by framing the first process, the one we have discussed so far, as the following thought experiment. One observer, whom we shall all Alice, is located at position~0 and prepares the particle in the state which is localized at~0. Alice could do this by measuring the position of the particle and we assume that the result of the measurement is that the particle is at position~0. We assume that Alice can do this measurement with arbitrarily good precision. Immediately after the measurement, the particle is allowed to evolve by its natural time evolution, the one which we have described above, so that after time t, its quantum state is given by equation (4.9) and its wave-function by equation (4.10). Then, at time t, another observer, Bob, who is located at point ~x does an experiment to detect the particle. Of course, in a given experiment, Bob might or might not find the particle at~x. But, given that the particle is has non-zero amplitude to propagate there, if Alice and Bob repeat this experiment sufficiently many times, Bob will eventually detect the particle at~x. The result of the experiment is to collapse the particle’s wave function to one which is localized at~x. The amplitude for the particle to propagate to~x is given by (4.10). If this were all there is to it, the result of the experiment violates causality. The second process that we will superpose with the one that we have described will require other states to be introduced. It then clearly involves an extension of single particle quantum mechanics. In the second process, the attempt by Bob, the observer who is located at~x, to observe a paarticle’s position creates a pair consisting of a particle and an anti-particle. The position measure collapses the wave function of the particle into the position eigenstate localized at ~x, the position which was the final state of the particle in the first experiment. The anti-particle is interpreted as a particle which moves backward in time, from time t to time 0. After time −t it has an amplitude to arrive at position~0 where it annihilates the particle that Alice, the observer at~0, has prepared in the localized state. The result of this second process is the same as that of the first process, a particle begins in a state localized at 0 and after a time t it is detected in a state localized at~x. The amplitude for the second process is similar but not identical to that of the first process, due to the fact 4.2 Coordinates 73

that the positron propagates backward in time. It is √ Z ei m2c4+~p2c2t/h¯−i~p·~x/h¯ <~0|eiHt/h¯ |~x > = d3 p (4.12) antiparticle (2πh¯)3

The total amplitude is the sum of amplitudes of the two processes,

−iHt/h¯ iHt/h¯ A =<~x|e |~0 >particle + <~0|e |~x >antiparticle (4.13) i~p·~x/h¯ √ √ Z e h 2 4 2 2 2 4 2 2 i = d3 p ei m c +~p c t/h¯ + e−i m c +~p c t/h¯ (4.14) (2πh¯)3

Now, the total expression can have destructive interference. We will not demonstrate it in the general case, but in the case where the mass of the particle and antiparticle is zero. There, we can perform integral in (4.12) explicitly,

∂ 1 P <~0|eiHt/h¯ |~x > = −iπδ((ct)2 −~x2)sign(t) − (4.15) antiparticle ∂(ct) 2π2i (ct)2 − |~x|2

We see that, like the amplitude for the particle, the amplitude for the antiparticle also spreads outside of its light cone. However, when we add the amplitudes of the two processes together, their sum is

∂ 1 <~x|e−iHt/h¯ |~0 > + <~0|eiHt/h¯ |~x > = −iπδ((ct)2 − |~x|2)sign(t) (4.16) particle antiparticle ∂(ct) π2

We see that the principal value part of the expression, which was nonzero outside of the light cone, has canceled. What remains describes the wave function of the initial particle spreading along its light cone, as we might expect for a massless particle, which travels at the speed of light. The upshot of the above development is that a correct treatment of a quantum mechanical particle which also obeys the laws of special relativity requires more than just single particle quantum mechanics. The resolution of the difficulty that we have suggested needs an anti-particle. Quantum field theory will supply us with an anti-particle. Another lesson is that the properties of the anti-particle must be finely tuned to be very similar to that of the particle. Otherwise the exact cancellation of the amplitude outside of the light cone would not happen. We will eventually see that this fine-tuning is generally a property of the relativistic wave equations which replace the Schrödingier equation. They have both positive and negative energy solutions which we shall interpret as belonging to the particle and the anti-particle that the wave equation simultaneously describes. We will put off further discussion of this fact until we study relativistic fields and their wave equations.

4.2 Coordinates

The non-relativistic classical and quantum fields ψσ (~x,t) which we have dealt with so far are functions of both the time, t and the space coordinates, labeled by the vector~x. So far,~x label points in three dimensional Euclidean space and time is parameterized by the variable t. In the following, when we proceed to study relativistic field theories, we will find it convenient to think of space and time coordinates from a unified point of view and include time to form a four-vector (ct,~x). The time is defined with a factor of the speed of light, so that if t is measured in time units, the x0 = ct is measured in distance units. Points in the four-dimensional space-time are called events. In the remainder of this chapter, we will introduce some of the notation which we will use to describe the relevant properties of spacetime when we are discussing relativistic field theories. We will also introduce scalar, vector and tensor fields. Then, we will discuss the symmetries of space-time and our four-dimensional Minkowski space in particular. For the most part we will be interested in infinite flat four-dimensional Minkowski space. However, at the outset, it is useful to briefly consider more general space-time. In any spacetime geometry, our basic need is a coordinate system which labels the events of the space-time. A coordinate system assigns a unique sequence of real numbers x0,x1,x2,...,xD−1 to each event in space-time. The number of entries in the sequence, D, is the dimension. We will usually deal with the physical case of four dimensions. The four real numbers x0,x1,x2,x3 contain the four bits of data that are necessary for locating an event. The component 74 Chapter 4. Space-time symmetry and relativistic field theory x0 = ct is associated with time, the other three components x1,x2,x3 are said to be the spatial coordinates. For short, we denote the array x0,x1,x2,x3 by an indexed object, xµ , where the index µ runs over the values µ = 0,1,2,3. Each distinct event in space-time should be associated with a distinct set of four numbers. Conversely, each distinct set of four numbers should label a unique event. Note that the position of the index of xµ is up. In the following, this will be important. An up index will be different from, and must be distinguished from, a down index. Also, we will use the convention that an index which is a Greek letter, such as µ,ν,λ,σ,ρ,α,β,..., typically runs over the range 0,1,2,3 and is used to denote a four-component object, whereas an index which is a letter a.b,c,... runs over the range 1,2,3 and is used to denote the three spatial dimensions. We will sometimes denote the spatial part of xµ by xa or~x and the time component as x0 or ct. We will sometimes consider dimensions D other than four. In that case, there is always one time dimension and D − 1 space dimensions. A useful idea is that of changing between different coordinate systems. To some extent the labelling of events in space-time is arbitrary. As well as the coordinate system with labels xµ , we could use an alternative, say one with some different labels, x˜µ . To be precise, if the four numbers x0,x1,x2,x3 label a specific event in space-time in the old coordinate system, the same event has the label x˜0,x˜1,x˜2,x˜3 in the new coordinate system. We could build up a dictionary for translating between the old and new coordinate systems. This dictionary is encoded in transformation functions x˜µ (x). These are four functions, x˜0(x),x˜1(x),x˜2(x),x˜3(x), each one a function of four variables, xµ = x0,x1,x2,x3. If we plug the old coordinates of a space-time event, xµ , into these functions, they give us the new coordinates, x˜µ , of the same event. We assume that such a coordinate transformation is invertible. This means that, if we know the functions x˜µ (x), we could at least in principle find the inverse transformation xµ (x˜). We also assume that we can take derivatives of the transformation functions so that ∂x˜µ ∂xµ ≡ ∂ x˜µ , ≡ ∂˜ xν (4.17) ∂xν ν ∂x˜ν ν are both non-singular 4 × 4 matrices, at least in the ranges of coordinates of interest. ∂ Note that we have defined derivatives, ∂µ ≡ ∂xµ , with a down index. The difference between down indices and up indices occurs in the way in which the objects carrying the indices transform under a general coordinate transformation, for example, the four-gradient, ∂µ , which has a down index transforms as

∂ ∂xν ∂ ∂xν ∂˜ = = = ∂ µ ∂x˜µ ∂x˜µ ∂xν ∂x˜µ ν where we have used the chain rule for differentiation. We also remind the reader that we are using the Einstein summation convention by which, unless it is explicitly stated otherwise, repeated up and down indices are assumed to be summed over their range. Thus,

∂xν ∂ D−1 ∂xν ∂ ≡ µ ν ∑ µ ν ∂x˜ ∂x ν=0 ∂x˜ ∂x

An inﬁnitesimal increment of the coordinates, dxµ , has an up index and transforms as

∂x˜µ dx˜µ = dxν ∂xν We will often see expressions where an index is set equal to a down index and then summed over all values of the index. This creates an object with a simpler transformation law, for example

∂x˜µ ∂xσ dx˜µ ∂˜ = dxρ ∂ = dxσ ∂ µ ∂xρ ∂x˜µ σ σ where we have used the chain rule of differential calculus,

∂x˜µ ∂xσ ∂xσ = = δ σ ∂xρ ∂x˜µ ∂xρ ρ

σ Here, δ ρ is the Kronecker delta symbol, which is equal to one when the up and down indices are equal, ρ = σ, and zero otherwise. 4.3 Scalars, vectors, tensors 75

It will often be useful to consider infinitesimal coordinate transformations. An infinitesimal transformation is one where the new coordinates differ from the old coordinates by an infinitesimal amount, which can be encoded in four functions f µ (x) of infinitesimal magnitude and arbitrary profile, so that

x˜µ = xµ + f µ (x) (4.18)

To linear order in inﬁnitesimals, it is easy to ﬁnd the inverse of this transformation

xµ = x˜µ − f µ (x˜) (4.19)

For these inﬁnitesimal transformations, ∂x˜µ ∂xµ = δ µ + ∂ f µ (x) , = δ µ − ∂ f µ (x) (4.20) ∂xν ν ν ∂x˜ν ν ν where we have written the right-hand-sides only to the linear order in the inﬁnitesimal transformation.

4.3 Scalars, vectors, tensors So far, the only structure which we have given space-time is the existence of a coordinate system and the possibility of transforming between different coordinate systems. This is already sufficient structure to define some fields. A relativistic field is a function of the space-time coordinates that transforms in a certain way. The simplest example of a field is a scalar field. A scalar field is a function of the space-time coordinates whose values at particular space-time events specify the values of some physical quantity. The field should have the same value at the same event of space-time when that event is described in any coordinate system. If xµ and x˜µ are coordinates which label the same event in two different coordinate systems, and the scalar field has functional form φ(x) in the xµ coordinates and φ˜(x˜) in the x˜µ coordinates, then the statement that the scalar field has the same value at the same event of space-time in the two coordinate systems gives the scalar field transformation law,

φ˜(x˜) = φ(x) (4.21)

In terms of inﬁnitesimal transformations (4.18), equation (4.21) is

µ φ˜(x˜) = φ(x) + δφ(x) + f (x)∂µ φ(x) + ... = φ(x) In the above equation, we see that φ˜(x˜) differs from φ(x) in two ways. First of all, its functional form µ changes. This is δφ(x). Secondly, the coordinate which it depends on change, this is the term f (x)∂µ φ(x). Canceling the untransformed φ(x) from each side of the above equation, we obtain, to linear order, the transformation law for the scalar ﬁeld,

λ δφ(x) = − f (x)∂λ φ(x) (4.22) Like an increment of the coordinates, dxµ , a vector field V µ (x) has a direction at a given space-time point. The components of the vector field, V µ (x) transform in a similar way, ∂x˜µ V˜ µ (x˜) = V ν (x) (4.23) ∂xν and, for an infinitesimal transformation,

µ λ µ µ λ δV (x) = − f (x)∂λV (x) + ∂λ f (x)V (x) (4.24)

We could also consider a vector ﬁeld with a lower index, Aµ (x) which transforms like the gradient ˜ ∂xν operator ∂µ = ∂x˜ν ∂ν , ∂xµ A˜ (x˜) = A (x) (4.25) µ ∂x˜ν ν and the inﬁnitesimal transformation

λ λ δAµ (x) = − f (x)∂λ Aµ (x) − ∂ν f (x)Aλ (x) (4.26) 76 Chapter 4. Space-time symmetry and relativistic ﬁeld theory

µ1...µk By similar reasoning, a tensor ﬁeld with any number of up and down indices, T ν1...ν` (x), has the transformation law,

µ1 µk σ1 σ` µ1...µk ∂x˜ ∂x˜ ∂x ∂x ρ1...ρk T˜ ν ...ν (x˜) = ...... T σ ...σ (x) (4.27) 1 ` ∂xρ1 ∂xρk ∂x˜ν1 ∂x˜ν` 1 ` and

µ1...µk λ µ1...µk µ1 λ...µk δT ν1...ν` (x) = − f (x)∂λ T ν1...ν` (x˜) + ∂λ f (x)T ν1...ν` (x˜) + ...

µk µ1...λ λ µ1...µk λ µ1...µk + ∂ f (x)T ... (x˜) − ∂ν f (x)T (x˜) + ... + ∂ν f (x)T (x˜) (4.28) λ ν1 ν` 1 λ...ν` ` ν1...λ Any physical scalar, vector or tensor field should transform in the way which we have outlined if they are to have physical meaning. When we set the indices in a pair equal, where one is an upper index and one is a lower index, and then we sum over all values of the index, the transformation law acting on those indices cancel. For example, the composite of two vector fields µ V (x)Aµ (x) transforms like a scalar field

µ λ µ δ V (x)Aµ (x) = − f (x)∂λ V (x)Aµ (x)

4.4 The metric Now that we have introduced coordinates of space-time, we must discuss how some fundamental quantities, like time and distance, for example, are to be computed. The geometry of space-time is encoded in a symmetric two-index tensor ﬁeld called the metric, gµν (x). It contains all of the information that we need in order to understand the geometry of a space-time. The metric transforms like a tensor ﬁeld with lower indices ∂x˜ρ ∂x˜σ g˜ (x˜) = g (x) (4.29) µν ∂xµ ∂xν ρσ The metric is usually assumed to be non-singular, so that it can be inverted and its inverse is denoted by the same symbol, but with up-indices, gµν (x), so that

µν µ νλ λ g (x)gνλ (x) = δ λ , gµν (x)g (x) = δµ The inverse of the metric transforms like a tensor with two up-indices, ∂xµ ∂xν g˜µν (x˜) = gρσ (x) (4.30) ∂x˜ρ ∂x˜σ The inﬁnitesimal transformations are

λ λ λ δgµν (x) = − f (x)∂λ gµν (x) − ∂µ f (x)gλν (x) − ∂ν f (x)gµλ (x) (4.31) µν λ µν µ λν ν µλ δg (x) = − f (x)∂λ g (x) + ∂λ f (x)g (x) + ∂λ f (x)g (x) (4.32) For the most part, we will not be interested in general space-times, but will focus on Minkowski space. Minkowski space is deﬁned as that space-time where one can ﬁnd a coordinate system so that the metric has the special form

−1 0 0 0 −1 0 0 0  0 1 0 0 µν  0 1 0 0 µν µ νλ λ ηµν =  , η =  , η η = δ , ηµν η = δ (4.33)  0 0 1 0  0 0 1 0 νλ λ µ 0 0 0 1 0 0 0 1

µν where we denote this special metric by the symbol ηµν and its inverse by η . Let us return to the case of a generic metric. Given an inﬁnitesimal increment of the coordinates, dxµ , the proper time is deﬁned by 2 µ ν −dτ = gµν (x)dx dx 4.5 Symmetry of space-time 77

(The minus sign on the right-hand-side of this equation is a matter of convention.) This proper time is the time which elapses on a clock which moves with an object along a trajectory. Here, the trajectory is given by the parametric equation xµ (s) = xµ + sdxµ , 0 ≤ s ≤ 1 The proper time is a well-deﬁned physical quantity and it should not depend on the coordinate system which is used. This is guaranteed by the coordinate transformation of the increment dxµ and the metric (??) which combine to µ ν µ ν g˜µν (x˜)dx˜ dx˜ = gµν (x)dx dx Finally, we observe that the metric can be used to raise and lower indices. If we take a vector ﬁeld, V µ (x), we can create a vector with a lower index by contracting it with the metric tensor

ν Vµ (x) = gµν (x)V (x)

and raise the index with the inverse of the metric,

µ µν V (x) = g (x)Vν (x)

We can use the coordinate transformation laws for the metric and for vector ﬁelds to see that, indeed, V µ (x) and Vµ (x) transform as vector ﬁelds with an upper and a lower index, respectively, when the metric and its inverse transform like tensors of the appropriate type.

4.5 Symmetry of space-time Now that we have introduced the concept of metric, we can discuss the idea of a symmetry of a space-time. We define a symmetry transformation of a space-time as a general coordinate transformation under which the metric remains unchanged. An infinitesimal transformation, which is implemented with a vector field fˆµ (x) is a symmetry of space-time if δgµν (x) = 0 where δgµν (x) is given in equation (??). We will use a hat on a vector field which corresponds to a symmetry, fˆµ (x), in order to distinguish it from a general coordinate transformation which we shall still denote by f µ (x). The condition that the coordinate transformation does not change the metric gives us a partial differential equation which the four functions fˆµ (x) must obey,

ˆλ ˆλ ˆλ ∂µ f (x)gλν (x) + ∂ν f (x)gµλ (x) + f (x)∂λ gµν (x) = 0 (4.34)

This equation is called the Killing equation. The solutions, fˆµ (x), of the Killing equation are called Killing vectors. Each linearly independent Killing vector generates a symmetry of space-time. Different space-times can have different symmetries, varying both in the number and the nature of the symmetry transformations. As one can imagine, a generic space-time might have no symmetry at all. There turns out to be a maximum number of symmetries that a space-time can have. The space-time that we will be the most interested in, Minkowski space which we introduce in the next section, is one of the maximally symmetric four dimensional spaces.

4.6 The symmetries of Minkowski space I Minkowski space is a maximally symmetric space-time. The largest number of Killing vectors that a four dimensional space-time can have is ten. The Killing equation on Minkowski space is

∂µ fˆν (x) + ∂ν fˆµ (x) = 0 (4.35)

where the index on fˆµ is lowered by the Minkowski metric,

ν fˆµ (x) ≡ ηµν fˆ (x) (4.36)

The ten solutions of this equation are: 1. four constants fˆµ = cµ corresponding to constant translations of the space-time coordinates ˆµ µ ν ρ 2. f = ω ν x with constants ωµν = −ωνµ , where ωµν = gµρ ω ν . There are six independent components of this 4 × 4 antisymmetric tensor which correspond to three infinitesimal spatial rotations and three infinitesimal Lorentz boosts. 78 Chapter 4. Space-time symmetry and relativistic field theory

Given that we have found the infinitesimal transformations, it is easy to find the finite transformation, it is the linear transformation µ µ ν µ x˜ = Λ ν x + c where cµ are constants and the constant matrix Λ obeys the equation

ρ σ Λ µ Λ ν ηρσ = ηµν

µ µ µ We can see from this equation that, to linear order, it is indeed solved by the expression Λ ν = δ ν + ω ν where ωµν is anti-symmetric. These are the matrices which implement Lorentz transformations. Here, we use the term Lorentz transformation to refer to both the change between reference frames moving at different constant velocities and rotations of the spatial coordinates. When they are combined with the constant translations of the coordinates cµ they are called Poincare transformations. 5. The Dirac Equation

So far, we have formulated an approach to the quantum mechanics of a many-particle system which led us to the non-relativistic ﬁeld equation, in the absence of interactions,

2 ∂ h¯ ~ 2 ih¯ ψ(~x,t) = − ∇ + εF ψ (~x,t) = 0 (5.1) ∂t 2m σ

In this section we shall discuss the equation which replaces this one in a relativistic theory. From our point of view, the main difference between the two is symmetry. The field equation above has Galilean symmetry. We want to trade it for an equation that has Lorentz symmetry. We will continue to discuss a many-Fermion system and to give the discussion a physical context, we will sometime call the Fermions “electrons” with the idea that they will eventually become the electrons of quantum electrodynamics. We have ignored the interaction terms in the above equation. We will continue to do this, to assume that the Fermions are noninteracting. Later on, once we have a relativistic field equation, we will let the fields interact. To seek the appropriate relativistic wave equation, we could recall our discussion of the previous chapters. If we simply postulate a Hamiltonian with a relativistic dispersion relation, so that the wave equation (5.1) is replaced by

∂ q ih¯ ψ (~x,t) = m2c4 − c2h¯ 2~∇2ψ (~x,t) (5.2) ∂t σ σ the resulting theory has difficulties with causality. There is a finite probability of the particle propagating faster than the speed of light. The difficulty lies in the fact that the “Hamiltonian” operator on the right- hand-side of this equation is not a polynomial in derivatives. Having all orders in derivatives, it is not a local differential operator. Dirac found a way to replace this equation by one where the Hamiltonian has the same spectrum, but the operator is a polynomial in derivatives. We also have this goal. In our discussion of single particle relativistic quantum mechanics, the problems with causality had a potential solution if, besides the particle, ther theory contained an anti-particle. Let us address the problems with causality by postulating the existence of an anti-particle which would satisfy and equation similar to (5.4) but with negative energy, q ∂ 2 4 2 2 2 ih¯ ψ˜ ˜ (~x,t) = − m c − c h¯ ~∇ ψ˜ ˜ (~x,t) (5.3) ∂t σ σ Assuming that both the particle and the anti-particle occur in the same theory, we could combine the two into 80 Chapter 5. The Dirac Equation the same multi-component field to find " " p 2 4 2 2~ 2 ∂ ψσ (~x,t) m c − c h¯ ∇ 0 ψσ (~x,t) ih¯ = p (5.4) ∂t ψ˜σ˜ (~x,t) 0 − m2c4 − c2h¯ 2~∇2 ψ˜σ˜ (~x,t) p One might wonder whether there is a matrix Hamiltonian which has eigenvalues ± m2c2 + c2h¯ 2~k2 and which is polynomial in derivatives. There is no such 2 × 2 matrix. We thus need to involve the spin indices ψ (~x,t) and consider σ as a four-component object. It is Dirac’s great insight that our problem can be solved ψ˜σ˜ (~x,t) with a 4 × 4 matrix. (We caution the reader that equation (5.4) is still not quite correct.) Consider the four Hermitian 4 × 4 matrices

β, α1, α2, α3 which have the algebraic properties

ββ = 1, βαi + αiβ = 0, αiα j + α jαi = 2δ i j1 where 1 is the 4 × 4 unit matrix. (Alternatively, if we consider matrices with the above properties, there is a way to show that the minimal size of such matrices is 4 × 4. Then, we consider the wave equation     ψ1(~x,t) ψ1(~x,t) ∂ ψ2(~x,t) h 2 iψ2(~x,t) ih¯   = βmc + ihc¯ ~α ·~∇   ∂t ψ3(~x,t) ψ3(~x,t) ψ4(~x,t) ψ4(~x,t) The “Dirac Hamiltonian” h 2 ~ i hD = βmc + ihc¯ ~α · ∇ (5.5) is a hermitian operator. It must have real eigenvalues. What is more 2 2 4 2 2~ 2 hD = m c − h¯ c ∇ (5.6)

~ 2 2 p 2 4 2 2 2 so, since the eigenvalues of ∇ are −~k , hD has eigenvalues ± m c + h¯ c ~k , which is the desired property. Moreover, hD is a polynomial, in fact it is at most linear in derivatives, and it is therefore a local operator. To make the Dirac equation look more covariant, we can deﬁne the matrices

β = iγ0 , α = γ0~γ (5.7) γ0 = −γ0† , γi = γi† (5.8) {γ µ ,γν } = 2η µν (5.9)

γ µ are called the Dirac gamma-matrices. Using them, the Dirac equation is the matrix differential equation

4 h µ mc i ∑ γab∂µ + δab ψb(~x,t) = 0 (5.10) b=1 h¯ or, with implicit summations over indices,

∂/ + mψ(x) = 0 (5.11a) where we shall use the slash notation for the product to the Dirac matrices with any other four-vector

µ A/ ≡ γ Aµ (5.12a) 5.1 Solving the Dirac equation 81

We will hereafter assume that we are using a system of units where h¯ = 1 and where c = 1. We shall ﬁnd it useful to deﬁne

ψ¯ (x) ≡ ψ†(x)γ0, ψ†(x) ≡ −ψ¯ (x)γ0 (5.13a)

Using this deﬁnition and taking a hermitian conjugate of the equation in (5.14a), we obtain

h ←− i ψ¯ (x) − ∂/ + m = 0 (5.14a)

where the left oriented arrow above the derivative indicates that it operators on whatever is to the left of it. The Dirac equation has a structure similar to the Schrödinger wave equation with the difference that, what we would call the single-particle Hamiltonian, hD, is a matrix and it is linear, rather than quadratic, in derivatives. We might expect that the Hamiltonian of the quantum ﬁeld theory is given by Z Z 3 † 3 H0 = d xψ (x)hDψ(x) = d xH (x) (5.15) h i H (x) = iψ¯ (x) ~γ ·~∇ + m ψ(x) (5.16)

and that the time derivative of ψ(~x,t) is generated by this Hamiltonian by taking a commutator,

∂ i ψ(x) = [ψ(x),H ] ∂t 0 In fact this will indeed be the case if the ﬁeld operator obeys the equal-time anti-commutation relations

n † o 3 ~ ψa(~x,t),ψb (~y,t) = δabδ (k −~y) n † † o {ψa(~x,t),ψb(~y,t)} = 0 , ψa (~x,t),ψb (~y,t) = 0 (5.17)

Our task in the following will be to assume that ψ(~x,t) indeed obeys the Dirac equation and this anti- commutation relation and to ﬁnd a solution of them. We have not discussed why the Dirac equation is relativistic. It is straightforward to see this, but we will put off the details until later. Here, we observe that, given the anti-commutation algebra of the Dirac matrices, 1 (γ µ ∂ )2 = γ µ γν ∂ ∂ = {γ µ ,γν }∂ ∂ = ∂ 2 µ µ ν 2 µ ν ν Using this identity, we can operate the matrix valued differential operator (−γ ∂ν +m) on the Dirac equation from the left to obtain

ν µ 2 2 (−γ ∂ν + m)(γ ∂µ + m)ψ = 0 → (−∂ + m )ψ = 0

We see that, if ψ(x) obeys the Dirac equation, it also obeys the relativistic wave equation (−∂ 2 +m2)ψ(x) = 0. This implies that the solutions of the Dirac equation also obey this relativistic wave equation, and must therefore propagate like relativistic matter waves.

5.1 Solving the Dirac equation To see how the Dirac equation is solved, it is useful to choose a speciﬁc form for the Dirac matrices

0 σ i 0 1 γi = , γ0 = (5.18) σ i 0 −1 0 82 Chapter 5. The Dirac Equation where we use 1 to denote the 2 × 2 unit matrix which appears in the upper and lower triangle of γ0. Also, σ i are the 2 × 2 Pauli matrices, which we remind the reader are given by 0 1 0 −i 1 0 σ 1 = , σ 2 = , σ 3 = (5.19) 1 0 i 0 0 −1 The Pauli matrices have the properties

σ iσ j + σ jσ i = 2δ i j1 , σ iσ j − σ jσ i = 2iεi jkσ k where εi jk is the totally antisymmetric tensor with ε123 = 1. It is easy to conﬁrm that the explicit form (5.18) indeed have the correct anti-commutation algebra for Dirac matrices. With the matrices in (5.18), the Dirac equation is " ~ # m ∂0 +~σ · ∇ u(x) ~ = 0 (5.20) −∂0 +~σ · ∇ m v(x) where we have split the four-component Dirac spinor into u(x) an v(x) which are two 2-component objects. To solve the differential equation, we use the ansatz u(x) µ ν u ~ u = eip ηµν x = e−iωt+ik·~x (5.21) v(x) v v

mu − i[ω −~σ ·~k]v = 0 (5.22) mv + i[ω +~σ ·~k]u = 0 (5.23)

We have now reduced the Dirac equation to two matrix equations. Equation (5.23) determines the 2- component object v in terms of u, that is, if u were known, we could determine v as i v = − [ω +~σ ·~k]u (5.24) m Plugging this into (5.22) yields the condition

ω2 =~k2 + m2 (5.25) p p which has two solutions for ω, ω = ~k2 + m2 and ω = − ~k2 + m2, the positive and negative energy solutions, respectively. In the following, we will use the notation where ω is the frequency which can be p either positive or negative, and E(~k) = ~k2 + m2 is positive, and sometimes abbreviated by E. Next, we note that ~σ ·~k is a Hermitian matrix which can be diagonalized. Once diagonal, it has real eigenvalues. The eigenstates of this matrix are said to be “eigenstates of helicity”. It is left as an exercise to the reader to show that there exist two eigenvectors, ~ ~ ~ ~ † † † † ~σ · ku+ = |k|u+ , ~σ · ku− = −|k|u− u+u+ = 1 = u−u− , u+u− = 0 = u−u+ We shall find the following identities very useful ~ ~ ~ ~ † |k| +~σ · k † |k| −~σ · k u+u+ = , u−u− = 2|~k| 2|~k| Then, putting it all together, we have four linearly independent solutions which we can superimpose to form the Dirac field,  q   q   Z i~k·~x−iEt  i 1 − |~k|/Eu i 1 + |~k|/Eu  3 e + ~ − ~ ψ(x) = d k √ 3  q a+(k) +  q a−(k) 2(2π) 2  1 + |~k|/Eu+ 1 − |~k|/Eu−   q   q   Z −i~k·~x+iEt  i 1 − |~k|/Eu i 1 + |~k|/Eu  3 e + † ~ − † ~ + d k √ 3  q b+(k) +  q b−(k) (5.26) 2(2π) 2  − 1 + |~k|/Eu+ − 1 − |~k|/Eu−  5.1 Solving the Dirac equation 83 p where E(~k) = ~k2 + m2. This solution will obey the anti-commutation relation for the Dirac field (5.17) if the Fourier coefficients satisfy the non-vanishing anti-commutation relations are

n ~ † ~ 0 o ~ ~ 0 n ~ † ~ 0 o ~ ~ 0 a+(k),a+(k ) = δ(k − k ) , a−(k),a−(k ) = δ(k − k ) (5.27) n ~ † ~ 0 o ~ ~ 0 n ~ † ~ 0 o ~ ~ 0 b+(k),b+(k ) = δ(k − k ) , b+(k),b+(k ) = δ(k − k ) (5.28)

All other combinations have vanishing anti-commutators. We can easily check that, with these anti- commutation relations for creation and annihilation operators, the solution in equation (5.26) obeys equations (5.17). With this solution, the Hamiltonian (5.16) is

Z q 3 ~ 2 2 † ~ ~ † ~ ~ † ~ ~ † ~ ~ H0 = d k k + m a+(k)a+(k) + a−(k)a−(k) + b+(k)b+(k) + b−(k)b−(k) (5.29) and the number operator is

Z N = d3xψ†(x,t)ψ(~x,t) Z 3 † ~ ~ † ~ ~ † ~ ~ † ~ ~ = d k a+(k)a+(k) + a−(k)a−(k) − b+(k)b+(k) − b−(k)b−(k) (5.30)

In both of the above expressions, we have dropped infinite constants. Unlike in the non relativistic theory, the vacuum energy density and the vacuum charge density both contain infinite constants which we have to simply drop in order to have a sensible Hamiltonian and number operator. We see from the expression (5.30), that, in direct analogy to the non relativistic system that we have ~ † ~ studied, electrons, which are associated with a±(k) and a±(k) contribute positively to the particle number ~ † ~ whereas holes, or positrons, which are associated with b±(k) and b±(k), have negative particle number. What differs from the non-relativistic theory is the fact that electrons and positrons have the same energy p spectrum. Since the energy of a single electron, ~k2 + m2, can be arbitrarily large, for large values of |~k, it is also so for holes (or positrons). This means that, as we shall seen, unlike the Fermi sea, the relativistic analog, which we can call the “Dirac sea”, is infinitely deep. This is what leads to the infinite values of the energy and number densities (which we have dropped). Another difference with the nonrelativistic theory, where the electron had a state of well-defined spin is that in the relativistic theory, it is the helicity (the states labeled by subscripts (+) and (-)), which are important. The helicity can be thought of as the projection of the spin in the direction of motion of the Fermion. We construct the basis of the Fock space beginning with the vacuum |O > which we assume is normalized, < O|O >= 1 and has the property that it is annihilated by all of the annihilation operators,

a+(~k)|O >= 0 , a−(~k)|O >= 0 , b+(~k)|O >= 0 , b−(~k)|O >= 0 for all values of~k. Then, multi particle and anti-particle states are created by operating creation operators

† ~ † ~ † ~ 0 † ~ 0 † ~ † ~ † ~0 † ~0 a+(k1)...a+(km)a−(k1)...a−(km0 )b+(`1)...b+(`n)b−(`1)...b−(`n0 )|O >

These states are eigenstates of particle number, N , with eigenvalue

N = m + m0 − n − n0 and they are eigenstates of the Hamiltonian, H0, with eigenvalue the total energy,

m q m0 q n q n0 q ~ 2 2 ~ 0 2 2 ~2 2 ~0 2 2 E = ∑ ki + m + ∑ (ki) + m + ∑ `i + m + ∑ (`i) + m 1 1 1 1 84 Chapter 5. The Dirac Equation

5.2 Lorentz Invariance of the Dirac equation The Dirac equation is clearly invariant under translations of the space-time coordinates. If ψ(~x,t) is a solution of the Dirac equation, then ψ(~x +~a,t + τ), with constants ~a and τ, is also a solution. What about Lorentz transformations? Let us begin by recalling how a Lorentz transformation of a scalar ﬁeld is implemented. Recall that a Lorentz transformation is the linear transformation on the coordinates

µ µ µ ν x → x˜ = Λ ν x

µ ν where the matrices Λ ν x satisfy the equation

µ ρ Λ ν Λ σ ηµρ = ηνσ

We are often interested in inﬁnitesimal transformations. For a Lorentz transformation

µ µ µ Λ ν = δ ν + ω ν + ... −1 µ µ µ (Λ ) ν = δ ν − ω ν + ...

In our discussion of coordinate transformations, we have derived the transformation property of the scalar ﬁeld,

φ˜(x˜) = φ(x)

which, for the Lorentz transformation, we can rewrite as

φ˜(x) = φ(Λ−1x)

The inﬁnitesimal transformation is then

ν µ δφ(x) = −ωµν x ∂ φ(x)

We would expect this transformation to be a symmetry of a relativistic wave equation that a scalar field would obey. We expect that, under a Lorentz transformation, the argument of the Dirac field also changes. However, the Dirac field has four components and the Lorentz transformation could also mix the components. Thus, we could make the ansatz that the Lorentz transformation of the Dirac field involves multiplication of it by a matrix,

−1 µ ν ψ˜ (x) = S(Λ)ψ(Λ x) = ψ(x) − ω ν x ∂µ ψ(x) + sψ(x) + ...

Here, S(Λ) + 1 + s + ... is a 4 × 4 matrix which depends on the Lorentz transformation. Since a Lorentz transformation should be invertible, we expect that S is an invertible matrix, that is, that detS 6= 0. This transformation is a symmetry of the Dirac equation if the transformed ﬁeld also satisﬁes the equation, that is, if

0 = [∂/ + m]ψ˜ (x) µ ν = [∂/ + m]ψ(x) + [∂/ + m] −ω ν x ∂µ + s ψ(x) µ ν µ ν µ = −ω ν γ ∂µ + ∂/,s ψ(x) = −ω ν γ + [γ ,s] ∂µ ψ(x) = 0 µ ν µ if − ω ν γ + [γ ,s] = 0

µ We need to ﬁnd s as a function of ω ν such that

µ µ ν [γ ,s] = ω ν γ

We can easily see that this equation is solved by

1 s = [γρ ,γσ ]ω 8 ρσ 5.2 Lorentz Invariance of the Dirac equation 85

To see this, consider

1 1 [γ µ ,s] = [γ µ ,[γρ ,γσ ]]ω = [γ µ ,γρ γσ ]ω 8 ρσ 4 ρσ 1 = ({γ µ ,γρ }γσ − γρ {γ µ ,γσ })ω 4 ρσ 1 = (2η µρ γσ − γρ 2η µσ )ω = ω µ γσ 4 ρσ σ

Thus, we have found the inﬁnitesimal Lorentz transformation of the Dirac ﬁeld,

1 δψ(x) = ω xµ ∂ ν + [γ µ ,γν ] ψ(x) (5.31a) µν 8 ←−ν 1 δψ¯ (x) = ψ¯ (x) xµ ∂ − [γ µ ,γν ] ω (5.31b) 8 µν

Although we shall not need it, we could also consider ﬁnite, rather than inﬁnitesimal Lorentz transformations. They are a symmetry of the Dirac equation if the matrix S satistifes

−1 µ −1ν ν S γ SΛ µ = γ

1 µ ν and this should be solved by S = 1 + s + ... = 1 + 8 [γ ,γ ]ωµν + .... For an inﬁnitesimal spatial rotation in the 1-2 plane (or, about the 3-axis), the only non-zero components of ωµν are ω12 and ω21 = −ω12 and

1 δψ(x) = ω x1∂ 2 − x2∂ 1 + γ1γ2 ψ(x) (5.32) 12 2 1 = ω (~x ×~∇)3 + γ1γ2 ψ(x) (5.33) 12 2 1 σ 3 0 = iω (~x × (−i~∇))3 + ψ(x) (5.34) 12 2 0 σ 3 ←− 1 σ 3 0 δψ¯ (x) = −iω ψ¯ (x) (~x × (−i ∇ ))3 + (5.35) 12 2 0 σ 3

Indeed, for a rotation by an axis in the direction of the inﬁnitesimal vector ~θ by angle |~θ|,

~ 1~ 1  θ · ~x × i ∇ + 2~σ 0 δψ(x) = i ψ(x)  ~ 1~ 1  0 θ · ~x × i ∇ + 2~σ ←− ~ 1 1  θ · ~x × i ∇ + 2~σ 0 δψ¯ (x) = −iψ¯ (x) ←−  ~ 1 1  0 θ · ~x × i ∇ + 2~σ that is, a rotation is a combination of a rotation of the spatial coordinates, implemented by the angular ~ 1~ momentum operator L =~x ×~p =~x × i ∇ and a rotation of the Dirac field spin, implemented by the Pauli 1 matrices 2~σ. Then, in particular, under infinitesimal rotations, the electron density and the mass operator transform like a scalar fields, δ ψ†(x)ψ(x) = ~θ ·~x ×~∇ ψ†(x)ψ(x) δ (ψ¯ (x)ψ(x)) = ~θ ·~x ×~∇ (ψ¯ (x)ψ(x)) 86 Chapter 5. The Dirac Equation

For an Lorentz transformation with infinitesimal velocity~v, the only non-zero components of ωµν are ω0a = va and ωa0 = −vv and 1 δψ(x) = x0~v ·~∇ −~v ·~x∂ 0 + γ0~v ·~γ ψ(x) (5.36) 2 ←− ←−0 1 δψ¯ (x) = ψ¯ (x) x0~v · ∇ −~v ·~x ∂ − γ0~v ·~γ (5.37) 2 ←− ←−0 1 δψ†(x) = ψ†(x) x0~v · ∇ −~v ·~x ∂ + γ0~v ·~γ (5.38) 2 The mass operator transforms like a scalar field, δ (ψ¯ (x)ψ(x)) = (x0~v ·~∇ −~v ·~x∂ 0)(ψ¯ (x)ψ(x)) However, the density is not a scalar, but has the transformation law δ ψ†(x)ψ(x) = (x0~v ·~∇ −~v ·~x∂ 0)ψ†(x)ψ(x) +~v · ψ¯ (x)~γψ(x) (5.39) Of course, it transforms like the time-component of a vector field, consistent with the fact that it is the time-component of jµ (x) = ψ†(x)ψ(x),−ψ¯ (x)~γψ(x) (5.40) Indeed, we could examine the full Lorentz transformation law and see that jµ (x) transforms like a vector field.

5.3 Phase symmetry and the conservation of electric current In the above, we have examined the transformation law for the Dirac field density and we found that it transforms like the time-component of a vector field (5.40). We will see shortly that this vector field obeys µ the continuity equation, ∂µ j (x) = 0, which we would expect for a “conserved current” and, when our Dirac Fermions are coupled to photons, it, scaled by a unit of electric charge e, will be identified with the electric charge and current densities. In non-relativistic terminology, ej0(x) is the “charge density” and e~j(x) is the “current density”. In relativistic physics, we simply call jµ (x) a “current” or a “conserved current”. We consider the Dirac equation and its conjugate

† h †µ ←− i ∂/ + m ψ = 0 , ψ γ ∂ µ + m = 0

Then, we note that, γ†µ γ0 = −γ0γ µ , so that, if we multiply the second equation above from the right by γ0 we obtain ←− † 0 h i ∂/µ + m ψ = 0 , ψ γ − ∂/ + m = 0 We shall shorten the notation by defining ψ¯ (x) ≡ ψ†(x)γ0 (5.41) Then, we can write the current as jµ (x) = −ψ¯ (x)γ µ ψ(x) (5.42) Now, we are ready to use derive the continuity equation. We form the current and we assume that the Dirac spinor satisfies the Dirac equation. Then, ←− µ µ h i −∂µ j (x) = ∂µ (ψ¯ (x)γ ψ(x)) = ψ¯ ∂/ + ∂/ ψ(x) = ψ¯ (x)[−m + m]ψ(x) = 0 This continuity equation for the current implies the conservation of charge, which we have so far called the number operator, d d Z Z ZZ N = d3xψ†(x)ψ(x) = d3x~∇ · ψ¯ (x)~γψ(x) = d2σnˆ · ψ¯ (x)~γψ(x) dt dt where we have used Gauss’ theorem to write the last term as a surface integral at the infinite boundary of three dimensional space. If our boundary conditions are such that the quantum expectation value of the current density goes to zero sufficiently rapidly there, the number operator is conserved. 5.4 The Energy-Momentum Tensor of the Dirac Field 87

5.4 The Energy-Momentum Tensor of the Dirac Field The Dirac equation can be derived from an action Z S = d4xL (x)

where the Lagrangian density is

1−→ 1←− L (x) = −iψ¯ (x) ∂/ − ∂/ + m ψ(x) 2 2

Here we have defined the action as the integral of the Lagrangian density. It should be kept in mind, of course, that the Dirac field describes Fermions and therefore the Lagrangian density depends on anti-commuting classical fields ψ(x) and ψ¯ (x). The “−i” in front and the symmetrization of the derivative operators are present to make the Lagrangian density real. Moreover, the leading terms, up to total derivatives are (x) = i †(x) ∂ (x) + ... which is compatible with the equal-time anti-commutation relations that we L ψ ∂x0 ψ have used for the Dirac field. The Dirac equation is easily recovered from this Lagrangian density using the Euler-Lagrange equations. Now, consider a space-time translation xµ → xµ + ε µ , where ε µ is an infinitesimal constant four-vector. The Dirac field transforms as

µ µ δψ(x) = −ε ∂µ ψ(x) , δψ¯ (x) = −ε ∂µ ψ¯ (x)

By inspection, we see that, when the Dirac ﬁeld transforms this way, the Lagrangian density transforms as

µ δL (x) = ∂µ [−ε L (x)]

The fact that the Lagrangian density varies by a total derivative term means that the inﬁnitesimal translation is a symmetry of the theory. Of course, we already knew that this should be the case, since we have already seen that the equation of motion has this symmetry. To ﬁnd the Noether current, we use Noether’s theorem which tells us that

µ ν ∂L (x) ν ∂L (x) µ Jε (x) = −ε ∂ν ψ(x) − ε ∂ν ψ¯ (x) + ε L (x) ∂(∂µ ψ(x) ∂(∂µ ψ¯ (x) µν = εν T0 (x) where i ←−µ Tµν (x) = ψ¯ (x) γν ∂ µ − γν ∂ ψ(x) (5.43) 0 2 is the canonical energy-momentum tensor. Using the Dirac equation (∂/ + m)ψ(x) = 0, which implies that ←−2 0 = (−∂/ + m)(∂/ + m)ψ(x) = (−∂ 2 + m2)ψ(x) and also ψ¯ (x)(− ∂ + m2) = 0, It is easy to check that µν T0 (x) obeys the continuity equation

i ←−2 ∂ Tµν (x) = ψ¯ (x) γν ∂ 2 − γν ∂ ψ(x) = 0 µ 0 2 . Another, alternative way to derive the Noether current is to begin with an inﬁnitesimal translation, as we have just done, but to assume that the transformation parameter now depends on the coordinates, ε µ (x). The variation of the Lagrangian now has the form

i ←−µ δL (x) = ∂ [ε µ (x)L (x)] − ∂ ε (x) ψ¯ (x) γν ∂ µ − γν ∂ ψ(x) µ µ ν 2 This equation makes no assumptions about the nature of ε µ (x), other than that it is inﬁnitesimal. If we allow it to go to a constant four-vector, we recover the fact that the Lagrangian density transforms as a total derivative. Now, instead, we assume that ε µ (x) vanishes sufﬁciently rapidly at the boundaries of the integral so that boundary terms which are generated upon integration by parts can be ignored. Moreover, we assume 88 Chapter 5. The Dirac Equation that the Euler-Lagrange equations (i.e. the Dirac equation for ψ(x) and ψ¯ (x)) are obeyed. We recall that, if the Euler-Lagrange equations are obeyed, any such variation of the action must vanish. Then, we must have

i ←−µ ∂ ψ¯ (x) γν ∂ µ − γν ∂ ψ(x) = 0 µ 2

This is just the conservation law for the Noether current that we obtained and conﬁrmed above in our ﬁrst derivation. This is a second route to Noether’s theorem. The space integral of the time-component of this energy-momentum tensor is the generator of space-time translations,

Z i Z ←−0 Z P0 = d3xT00(x) = d3xψ†(x) ∂ 0 − ∂ ψ(x) = d3xψ†(x)h ψ(x), h = −γ0~γ ·~∇ − iγ0m 0 2 D D Z i Z ←−0 Pa = d3xT0a(x) = d3xψ†(x) γ0γa∂ 0 − γ0γa ∂ ψ(x) 0 2 1 Z = d3xψ†(x)γ0γah + h γ0γaψ(x) 2 D D Z = d3xψ†(x)(−i∇a)ψ(x)

In both of the equation above, we have eliminated the time derivatives from the integrand by using the Dirac 0 equation. This means replacing i∂ ψ by hDψ where hD is the single-particle Dirac Hamiltonian defined in equation (5.5). The result is that P0 is the Dirac field Hamiltonian and we recognize Pa as the Dirac field linear momentum. We note that the four-divergence of the energy-momentum tensor on its second index also vanishes,

i ←−µ i ←− ←−←−µ ∂ Tµν (x) = ψ¯ (x) ∂∂/ µ − ∂/ ∂ ψ(x) + ψ¯ (x) ∂/ ∂ µ − ∂/ ∂ ψ(x) = 0 ν 0 2 2 ←− where we have used the Dirac equation ∂ψ/ (x) = −mψ(x) and ψ¯ (x) ∂/ = ψ¯ (x)m. However, clearly, the energy-momentum tensor that we have found is not symmetric. It therefore cannot immediately be used to construct the Noether current for Lorentz transformations. The fact that its four-divergence vanishes for either of its indices means that it can be decomposed into a symmetric and anti-symmetric part 1 Tµν (x) = Tµν (x) + Tνµ (x) (5.44) 2 0 0 1 Tµν (x) = Tµν (x) − Tνµ (x) (5.45) A 2 0 0 and that both of these tensors are conserved,

µν µν ∂µ T (x) = 0, ∂µ TA (x) = 0 This fact will be important to us shortly when we discuss improving this energy-momentum tensor. µ µ µ ν Now, consider an inﬁnitesimal Lorentz transformation, x → x + ω ν x where the transformation of the Dirac ﬁeld is given in equations (5.31a) and (5.31b), which we copy here

1 δψ(x) = ω xµ ∂ ν + [γ µ ,γν ] ψ(x) µν 8 ←−ν 1 δψ¯ (x) = ψ¯ (x) xµ ∂ − [γ µ ,γν ] ω 8 µν

Under this transformation, the Lagrangian density varies as

µ ν ν µ δL (x) = ωµν x ∂ L (x) = ∂ ωµν x L (x)

The fact that the Lagrangian density varies by a total derivative term means that the Lorentz transformation is a symmetry of the theory. Of course, we already knew that this should be the case, since we have already demonstrated that the equation of motion has this symmetry. 5.4 The Energy-Momentum Tensor of the Dirac Field 89

To ﬁnd the Noether current, let us assume that the transformation parameter depends on the coordinates, ωµν (x). The variation of the Lagrangian now has the form ν µ δL (x) = ∂ ωµν (x)x L (x) i 1 ←−σ 1 − ∂ ω ψ¯ (x) γλ xρ ∂ σ + [γρ ,γσ ] − ∂ xρ − [γρ ,γσ ] γλ ψ(x) λ ρσ 2 8 8

Now, if ωµν (x) vanishes sufﬁciently rapidly at the boundaries of the integral so that boundary terms can be ignored, and if we assume that the equations of motion are obeyed, the action must vanish for any variation, in particular, with any proﬁle of ωρσ (x). Then, we must have

λρσ ∂λ M (x) = 0 (5.46) where the Noether current is given by i n o Mλρσ (x) = Tλσ (x)xρ − Tλρ (x)xσ + ψ¯ (x) γλ ,[γρ ,γσ ] ψ(x) (5.47) 0 0 16 µν where T0 (x) is the energy-momentum tensor which was associated with translations. We could also ﬁnd this current by using the more conventional Noether theorem. The conservation law (5.46) is a result of µν Noether’s theorem. Doing the derivatives explicitly, and remembering that ∂µ T0 (x) = 0, we get the identity i n o Tσρ (x) − Tρσ (x) = ∂ ψ¯ (x) γλ ,[γρ ,γσ ] ψ(x) (5.48) 0 0 λ 16

µν This is an equation for the anti-symmetric part of T0 (x), which states that it is given by the four-divergence i λ ρ σ of 16 ψ¯ (x) γ ,[γ ,γ ] ψ(x). The latter quantity is clearly anti-symmetric in the indices ρ and σ. We can use the algebraic properties of the gamma-matrices to see that the combination γλ [γρ ,γσ ] is equal to zero unless the three indices λ,ρ,σ are all different. Therefore γλ [γρ ,γσ ] = −4iελρσν γ5γν where ελρσν is the totally anti-symmetric tensor with ε0123 = 1. µν µν Then, recalling that we can write T0 (x) as its symmetric part T (x) plus its anti-symmetric part which we have found above, 1 Tρσ (x) = Tρσ (x) + ∂ ελρσν ψ¯ (x)γ5γν ψ(x) (5.49) 0 λ 4 Moreover 1 Mλρσ (x) = Mλρσ (x) + ∂ εγλσν xρ − εγλρν xσ ψ¯ (x)γ5γν ψ(x) . (5.50) 0 0 γ 4 Mλρσ (x) = Tλσ (x)xρ − Tλρ (x)xσ labelm (5.51)

Now, we see that, for the improved energy momentum tensor, we could simply use its symmetric part i ←− ←− Tµν (x) = ψ¯ (x)[γ µ ∂ ν + γν ∂ µ − γ µ ∂ ν − γν ∂ µ ]ψ(x) , ∂ Tνµ (x) = 0 (5.52) 4 λ It obeys

νµ ∂λ T (x) = and the spatial integral of the energy-momentum charge density Z Z Z 3 0µ 3 0µ 3 a0µc 5 c d xT0 (x) = d xT (x) + d x∇aε ψ¯ (x)γ γ ψ(x)

R 3 a0µc 5 c where Gauss’ theorem could be used to write d x∇aε ψ¯ (x)γ γ ψ(x) as a surface integral on the sphere at inﬁnity, where we could assume that the integrand vanishes sufﬁciently rapidly that the surface integral is zero. Then we conclude that Z Z µ 3 0µ 3 0µ P = d xT0 (x) = d xT (x) 90 Chapter 5. The Dirac Equation

In addition Z Mρσ = d3xM0ρσ (x)

and, we can us the improved energy-momentum tensor Tµν (x) as the energy-momentum tensor whose integrals give the current and charge densities and as well, the current associated with the Lorentz transformation can be constructed from this improved energy-momentum tensor as is shown in equation (??). There are good reasons why it is convenient to have a symmetric energy-momentum tensor. By modifying Tρσ (x) to make it symmetric, we will be able to unify the generator of translations and Lorentz transformations. If we recall that a space-time symmetry is a coordinate transformation which is generated by a Killing vector fˆµ (x), we might make a candidate for a conserved current by contracting the energy- momentum tensor with the vector ﬁeld which generates the co-ordinate transformation,

µ µ ν T f (x) ≡ T ν (x) f (x)

Then, to have a conservation law, we need

µ ∂µ T f (x) = 0

With a vector field f µ (x), we will have such a conservation law if: 1. Tµν (x) is conserved, i.e. ∂ Tµν (x) = 0. Then ∂ Tµ (x) = 0 if fˆµ = aµ , a constant vector. A µ µ fˆ translation invariant field theory should have a conserved energy-momentum tensor. A field theory can be translation invariant without being Lorentz invariant. It would still have a conserved energy- momentum tensor. We know an example from our study of non-relativistic many particle theory. However, if the theory is not Lorentz invariant, it should not be expected to have a symmetric energy-momentum tensor. 2. Tµν (x) is conserved and Tµν (x) = Tνµ (x) is symmetric. Then Tµ (x) is conserved when fˆµ (x) obeys fˆ the Killing equation ∂µ fˆν (x) + ∂ν fˆµ (x) = 0 A conserved, symmetric energy-momentum tensor can thus be used to generate all of the symmetries of Minkowski space. A translation and Lorentz invariant field theory should have a conserved and symmetric energy-momentum tensor. µν µ µ 3. Finally, T (x) is conserved, symmetric and has vanishing trace, T µ (x) = 0. Then T f (x) is conserved when f µ (x) satisfies the conformal Killing equation η ∂ f (x) + ∂ f (x) − µν ∂ f λ (x) = 0 µ ν ν µ 2 λ and it generates a conformal transformation. Notice that all solutions of the Killing equation are also solutions of the conformal Killing equation. However, the conformal Killing equation is less restrictive. It has more solutions than the Killing equation. The extra solutions correspond to conformal transformations. A conformal field theory should have a conserved, symmetric and traceless energy- momentum tensor. Our example of the non-interacting Dirac field is a conformal field theory when m = 0.

5.5 Summary of this chapter The Dirac theory of spinor ﬁelds ψ(x) and ψ¯ (x) is described by an action Z S[ψ,ψ¯ ] = dxL (x)

which is a space-time volume integral of the Lagrangian density

1−→ 1←− L (x) = −iψ¯ (x) ∂/ − ∂/ + m ψ(x) 2 2 5.5 Summary of this chapter 91

Applying the Euler-Lagrange equation to the Lagrangian density yields the Dirac equation h ←− i ∂/ + mψ(x) = 0, ψ¯ (x) − ∂/ + m = 0

µ µ ν µν The Dirac matrices γ are 4×4 and they obey {γ ,γ } = 2η with ηµν the metric of Minkowski space-time. The equal time anti-commutation relations are

n †b o 0 0 b 4 ψa(x),ψ (y) δ(x − y ) = δa δ (x − y)

0 0 n †a †b o 0 0 {ψa(x),ψb(y)}δ(x − y ) = 0 , ψ (x),ψ (y) δ(x − y ) = 0

The Dirac theory has a phase symmetry which results in the conserved Noether current corresponding to particle number,

µ µ jV (x) = −ψ¯ (x)γ ψ(x) The Dirac theory is Poincare invariant. The Noether currents associated with this symmetry, Tµ (x) = µν −T (x) fν (x), can be formed from the appropriate Killing vectors fµ (x) and the symmetric, conserved energy-momentum tensor i −→ −→ ←− ←− Tµν (x) = ψ¯ (x)[γ µ ∂ ν + γν ∂ µ − γ µ ∂ ν − γν ∂ µ ]ψ(x) 4 With the explicit representation 0 σ i 0 1 γi = , γ0 = (5.53) σ i 0 −1 0 and the ortho-normal helicity eigenvectors,

~σ ·~ku+ = |~k|u+ ~σ ·~ku− = −|~k|u− the Dirac ﬁeld is  q   q   Z i~k·~x−iEt  i 1 − |~k|/Eu i 1 + |~k|/Eu  3 e + ~ − ~ ψ(x) = d k √ 3  q a+(k) +  q a−(k) 2(2π) 2  1 + |~k|/Eu+ 1 − |~k|/Eu−   q   q   Z −i~k·~x+iEt  i 1 − |~k|/Eu i 1 + |~k|/Eu  3 e + † ~ − † ~ + d k √ 3  q b+(k) +  q b−(k) 2(2π) 2  − 1 + |~k|/Eu+ − 1 − |~k|/Eu−  p where E(~k) = ~k2 + m2 and the non-vanishing anti-commutation relations are

n ~ † ~ 0 o ~ ~ 0 n ~ † ~ 0 o ~ ~ 0 a+(k),a+(k ) = δ(k − k ) a−(k),a−(k ) = δ(k − k ) n ~ † ~ 0 o ~ ~ 0 n ~ † ~ 0 o ~ ~ 0 b+(k),b+(k ) = δ(k − k ) b+(k),b+(k ) = δ(k − k )

The vacuum is deﬁned by

a±(k)|O >= 0 , b±(k)|O >= 0 ∀k,±

† † and particle and anti-particle states are created by a±(k) and b±(k), respectively. The Noether charges for the phase and space-time translation symmetries are given by Z Z 3 0 3 † † N = d xjV (x) = d k ∑ as (k)as(k) − bs (k)bs(k) s=± Z Z q 3 00 3 ~ 2 2 † † H = − d xT (x) = d k ∑ k + m as (k)as(k) + bs (k)bs(k) s=± Z Z a 3 0i 3 a † † P = − d xT (x) = d k ∑ k as (k)as(k) + bs (k)bs(k) s=±

6. Photons

We must now turn from our treatment of non-relativistic and relativistic Fermions to a Bosonic degree of freedom, the electromagnetic field, whose physical manifestation is familiar to us as the electric field ~E(x) and the magnetic field ~B(x). These are both spatial three-vector fields. They appear in abundance in the physical world. We wish to study how these arise from a quantum theory. To do so, we begin by remembering that the low energy states of weakly interacting quantum Fermi and Bose gases are very different. Fermions have a Fermi surface, particles and holes and we have made use of these concepts to construct quantum field theories of non-relativistic Fermions and their relativistic analog, the Dirac theory. Bose gases, on the other hand, exhibit a Bose-Einstein condensate. The condensate was described by a classical part of the quantum field, in terms of our non-relativistic Bose field, we had ψ(~x,t) = η(~x,t) + ψ˜ (~x,t) where ψ(~x,t) and ψ˜ (~x,t) were quantized fields obeying equal time commutation relations and η(~x,t) was a classical field. Moreover, when quantum fluctuations are small, the classical part of the Bose field obeys the same field equation as the quantum field. For quantum electrodynamics, we can invert this logic. We know the field equations that classical electric and magnetic fields must obey, they are Maxwell’s equations. If these classical fields are the classical parts of quantum fields, and the quantum field theory is weakly coupled, in that the effects of quantum corrections are a small, we might expect that the quantum field theory is simply a theory where the quantized electric and magnetic fields have Maxwell’s equations as their field equations. This expectation will turn out to be correct. In fact, Maxwell’s equations as the field equations for the quantum theory turns out to be the only mathematically consistent formulation of quantum electrodynamics in four space-time dimensions. This is a result of symmetry and the requirement of renormalizability, which we shall learn about later in this chapter, and in later chapters, respectively.

6.1 Relativistic Classical Electrodynamics Classical electrodynamics is governed by Maxwell’s equations which are partial differential equations for the electric and magnetic ﬁelds, ~E and ~B, respectively,

~∇ ·~E(~x,t) = ρ(~x,t) (6.1) −~E˙(~x,t) +~∇ ×~B(~x,t) =~j(~x,t) (6.2) ~∇ ·~B(~x,t) = 0 (6.3) ~B˙(~x,t) +~∇ ×~E(~x,t) = 0 (6.4) 94 Chapter 6. Photons

We are working in a system of units where the constants ε0 and µ0 that sometimes appear in these equations are set equal to one. (This requires that the speed of light be set equal to one.) We are quoting Maxwell’s equations with sources, a charge density rho(~x,t) and a current density~j(~x,t) which we will leave unspecified for now. Maxwell’s equations are internally consistent only when the charge and current densities satisfy the continuity equation ∂ ρ(~x,t) +~∇ ·~j(~x,t) = 0 (6.5) ∂t To put Maxwell’s equations into relativistic notation, we identify the electric and magnetic fields with the components of an anti-symmetric two-index tensor field, F µν (x) as

0i Ei(~x,t) ≡ F (6.6) i jk i j ε Bk(~x,t) ≡ F (x) (6.7) and the charge and current densities as a four-current ρ(~x,t),~j(~x,t) = j0(~x,t),~j(~x,t) ≡ jµ (x) (6.8)

and the continuity equation is

µ ∂µ j (x) = 0 (6.9) With this notation, Maxwell’s equations become

µν µ ∂ν F (x) = j (x) (6.10a)

∂µ Fνλ (x) + ∂ν Fλ µ (x) + ∂λ Fµν (x) = 0 (6.10b)

These are the relativistic form of Maxwell’s equations. They are the ﬁeld equation of classical electrodynamics.

6.2 Covariant quantization of the photon In this section, we will outline a scheme for identifying the correct quantum ﬁeld theory of the photon and then, in maintaining explicit Lorentz covariance in solving the quantum ﬁeld theory.

6.2.1 Field equations and commutation relations Equations (6.10a) and (6.10b) are the field equations of classical electrodynamics and they are also the equations that we expect that the quantized electromagnetic fields must obey. However, we still do not have a guide to determining the operator nature of the fields. The operator nature of the fields is defined by the commutation relations which we have yet to find. Our strategy for finding commutation relations will be to construct a Lagrangian density from which the field equations can be derived by using a variational principle. Then we will deduce the commutation relations by examining the time derivative terms in the Lagrangian density. Finding a Lagrangian density requires an important preliminary step which involves identifying the appropriate dynamical variable. This is the variable which should be varied when we use the variational principle to find the field equation. This variable turns out to be the vector potential field Aµ (x). It is introduced to solve equation (6.10b). Consider the ansatz

Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x) (6.11)

By plugging it into equation (6.10b) we can see that equation is satisﬁed identically for any Aµ (x). Then, equation (6.10a) becomes an equation for Aµ (x), it is

2 µ µ ν µ −∂ δ ν + ∂ ∂ν A (x) = ej (x) (6.12) In classical electrodynamics, this is a partial differential wave-equation which we must solve in order to find the four-vector potential Aµ (x) which we then plug into formula (6.11) to find the field strengths ~E(x) and ~B(x). 6.2 Covariant quantization of the photon 95

Now, we are ready to write down an action and a Lagrangian density from which equation (6.10a) or (6.12) can be derived. The action is, as usual the space-time volume integral of the Lagrangian density Z S[A] = dxL (x) where the Lagrangian density is 1 L (x) = − F (x)F µν (x) + eA (x)jµ (x) (6.13) 4 µν µ

In the Lagrangian density, Aµ (x) is bhe basic dynamical variable and Fµν (x) is assumed to be constructed from Aµ (x) as in equation (6.11). It is easy to check that the Euler-Lagrange equations, applied to the ﬁeld Aµ (x),

∂L ∂L ∂µ = = 0 ∂(∂µ Aν (x)) ∂Aν (x) reproduce the ﬁeld equation (6.10a) or (6.12). Now, to implement the next step, which ﬁnds the commutation relations, we examine the time derivative terms in the Lagrangian density,

1 ∂ 2 ∂ L (x) = ~A(x) − ~A(x) ·~∇A (x) + ... 2 ∂t ∂t 0 Here, we see the first complication in the quantization of the photon. The Lagrangian density does not contain the time derivative of the temporal component of the vector potential field A0(x). The momentum conjugate to the time component of the gauge field vanishes. Whenever the relationship between the generalized velocities, ∂ A (x), and the generalized momenta, Πµ (x) = ∂L , cannot be solved for the velocities, ∂t µ ( ∂ A (x)) ∂ ∂t µ the dynamical system has constraints. We might have expected that this description of electrodynamics is as a constrained system. After all, in four dimensions, we expect that the photon has two physical polarizations. However, the vector potential field contains four functions, it would seem to be too many to describe the photon. Indeed, this is so and the manifestation is the constraint that we find when we begin quantizing the theory. There are systematic ways to study constrained systems which could be, and which routinely are exploited at this point. However, we will choose a technically simpler (albeit perhaps logically not quite as clear) approach which exploits the gauge invariance of the theory. Gauge invariance is the fact that, if we make the replacement

Aµ (x) → A˜µ (x) = Aµ (x) + ∂µ χ(x) (6.14) the ﬁeld strength tensor Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x) is left unchanged. Then, Maxwell’s equations (6.10a) or (6.12) are left unchanged. The Lagrangian density changes by a total derivative,

µ L → L + ∂µ (χ(x)j (x))

Gauge symmetry reﬂects the redundancy of the dynamical variable Aµ (x). We can make use of this redundancy to enforce a constraint on Aµ (x) called a gauge condition. For various reasons, mainly to do with technical simplicity of calculations that we shall do later on, we are mostly interested in Lorentz invariant gauge conditions such as

µ ∂µ A (x) = 0 (6.15) which can always be imposed by exploiting the gauge symmetry. If a generic Aµ (x) did not obey this condition, we could find a new field A˜µ (x) = Aµ (x) + ∂µ χ(x) which does satisfy it by choosing χ(x) which 2 µ satisfies the equation ∂ χ(x) + ∂µ A (x) = 0. When we assume that the gauge field obeys this gauge fixing condition, the Lagrangian density becomes 1 L (x) = − ∂ A (x)∂ µ Aν (x) + A (~x)jµ (x) + total derivatives (6.16) 2 µ ν µ 96 Chapter 6. Photons

(As we have previously argued, the total derivative terms have no effect on either the ﬁeld equations or the commutation relations that we will ﬁnd, so they can be ignored.) The time derivative terms in the Lagrangian density are

1 ∂ ∂ L (x) = A (x)η µν A (x) + ... (6.17) 2 ∂t µ ∂t ν From these time-derivative terms, we can identify the generalized momenta

∂L ∂ Πµ (x) = = A (x) ∂ ∂t µ ∂( ∂t Aµ (x)) recall that the Poisson bracket, which would be Aµ (~x,t),Πν (~y,t) PB = ηµν δ(~x −~y) Aµ (~x,t),Aν (~y,t) PB = 0, Πµ (~x,t),Πν (~y,t) PB = 0 once written with the generalized velocities is

∂ Aµ (~x,t), Aν (~y,t) = ηµν δ(~x −~y) ∂t PB ∂ ∂ Aµ (~x,t),Aν (~y,t) PB = 0, Aµ (~x,t), Aν (~y,t) = 0 ∂t ∂t PB In the quantized theory, these Poisson brackets are replaced by the equal-time commutation relations

∂ A (~x,t), A (~y,t) = iη δ(~x −~y) (6.18) µ ∂t ν µν ∂ ∂ .A (~x,t),A (~y,t) = 0, A (~x,t), A (~y,t) = 0 (6.19) µ ν ∂t µ ∂t ν

(We have set h¯ = 1, otherwise it would be a factor on the right-hand-side of equation (6.18).) This leaves us with the quantum field theory specified by the field equation, obtained by applying the Euler-Lagrange equation to (6.16), and the constraint,

− ∂ 2Aµ (x) = jµ (x) (6.20) µ ∂µ A (x) = 0. (6.21) together with the equal-time commutation relations in equations (6.18) and (6.19). µ The equation ∂µ A (x) = 0 is linear in derivatives and it should properly be regarded as a constraint, rather than an equation of motion. There are two approaches to quantizing a constrained system. In the first approach, one can solve the constraints at the classical level and reduce the system to its basic set of dynamical variables and then proceed to quantize the system with a minimal number of variables. The second approach is to quantize the system without imposing the constraints and then, once we find the quantum states of the unconstrained system, to impose the constraints as physical state conditions. These are conditions which pick out a subspace of the full quantum Hilbert space of the theory, and declares that subspace as containing “physical states”. In that subspace the constraints are obeyed by the Hilbert space matrix elements of the quantum mechanical operators. It is this second approach that we will take here in discussing covariant quantization. The main reason for this is to maintain Lorentz covariance of the procedure as far as possible. The approach that we will now take is to quantize the theory using the field equation (6.20) and the equal time commutation relations (6.18) and (6.19). Then, once we have constructed the Hilbert space on which the field operators act, we will impose the constraint (6.21) as a physical state condition. To begin, we will solve the wave equation (6.20) with the sources set to zero,

−∂ 2Aµ (x) = 0 6.2 Covariant quantization of the photon 97

The solution is

3 Z d k h ν ν i ikν x ~ −ikν x † ~ Aµ (x) = e aµ (k) + e aµ (k) (6.22) (2π)32|~k| where, inside the integral, the temporal component of the wave-vector is deﬁned so that the plane waves ν ±ikν x ~ ~ e satisfy the wave equation, that is, kµ = (−|k|,k). This solution of the wave equation obeys the equal-time commutation relations (6.18) and (6.19) if the creation and annihilation operators obey the commutation relations

h ~ † ~ i ~ ~ aµ (k),aν (`) = ηµν δ(k − `) (6.23) h ~ ~ i h † ~ † ~ i . aµ (k),aν (`) = 0, aµ (k),aν (`) = 0 (6.24)

The wave-functions of the positive and negative energy states are

ν ν ikν x −ikν x + e − e ~ ~ φ (x) = , φ (x) , kµ = (|k|,k) k (2π)32|~k| k (2π)32|~k| respectively, and the annihilation and creation operators are projected out of the ﬁeld operator by the integrals ←−! Z ∂ ∂ a (~k) = d3xφ +∗(x) i − i A (x) µ k ∂t ∂t µ ←−! Z ∂ ∂ a (~k) = d3xφ −∗(x) i − i A (x) µ k ∂t ∂t µ

~ † ~ The operators aµ (k) and aµ (k) have the properties of annihilation and creation operators and we can construct a Fock space by beginning with a vacuum state, |O > which has the property ~ ~ aµ (k)|O >= 0, ∀k, µ and then creating a basis for multi-photon states, n o | >,a† (~k)| >,...,a† (~k )a† (~k )...a† (~k )| >,... O µ O µ1 1 µ2 2 µn n O

Now, we come to one of the problems which plagues this, and any Lorentz covariant quantization. The natural dual of a Fock space basis vector is gotten by conjugation.For example the dual of the one-photon † ~ ~ state aµ (k)|O > is < O|aµ (k). Here, < O| is the dual of the vacuum state and < O|O >= 1. Then, we can use the commutation relations to compute the norm of states. For example, consider the generic one-photon state Z 3 µ ~ ~ |ζ >= d kζ (k)aµ (k)|O >

Its norm is Z 3 ∗µ ~ ν ~ 0 ~ ~ 0 < ζ|ζ > = d kζ (k)ζ (k ) < O|aµ (k)aν (k )|O > Z 3 ∗ ~ µ ~ = d kξµ (k)ξ (k)

∗ ~ µ ~ µ ~ 0 ~ ~ If ζµ (k) is (on the average) a time-like vector, ζµ (k)ξ (k) < 0 (all we would need is that ζ (k) = (ξ (k),0) for example), the state |ζ > has negative norm. We expect that this problem persists with multi-photon states. The Fock space has an indeﬁnite metric. There are states with negative norm, which is not physically acceptable, as they would imply the nonsensical result of negative probabilities as the answers to some physical questions. However, we have not enforced the gauge ﬁxing constraint yet. The entire Fock space are not physical states. Only a subspace of the Fock space are physical states and it is possible that the subspace has a 98 Chapter 6. Photons

µ non-negative norm. We will impose the gauge ﬁxing constraint ∂µ A (x) = 0 on the Fock space as a physical µ state condition. If we attempt to impose the condition in its strongest form, ∂µ A (x)|phys >= 0, where we would then distinguish |phys > as a “physical state”, we discover that this is too many constraints. There would be no physical states at all, besides the zero vector in the Fock space. We need to impose a weaker condition. The weaker condition is that the matrix element of the constraint between any two physical states vanishes,

0 µ < phys |∂µ A (x)|phys > = 0 (6.25) so that, in the physical subspace of the Fock space, the expectation value of the physical state condition is always obeyed. We can do this as follows. Let us, for the moment, return to the ﬁeld equation with a source (6.20), which we did not solve, and make an observation. We note that, by taking a four-divergence of the equation of motion in (6.20), even in the the presence of the source current, conservation of the current implies that the constraint operator obeys the free wave equation,

2 µ µ 2 µ ∂µ −∂ A (x) = ∂µ j (x) = 0 → −∂ ∂µ A (x) = 0 independently of the existence of currents (and interactions which will be introduced through the currents), as long as the current is still conserved. The constraint therefore always satisﬁes a free wave equation 2 µ ∂ ∂µ A (x) = 0, even in the interacting theory. A general solution of the free wave equation is

3 Z d k h µ µ i µ ikµ x ~ −ikµ x † ~ ~ ~ ∂µ A (x) = q e ξ(k) + e ξ (k) , kµ = (−|k|,k) (6.26) (2π)32|~k| µ(+) µ(−) = ∂µ A (x) + ∂µ A (x) (6.27) where we have made the decomposition into positive and negative frequency parts

3 3 Z d k ν Z d k ν µ(+) ikν x ~ µ(−) −ikν x † ~ ∂µ A (x) = q e ξ(k), ∂µ A (x) = q e ξ (k) (2π)32|~k| (2π)32|~k|

† † µ(+) µ(−) µ(−) µ(+) Moreover, ∂µ A (x) = ∂µ A (x) and ∂µ A (x) = ∂µ A (x). We can then impose the physical state condition as

µ(+) ~ ~ ∂µ A (x) |phys >= 0 or ξ(k)|phys >= 0 ∀k (6.28) even when the current is nonzero. Then, taking the conjugate yields

µ(−) † ~ ~ < phys| ∂µ A (x) = 0 or < phys|ξ (k) = 0 ∀k (6.29) and, thus, we see that for any two physical states

0 µ 0 µ(−) µ(+) < phys |∂µ A (x)|phys >=< phys | ∂µ A (x) + ∂µ A (x) |phys >= 0 (6.30)

This is the best that we can do for the constraint. Its expectation value in any physical state will be zero. In our present example, when the source current is set to zero, and the photon ﬁeld is decomposed into † ~ ~ ~ µ ~ † ~ µ † ~ creation and annihilation operators, aµ (k) and aµ (k), ξ(k) = ik aµ (k), ξ (k) = −ik aµ (k) and the physical state condition is

µ ~ µ † ~ ~ k aµ (k)|phys >= 0, < phys| k aµ (k) = 0, ∀k (6.31)

As another example, consider the generic one-photon state, which is Z 3 µ † |ζ >= d kζ (k)aµ (k)|O >

If we require that

(+)µ ∂µ A (x)|ζ >= 0 we get the condition

µ kµ ζ (k) = 0

We can use this equation to solve for the time-component of ζ µ (k) to get 1 ζ 0(k) = ~k ·~ζ(k) |~k|

The physical one-photons states have norm,

Z Z a b 3 ∗ν ~ ~ 3 ~ ∗ ~ ab k k b ~ < ζ|ζ >= d k ζ (k)ζν (k) = d kζa (k) δ − ζ (k) ≥ 0 ~k2 which is non-negative since the quadratic form kakb Pab = δab − (6.32) ~k2 has non-negative eigenvalues (1,1,0). Thus we see that, amongst one-photon states, we have solved the problem of negative norm by restricting ourselves to physical states, whose norms are greater than equal to zero. We still have the possibility that states can have zero norm. We will have to deal with this issue separately. First, we ask the question as to whether all of the physical states have a non-negative metric. To see how this works, let us consider a generic n-photon state, 1 Z 3 3 µ1...µn † † |ζn >≡ √ d k1 ...d knζ (~k1,...,~kn)a (~k1)...a (~kn)|O > n! µ1 µn

µ ...µn where ζ 1 (~k1,...,~kn) is completely symmetric function under permutations of its labels µi,ki. This state has norm Z < | >= d3k ...d3k µ1...µn (~k ,...,~k ) ∗ (~k ,...,~k ) ζn ζn 1 nζ 1 n ζµ1...µn 1 n

µ ~ When we apply the physical state condition k aµ (k)|ζ >= 0, we learn that the state |ζ > is a physical state if and only if

µ1...µn ~ ~ k1µ1 ζ (k1,...,kn) = 0 q 0 ~ ~ 2 where k1 = |k1| = k1 and, we obtain 1 0a2...an ~ ~ a1a2...an ~ ~ ζ (k1,...,kn) = k1a1 ζ (k1,...,kn) |~k1| By similar reasoning, we can ﬁnd 1 00a3...an ~ ~ a1a2a3...an ~ ~ ζ (k1,...,kn) = k1a1 k2a2 ζ (k1,...,kn) |~k1||~k2| 1 000a4...an ~ ~ a1a2a3...an ~ ~ ζ (k1,...,kn) = k1a1 k2a2 k3a3 ζ (k1,...,kn) |~k1||~k2||~k3| 100 Chapter 6. Photons

µ ...µn and so on. These equations determine all of the time components of ζ 1 (~k1,...,~kn) in terms of a a a ...an ζ 1 2 3 (~k1,...,~kn). Using these, it is easy to prove that, if |ζn > is a physical state, Z < | > = d3k ...d3k µ1...µn (~k ,...,~k ) ∗ (~k ,...,~k ) ζ ζ 1 nζ 1 n ζµ1...µn 1 n Z 3 3 a1...an ~ ~ b1...bn∗ ~ ~ = d k1 ...d knζ (k1,...,kn)Pa1b1 ...Panbn ζ (k1,...,kn) where the intermediate matrices are the Pab deﬁned in equation (6.32). Since, as was noted there, this is a non-negative matrix in that it has non-negative eigenvalues: (1,1,0) the right-hand-side of the equation above is non-negative. This fact is sufﬁcient to prove that any physical state is has non-negative norm. In order to remove the zero norm states, we must modify our notion of physical states. We will call a physical state which has zero norm a null state. We will add the condition that a state is an equivalence class µ of states, all of which satisfy the physical state condition k aµ (k)|phys >= 0 and which are further related by the equivalence relation

|phys > ∼ |phys0 > if and only if || |phys > − |phys0 > || = 0 (6.33)

In words, two physical states are in the same equivalence class if and only if they differ by a null state. The reader might recall that an equivalence relation, ∼, imposed on a set S = {a,b,c,...} must have the properties that it is decidable: For any a,b ∈ S, either a ∼ b or a b; it is reﬂexive: a ∼ a, ∀a ∈ S; it is symmetric a ∼ b implies b ∼ a for all a,b ∈ S; and it is transitive a ∼ b and b ∼ c implies a ∼ c. It is easy to check that our correspondence in (6.33) is indeed an equivalence relation. The beautiful property that an equivalence relation has is that it sections a set into distinct equivalence classes. Any two objects are either equivalent to each other, in which case they are in the same class, or they are not equivalent to each other, in which case they are in different equivalence classes. Every object in the set is in some equivalence class. We can immediately see that our equivalence relation solves the problem of zero norm states. Consider a physical state |phys >. Since there are no states of negative norm, it must have either zero norm, or it must have positive norm. Let us assume that it has zero norm. Then, since

|| |phys > −0 || = || |phys > || = 0 the state |phys > is in the same equivalence class as the zero vector in Fock space. Conversely, if it is in the same equivalence class as the zero vector, it has zero norm. This implies that, if it is not in the equivalence class of the zero vector, it must have positive norm. Therefore, all of the other equivalence classes have positive norm. The equivalence relation has given us a positive inner product on equivalence classes. Now, we must also show that, when we calculate the matrix element of an operator, any state in the same equivalence class will give the same result. Consider the three linear combinations of the creation operators which commute with the physical state condition,

s† ~ µ ~ † ~ a (k) ≡ εs (k)aµ (k), s = 1,2, †µ ~ ~ ~ kµ a (k) , kµ = (−|k|,k)

µ (where εs (~k) = (0,~ts) where~ts are two orthogonal unit 3-vectors which are also orthogonal to~k) and therefore convert physical states to physical states and a fourth operator

˜µ † ~ ˜ ~ ~ k aµ (k) , kµ = (|k|,k) which converts a physical state to an unphysical state. Using these operators, we can see that the Fock space of all states decomposes into a direct sum of physical and un-physical states. As we have already observed, the vacuum |O > is physical. Amongst the basis for one-photon states, n o µ † ~ µ † ~ ˜µ † ~ ~ εs aµ (k)|O >, k aµ (k)|O >, k aµ (k)|O > ∀s,k the subset

n µ † ~ µ † ~ ~ o εs aµ (k)|O >, k aµ (k)|O > ∀s,k 6.2 Covariant quantization of the photon 101 span the subspace of physical states and the remaining set n o ˜µ † ~ ~ k aµ (k)|O > ∀k span the space of unphysical states. Moreover, amongst the physical one-photon states, the sub-basis

n µ † ~ ~ o εs aµ (k)|O > ∀s,k

µ † ~ ~ spans the set of all physical positive norm states and the basis vector k aµ (k)|O > ∀k are null states. Amongst all possible multi-photon states, as soon as a multi-photon state has, amongst the creation ˜µ † ~ operators which create it from the vacuum, even one of the operators k aµ (k), it is an unphysical state. If it µ † ~ has no such operators, and it is therefore a physical state, as soon as it has one of the operators k aµ (k), it is a null state. This latter fact allows us to characterize the equivalence classes of the physical n-photon states as 1 Z Z 2 3 ~ ~ s1† ~ sn† ~ 3 ~ †µ ~ |ζn >≡ √ d k1 ...d knζs ...s (k1,...,kn)a (k1)...a (kn)|O > + d kφ(k)kµ a (k)|physical state > n! 1 n where, in the last term, φ(~k) is any square-integrable function and |physical state > is any physical state. For each distinct physical state, there is an element of the equivalence class. We can easily see that the norm of these states are positive Z < | >= d3k ...d3k ∗ (~k ,...,~k ) (~k ,...,~k ) > 0 ζn ζn 1 nζs1...sn 1 n ζs1...sn 1 n

R 3 ~ †µ ~ The physical state condition and the fact that the state in d kφ(k)kµ a (k)|physical state > is physical are important in getting the above equation. We could simply describe the equivalence class by a representative, one member of the class, in this case, most conveniently, the state with n transversely polarized photons 1 Z 2 3 s1† sn† √ d k1 ...d knζs ...s (~k1,...,~kn)a (~k1)...a (~kn)|O > n! 1 n We have thus eliminated the negative and zero norm states and we have a Fock space for the transverse photons, which have two polarizations, just as we expected. However, there is still one ﬂy in the ointment. We must show that the equivalence relation is internally consistent in that the expectation value of operators of interest do not depend on the representative of an equivalence class that we have chosen. We need to show that the result would be the same for any member of the class. In fact, at ﬁrst glance, it is not the case. For example, a matrix element between two of the states Z 1 3 3 3 3 ∗ ~ ~ ~ ~ < ζm|Aµ (x)|ζn >= √ d k1 ...d kmd `1 ...d `nζ (k1,...,km)ζr ...r (`1,...,`n)· m!n! s1...sm 1 n s1 ~ sn ~ r1† ~ rm† ~ . · < O|a (k1)...a (kn)Aµ (x)a (`1)...a (`n)|O > 1 Z Z 3 3 3 ~ ∗ ~ ~ s1 ~ sn ~ †ν ~ 0 + √ d k1 ...d km d kφ(k)ζ (k1,...,km) < O|a (k1)...a (kn)Aµ (x)kν a (k)|physical state > m! s1...sm Z 3 ∗ ~ 3 3 ~ ~ ν ~ r1† ~ rm† ~ + d kφ (k)d `1 ...d `nζr1...rn (`1,...,`n) < physical state|kν a (k)Aµ (x)a (`1)...a (`n)|O > Z 3 3 0 ∗ ~ ~ 0 ν †λ ~ 0 + d kd k φ (k)φ(k ) < physical state|kν a Aµ (x)kλ a (k)|physical state > 1 Z 3 3 ∗ ~ ~ ~ ~ s1 ~ sn ~ r1† ~ rm† ~ = √ d k1 ...d `nζ (k1,...,km)ζr ...r (`1,...,`n) < O|a (k1)...a (kn)Aµ (x)a (`1)...a (`n)|O > m!n! s1...sm 1 n 1 Z 3 3 ∗ ~ ~ s1 ~ sn ~ 0 + i∂µ φ(x)√ d k1 ...d kmζ (k1,...,km) < O|a (k1)...a (kn)|physical state > m! s1...sm 1 Z ∗ 3 3 ~ ~ r1† ~ rm† ~ − i∂µ φ (x)√ d `1 ...d `nζr ...r (`1,...,`n) < physical state|a (`1)...a (`n)|O > n! 1 n

We see that the expectation value of Aµ (x) can depend on the representative in the equivalence class. The only way out of this is to restrict the operators that we can take expectation values of to those operators whose matrix elements do not depend on which representative we choose. First of all, we can project Aµ (x) 102 Chapter 6. Photons

µ onto those states which are orthogonal to kµ , either ∂µ A (x) which is zero in any physical states and the two polarizations

µ ∗ ←−! Z eikµ x ε µ (~k) ∂ ∂ ε∗µ (~k)a† (~k) = − d3x s i − i A (x), s = 1,2 s µ q ∂t ∂t µ (2π)32|~k|

Indeed we will use these projections onto the wave-functions of physical photons when we discuss the S-matrix. The other possibility is with local operators – instead of computing matrix elements of Aµ (x), we could insist on the gauge invariant combination Fµν (x). Its expectation value is also independent of the choice of representative. Thus, if we restrict ourselves to computing the expectation values of gauge invariant operators, or gauge ﬁelds projected onto the physical state wave-functions, the choice of representatives of the equivalence classes are irrelevant, and for the n-photon state we could simply choose the representative which is constructed from the physical polarizations, 1 Z 2 3 s1† sn† √ d k1 ...d knζs ...s (~k1,...,~kn)a (~k1)...a (~kn)|O > n! 1 n

6.2.2 Massive photon (Optional reading) Maxwell’s equations are the ﬁeld equations which describe a massless photon. However, we could ask, the question “what if the photon had a small mass”. To introduce a photon mass, we would modify the ﬁrst of Maxwell’s equations (6.10a) so that it has the form

µν 2 µ µ ∂ν F (x) + m A (x) = j (x) (6.34) From the point of view of physics, this is a perfectly reasonable thing to do. After all, we do not know that the photon is precisely massless.1 All we have is an experimental lower bound on the photon mass which is currently very small, but it will never be zero. Now, with a photon mass, the Maxwell equation has an explicit dependence on the vector field Aµ (x). The Maxwell equation is no longer gauge invariant. This means that the vector potential is less redundant than it was for a strictly massless photon. However, we will not be able to completely avoid constraints. The photon has spin one. A massive photon therefore must have three spin states. The four components of Aµ (x) are still too many variables to describe it. There must still be a constraint. In fact, we can easily find out what the constraint is. We operate ∂µ on the photon field µν µν equation (6.34), remember that, since F (x) is anti-symmetric, ∂µ ∂ν F (x) = 0, assume that the current is µ conserved, ∂µ j (x) = 0, and we obtain µ ∂µ A (x) = 0 (6.35) as a consequence of the field equation. This is the same equation as we imposed as a gauge condition in the previous section. As we did there, we will treat it as an equation of constraint. Once Aµ (x) satisfies this constraint, its field equation (6.34) is

(−∂ 2 + m2)Aν (x) = jν (x) (6.36)

Which contains the relativistic wave operator −∂ 2 + m2, and which will describe a massive particle where m is the mass. As with the massless photon, we will begin by quantizing the theory which has the field equation (6.36), irrespective of the constraint (6.35). For this, we need to identify the commutation relations. The identification of the commutation relations follows the same logic as it did there, as well, the solution of the free wave equation with the difference that the wave-four-vector which appears in the decomposition fo the ~ ~ field into creation and annihilation operators and the photon wave-functions is kµ = (−E(k),k). The physical state condition again requires that polarization tensors satisfy the equation (??) and the inner product Z < | > = d3k ...d3k µ1...µn (~k ,...,~k ) ∗ (~k ,...,~k ) ζ ζ 1 nζ 1 n ζµ1...µn 1 n Z 3 3 a1...an ~ ~ ˜ ˜ b1...bn∗ ~ ~ = d k1 ...d knζ (k1,...,kn)Pa1b1 ...Panbn ζ (k1,...,kn)

1The current upper bound on the photon mass, according to the Particle Data Group tables is m < 2 × 10−16eV. 6.3 Space-time symmetries of the photon 103

where, now kakb Pãb = δab − ~k2 + m2 is now a positive definite matrix with eigenvalues (1,1,m2/(m2 +~k2)). Once the physical state condition has been imposed, the Hilbert space metric is positive definite and there is no need for the further equivalence relation. The cost of this simplicity is that the photon has an extra mode, it is a massive spin-1 particle and it has three instead of two polarization. The extra mode should decouple when m → 0 whence the states which contain it become null states.

6.3 Space-time symmetries of the photon Once we can obtain the ﬁeld equations from an action principle, we can use Noether’s theorem to identify conserved currents which result from symmetries of the theory. If we set the source, jµ (x) to zero, the free Maxwell theory is invariant under space-time translations. If we consider an inﬁnitesimal translation

δµ Aλ (x) = −∂µ Aλ (x) (where we have omitted the inﬁnitesimal parameter) we ﬁnd that the Lagrangian density varies as ν δµ L = −∂ν δµ L (x) and the Noether current density is

ν ∂L ν Jµ (x) = −δµ Aλ (x) + δµ L (x) ∂(∂ν Aλ (x)) νλ ν = ∂µ Aλ (x)F (x) + δµ L (x) From this Noether current, we can deduce the energy-momentum tensor 1 Tµν (x) = F µλ (x)∂ ν A (x) − η µν F (x)Fρσ (x) λ 4 ρσ We know from Noether’s theorem that this energy-momentum tensor must be conserved, µν ∂µ T (x) = 0 and it is easy to check that this is so by using Maxwell’s equations. This energy-momentum tensor is not gauge invariant. Its physical interpretation is therefore problematic. Moreover it is not symmetric and it cannot be used to form the conserved current corresponding to Lorentz transformations. The situation with gauge invariance can be improved by realizing that, as well as a space-time transformation, we are free to do gauge transformations. The gauge field transforms under a coordinate transformation as ρ ρ δ f Aλ (x) = − f (x)∂ρ Aλ (x) − ∂λ f (x)Aρ (x) the right-hand-side of this equation is not gauge invariant. We can fix this by augmenting this coordinate transformation with a gauge transformation ˜ ρ δ f Aλ (x) = ∂λ f (x)Aρ (x) so that the total transformation is, to linear order, the sum of the two transformations, ˆ ˜ ρ δ f Aλ (x) ≡ δ f Aλ (x) + δ f Aλ (x) = − f (x)Fλρ (x) which is manifestly gauge invariant. Note that, this is an infinitesimal general coordinate transformation. If we now specialize to a transformation that we expect to be a symmetry, where f ρ (x) is a Killing vector, and if we use this symmetry and apply Noether’s theorem, we find the energy-momentum tensor 1 Θµν (x) = F µλ (x)Fν (x) − η µν F (x)Fρσ (x) (6.37) λ 4 ρσ The resulting tensor, Θµν (x), is 104 Chapter 6. Photons

1. Gauge invariant µν 2. Conserved, ∂µ Θ (x) = 0 3. Symmetric, Θµν (x) = Θνµ (x) µ 4. Traceless, Θ µ (x) = 0 These properties allow us to ﬁnd a Noether current corresponding to any of the space-time symmetries of the theory by contracting the tensor with a conformal Killing vector,

µ µν J f (x) = Θ (x) fν (x) This current satisﬁes

µ ∂µ J f (x) = 0

if fµ (x) satisﬁes the conformal Killing equation

1 ∂ f (x) + ∂ f (x) − η ∂ f λ (x) = 0 µ ν ν µ 2 µν λ In the absence of sources, classical electrodynamics has not only translation and Lorentz invariance, it is conformally invariant. The energy density is the time-time component of the energy-momentum tensor

1 Θ00(x) = ~E2(~x,t) +~B2(~x,t) 2 and the momentum density is the Poynting vector

i Θ0i = ~E(~x,t) ×~B(~x,t)

These can be used to study the energy and momentum that are stored in a classical conﬁguration of electric and magnetic ﬁelds. Using

3 Z d k h µ µ i a ikµ x a 0 ~ ~ a ~ −ikµ x ~ 0 ~ ~ ~ E (x) = q e ik a (k) − i|k|a (k) + e −ika (k) + i|k|~a(k) (2π)32|~k| 3 Z d k h µ µ i ikµ x ~ a b ~ −ikµ x ~ a †b ~ = q e i|k|P ba (k) − e i|k|P ba (k) + constraint (2π)32|~k|

we can see that the Hamiltonian and linear momentum operators are just what one would expect Z Z 3 ~ † ~ 3 ~ † H = d k|k| as (k)as(k) , P = d k k as (k)as(k)

where we have dropped terms which would vanish when we take matrix elements in physical states. These are just what we would expect for the energies and momenta of multi-photon states.

6.4 Quantum Electrodynamics Now that we have discussed the quantum ﬁeld theories of the Dirac ﬁeld and the photon, we are ready to put the two together. The way to do that is to identify the current jµ which occurs in Maxwell’s equations with the conserved Noether current for phase symmetry of the Dirac theory,

jµ (x) = −eψ¯ (x)γ µ ψ(x)

µ Also, we saw that, in the Maxwell action the current was coupled by adding the term Aµ (x)j (x) to the Lagrangian density. We could therefore add the term

µ Aµ (x)j (x) = −eψ¯ (x)A/(x)ψ(x) 6.4 Quantum Electrodynamics 105 to the sum of the Dirac and Maxwell actions to get h ←− i 1 L = −iψ¯ (x) 1 ∂/ − 1 ∂/ − ieA/(x) + m ψ(x) − F (x)F µν (x) (6.38) 2 2 4 µν This is the Lagrangian density of electrons which carry charge e interacting with massless photons. The ﬁeld theory which it describes has gauge invariance, that is, the Lagrangian density is strictly invariant under the substitution

Aµ (x) → Aµ (x) + ∂µ χ(x) (6.39) ψ(x) → eieχ(x)ψ(x) , ψ¯ (x) → ψ¯ (x)e−ieχ(x) (6.40)

Note that the phase symmetry of the Dirac field theory has been promoted to a local symmetry. Also, as a consequence, the Dirac field itself is not gauge invariant. Also, the derivative of the Dirac field now appears as the “covariant derivative” Dµ ψ(x) = [∂µ − ieAµ (x)]ψ(x). Under the gauge transformation ieχ(x) Dµ ψ(x) → e Dµ ψ(x), so that the covariant derivative Dµ ψ(x) transforms in the same way that ψ(x) does. Applying the Euler-Lagrange equation to (6.38) yields the coupled Dirac and Maxwell equations

∂/ − ieA/(x) + mψ(x) = 0 µν ν ∂µ F (x) = eψ¯ (x)γ ψ(x)

∂µ Fνλ (x) + ∂ν Fλ µ (x) + ∂λ Fµν (x) = 0 These are no longer linear equations. They are coupled nonlinear partial differential equations and it is not known how to solve them exactly, except in a few very special cases. The main approach to solving them is perturbative. In that approach, one begins by setting e = 0 and solving the free field theory of the Dirac field and the photon field, which we already know how to do. Then, we use perturbation theory to correct these solutions for the presence of the coupling which is of order e. The corrections to free field theory are governed by a dimensionless parameter which is called the fine structure constant,

e2 1 ≈ 4πhc¯ 137 which turns out to be small for quantum electrodynamics. This makes the perturbation theory exceedingly accurate. In the following chapters we will spend a large amount of time in learning how to do perturbation theory. However, as we have already discussed, to quantize the theory of the photon, we could begin by ﬁxing a gauge, that is, by using the gauge invariance of the Lagrangian density to impose a gauge condition on µ the photon ﬁeld. The condition that we will choose will be the Lorenta invariant ∂µ A (x) = 0. Then, the Lagrangian density becomes h −→ ←− i 1 L = −iψ¯ (x) 1 ∂/ − 1 ∂/ − ieA/ + m ψ(x) − ∂ A (x)∂ µ Aν (x) + total derivatives (6.41) 2 2 2 µ ν The non-zero equal-time canonical anti-commutation and commutation relations are

n † o 0 0 ψa(x),ψb (y) δ(x − y ) = δabδ(x − y) (6.42) ∂ A (x), A (y) δ(x0 − y0) = iη δ(x − y) (6.43) µ ∂y0 ν µν and the ﬁeld equations and constraint become

∂/ − ieA/ + mψ(x) = 0 − ∂ 2Aµ (x) = −eψ¯ (x)γ µ ψ(x) µ ∂µ A (x) = 0 It is this quantum ﬁeld theory, the one contained in the equations of motion (??)-(??), the constraint (??) and the equal time commutation relations (6.42)-(6.43) that we must solve. We will proceed to do this in later chapters using perturbation theory, 106 Chapter 6. Photons

6.5 Summary of this chapter The theory of the photon is governed by the Lagrangian density

1 L (x) = − F (x)F µν (x) + A (x)jµ (x) 4 µν µ

where Aµ (x) is the dynamical variable and

Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x)

The above formula solves the source-free Maxwell equations

∂µ Fνλ (x) + ∂ν Fλ µ (x) + ∂λ Fµν (x) = 0

Maxwell’s equations with sources,

µν µ ∂ν F (x) = j (x)

are obtained from Lagrangian density using the Euler-Lagrange equations. The Lagrangian density and Maxwell’s equations are invariant under the gauge transformation

Aµ (x) → A˜µ (x) = Aµ (x) + ∂µ χ(x)

A gauge transformation can be used to ﬁx the gauge

µ ∂µ A (x)

whereupon the equations of motion are

−~∂ 2Aµ (x) = jµ (x)

and the non-vanishing equal-time commutation relations are

∂ A (x), A (y) δ(x0 − y0) = η δ 4(x − y) µ ∂t ν µν ∂ ∂ A (x),A (y)δ(x0 − y0) = 0, A (x), A (y) δ(x0 − y0) = 0 µ ν ∂t µ ∂t ν

The gauge condition obeys a free wave equation

2 µ −∂ ∂µ A (x) = 0

µ which, even in the interacting theory, allows the decomposition of ∂µ A (x) into positive and negative frequency parts

µ µ (+) µ (−) ∂µ A (x) = ∂µ A (x) + ∂µ A (x)

and the physical state condition is

µ (+) ∂µ A (x) |phys >= 0

together with the equivalence relation

|phys >∼ |phys0 > if and only if || |phys > −|phys0 > || = 0

When we set the current equal to zero, Aµ (x) obeys a free wave equation whose solution is

3 Z d k h µ µ i ikµ x ~ −ikµ x † ~ Aµ (x) = e aµ (k) + e aµ (k) (2π)32|~k| 6.5 Summary of this chapter 107 where the creation and annihilation operators have the non-zero commutation relation

h ~ † ~ 0 i 3 ~ ~ 0 aµ (k),aν (k ) = ηµν δ (k − k )

The physical states of the photon are created by the operators

∗µ ~ † ~ εs (k)aµ (k), s = 1,2 †µ ~ ~ kµ a , kµ = (−|k|,k) where the two physical polarizations of the photon are represented by the vectors

0 a a ~ ∗b ~ ab kakb a ~ ∗a ~ ε = 0, kaε (k) = 0 , ε (k)ε (k) = δ − , ε (k)ε 0 (k) = δss0 s s ∑ s s 2 ∑ s s s ~k a The states

∗µ ~ † ~ εs (k)aµ (k)|O > are the the physical, transverse polarizations of the photon and

†µ ~ kµ a (k)|O > is a zero norm physical state called a null state. The equivalence class of a physical multi-photon state is given by

∗µ1 (~k )a† (~k ) ∗µ2 (~k )a† (~k ) ... ∗µn (~k )a† (~k ) | > +k a†µ (~k)|any physical state > εs1 1 µ1 1 εs2 2 µ2 2 εsn n µn n O µ In order to have a well-deﬁned equivalence relation, the observables that we take expectation values of must either be local gauge invariant operators (made from Fµν (x)) or the projections of Aµ (x) onto the wave-functions of the states with physical polarizations

ν ←−−! ν ←−−! Z e−ikν x ∂ ∂ Z eikν x ∂ ∂ d3x ε µ (~k) i − i A (x), − d3x ε∗µ (~k) i − i A (x) q s ∂x0 ∂x0 µ q s ∂x0 ∂x0 µ (2π)32|~k| (2π)32|~k|

The symmetric, traceless energy-momentum tensor is 1 Θµν (x) = F µλ (x)F ν (x) + η µν F (x)Fρσ (x) λ 4 ρσ obeys

µν λ ν ∂µ Θ (x) = −j (x)Fλ (x) and it is conserved when jµ = 0. In that case, the energy and momentum are given by Z Z 3 ~ † ~ 3 ~ † H = d k|k| as (k)as(k) , P = d k k as (k)as(k)

Free photons have a conformal symmetry. The Noether currents for space-time symmetry are

µ µν J f (x) = Θ (x) fν (x) where fν (x) is a conformal Killing vector obeying the conformal Killing equation 1 ∂ f (x) + ∂ f (x) − η ∂ f λ (x) = 0 µ ν ν µ 2 µν λ

Functional methods and IIIquantum electrodynamics

7 Functional Methods and Correlation Func- tions ...... 111 7.1 Functional derivative 7.2 Functional integral 7.3 Photon Correlation functions 7.4 Functional differentiation and integration for Fermions 7.5 Generating functionals for non-relativistic Fermions 7.6 The Dirac ﬁeld 7.7 Summary of this chapter

8 Quantum Electrodynamics ...... 135 8.1 Quantum Electrodynamics 8.2 The generating functional in perturbation theory 8.3 Wick’s Theorem 8.4 Feynman diagrams 8.5 Connected Correlations and Goldstone’s theorem 8.6 Fourier transform 8.7 Furry’s theorem 8.8 One-particle irreducible correlation functions 8.9 Some calculations 8.10 Quantum corrections of the Coulomb potential 8.11 Renormalization 8.12 Summary of this Chapter

9 Formal developments ...... 167 9.1 In-ﬁelds, the Haag expansion and the S-matrix 9.2 Spectral Representation 9.3 S-matrix and Reduction formula 9.4 More generating functionals

7. Functional Methods and Correlation Functions

We have now understood the theories of the Dirac field and the photon field when they do not interact with other fields. We have also discussed how they can interact, and we have formulated the interacting quantum field theory which is quantum electrodynamics, which is defined either by its Lagrangian density or by its field equations and equal-time commutation relations. It is now time to begin understanding how to extract physical information from this quantum field theory. In order to do that, we shall, once again, repackage the information that is contained in either the Lagrangian density or the field equations and equal-time commutation relations in a third form, the set of all correlation functions of the quantum field theory. The latter can be regarded as containing information which is equivalent to the first two. Of course, a given quantum field theory has an infinite number of correlation functions and our description of the theory in terms of them would not be useful without a compact form of presenting them. This compact form can be found in the functional methods which we shall introduce shortly. In this chapter, we will define what we mean by a correlation function. Then we will use functional methods to search for expressions which encode correlation functions of a given quantum field theory. This will involve studying generating functionals from which correlation functions can be found by taking functional derivatives and the formal expressions for the correlation functions themselves are most elegantly presented as functional integrals. For these, we shall have to introduce functional differentiation and functional integration.

7.1 Functional derivative In order to use functional methods, we must learn how to take functional derivatives and functional integrals. We will begin with functional derivatives which is the easier of the two. In order to understand the concept of functional derivative, consider, as a simple illustration, a real-number valued function φ(x) of one real variable x and a functional, Z[φ] of that function. Remember that a functional is a mathematical object into which we put a function, in this case φ(x). The output of the functional is a number, the value of the functional when it is evaluated on that function. A more precise way to deﬁne what we mean by a functional is to begin with a discrete, complete, inﬁnite set of square integrable functions,

{ f1(x), f2(x),...} which we can assume are orthonormal Z dx fm(x) fn(x) = δmn 112 Chapter 7. Functional Methods and Correlation Functions and obey a completeness relation

∑ fn(x) fn(y) = δ(x − y) n Here, we are not being speciﬁc about the dimension of the space, or whether it is space and time and we denote the coordinates by x. Any function can be expanded in the basis of square integrable functions as

φ(x) = ∑cn fn(x) n

The inﬁnite sequence of coefﬁcients cn are found by Z cn = dx fn(x)φ(x)

If φ(x) is itself a square-integrable function, it is completely specified by the coefficients in this expansion. If we know the infinite-component vector (c1,c2,...), we can reconstruct φ(x). . Alternatively, if we know the function f (x) we can, in principle, find the coefficients cn and construct the vector. If we plug the expansion of φ(x) into the functional, Z[φ], the functional becomes an ordinary function of the components of the vector (c1,c2,...).

Z[φ] = Z(c1,c2,...)

We deﬁne the functional derivative of Z[φ] by φ(x) in terms of its ordinary derivative by each of these coefﬁcients,

δZ[φ] ∂ ≡ ∑ fn(x) Z(c1,c2,...) δφ(x) n ∂cn This defines a functional derivative in terms of an infinite number of ordinary derivatives. In particular, we can use this definition of functional derivative to generalize the Liebnitz rules for derivatives to functional derivatives,

δ δ δ ((Z [φ])(Z [φ])) = Z [J] Z [φ] + Z [φ] Z [φ] δφ(x) 1 2 δφ(x) 1 2 1 δφ(x) 2 δ ∂ f (Z[φ]) δZ[φ] f (Z[φ]) = δφ(x) ∂Z[φ] δφ(x)

Another useful identity is

δφ(y) = δ(x − y) δφ(x)

Also, with K(y1,...,yn) a completely symmetric function of its arguments,

δ Z dy dy ...dynφ(y )φ(y )...φ(yn)K(y ,y ,...,yn) = δφ(x) 1 2 1 2 1 2 Z = n dy2 ...dynφ(y2)...φ(yn)K(x,y2,...,yn) (7.1)

These identities are all we will ever really need for a functional derivative. However, it is useful to note that the functional derivative is very similar to the variational derivative that we used when we studied the derivation of the Euler-Lagrange equations of motion from an action principle. We consider two functions which differ by an inﬁnitesimal amount, φ(x), and φ(x) + δφ(x). Then we expand the functional Z[φ + δφ] to linear order in δφ(x),

Z δZ[φ] Z[J + δφ] = Z[φ] + d4x δφ(x) + ... (7.2) δφ(x) 7.2 Functional integral 113

The coefficient of δφ(x) in the linear term is the functional derivative of Z[φ] by φ(x). When we are finding this coefficient, we are allowed to assume that δφ(x) has support in a compact region and goes to zero on the boundaries of the system so that we can integrate by parts as many times as is needed to get the functional into the form in equation (7.2). The similarity with the variational derivative makes it easy to demonstrate the following equations

δ R R ei dyφ(y) f (y) = i f (x) e.i dyφ(y) f (y) δφ(x) Z δ − 1 R dydy0 (y) (y,y0) (y0) 0 0 0 − 1 R dydy0 (y) (y,y0) (y0) e 2 φ ∆ φ = − dy ∆(x,y )φ(y ) e 2 φ ∆ φ δφ(x)

δ δ − 1 R dydy0 (y) (y,y0) (y0) e 2 φ ∆ φ = δφ(x1) δφ(x2) Z 0 00 0 00 0 00 − 1 R dydy0φ(y)∆(y,y0)φ(y0) = −∆(x1,x2) + dy dy ∆(x1,y )∆(x2,y )φ(y )φ(y ) e 2

These equations will be useful in the following.

7.2 Functional integral We can deﬁne a functional integral using similar ideas as those which we used for functional derivatives. In a functional integral, there will be integration measure which indicates an integral over a set of functions and there will be an integrand will be a functional. If we consider the example of a functional F[φ] of a single real function of one real variable, φ(x), we wish to deﬁne the expression Z [dφ(x)] F[φ]

We can expand the function φ(x) in inﬁnite series of square integrable functions, Z φ(x) = ∑cn fn(x) , cn = dx fn(x)φ(x) n

Then, any functional of φ(x) is equivalent to a function of the inﬁnite array of coefﬁcients, {c1,c2,...},

F[A] = F(c1,c2,...)

We deﬁne the functional integral as Z Z ∞ Z ∞ [dA(x)] F[A] ≡ dc1 dc2 ... F(c1,c2,...) (7.3) −∞ −∞

1 where each component is integrated over the entire real line −∞ < ci < ∞. Using this deﬁnition, we can ﬁnd some examples of functional integrals. For example, The functional delta function can be gotten from an integral over functional plane waves, Z R Z ∞ i dyφ(y)P(y) i∑a ca pa [dφ(x)]e = dc1dc2 ... e = ∏2πδ(pa) ≡ δ(P(x)/2π) 1

The Gaussian integral

Z 1 R Z 1 − dydxφ(y)K(y,z)φ(z) − ∑mn cmKmncn [dφ(x)] e 2 = dc1dc2 ...e 2

1In principle, one could consider definite integrals over other intervals, or even indefinite integrals. However, the only functional integrals which we will use are definite integrals over the entire real line. 114 Chapter 7. Functional Methods and Correlation Functions

where Z Z Kmn = dx dy fm(x)K(x,y) fn(y)

is a symmetric matrix. A symmetric matrix can be diagonalized by an orthogonal transformations,

t [OKO ]mn = kmδmn

where O is a real matrix, Ot is the transpose of O and it has the property that OOt = 1 = Ot O. Such a matrix is called “orthogonal”. What is more, we can change the integration variable by rotating the vector over t which we are integrating, cn → Onmcm, Since OO = 1 implies that |detO| = 1, the Jacobian for this change of variables is equal to one. The result is the Gaussian integral

∞ Z s ∞ 1 2 2π − 1 ∏ dcm exp − cmkm = ∏ = det 2 (K/2π) m=1 2 m=1 km

The integrals over each of the ci are ﬁnite only when the eigenvalues of K are all positive. We have also made use of the fact that the determinant of a matrix is equal to a product over its eigenvalues. The integral of a Gaussian is thus proportional to the inverse of the square root of the determinant of the quadratic form in the exponent, Then, we can also get a formula for the offset Gaussian Z − 1 R dydx (y)K(y,z) (z)+iR dyJ(y) (y) − 1 − 1 R dydzJ(y)K−1(y,z)J(z) [dφ(x)]e 2 φ φ φ = det 2 (K/2π)e 2 (7.4)

where K−1 is the inverse of the quadratic form K, Z Z dyK(x,y)K−1(y,z) = δ(x − z) = dyK−1(x,y)K(y,z)

We can also take functional derivatives of equation (7.5), we can ﬁnd the correlation function so of the integration variable. For example, Z − 1 R dydxφ(y)K(y,z)φ(z) − 1 −1 [dφ(x)]e 2 φ(x1)φ(x2) = det 2 (K/2π)K (x1,x2) (7.5)

and Z − 1 R dydxφ(y)K(y,z)φ(z) − 1 −1 [dφ(x)]e 2 φ(x1)...φ(xn) == det 2 (K/2π) ∑ ∏ K (xa,xb) (7.6) pairings pairs

This integration formula will be very useful in the following sections and chapters.

7.3 Photon Correlation functions A correlation function of quantum ﬁelds is an expectation value of a product of ﬁeld operators where, generally, each of the operators has a different space-time argument. We will consider such correlations in the ground state of the system, the vacuum |O >. Of most use to us will be time-ordered correlations such as

< O|T Aµ1 (x1)Aµ2 (x2)...Aµn (xn)|O > (7.7)

Time ordering is signiﬁed by the presence of the symbol T which indicates that the operators which occur immediately to the right of it are to be put in order such that their time arguments have decreasing value as we follow the operators from left to right. That is,

T Aµ1 (x1)Aµ2 (x2)...Aµn (xn) = AµP(1) (xP(1))AµP(2) (xP(2))...AµP(n) (xP(n))

0 0 0 if xP(1) > xP(2) > ... > xP(n) and where {P(1),P(2),...,P(n)} is a permutation of {1,2,...,n}. 7.3 Photon Correlation functions 115

Sometimes it is useful to use the Heavyside step function to indicate the possible relative times, for example, in the two-point correlation function 0 0 0 0 T Aµ1 (x1)Aµ2 (x2) = θ(x1 − x2)Aµ1 (x1)Aµ2 (x2) + θ(x2 − x1)Aµ2 (x2)Aµ1 (x1) The time ordered function has the distinct advantage that it is symmetric under the interchange of the arguments of the ﬁelds, for example

T Aµ1 (x1)Aµ2 (x2) = T Aµ2 (x2)Aµ1 (x1) It is symmetric because the time ordering symbol T will order the operators in the same way, irrespective of the order in which they appear in the above expression. This time-ordering operation and its properties should be familiar from time-dependent perturbation theory in quantum mechanics where it is widely used. In the following, we will use the photon field as an example. Later, we will generalize the development to include Fermions. In the previous chapter, we solved the theory of the non-interacting photon. Recall that the photon field obeys the wave equation, gauge fixing condition and equal-time commutation relations 2 µ − ∂ Aµ (x) = 0, ∂µ A (x) = 0 (7.8) h i A (x), ∂ A (y) (x0 − y0) = i (x − y) (7.9) µ ∂y0 ν δ ηµν δ h i A (x),A (y) (x0 − y0) = 0 , ∂ A (x), ∂ A (y) (x0 − y0) = 0 (7.10) µ ν δ ∂x0 µ ∂y0 ν δ The photon is expanded in creation and annihilation operators as 3 Z d k h µ µ i ikµ x ~ −ikµ x † ~ ~ ~ Aµ (x) = q .e aµ (k) + e aµ (k) , kµ = (−|k|,k) (7.11) (2π)32|~k| where the non-vanishing commutator of creation and annihilation operators is h ~ † ~ 0 i ~ ~ 0 aµ (k),aν (k ) = ηµν δ(k − k ) ~ and the vacuum was defined by aµ (k)|O >= 0 With the expansion in equation (7.11), the time-ordered two-point function is

0 −ik0(x0−y0) ~ 0 0 Z dk e θ(x0 − y0)e−i|k|(x −y ) = − lim (7.14) ε→0 2πi k0 − |~k| + iε where ε is an inﬁnitesimal positive real number. 2 Similarly

0 ik0(x0−y0) ~ 0 0 Z dk e θ(y0 − x0)ei|k|(x −y ) = − lim (7.15) ε→0 2πi k0 − |~k| + iε

2To understand this formula, we observe that, the integral over k0 in equation (7.14) is a line integral along the real −ik0(x0−y0) axis in the complex k0-plane. The integrand, − 1 e , is regarded as a function of the complex variable k0. It has 2πi k0−|~k|+iε 0 ~ 1 (−i|~k|−ε)(x0−y0) a pole at k = |k| − iε which is in the lower half of the complex plane. The residue at the pole is − 2πi e . If 0 0 0 x0 − y0 > 0, the factor in the integrand e−ik (x −y ) goes to zero on the half-circle at inﬁnity of the lower half-plane. The line integral along this half-circle is thus zero and it can be added to the integral in equation (7.14) without changing the value of the integral. The resulting integral is a line integral over a contour which encloses the entire lower half-plane including the position of the pole. The orientation of the contour is clockwise, thus the minus sign. It can then be evaluated using Cauchy’s integral formula, which evaluates it as −2πi times the residue of the pole which is enclosed by ~ 0 0 the contour. Here, the pole is at k0 = |~k| − iε and the net result is e(−i|k|−ε)(x −y ) which is the value of the left-hand-side of (7.14) when x0 − y0 > 0. On the other hand, if x0 − y0 < 0, the integrand goes to zero at the boundaries of the upper half-plane and, completing the contour there and using Cauchy’s integral formula we obtain zero. This also agrees with the left-hand-side of equation (7.14). 116 Chapter 7. Functional Methods and Correlation Functions

Putting the terms together, we have

" 0 0 0 0 0 0 # Z d4k ei~k·(~x−~y)−ik (x −y ) e−i~k·(~x−~y)+ik (x −y ) ∆µν (x,y) = −ηµν + (7.16) (2π)4 2i|~k|(k0 − |~k| + iε) 2i|~k|(k0 − |~k| + iε) or, upon combining the integrands,

µ Z d4k eikµ (x−y) (x,y) = −i ∆µν ηµν 4 µ (7.17) (2π) kµ k − iε

If we operate the wave operator on the above expression for the two-point function

µ µ Z d4k eikµ (x−y) Z d4k eikµ (x−y) − 2 (x,y) = −i (− 2) = −i k kµ ∂ ∆µν ηµν 4 ∂ µ ηµν 4 µ µ (2π) kµ k − iε (2π) kµ k − iε 4 Z d k µ = −iη eikµ (x−y) = −iη δ(x − y) µν (2π)4 µν we see that it is proportional to a Green function, that is

2 −∂ ∆µν (x,y) = −iηµν δ(x − y) (7.18)

A Green function for a wave operator must always be defined using a boundary condition. The “iε” which appears in the denominator of the integrand in (7.17) is where the information that ∆µν (x,y) must be the time ordered Green function is input. To get an alternative check on the validity of equation (7.18), we can use the field equation and commutation relations to obtain a formula for the two-point function. To begin, let us take its first time derivative,

∂ ∂ d 0 0 ∆ (x,y) =< O|T A (x)A (y)|O > + θ(x − y ) < O|A (x)A (y)|O > ∂x0 µν ∂x0 µ ν dx0 µ ν d 0 0 + θ(y − x ) < O|A (y)A (x)|O > dx0 ν µ ∂ 0 0 =< O|T A (x)A (y)|O > +δ(x − y ) < O| A (x),A (y) |O > ∂x0 µ ν µ ν ∂ =< O|T A (x)A (y)|O > ∂x0 µ ν d where we have used dx θ(x) = δ(x) and the commutation relation of the photon field (7.10). Then, consider its second time derivative, 2 2 ∂ ∂ d 0 0 ∂ ∆µν (x,y) =< O|T Aµ (x)Aν (y)|O > + θ(x − y ) < O| Aµ (x)Aν (y)|O > ∂x02 ∂x02 dx0 ∂x0 d ∂ + θ(y0 − x0) < O|A (y) A (x)|O > dx0 ν ∂x0 µ 2 ∂ 0 0 h ∂ i =< O|T Aµ (x)Aν (y)|O > +δ(x − y ) < O| 0 Aµ (x),Aν (y) |O > ∂x02 ∂x ∂ 2 =< O|T Aµ (x)Aν (y)|O > −iηµν δ(x − y) ∂x02 ~ 2 = ∇ < O|T Aµ (x)Aν (y)|O > −iηµν δ(x − y) where we have used the field equation and the equal-time commutation relation. This yields a derivation of equation (7.18). Thus, the two-point function of the free photon is proportional to a Green function for the 2 2 wave-operator −∂ . We see by operating −∂ on the expression for ∆µν (x,y) which we derived in equation (7.17) obeys equation (7.18). Of course, this derivation used the free wave equation for the photon field and it applies only to free photons. In an interacting field theory, the equation will be more elaborate and it will consequently be more difficult to solve. 7.3 Photon Correlation functions 117

We have found the 2-point correlation function for the free photon. We could proceed to find all of the higher multi-point functions in a similar way by plugging the expression for the photon in terms of creation and annihilation operators in equation (7.11) and evaluating the matrix element directly. However, we will take an alternative approach and find all of the higher order correlation functions at once by finding a generating functional.

7.3.1 Generating functional for correlation functions of free photons A generating functional is a functional of a source field which we can use to find correlation functions of quantum fields by taking functional derivatives by the source. In our example of the photon, we would like to find a functional Z[J] of a source field Jµ (x) so that the correlation functions are given by taking functional derivatives by Jµ (x) and then evaluating at Jµ = 0,

1 δ 1 δ < O|T Aµ (x1)...Aµ (xn)|O >= ... Z[J] (7.19) 1 n µ1 µn i δJ (x1) i δJ (xn) J=0 Here, the n-point correlation function of the photon ﬁeld is obtained by taking n functional derivatives of the generating functional and then putting the argument, Jµ (x) to zero. Another way to write the same information that is contained in equation (7.19) is ∞ in Z µ1 µn Z[J] =1 + ∑ dx1 ...dxnJ (x1)...J (xn) < O|T Aµ1 (x1)...Aµn (xn)|O > (7.20) n=1 n! By plugging the expression for Z[J] on the right-hand-side of the above equation (7.20) into the right-hand- side of equation (7.19) and using the rules for functional differentiation, speciﬁcally the example in equation (7.1), and then setting J = 0, we see that Z[J] is indeed the generating functional. We can write it in a shorthand notation by formally summing the series on its right-hand-side to form the expression with a time-ordered exponential

R µ Z[J] =< O|T ei dxJ (x)Aµ (x)|O > (7.21) We emphasize that this is a formal expression. It is always to be understood to be defined by its Taylor expansion in powers of the exponent. If we know the explicit form of the generating functional, then, in principle, we know all of the time- ordered correlation functions of the quantum field Aµ (x). This can be regarded as an exact solution of the quantum field theory. We will indeed be able to find and explicit formula for Z[J] and therefore an exact solution for the case of the free photon field. We will now proceed to find an explicit formula for the generating functional. Consider the functional derivative, 1 δ ∞ in 1 δ Z Z[J] = dx ...dx Jµ1 (x )...Jµn (x ) < | A (x )...A (x )| > µ ∑ µ 1 n 1 n O T µ1 1 µn n O i δJ (y) n=1 n! i δJ (y) ∞ in−1 Z µ1 µn−1 = ∑ dx1 ...dxn−1J (x1)...J (xn−1) < O|T Aµ (y) Aµ1 (x1)...Aµn−1 (xn−1)|O > n=1 (n − 1)! and operate the wave-operator on it, 1 δ − ∂ 2 Z[J] y i δJµ (y) ∞ in−1 Z µ1 µn−1 2 = − ∑ dx1 ...dxn−1J (x1)...J (xn−1)∂y < O|T Aµ (y) Aµ1 (x1)...Aµn−1 (xn−1)|O > n=1 (n − 1)!

2 Then, we use the fact that the time derivatives in the expression ∂y < O|T Aµ (y) Aµ1 (x1)...Aµn−1 (xn−1)|O > produce time delta-functions when they operate on the time-ordering theta-functions to get

2 ∂y < O|T Aµ (y) Aµ1 (x1)...Aµn−1 (xn−1)|O > n−1

= ∑ (−i)ηµµk δ(y − yk) < O|T Aµ1 (x1)...Aµk−1 (xk−1)Aµk+1 (xk+1)...Aµn−1 (xn−1)|O > k=1 118 Chapter 7. Functional Methods and Correlation Functions

So that we get

1 δ − ∂ 2 Z[J] y i δJµ (y) ∞ in−1 Z µ1 µn−1 = ∑ dx1 ...dxn−1J (x1)...J (xn−1)· n=1 (n − 1)! n−1

· ∑ (−i)ηµµk δ(y − xk) < O|T Aµ1 (x1)...Aµk−1 (xk−1)Aµk+1 (xk+1)...Aµn−1 (xn−1)|O > k=1 ∞ in−2 Z µ µ1 µn−2 = J (y) ∑ dx1 ...dxn−2J (x1)...J (xn−2) < O|T Aµ1 (x1)...Aµn−2 (xn−2)|O > n=2 (n − 2)! = Jµ (y)Z[J]

The ﬁnal result is the functional differential equation

1 δ −∂ 2 Z[J] = J (y) Z[J] (7.22) i δJµ (y) µ

If we divide each side of (7.22) by Z[J] and we note that ∆µν (x,y) is a Green function for the wave operator, we see that

δ Z lnZ[J] = − dx ∆ (y,z)Jν (z) (7.23) δJµ (y) µν from which we conclude that lnZ[J] must be a quadratic functional of J. The functional anti-derivative of equation (7.24) is

1 Z lnZ[J] = constant − dxdy Jµ (x)∆ (y,z)Jν (z) (7.24) 2 µν and, ﬁxing the constant so that Z[0] = 1, we have

1 Z Z[J] = exp − dxdyJµ (x)∆ (x,y)Jν (y) (7.25) 2 µν

This is our explicit solution for the generating functional. From it, we can deduce the general n-point correlation function by taking functional derivatives. It is clear that all correlation functions with an odd number of photon ﬁelds vanish. When there are an even number of photon ﬁelds, the result is

< O|T Aµ1 (x1)...Aµn (xn)|O >= ∑ ∏ ∆µaµb (xa,xb) (7.26) pairings pairs where the sum is over all pairings of the indices where each index is paired with another index. There are n! n n distinct pairings. The product is over all of the pairs in each pairing. Equation (7.26) is usually called 2 n 2 2 2 ! Wick’s theorem. As an example of how it works, consider the four-point correlation function

1 δ 1 δ 1 δ 1 δ < O|T Aµ (x1)Aµ (x2)Aµ (x3)Aµ (x4)|O >= Z[J] 1 2 3 4 µ1 µ2 µ3 µ4 i δJ (x1) i δJ (x2) i δJ (x3) i δJ (x4) J=0

= ∆µ1µ2 (x1,x2)∆µ3µ4 (x3,x4) + ∆µ1µ3 (x1,x3)∆µ2µ4 (x2,x4) + ∆µ1µ4 (x1,x4)∆µ3µ2 (x3,x2) (7.27) where we see that there are three distinct pairings of four indices and the result is a sum over products of two-point functions for each pair in the pairings. 7.3 Photon Correlation functions 119

7.3.2 Photon Generating functional as a functional integral In this section, we will use a representation of the generating functional which is a functional integral. The appropriate expression, for the photon, is

R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)+Aµ (y)Jµ (y)] [dAµ (x)]e 2 2 Z[J] = R 1 µ ν (7.28) R i dy[− ∂µ Aν (y)∂ A (y)] [dAµ (x)]e 2 By doing the Gaussian integral explicitly, we can conﬁrm that this equation gives back the generating functional which we obtained in equation (7.25). We have included the “iε” term so that (after one R 1 2 µ integration by parts) the quadratic terms in the exponent is − dy 2 Aµ (x)(−∂ − iε)A (x) so that, when we ﬁnd the Green function for the operator (−∂ 2 − iε), it will be the time-ordered one. Then, the change of integration variables that is needed is Z ν Aµ (x) → Aµ (x) + i dy∆µν (x,y)J (y)

2 and remembering that −∂ ∆µν (x,y) = −iηµν δ(x − y), we get Z 1 iε dy − ∂ A (y)∂ µ Aν (y) + A (y)Aν (y) + A (y)J (y) 2 µ ν 2 ν µ µ Z 1 iε i Z → dy − ∂ A (y)∂ µ Aν (y) + A (y)Aν (y) + dxdyAµ (x)∆ (x − y)Aν (y) 2 µ ν 2 ν 2 µν and the functional integral in equation (7.28) becomes

Z R 1 µ ν iε ν i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)+Aµ (y)Jµ (y)] [dAµ (x)]e 2 2

Z R 1 µ ν iε ν 1 R µ ν i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] − dxdyA (x)∆µν (x−y)A (y) = [dAµ (x)]e 2 2 e 2

and, canceling the denominator we see that the functional integral reproduces our explicit expression for the generating functional (7.25). Alternatively, we can show that the functional integral formula (7.28) obeys the functional differential equation (7.22). This, together with the input that the Green function that is used to invert the wave-operator is the time-ordered one, would also establish that it is the correct generating functional. To show that it obeys the differential equation, consider 1 δ − ∂ 2 Z[J] = i δJµ (x) R 1 µ ν iε ν µ R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)+Aµ (y)J (y)] 2 [dAµ (x)]e 2 2 (−∂ )Aµ (x) = R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] [dAµ (x)]e 2 2 R R µ R 1 µ ν iε ν i dy[Aµ (y)J (y)] iδ i dy[− 2 ∂µ Aν (y)∂ A (y)+ 2 Aν (y)A (y)] [dAµ (x)]e µ e = δA (x) R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] [dAµ (x)]e 2 2 R n R µ R 1 µ ν iε ν o iδ i dy[Aµ (y)J (y)] i dy[− 2 ∂µ Aν (y)∂ A (y)+ 2 Aν (y)A (y)] [dAµ (x)] δAµ (x) e e = R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] [dAµ (x)]e 2 2 R R µ R 1 µ ν iε ν iδ i dy[Aµ (y)J (y)] i dy[− 2 ∂µ Aν (y)∂ A (y)+ 2 Aν (y)A (y)] [dAµ (x)] µ e e − δA (x) R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] [dAµ (x)]e 2 2

R 1 µ ν iε ν µ R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)+Aµ (y)J (y)] [dAµ (x)]e 2 2 = Jµ (x) R 1 µ ν iε ν R i dy[− ∂µ Aν (y)∂ A (y)+ Aν (y)A (y)] [dAµ (x)]e 2 2

= Jµ (x) Z[J] (7.29) 120 Chapter 7. Functional Methods and Correlation Functions where we have dropped the total functional derivative term. We recover equation (7.22), the functional differential equation that the generating functional must obey. The functional integral would be a solution of this equation if it produces time ordered correlation functions. Indeed this has been mandated by inserting the “iε00 term in the appropriate place. We will stop writing the iε term explicitly from now on, but we will always assume that it is there implicitly. We conclude that equation (7.28) is the appropriate representation of the generating functional. We can restore its more gauge invariant form by unﬁxing the relativistic gauge. If we do the change of the functional integration variable

Aµ (x) → A˜µ (x) = Aµ (x) + ∂µ χ(x) the exponent in the integrand in the numerator changes as Z 1 dy − ∂ A (y)∂ µ Aν (y) + A (y)Jµ (y) → 2 µ ν µ Z 1 1 dy − ∂ A (y)∂ µ Aν (y) + ∂ 2χ(y)∂ Aµ (y) − ∂ 2χ(y)∂ 2χ(y) + A (y)Jµ (y) + ∂ χ(y)Jµ (y) 2 µ ν µ 2 µ µ

R µ Integration by parts puts the last term in this equation in the form dyχ(y)∂µ J (y) and it would vanish µ µ if J (x) obeyed the continuity equation ∂µ J (x) = 0. We recall that the consistency of the physical state conditions required that we only ever take correlation functions of gauge invariant operators or else we only ever use the general correlation function for components of the four-vector field projected onto its transverse polarizations. So far, in our treatment of correlations of the photon, we have ignored this requirement and the generating functional that we have found computes arbitrary correlations and Jµ (x) are simply four un-constrained test functions. If we will only ever compute correlation functions of gauge invariant or transverse fields, it is sufficient to take derivatives by a constrained, rather than free Jµ (x), where the µ constraint is ∂µ J (y) = 0. In that case, the last term in the equation above can be put to zero. Then, since χ(x) was introduced by a simple change of the integration variable, and the result of the functional integral cannot not depend on the field χ(x), we can do the Gaussian integral over it (and divide by an infinite constant which will cancel with a similar factor from the denominator in Z[J]). We obtain

R 1 µν ν R i dy[− Fµν (x)F (x)+Aν (y)J (y)] [dAµ (x)]e 4 Z[J] = R 1 µν R i dy[− Fµν (x)F (x)] [dAµ (x)]e 4 We have gone from the gauge fixed to the gauge invariant form of the functional integral representation of the generating functional. It is actually the gauge-fixed functional integral which will be of most use to us later when we formulate perturbation theory. One can return to a more general fixed gauge by beginning with the Faddeev-Popov substitution Z µ 2 2 1 = [dχ]δ(∂µ A (x) − ∂ χ(x) − f )|det −∂ |

Into the functional integrals in the numerator and the denominator. In this expression, when the functional delta-function is used to evaluate the integral over χ(x) it produces a Jacobian which cancels the factor |det−∂ 2|. Then the numerator in Z[J] becomes Z R 1 µν µ µ 2 2 i dy[− Fµν (y)F (y)+Aµ (y)J (y)] [dAµ (x)][dχ(d)]δ(∂µ A (x) − ∂ χ(x))|det −∂ |e 4

Upon doing a gauge transformation Aµ (x) → Aµ (x)+∂µ χ(x), assuming that the integration measure [dAµ (x)] is invariant under this transformation, and using the gauge invariance of the exponent, R 1 µν µ µ dy − 4 Fµν (y)F (y) + Aµ (y)J (y) , (remember ∂µ J = 0), the expression becomes Z Z R 1 µν µ µ 2 i dy[− Fµν (y)F (y)+Aµ (y)J (y)]  [dχ(x)] [dAµ (x)]δ(∂µ A (x) − f (x))det −∂ e 4 7.4 Functional differentiation and integration for Fermions 121 R where the infinite factor of the volume of the set of gauge transformations [dχ(x)] has been extracted. This factor cancels when we take the ratio to find the generating functional. The expression in the equation ξ R dx f (x)2 above cannot depend on the function f (x). We multiply it by e 2 and then we integrate over f (x), using the functional delta function to evaluate the integral. We do this for both the numerator and the denominator in Z[J] so that the extra factors that we produce are identical in both places and cancel in the ratio. We obtain the gauge fixed functional generating functional

R h 1 µ ν 1−ξ µ ν µ i R 2 i dy − 2 ∂µ Aν (y)∂ A (y)+ 2 ∂µ A (x)∂ν A (x)+Aµ (y)J (y) [dAµ (x)]det −∂ e Z[J] = h i (7.30) R 1 µ ν 1−ξ µ ν R 2 i dy − 2 ∂µ Aν (y)∂ A (y)+ 2 ∂µ A (x)∂ν A (x) [dAµ (x)]det(−∂ )e Using the formula for the Gaussian integral of a correlation function, and the gauge-fixed generating functional in equation (7.30), we can find the two-point function of the photon in this covariant gauge fixing

4 Z d k µ µ η 1 k k (x,y) = −i eikµ (x −y ) µν − − µ ν ∆µν 4 ν 1 ν 2 (7.31) (2π) kν k − iε ξ (kν k − iε)

and the generating functional for would use this version of ∆µν (x,y). Computations of gauge invariant correlation functions should obtain results which are independent of the parameter ξ. As a result, the parameter ξ can be chosen for our convenience. Some convenient choices are 1. ξ = 1 the “Feynman gauge”, which is used for most perturbative computations 2. ξ = ∞ the “Landau gauge”, which is sometimes convenient when demonstrating gauge invariance is important. 1 3. ξ = 3 the “Fried-Yennie gauge”, where the electron self-energy will not require an infrared regularization. Finally, we must return to the point that our various expressions for the generating functionals are µ equivalent only if ∂µ J (x) = 0. This means that the computation of correlations of gauge invariant operators should give identical results. For example, 4 Z d k µ µ k k η − k k η − k k η + k k η h | F (x)F (y)| i = −i eikµ (x −y ) µ ρ νσ ν ρ µσ µ σ νρ ν σ µρ O T µν ρσ O 4 ν (2π) kν k − iε (7.32) is independent of the gauge parameter.

7.4 Functional differentiation and integration for Fermions In order to find functional integral representations of the generating functionals for correlations of Fermions, we shall need a generalization of the definitions of functional derivative and functional integral. We recall that, even at the level of classical field theory, it was convenient to describe Fermions using anti-commuting functions. This was an essential part of the use of the Lagrangian density, the Euler-Lagrange equations and Noether’s theorem when they applied to field theories describing Fermions. It will turn out that anti- commuting functions are essential for the generating functional and the functional integral representation of a quantum field theory of Fermions. For this, we have to generalize our functional calculus to anti-commuting functions. Let us begin with some of the properties of anti-commuting numbers. Consider two numbers, η1 and η2 which have the property 2 2 η1η2 = −η2η1, η1 = 0, η2 = 0

Given these properties, functions of η1 and η2, g(η1,η2) can only have a very simple form. Their Taylor expansion in the coordinates η1 and η2 can only have two terms

g(η1,η2) = g0 + g1η1 + g2η2 + g12η1η2

where (g0,g1,g2,g12) are four real numbers (or complex numbers if we were considering complex values funcitons). The derivative of the function is deﬁned as ∂ g(η1,η2) ≡ g1 + g12η2 ∂η1 122 Chapter 7. Functional Methods and Correlation Functions

This is just what we would expect a derivative to do. Moreover

∂ g(η1,η2) ≡ g2 − g12η1 ∂η2 and ∂ ∂ ∂ ∂ g(η1,η2) = g12, g(η1,η2) = −g12 ∂η2 ∂η1 ∂η1 ∂η2 which contains all of the information that we need to know about derivatives by anti-commuting numbers, including that they anti-commute, so that

∂ ∂ ∂ ∂ = − , ∂η1 ∂η2 ∂η2 ∂η1 ∂ ∂ ∂ ∂ = 0, = 0 ∂η1 ∂η1 ∂η2 ∂η2 and ∂ ∂ η2 = −η2 , ∂η1 ∂η1 ∂ ∂ ∂ ∂ η1 = −η1 (η1g(η1,η2)) = g(η1,η2) − η1 g(η1,η2), ∂η2 ∂η2 ∂η1 ∂η1 ∂ ∂ (η2g(η1,η2)) = g(η1,η2) − η2 g(η1,η2) ∂η2 ∂η2 Besides derivatives, we need to know how to integrate. It turns out that the mathematically consistent way of deﬁning an integral is to simply say that it does exactly the same thing as a derivative, thus

Z ∂ Z ∂ Z ∂ ∂ dη1g(η1,η2) = g(η1,η2), dη2g(η1,η2) = g(η1,η2), dη2dη1g(η1,η2) = g(η1,η2) ∂η1 ∂η2 ∂η2 ∂η1 This turns out to be a consistent calculus for anti-commuting numbers. For example, the deﬁnition of integral has the consequence

Z ∂ Z ∂ dη1 g(η1,η2) = 0, dη2dη1 g(η1,η2) = 0 ∂η1 ∂η1 so that we can integrate by parts without surface terms, Z Z ∂ 0 ∂ 0 dη2dη1g(η1,η2) g (η1,η2) = − dη2dη1 g(−η1,−η2)g (η1,η2) ∂η1 ∂η1 Also, the integration is translation invariant in that Z Z dη1g(η1,η2) = dη1g(η1 + ζ,η2) where ]zeta is a third anti-commuting variable. Also, for a change of variables as Z Z dη1dη2g(η1,η2) = dη1dη2(detm)g(m11η1 + m12η2,m21η1 + m22η2)

We will need to be able to do Gaussian integrals. Of most interest will be integral of the form Z dηdη¯ eη¯ Dη = D or its generalization to higher dimensions numbers of variables. It is easy to conﬁrm that Z ∑i, j η¯iDi jη j dη2dη¯2dη1dη¯1e = D11D22 − D12D21 = detD 7.4 Functional differentiation and integration for Fermions 123

More generally with two sets of n anti-commuting variables, η1,...,ηn and η¯1,...,η¯n and the integral

Z ! dηndη¯n ...dη1dη¯1 exp ∑η¯ jD jkηk jk Z ! n(n−1)/2 = (−1) dηn ...dη1dη¯n ...dη¯1 exp ∑η¯ jD jkηk jk Z ! n(n−1)/2 = (−1) dηn ...dη1dη¯n ...dη¯1 exp ∑η¯1D1kηk + ∑ η¯ jD jkηk k j6=1,k Z n ( !) n(n−1)/2 = (−1) dηn ...dη1dη¯n ...dη¯2dη¯1 ∑ (1 + η¯1D1`η`)exp ∑ η¯ jD jkηk `=1 j6=1,k Z n ( !) n(n−1)/2 = (−1) dηn ...dη1dη¯n ...dη¯2 ∑ D1`η` exp ∑ η¯ jD jkηk `=1 j6=1,k6=` n ( Z !) `−1 (n−1)(n−2)/2 = ∑ (−1) D1` (−1) dηn ...dη`+1dη`−1 ...dη1dη¯n ...dη¯2 exp ∑ η¯ jD jkηk `=1 j6=1,k6=`

The expression above is precisely the expansion of the determinant of the matrix D using minors and co-factors3 which is normally used to calculate a determinant and we must conclude that, for any n,

Z ! dηndη¯n ...dη1dη¯1 exp ∑η¯ jD jkηk = detD (7.33) jk

Now that we have introduced derivatives and integrals which use anti-commuting numbers, we must generalize what we have learned to a discussion of functional dervatives and integrals for anti-commuting functions. As was the case with functional derivatives by ordinary, commuting functions, which could be defined in terms of ordinary derivatives by ordinary, commuting variables, functional derivatives by anti-commuting functions can be defined as derivatives by anti-commuting varanbles. Let us consider an anti-commuting function η(x). Here, x denotes the coordinates of the space on which the function is defined, which are an array of real variables. An anti-commuting function is one which obeys

η(x)η(y) = −η(y)η(x) for any arguments x and y. In particular, this implies that η2(x) = 0. We can deﬁne an anti-commuting function in terms of anti-commuting numbers. Consider a complete orthonormal set of normalized square-integrable functions, { f1(x), f2(x),...} Z dx fm(x) fn(x) = δmn , ∑ fn(x) fn(y) = δ(x − y) n Note that these are ordinary commuting, real-number-valued functions or ordinary real variables. We can expand η(x) in a series of these functions

∞ Z η(x) = ∑ ηn fn(x) , ηn = dx fn(x)η(x) n=1 where the coefﬁcients, {η1,η2,...} are anti-commuting numbers,

2 ηmηn = −ηnηm,ηn = 0, ∀m,n

A functional Z[η] of η(x), can always be written as an ordinary function of the inﬁnite set anti-commuting numbers Z(η1,η2,...) by plugging the expansion of η(x) into the Z[η].

3 The co-factor here is D1` and the minor is the the determinant of the matrix that would be obtained by removing the ﬁrst row and the `’th column of D. 124 Chapter 7. Functional Methods and Correlation Functions

Then the functional derivative by η(x) can be defined as δZ[η] ∞ ∂ ≡ ∑ fn(x) Z(η1,η2,...) δη(x) n=1 ∂ηn The derivatives by the anti-commuting variables themselves must be anti-commuting which leads to similar formulas for the functional derivatives δ δ δ δ δ 2 = − , = 0 δη(x) δη(y) δη(y) δη(x) δη(x) δ δ η(y)Z[η] = δ(x − y)Z[η] − η(y) Z[η] δη(x) δη(x) In addition, we can define the functional integral Z Z ∂ ∂ [dη(x)]Z[η] ≡ dη1dη2...Z(η1,η2,...) = ...Z(η1,η2,...) ∂η1 ∂η2 We will be particularly interested in Gaussian integrals which are of a form similar to those which we studied above. For this purpose, we introduce a second, independent anti-commuting function η¯ (x). Then, we consider the Gaussian Z Z Z ! [dη(x)dη¯ (x)]exp dydzη¯ (y)D(y,z)η(z) ≡ dη1dη¯1dη2dη¯2 ... exp ∑ η¯mDmnηn m,n where we have defined [dη(x)dη¯ (x)] ≡ dη1dη¯1dη2dη¯2 ... and Z Dmn = dydz fm(y)D(y,z) fn(z)

Then, the infinite-dimensional generalization of our Gaussian integral formula (7.33) gies a formal definition of the Gaussian functional integral Z Z [dη(x)dη¯ (x)]exp dydzη¯ (y)D(y,z)η(z) = detD where detD is defined as the determinant of the infinite matrix Dmn. Note that, unlike the case of a Bosonic Gaussian integral, the determinant appears with a positive power. We will also be particularly interested in an off-set Gaussian integral of the form Z Z Z [dη(x)dη¯ (x)]exp dydzη¯ (y)D(y,z)η(z) + dy(ξ¯(y)η(y) + η¯ (y)ξ(y))

This integral can be done by a change of variables, Z Z η(x) → η(x) − dyD−1(x,y)ξ(y), η¯ (x) → η¯ (x) − dyξ¯(y)D−1(x,y)

The functional integration measure is invariant under such a change. It also requires that the inverse of the quadratic form exists so that Z Z dyD(x,y)D−1(y,z) = δ(x − z), dyD−1(x,y)D(y,z) = δ(x − z)

Thus, plugging in this change of variables and doing the Gaussian integral yields Z Z Z [dη(x)dη¯ (x)]exp dydzη¯ (y)D(y,z)η(z) + dy(ξ¯(y)η(y) + η¯ (y)ξ(y))

Z = detD exp − dxdyξ¯(x)D−1(x,y)ξ(y) (7.34)

We shall make extensive use of the above formula in the following sections. 7.5 Generating functionals for non-relativistic Fermions 125

7.5 Generating functionals for non-relativistic Fermions While we are on the subject of propagators, let us revisit the non-relativistic theory and discuss the computation of correlation functions in that theory. The non relativistic theory is deﬁned by the ﬁeld equation and anti-commutation relations ! ∂ h¯ 2~∇2 ih¯ + + µ ψ (~x,t) = 0 (7.35) ∂t 2m σ

†ρ ρ {ψσ (~x,t),ψ (~y,t)} = δσ δ(~x −~y) †σ †ρ {ψσ (~x,t),ψρ (~y,t)} = 0 , {ψ (~x,t),ψ (~y,t)} = 0

Consider the ground state |O > and the correlation functions

ρ 0 0 0 †ρ 0 0 0 †ρ 0 0 gσ (~x,t;~x ,t ) ≡< O|θ(t −t )ψσ (~x,t)ψ (~x ,t ) − θ(t −t)ψ (~x ,t )ψσ (~x,t)|O > †ρ 0 0 ≡< O|T ψσ (~x,t)ψ (~x ,t )|O > (7.36)

Often important for applications are the retarded and advanced correlation functions

ρ 0 0 0 †ρ 0 0 gRσ (~x,t;~x ,t ) ≡< O|θ(t −t ) ψσ (~x,t), ψ (~x ,t ) |O > (7.37) ρ 0 0 0 †ρ 0 0 gAσ (~x,t;~x ,t ) ≡ − < O|θ(t −t) ψσ (~x,t),ψ (~x ,t ) |O > (7.38) We can get an explicit expression for the two-point correlation function. To do this, we can substitute the expressions for the operators in terms of creation and annihilation operators into the expression (7.36) and evaluate the resulting Green function explicitly. Recall that the expression for the quantum ﬁeld in terms of creation and annihilation operators is

3 2~ 2 3 2~ 2 Z d k i~k·~x−i h¯ k − t/h Z d k −i~k·~x−i h¯ k − t/h 2m µ ¯ ~ 2m µ ¯ † ~ ψσ (~x,t) = e ασ (k) + e β (k) ~ 3 ~ 3 σ |k|>kF (2π) 2 |k|≤kF (2π) 2 where the creation and annihilation operators anti-commute, with the non-vanishing anti-commutators being

n ~ ρ† ~ 0 o ρ ~ ~ 0 ασ (k),α (k ) = δσ δ(k − k ) (7.39) n σ ~ † ~ 0 o σ ~ ~ 0 β (k),βρ (k ) = δ ρ δ(k − k ) (7.40)

Then we obtain

Z 3 ~ 0 h¯2~k2 0 ρ 0 0 0 ρ d k ik·(~x−~x )−i 2m −µ (t−t )/h¯ gσ (~x,t :~x ,t ) = θ(t −t )δσ 3 e k>kF (2π) Z 3 ~ 0 h¯2~k2 0 0 ρ d k ik·(~x−~x )+i 2m −µ (t−t )/h¯ +θ(t −t)δσ 3 e (7.41) k

2 2 h¯ kF Here, kF is the Fermi wave-number which gives the Fermi energy, εF = 2m and, in the absence of interactions or temperature, the Fermi energy is equal to the chemical potential εF = µ. We note that the dependence on the indices σ and ρ is trivial. To streamline the notation, we deﬁne

ρ 0 0 ρ 0 gσ (~x,t :~x ,t ) ≡ δσ g(x − x )

where we are not denoting (~x,t) ≡ x. We will also denote the spacetime volume integral R dtd3x ≡ R dx. Now, as we did with the photon, we can use the contour integral representation of the theta-function to introduce a frequency integral and we obtain the expression

~ 0 0 Z dωd3k ieik·(~x−~x )−iω(t−t ) g(x − x0) = (2π)4 h¯ 2~k2 ω − 2m − µ /h¯ + iεsign(k − kF ) 126 Chapter 7. Functional Methods and Correlation Functions

We can also easily ﬁnd the retarded and advanced green functions,

Z 3 i~k·(~x−~x0)−iω(t−t0) 0 dωd k ie gR(x − x ) = (2π)4 h¯ 2~k2 ω − 2m − µ /h¯ + iε Z 3 i~k·(~x−~x0)−iω(t−t0) 0 dωd k ie gA(x − x ) = (2π)4 h¯ 2~k2 ω − 2m − µ /h¯ − iε The above expressions obey the equation for a Green function. For example, ! ∂ h¯ 2~∇2 ih¯ + + µ g(x − y) = ih¯ δ(x − y) ∂t 2m

It is easy to use the field equation and the anti-commutation relations to confirm that g(x − y) should indeed obey this equation. Now, we shall consider the following generating functional for the correlation functions of the Fermion fields, ∞ Z m Z n † 1 †ρ †σ Z[η,η ] ≡ ∑ < O|T i dx η (x)ψρ (x) i dy ψ (y)ησ (y) |O > (7.42) m,n=0 m!n! where the time ordering means that the individual terms in the product are ordered according to the values of their time arguments. As with the photon field which we studied earlier, the ordering is such that the times are always decreasing as we read the operator product from left to right. Under the time-ordering symbol, the source functions, the functional derivatives by the source functions, and the fields anti-commute with each other. We have defined the product as above with the source functions paired with the operators. We can make a formal summation of the series to write the functional in the more compact notation,

R †ρ †σ Z[η,η†] =< O|T ei dx [η (x)ψρ (x)+ψ (x)ησ (x)]|O > (7.43) As in the case of the photon, it is easy to derive a simple functional differential equation that this generating functional should satisfy. Also, as with the photon, the solution of that equation is the exponential of a quadratic in the sources,

Z † σ† Z[η,η ] = exp − dxdy η (x) g(x,y) ησ (y) (7.44)

Functional derivatives of Z[η,η†] by the anti-commuting functions η(x) and η†(x) yield the time-ordered correlation functions of non-relativistic Fermions

†ρ1 †ρk < O|ψσ1 (x1)...ψσk (xk)ψ (y1)...ψ (yk)|O >

δ δ δ δ † = ...... Z[η,η ] †σ1 †σk δη (x1) δη (xk) δηρ1 (x1) δηρk (xk) η=0=η†

These correlation functions vanish unless the number of ψ’s and ψ†’s are equal. As we shall learn later on, 1 1 this is a consequence of symmetry. We have then also anticipated that the factors of i and − i that would accompany the individual functional derivatives also cancel. We would now like to present the generating functional as a functional integral. The functional integral is

i R †σ †σ R †ρ S+i dx[η (x)ψσ (x)+ψ (x)ησ (x)] [dψρ (x)dψ (x)] e h¯ Z[η,η†] = (7.45) i R †ρ S [dψρ (x)dψ (x)] e h¯ where the action is Z S = dxL (x) 7.5 Generating functionals for non-relativistic Fermions 127

and the Lagrangian density is ←−! ih¯ ∂ ih¯ ∂ h¯ 2 L (x) = ψ†σ (x) − ψ (x) − ~∇ψ†σ (x) ·~∇ψ (x) + µψ†σ (x)ψ (x) 2 ∂t 2 ∂t σ 2m σ σ We are now denoting (~x,t) ≡ x. Inside the functional integration, and the action and Lagrangian density †σ written above, ψσ (x) and ψ (x) are anti-commuting functions. Even though we use the same notation for them as for the quantum fields, they are not quantum fields, when they appear inside a functional integral they are just anti-commuting functions. The integration is over the space of all such functions. We need to know very little about the precise definition of this integration in order to show that equation (7.45) is equivalent to (7.44). All that we need to know is that the integration measure is invariant under translations in function space. That allows us to do the transformation of variables in the integral in the numerator, 1 Z 1 Z ψ (x) = ψ˜ (x) − d4yg(x,y)η (y) , ψ†σ (x) = ψ˜ †σ (x) − d4y η†σ (y)g(y,x) σ σ i σ i 1 where we use the fact that i g(x − y) is the Green function for the non-relativistic wave operator. Then, we also need the property that integrations by parts within the exponent are allowed and they do not generate any residual surface terms. Our transformation of variables completes the square in the exponent. The functional integral in the numerator becomes Z i R †σ †σ †ρ S+i dx[η (x)ψσ (x)+h¯ψ (x)ησ (x)] [dψρ (x)dψ (x)] e h¯ Z i R †σ †ρ S − dxdy η (x)g(x,y)ησ (y) = [dψρ (x)dψ (x)] e h¯ e

and the second factor in the integrand, which does not depend on the integration variables can be factored out. It is identical to the functional in equation (7.44). The integral is then identical to the one in the denominator of equation (7.45) and it cancels. Therefore (7.45) and (7.44) are identical. There are a few more interesting functional integral formulae which are implied by the above development. If we use the functional integral for a generating functional, taking functional derivatives of it by the sources and then setting the sources equal to zero yields the following integral for time-ordered correlation functions:

†σ1 †σs < O|T ψρ1 (x1)...ψρr (xr)ψ (y1)...ψ (ys)|O >

i R † S †σ †σs [dψ(x)dψ ](x)e h¯ ψρ (x1)...ψρ (xr)ψ 1 (y1)...ψ (ys) = 1 r (7.46) i R †ρ S [dψρ (x)dψ (x)] e h¯ This has the implication that all correlation functions can be found by simply integrating the classical fields with the appropriate functional integration measure. Here, for the case of free field theory, this is a Gaussian integral (the exponent is quadratic in the fields) and the factorization into two-point functions is also a property of a Gaussian functional integral.

†σ1 †σs deg σ j < O|T ψρ1 (x1)...ψρr (xr)ψ (y1)...ψ (ys)|O >= ∑ (−1) ∏ δρi g(xi − y j) pairings pairs (ρixi,σiy j) (7.47) where deg is the number of neighbours which need to be interchanged in order to put the ﬁelds in the pairing adjacent to each other.

7.5.1 Interacting non-relativistic Fermions Now, what about the interacting non-relativistic ﬁeld theory, where the Lagrangian density has an interaction term,

←−! 2 †σ ih¯ ∂ ih¯ ∂ h¯ †σ †σ λ †σ 2 L (x) = ψ (x) − ψ (x) − ~∇ψ (x) ·~∇ψ (x) + µψ (x)ψ (x) − ψ (x)ψ (x) 2 ∂t 2 ∂t σ 2m σ σ 2 σ 128 Chapter 7. Functional Methods and Correlation Functions

The action is the space-time integral of the Lagrangian density, S = R dxL (x), and the equation of motion is ! δ ∂ h¯ 2~∇2 S = ih¯ + + µ ψ (x) − λψ†ρ (x)ψ (x)ψ (x) = 0 δψ†σ (x) ∂t 2m σ ρ σ

where we have observed that the ﬁeld equation can be derived by setting a functional derivative of the action to zero. The natural conjecture is that the correlation functions of this interacting theory is described by the functional integral

i R †σ †σ R †ρ S+i dx[η (x)ψσ (x)+ψ (x)ησ (x)] [dψρ (x)dψ (x)] e h¯ Z[η,η†] = (7.48) i R †ρ S [dψρ (x)dψ (x)] e h¯

where we simply use the action of the interacting field theory. This guess turns out to be correct. The functional integral in equation (7.48) represents the full interacting quantum field theory when we insert the action which is the space-time integral of L (x) for the interacting field theory. In the tutorial we shall basically give a proof of this statement by finding a functional differential equation for the interacting theory and showing that the functional integral is a solution of that equation.

7.6 The Dirac field In this section, we shall include Dirac fields in correlations functions. Let us begin with the non-interacting Dirac fields which we denote by ψ(x) and ψ¯ (x) and we shall examine the quantitites

< O|T ψa1 (x1)...ψan (xn)ψ¯b1 (y1)...ψ¯bn (yn)|O > with the same anti-symmetric time ordering that we used for non-relativistic Fermions. The non-interacting ﬁelds obey the Dirac equation h ←− i ∂/ + mψ(x) = 0 , ψ¯ (x) − ∂/ + m = 0

and the equal time anti-commutation relations

† 0 0 {ψa(x),ψb (y)}δ(x − y ) = δabδ(~x −~y) 0 0 † † 0 0 {ψa(x),ψb(y)}δ(x − y ) = 0 , {ψa (x),ψb (y)}δ(x − y ) = 0 The generating functional for correlation functions of Dirac ﬁelds is Z Z[η,η¯ ] =< O|T exp i dx [η¯ (x)ψ(x) + ψ¯ (x)η(x)] |O > (7.49)

and correlation functions are obtained from it by functional derivatives by the anti-commuting functions η¯ (x) and η(x),

δ δ δ δ < O|T ψa (x )...ψa (xn)ψ¯ (y )...ψ¯ (yn)|O > = ...... Z[η,η¯ ] 1 1 n b1 1 bn ¯ ¯ δηa1 (x1) δηan (xn) δηb1 (y1) δηbn (yn) η=0=η¯ In the following we will get an explicit form for this generating functional and we will discuss its properties.

7.6.1 Two-point function for the Dirac ﬁeld The time-ordered two-point correlation function of the Dirac theory is

0 0 0 0 gab(~x,t;~x ,t ) ≡< O|T ψa(~x,t)ψ¯b(~x ,t )|O > 0 0 0 0 0 0 =< O|θ(t −t )ψa(~x,t)ψ¯b(~x ,t ) − θ(t −t)ψ¯b(~x ,t )ψa(~x,t)|O > (7.50)

We shall use the same notation, g(x,y) for the two-point function in the Dirac theory as we used for non- relativistic Fermions. This should not cause confusion as we shall never discuss both theories in the same 7.6 The Dirac ﬁeld 129 context. The two-point function is proportional to a Green function for the Dirac wave operator. To see this, we operator the Dirac wave operator on the time-ordered function to obtain

0 0 0 0 [∂/ + m]g(x,y) = [∂/ + m]ac < O|θ(x − y )ψc(x)ψ¯b(y) − θ(y − x )ψ¯b(y)ψc(x)|O > 0 0 = γ δ(x − y0) < O|{ψa(x),ψ¯b(y)}|O > + < O|T [∂/ + m]acψc(x),ψ¯b(y)|O > 0 2 = (γ )abδ(x − y) = −δabδ(x − y) where we have used the ﬁeld equation, the anti-commutation relation and the observation that the derivative d 0 of a Heavyside function is a delta function dx θ(x) = δ(x) We see that g(x,y ) is proportional to a Green function for the Dirac wave equation

[∂/ + m]g(x,y) = −δ(x − y) (7.51) where we have suppressed the Dirac spinor indices. In order to ﬁnd the Green function explicitly, we can plug our solution of the Dirac equation into the equation for the Green function. The explicit solution was

3 Z d k h µ n o µ n oi ikµ x ~ ~ −ikµ x † ~ † ~ ψ(x) = 3 e ψ++a+(k) + ψ+−a−(k) + e ψ−+b+(k) + ψ−−b−(k) (7.52) (2π) 2 where q ~ ~ 2 2 kµ = (−E,k) , E = k + m where the ﬁrst ± in the subscript on the wave-functions denote positive and negative energy solutions whereas the second ± denote positive and negative helicity. We found these spinors explicitly when we discussed the non-interacting Dirac theory,  q   q  i − |~k|/Eu i + |~k|/Eu ~ 1 + ~ 1 1 − ψ++(k) =  q  ψ+−(k) = √  q  1 + |~k|/Eu+ 2 1 − |~k|/Eu−  q   q  i − |~k|/Eu i + |~k|/Eu ~ 1 1 + ~ 1 1 − ψ−+(k) = √  q  ψ−−(k) = √  q  2 − 1 + |~k|/Eu+ 2 − 1 − |~k|/Eu−

Then, we can see explicitly that

i/k − m ψ ψ¯ + ψ ψ¯ = ++ ++ +− +− −2iE i/k + m ψ ψ¯ + ψ ψ¯ = −+ −+ −− −− −2iE and

3 Z d k µ µ i/k − m < O|ψ(x)ψ¯ (y)|O >= eikµ (x −y ) (2π)3 −2iE 3 Z d k µ µ i/k + m < O|ψ¯ (y)ψ(x)|O >= e−ikµ (x −y ) (2π)3 −2iE

We can use equation (7.14) to represent the Heavyside step function as a contour integral,

4 0 Z d k µ iγ E − i~γ ·~k + m θ(x0 − y0) < O|ψ(x)ψ¯ (y)|O >= eikµ ·(x−y) (2π)4 2E(k0 + E − iε) 4 0 Z d k µ iγ E − i~γ ·~k + m θ(y0 − x0) < O|ψ¯ (y)ψ(x)|O >= eikµ (x−y) (2π)4 2E(−k0 + E − iε) 130 Chapter 7. Functional Methods and Correlation Functions

so that

4 Z d k µ i/k − m g(x,y) =< | (x) ¯ (y)| >= eikµ (x−y) O T ψ ψ O 4 µ 2 (7.53) (2π) kµ k + m − iε It can easily be checked that this expression satisﬁes the equation for a Green function for the Dirac operator. In the discussion above, we have computed < O|T ψ(x)ψ¯ (y)|O > for the Dirac ﬁeld theory. Here, we also note that the other possible two-point correlation functions < O|T ψ(x)ψ(y)|O > and < O|T ψ¯ (x)ψ¯ (y)|O > must both vanish. This is due to the phase symmetry of the theory. In fact, phase symmetry implies that any correlation function must vanish unless it has the same number of ψ’s and ψ¯ ’s. We will see this explicitly when we derive the generating functional in the next section.

7.6.2 Generating functional for the Dirac field Now that we have found the two-point correlation function for the Dirac field theory in the previous section, we are ready to find the full generating functional for all of the time-ordered correlation functions. Z Z[η,η¯ ] =< O|T exp i dx [η¯ (x)ψ(x) + ψ¯ (x)η(x)] |O > (7.54)

As we did for the photon, we can easily derive a functional differential equation that th generating functional in equation (7.54) must satisfy. Then, we can solve the equation with the boundary condition that Z[η = 0,η¯ = 0] = 1 (7.55) The functional differential equation for Z[η,η¯ ] is obtained by operating the Dirac wave operator on the functional derivative, h−→ i δ ∂/ + m Z[η,η¯ ] = δη¯ (x)

R ∞ 0 R 3 R x0 0 R 3 0 i 0 dy d y [η¯ (y)ψ(y)+ψ¯ (y)η(y)] i dy d y [η¯ (y)ψ(y)+ψ¯ (y)η(y)] γ ∂0 < O|T e x iψ(x)e −∞ |O > 0 0 iR ∞ dy0 R d3y [η¯ (y)ψ(y)+ψ¯ (y)η(y)] iR x dy0 R d3y [η¯ (y)ψ(y)+ψ¯ (y)η(y)] − γ < O|T e x0 iψ˙ (x)e −∞ |O > Z 0 iR ∞ dy0 R d3y [η¯ (y)ψ(y)+ψ¯ (y)η(y)] 3 = γ < O|T e x0 ψ(x), d y[η¯ (y)ψ(y) + ψ¯ (y)η(y)] · y0=x0 0 R x 0 R 3 ¯ · ei −∞ dy d y [η(y)ψ(y)+ψ¯ (y)η(y)]|O > = η(x)Z[η,η¯ ] (7.56) where we have used the equation of motion and the equal-time anti-commutation relations of the free Dirac ﬁeld. The upshot of the above is the functional differential equation

h−→ i δ ∂/ + m Z[η,η¯ ] = η(x)Z[η,η¯ ] (7.57) δη¯ (x)

This equation, together with its complex conjugate and the boundary condition (7.55) has the solution

Z Z[η,η¯ ] = exp − dxdy η¯ (x)g(x,y)η(y) (7.58)

which is our explicit solution for the generating functional for correlation functions of the Dirac ﬁeld theory. It is easy to use equation (7.51) to check that this generating functional indeed obeys the equation (7.57). Also, we now see explicitly that a non-vanishing multi-point correlation function must have the same number of ψ’s as ψ¯ ’s, since taking n functional derivatives by η¯ will generate the exponential in (7.58) times an n’th order monomial in η. It is only n further functional derivatives by η which will give a non-zero result when η and η¯ are both set equal to zero. We also see that taking repeated functional derivatives of equation (7.58) will an equation for a multi-point correlation function < | (x )... (x ) ¯ (y )... ¯ (y )| >= (− )#perm g (x − y ) O T ψa1 1 ψan n ψb1 1 ψbn n O ∑ 1 ∏ aib j i j (7.59) pairings pairs 7.7 Summary of this chapter 131

where the pairings are of x’s with y’s and the integer #perm is the number of permutations of neighbours which is required to put the operators in the product in the order of the pairings. For example, the four-point function is given by

< O|T ψa1 (x1)ψa2 (x2)ψ¯b1 (y1)ψ¯b2 (y2)|O >= − ga1b1 (x1 − y1)ga2b2 (x2 − y2)

+ ga1b2 (x1 − y2)ga2b1 (x2 − y1) (7.60) The relative minus sign of the two terms comes from the difference of the number of interchanges of neighbours that would be needed to put the operators in the appropriate order in each term.

7.6.3 Functional integral for the Dirac field We have developed a functional integral formulation of the non-relativistic quantum field theory. It is straightforward to generalize this development to the Dirac field. The result uses the Dirac action Z h −→ ←− i 4 1 / 1 / S = −i d xψ¯ (x) 2 ∂ − 2 ∂ + m ψ(x) (7.61)

The development is identical to that of the non-relativistic theory with ψ† and η† replaced by ψ¯ and η¯ and the non-relativistic action is replaced by the Dirac action. The functional integral representation of the generating function for time-ordered correlation functions is

R R 4 [dψdψ¯ ]eiS[ψ,ψ¯ ]+i d x[η¯ (x)ψ(x)+ψ¯ (x)η(x)] Z[η,η¯ ] = R (7.62) [dψdψ†]eiS[ψ,ψ¯ ] or Z Z[η,η¯ ] = exp − d4x η¯ (x)g(x,y)η(y) (7.63)

We can find this by translating the variables in the functional integral, as we did for the non-relativistic field theory. Now, g(x,y) is a matrix. For brevity of notation, we have suppressed the indices. In this free field theory, all correlation functions factorize into products of two-point correlation functions. As in the non-relativistic case, phase symmetry implies that a correlation function will vanish unless it contains the same number of ψ’s and ψ¯ ’s. The general formula is

< O|T ψa1 (x1)...ψar (xr)ψ¯b1 (y1)...ψ¯br (yr)|O > δ δ δ δ = ...... Z[η,η¯ ] ¯ ¯ δηa1 (x1) δηar (xr) δηb1 (y1) δηbr (ys) η=η¯ =0

R iS[ψ,ψ¯ ] [dψ(x)dψ¯ (x)]e ψa (x1)...ψar (xr)ψ¯b (y1)...ψ¯b (yr) = R 1 1 r [dψ(x)dψ¯ (x)]eiS[ψ,ψ¯ ] r = (− )# g (x − y ) ∑ 1 ∏ aibP(i) i P(i) (7.64) P i=1 In the last formula, the sum is over permutations and # is the number of exchanges of neighbours needed to put the operators in each pair next to each other.

7.7 Summary of this chapter

Consider a complete set of real, square integrable functions { fn(x)} such that Z ∑ fn(x) fn(y) = δ(x − y) , dx fm(x) fn(x) = δmn n Any function φ(x) can be expanded in this set

φ(x) = ∑cn fn(x) n 132 Chapter 7. Functional Methods and Correlation Functions

Here, φ(x) is a real field, fn(x) are real-valued functions and cn are real coefficients. A functional F[φ] maps a function φ(x) onto a number F[φ]. Using the expansion of φ(x), we can present F as a function of the infinite set of numbers cn, F(c1,c2,...).

F[φ] = F(c1,c2,...)

Then, the functional derivative is deﬁned as δF[φ] ∂ ≡ ∑ fn(x) F(c1,c2,...) δφ(x) n ∂cn The functional integral is deﬁned as Z Z [dφ(x)]F[φ] ≡ dc1dc2 ...F(c1,c2,...)

where each cn is integrated on the real line from −∞ to ∞. For anti-commuting functions ψ(x), we consider the expansion

ψ(x) = ∑ψn fn(x) (7.65) n where ψn are anti-commuting numbers, ∂ ∂ ∂ ∂ ψmψn = −ψnψm , = − ∂ψn ∂ψm ∂ψm ∂ψn ∂ ∂ [ψmF(ψ1,ψ2,...)] = δmnF(ψ1,ψ2,...) − ψm F(ψ1,ψ2,...) ∂ψn ∂ψn The functional derivative is deﬁned by as

δF[ψ] ∂ ≡ ∑ fn(x) F(ψ1,ψ2,...) δψ(x) n ∂ψn The functional integral is deﬁned as Z Z [dψ(x)]F[ψ] ≡ dψ1dψ2 ...F(ψ1,ψ2,...) where the integral Z ∂ dψnF[ψ1,ψ2,...] ≡ F(ψ1,ψ2,...) ∂ψn for each variable ψn. For real anti-commuting variables, the coefﬁcients ψn in (7.65) are real in that each ψn is an independent anti-commuting number obeying the above rules. We will generally be interested in complex Fermions. In that case, for each value of the index n there are two anti-commuting number, ψn and ψ¯n which we can consider independent variables, so that each of them have all of the properties that we have described for one of them above and, in addition, ∂ ∂ ∂ ∂ ψmψ¯n = −ψ¯nψm , = − ∂ψn ∂ψ¯m ∂ψ¯m ∂ψn ∂ ∂ ∂ ∂ ψ¯m = −ψ¯m , ψm = −ψm ∂ψn ∂ψn ∂ψ¯n ∂ψ¯n Z ∂ dψnF(ψ1,ψ2,...,ψ¯1,ψ¯2,...) ≡ F(ψ1,ψ2,...,ψ¯1,ψ¯2,...) ∂ψn Z ∂ dψ¯nF(ψ1,ψ2,...,ψ¯1,ψ¯2,...) ≡ F(ψ1,ψ2,...,ψ¯1,ψ¯2,...) ∂ψ¯n 7.7 Summary of this chapter 133 and ψ(x) = ∑ψn fn(x) , ψ¯ (x) = ∑ψ¯n fn(x) n n The Gaussian integrals for real Bosons and complex Fermions are Z i Z Z [dφ(x)]exp dydxφ(x)D(x,y)φ(y) + i dwJ(x)φ(w) 2 Z − 1 i = [detD] 2 exp − dydzJ(y)D−1(y,z)J(z) 2 Z Z Z [dψ(x)dψ¯ (x)]exp i dydxψ¯ (x)D(x,y)ψ(y) + i dw(ψ¯ (x)ψ(w) + ψ¯ (w)ψ(w))

 Z = [detD] exp −i dydzψ¯ (y)D−1(y,z)ψ(z) respectively, where we have absorbed undetermined constants into the deﬁnition of the measure.

The generating functional for the free photon and the Dirac ﬁeld are deﬁned by the equations Z 4 µ Z[J] = hO|T exp i d xJ (x)Aµ (x) |Oi Z Z[η,η¯ ] = hO|T exp i d4x[η¯ (x)ψ(x) + ψ¯ (x)η(x)] |Oi so that

1 δ δ < O|T Aµ1 (x1)...Aµn (xn)|O >= n ... Z[J] i δJµ1 (x1) δJµn (xn) J=0

< O|T ψa1 (y1)...ψar (yr)ψ¯b1 (z1)...ψ¯bs (zs)|O > 1 δ δ δ δ = ...... Z[η,η¯ ] r−s ¯ ¯ i δηa1 (y1) δηar (yr) δηb1 (z1) δηbs (zs) η=¯ 0=η (which vanishes unless r = s). The generating functionals have covariant functional integral representations

R R µ [dAµ (x)] exp iS[Aν ] + i dyAµ (y)J (y) Z[J] = R [dAµ (x)] exp(iS[Aν ]) R [dψ(x)dψ¯ (x)] exp(iS[ψ,ψ¯ ] + iR dy[ηψ¯ (y) + ψη¯ (y)]) Z[η,η¯ ] = R [dψ(x)dψ¯ (x)] exp(iS[ψ,ψ¯ ]) where the actions are Z 1 ξ S[A ] = d4x − F (x)F µν (x) − (∂ Aµ )2 ν 4 µν 2 µ Z S[ψ,ψ¯ ] = d4x−iψ¯ (x)(∂/ + m)ψ(x)

They also have the explicit representations

1 Z Z[J] = exp − dydzJµ (y)∆ (y,z)Jν (z) 2 µν Z Z[η,η¯ ] = exp − dydzη¯ (y)g(y,z)η(z) 134 Chapter 7. Functional Methods and Correlation Functions where the time-ordered Green functions obey

2 µν µ ν µ −∂ η + (1 − ξ)∂ ∂ ∆νρ (x,y) = −iδ ρ δ(x − y)

(∂/ + m)abgbc(x,y) = −δacδ(x − y) and they have the explicit forms

k k 4 µ ν Z d k ηµν − (1 − 1/ξ) 2 ∆ (x,y) = hO|T A (x)A (y)|Oi = −i eik(x−y) k −iε µν µ ν (2π)4 k2 − iε Z 4 d k ik(x−y) [−i/k + m]ab g (x,y) = hO|T ψa(x)ψ¯ (y)|Oi = − e ab b (2π)4 k2 + m2 − iε 8. Quantum Electrodynamics

8.1 Quantum Electrodynamics The quantum field theory which is composed of the coupled Maxwell and Dirac theories is called quantum electrodynamics. The Dirac field is usually associated with the electron and positron but of course, they could be any charged Dirac spinor field, so we will generally refer to it as the “Dirac field”. Classical electrodynamics coupled to the relativistic Dirac field theory has the Lagrangian density Z S[A,ψ,ψ¯ ] = d4xL (x) (8.1) h ←− i 1 L (x) = −iψ¯ (x) 1 ∂/ − 1 ∂/ − ieA/(x) + m ψ(x) − F (x)F µν (x) (8.2) 2 2 4 µν The field equations that follow from the application of the Euler-Lagrange equations to this Lagrangian density are those of the coupled Maxwell-Dirac theory

µν ν ∂µ F (x) = eψ¯ (x)γ ψ(x) (8.3)

Fµν (x) = ∂µ Aν (x) − ∂ν Aµ (x) (8.4) ∂/ − ieA/(x) + mψ(x) = 0 (8.5)

These equations deﬁne a ﬁeld theory which is invariant under the gauge transformation

ieχ(x) −ieχ(x) Aµ (x) → Aµ (x) + ∂µ χ(x), ψ(x) → e ψ(x), ψ¯ (x) → ψ¯ (x)e ψ¯ (x)

As in the case of the free photon ﬁeld which we studied in earlier chapters, we shall need to ﬁx a gauge. It will be most convenient to use the same relativistic gauge condition

µ ∂µ A (x) = 0

This constraint is used to simplify the Maxwell equation and the Lagrangian so that we can quantize the theory. Then, it is imposed as a physical state condition

(+)µ ∂µ A (x)|phys >= 0 (8.6)

together with an equivalence relation which eliminates null states. Even in the interacting theory, we can consistently impose this physical state condition. This is due to the fact that, upon taking a four-divergence 136 Chapter 8. Quantum Electrodynamics of equation (8.3) and noting that when the Dirac equation (8.5) is satisﬁed, the current (being a Noether current for phase symmetry) is conserved, the gauge condition obeys the free wave equation, 2 µ −∂ ∂µ A (x) = 0 Since it obeys this wave equation, it can be uniquely decomposed into positive and negative frequency parts and then we can impose the physical state condition above using the positive frequency part. We will assume that the vacuum state |O > is a unit normalized physical state, that is,

(+)µ ∂µ A (x)|O >= 0, < O|O >= 1 (8.7) As well as the physical state condition, we shall assume that the physical states include states with zero norm and that an equivalence relation, identical to the one that we introduced for the free photon, must be imposed in the interacting theory too and, as with the free photon, it is made internally consistent (and can hereafter ignored) if we impose the proviso that we must limit our computations of correlation functions to those of gauge invariant operators or a gauge invariant quantities which project physical polarizations from the photons or wave-functions of the Dirac fields. Once we have fixed the gauge, the field equations become

2 − ∂ Aµ (x) = −eψ¯ (x)γµ ψ(x) (8.8) ∂/ − ieA/(x) + mψ(x) = 0 (8.9) These equations can be obtained from the gauge-ﬁxed action with the Lagrangian density Z Sgf[A,ψ,ψ¯ ] = dxL (x) (8.10)

1 µ ν µ L (x) = −iψ¯ (x) ∂/ + m ψ(x) − ∂ A (x)∂ A (x) − eA (x)ψ¯ (x)γ ψ(x) (8.11) 2 µ ν µ We could also have gotten this Lagrangian density, up to some total derivative terms, by imposing the gauge condition in the electrodynamics Lagrangian density (8.2). The non-vanishing equal-time (anti-)commutation relations, deduced from the time derivative terms in the Lagrangian density (8.11) are

† 0 0 4 {ψa(x),ψb (y)}δ(x − y ) = δabδ (x − y) (8.12) h i A (x), ∂ A (y) (x0 − y0) = i 4(x − y) (8.13) µ ∂y0 ν δ ηµν δ The equations of motion (8.8) and (8.9), the commutation relations (8.12) and (8.13), and the physical state condition (8.6) define a quantum field theory. Important for this definition is the existence of a vacuum |O > which is a physical state as in (8.7). Aside from being physical, the vacuum is also defined as being an eigenstate of the Hamiltonian of the quantum field theory and as being the lowest energy physical state of the theory. In the following, we shall study this theory in detail using functional methods. A generating functional for time-ordered correlation functions is given by Z µ Z[J,η,η¯ ] =< O|T exp i dx Aµ (x)J (x) + η¯ (x)ψ(x) + ψ¯ (x)η(x) |O > (8.14)

It is straightforward to use the ﬁeld equations and the commutation and anti-commutation relations for the ﬁeld operators to show that this generating functional must obey the functional differential equations 1 δ 1 δ ∂/ + m − ieγ µ Z[J,η,η¯ ] = η(x)Z[J,η,η¯ ] (8.15) i δJµ (x) i δη¯ (x) 1 δ δ δ −∂ 2 + e γ Z[J,η,η¯ ] = J (x)Z[J,η,η¯ ] (8.16) i δJµ (x) δη(x) µ δη¯ (x) µ These equations for the generating functional are solved by the functional integral

R R µ [dAµ (x)dψ(x)dψ¯ (x)]exp iSgf[A,ψ,ψ¯ ] + i dx Aµ (x)J (x) + η¯ (x)ψ(x) + ψ¯ (x)η(x) Z[J,η,η¯ ] = R [dAµ (x)dψ(x)dψ¯ (x)]exp iSgf[A,ψ,ψ¯ ] (8.17) 8.1 Quantum Electrodynamics 137 where the gauge-fixed action Sgf[A,ψ,ψ¯ ] is given in equation (8.10). Now, we could to do what we did for the free photon. We could unfix the relativistic gauge to find a gauge invariant functional integral. We could also re-fix the gauge with a gauge parameter. The equations above would be just that gauge fixing with a special choice of the parameter which gives the Feynman gauge, ξ = 1. However, we caution the reader that the coupling of the functional integration variables to the Bosonic and Fermionic sources, Jµ (x), η(x) and η¯ (x) in (8.17) is not gauge invariant. The gauge transformation that we have to do would transform the terms R µ (Aµ J + ψη¯ + ηψ¯ ) in the exponent of the integrand in the numerator. The consequence is that photon and Fermion correlation functions are gauge dependent. Indeed, generic correlation functions will depend on how the gauge is fixed. The exception occurs when the photons and Fermions are put together to make µ gauge invariant composite operators, for example Fµν (x), ψ¯ (x)ψ(x), ψ¯ (x)γ ψ(x) or ψ¯ (x)[∂µ − ieAµ ]ψ(x). In such cases, if all of the operators in a correlation function are gauge invariant, the correlation function is independent of the way in which the gauge is fixed. We deal with this subtlety by ignoring it for the time being, and remembering later that the correlations functions that we compute have physical meaning only when they are used to compute gauge invariant quantities, such as correlation functions of gauge invariant operators. Another object that we shall use correlation functions for is the S matrix of scattering theory. It uses the projections of the field operators onto the wave-functions of the Dirac field and the physical polarizations of the photon. Generally, once theae projections have beendone for all of the operators in a correlation function, it is gauge invariant. In a generic covariant gauge,

R R µ [dAµ (x)dψ(x)dψ¯ (x)]exp iS[A,ψ,ψ¯ ] + i dx Aµ (x)J (x) + η¯ (x)ψ(x) + ψ¯ (x)η(x) Z[J,η,η¯ ] = R [dAµ (x)dψ(x)dψ¯ (x)]exp(iS[Aν ,ψ,ψ¯ ]) (8.18) Z S[A,ψ,ψ¯ ] = dx L (x)

1 µν ξ µ 2 L (x) = −iψ¯ (x) ∂/ − ieA/(x) + m ψ(x) − F (x)F (x) − (∂ A (x)) (8.19) 4 µν 2 µ To be sure, we repeat, in spite of the notation, (8.18) is not equal to (8.17). However, if we use either (8.18) or (8.17) to generate the correlation functions of gauge invariant operators, they will yield identical results. They will also give us identical scattering matrix elements, as those quantities are also gauge invariant. We can therefore use either functional to do intermediate computations. Normally, computations using the manifestly covariant formalism of (8.18) will be more convenient. Before we proceed, there is one more observation which will be needed in the following. First of all, in four spacetime dimensions, the quantum field theory with Lagrangian density (8.2) or gauge-fixed version thereof, contains all of the terms that are allowed in a mathematically consistent model which has Lorentz invariance. All other Lorentz invariant terms that we could add would eventually either have to be tuned to zero or they would render the model inconsistent. However, even in (8.2), there is one more freedom of choice that we will have to make use of. We can scale the coefficients of all of the integration variables in the functional integral so that

1 1 2 2 ψ(x) → Z2 ψ(x), ψ¯ (x) → Z2 ψ¯ (x) 1 2 Aµ (x) → Z3 Aµ (x) The Jacobians for this change of variables which would appear in the functional integrals are constants independent of the integration variables which cancel in the ratio of functional integrals that we use to compute the generating functional. In addition, we recognize that the parameters m,e,ξ which appear in the Lagrangian density should be treated as parameters of the model rather than constants equal to the physical mass and charge of the electron. In that case, we can regard them as functions of the physical mass and charge in the following sense

m → Zm(m,e)m Z1 e → 1 e 2 Z2Z3

ξ → Zξ ξ 138 Chapter 8. Quantum Electrodynamics where, now we regard all of the constants that we have introduced, Z1,Z2,Z3,Zm,Zξ as functions of the electron mass m, the electron charge e and, generally, on the gauge-ﬁxing parameter ξ. The individual terms which appear in the action which is inserted into the functional integral are thus to be modiﬁed as

−eψ¯ A/ψ → −Z1eψ¯ A/ψ = −(1 + δZ1)eψ¯ A/ψ Z1 = 1 + δZ1 (8.20)

−iψ¯ ∂ψ/ → −iZ2ψ¯ ∂ψ/ = −i(1 + δZ2)ψ¯ ∂ψ/ Z2 = 1 + δZ2 (8.21)

−imψψ¯ → −iZmmψψ¯ = −i(m + δm)ψψ¯ Zmm = m + δm (8.22) 1 1 1 − F F µν → − Z F F µν = − (1 + δZ )F F µν Z = 1 + δZ (8.23) 4 µν 4 3 µν 4 3 µν 3 3 ξ ξ ξ + δξ (∂ Aµ )2 → Z (∂ Aµ )2 = (∂ Aµ )2 Z ξ = ξ + δξ (8.24) 2 µ ξ 2 µ 2 µ ξ

We shall see that symmetry considerations will set δξ = 0 and δZ1 = δZ2. We will assume these conditions and we will justify them later, a posteriori, when we discuss the Ward-Takahashi identities. The gauge-ﬁxed action and Lagrangian density now appear as

S[A,ψ,ψ¯ ] = S0[A,ψ,ψ¯ ] + Sint[A,ψ,ψ¯ ] (8.25) Z S0[A,ψ,ψ¯ ] = dxL0(x) (8.26) Z Sint[A,ψ,ψ¯ ] = dxLint(x) (8.27) 1 ξ L (x) = −iψ¯ [∂/ + m]ψ − F F µν − (∂ Aµ )2 (8.28) 0 4 µν 2 µ δZ3 L (x) = −eψ¯ A/ψ − iδZ ψ¯ ∂ψ/ − iδmψψ¯ − F F µν − δZ eψ¯ A/ψ (8.29) int 2 4 µν 2 where we have included the terms with δZ2,δZ3,δm in the interactions. The δZ2,δZ3,δm are to be determined by comparing the predictions of the quantum field theory model described by equations (8.25)- (8.29) with the physical system that it is intended to describe. For quantum electrodynamics, we shall determine δZ2,δZ3,δm so that m is the physical mass of the Dirac field and e is its electric charge. In perturbative calculations we shall encounter infinities which will have to be defined with a regularization, which is tantamount to imposing a high energy cutoff on the theory, that is, of assuming that the fields contain only modes with wave-numbers and frequencies smaller than such a cutoff. The “renormalization constants” will then also depend on this cutoff and one of their roles will be to cancel the singularities. It is useful to think of the procedure of adjusting the renormalization constants in a broader sense, as a tuning of the parameters of the quantum field theory model which we are constructing so that the model matches and describes phenomena in nature. The fact that this can be done at all is a property of the quantum field theory called “renormalizability” and it is only renormalizable quantum field theories that are mathematically consistent, at least in the context of interacting quantum field theory that we will develop in this monograph. Quantum electrodynamics as we have constructed it is a renormalizable quantum field theory. The property of renormalizability depends on the type of terms, or “local operators”, that we have included in the Lagrangian density. An operator is called a “local operator” if it is constructed from a product of the basic fields, here Aµ (x),ψ(x),ψ¯ (x) and finite numbers of derivatives of these fields, all evaluated at the same space-time point. The terms that appear in the Lagrangian density are classified by their classical scaling dimensions. The classical scaling dimension of a local operator is gotten by counting the dimensions of the derivatives and fields from which it is composed. In d spacetime dimensions, we assign classical scaling dimension (d − 2)/2 to each Aµ , (d − 1)/2 to ψ and ψ¯ and one for each derivative of these fields. We call this the “classical scaling dimension” since the true scaling dimension of an operator in an interacting field theory can differ from the classical dimension that we are discussing here. As a first pass at the subject, we can classify a quantum field theory as being “renormalizable” or “non-renormalizable” using these classical dimensions. We begin by classifying local operators according to their classical scaling dimensions. A local operator is called classically “relevant” if its total classical scaling dimension is less than d, classically “marginal” if it 1 µν is exactly d, and classically “irrelevant” if it is greater than d. With this counting, the terms − 4 Fµν (x)F (x) and −iψ¯ (x)∂ψ/ (x) in the Lagrangian density are always classically exactly marginal. Generally, mass 8.2 The generating functional in perturbation theory 139

terms such as −imψ¯ (x)ψ(x) are classically relevant and the interaction term −eψ¯ (x)A/(x)ψ(x) is classically marginal only in four space-time dimensions, d = 4. It is classically relevant when d < 4 and irrelevant when d > 4. A renormalizable field theory has a Lagrangian density containing only classically relevant and marginal operators. Inspection of the Lagrangian density for electrodynamics that we are using finds only classically marginal and relevant operators, so the quantum field theory which it describes is indeed renormalizable. In fact, aside from the gauge fixing terms, and a potential photon mass term, there is no flexibility to add other local operators to this Lagrangian density. Any other modification of the theory would be non- renormalizable. This has the beauty that, restricting our attention to renormalizable quantum field theories eliminates an infinite number of possibilities of additional local operators with arbitrary coefficients that one could otherwise potentially add to the Lagrangian density and then have to fit to describe physics. As we formulate it here, quantum electrodynamics is a three-parameter model of charged Dirac spinor fields interacting with photons. We could take a broader view of the construction of a quantum field theory model as the process of, first of all, identifying the basic fields, here for example, we are constructing a theory of a vector field and a Dirac spinor field, and then placing all of the possible classically relevant and marginal local operators in the Lagrangian density with arbitrary coefficients. These coefficients of the local operators become the parameters of the theory which should eventually be tuned in order that the theory describes the natural phenomena which are being modelled. Indeed, this gives us a rule for introducing counter-terms. In principle, there should be one counter-term for each possible relevant or marginal local operator in the quantum field theory. Aside from the identification of the basic fields and renormalizability, there is another constraint on construction of a quantum field theory model. This constraint is symmetry. In our electrodynamics model, there are three important symmetries, Lorentz invariance, phase symmetry of the Dirac fields which lead to the conservation law for the Noether current, which we identify with electric current, and gauge symmetry. There are also some discrete symmetries, charge conjugation, parity and time reversal invariance. As well as renormalizability, we impose a symmetry restriction on the operators that we put in the Lagrangian density. We only allow terms which respect the symmetries. This means that the counter-terms must also respect the symmetries, including gauge invariance. This constrains the counter-terms that we have been discussing so that

δξ = 0, δZ1 = δZ2 (8.30)

There is one important caveat to our discussion of symmetry. Generally, a quantum field theory will exhibit a symmetry only if all facets of the basic definition of the quantum field theory exhibit the symmetry. There are examples of symmetries which are manifest in the Lagrangian density, but which are violated by the functional integration or the regularization procedure which is needed in order to define the ultraviolet singular quantities which appear when we do computations. To preserve the symmetries, in particular, it will be needed that the regularization is compatible with the symmetries that we are preserving. The regularization is introduced in order to define the otherwise singular quantities that are encountered in explicit computations. In the version of quantum electrodynamics that we are considering, we have the good fortune that we shall be able to find a regularization which preserves Lorentz invariance, phase symmetry and gauge invariance. This is consistent with the expectation that all of the counter-terms that we introduce should be invariant under these symmetries. An example of a symmetry which is routinely violated by regularization is scale invariance. If we put the mass of the Dirac field to zero, the Lagrangian density exhibits a scale symmetry which extends to conformal symmetry. This symmetry is routinely violated by the introduction of a regularization, so that, even though it is a symmetry of the classical field theory, it is not a symmetry of quantum field theory. This phenomenon is called a “scale anomaly”. There are other examples of anomalies and they have been an important factor in building quantum field theory models of elementary particle physics.

8.2 The generating functional in perturbation theory Due to the presence of the cubic coupling term in the exponent in the integrand, it is not known how to take the functional integral of interacting quantum electrodynamics exactly. However, for quantum electrodynamics, the parameter in front of the cubic term is small and perturbation theory is a viable and important tool for 140 Chapter 8. Quantum Electrodynamics

studying this functional integral. To implement perturbation theory, we consider the Taylor expansion of the numerator and the denominator of equation (8.18) in powers of the interactions, which well be proportional to powers of e,

Z[J,η,η¯ ] = n R µ ∞ (i) R R iS0[A,ψψ¯ ]+i [Aµ J +ηψ¯ +ψη¯ ] ∑ dw1 ...dwn [dAµ (x)dψ(x)dψ¯ (x)]e Lint(w1)...Lint(wn) 0 n! (8.31) (i)n ∞ R R iS0[A,ψψ¯ ] ∑0 n! dw1 ...dwn [dAµ (x)dψ(x)dψ¯ (x)]e Lint(w1)...Lint(wn)

where S0[A,ψ,ψ¯ ] is the free-ﬁeld theory action given in equations (8.26) and (8.28) and the interaction action S0[A,ψ,ψ¯ ] and the interaction Lagrangian density Lint(x) are given in equations (8.27) and (8.29), respectively. For each term in the summations in (8.32), the integrands in the integrals have the form of an exponential of a quadratic functional times a monomial in the integration variable. As we have seen in our study of functional integrals, these integrations can be done exactly and we have an explicit formula for the result. It is usually easiest to do them for a speciﬁc correlation function

< O|T Aµ1 (x1)...ψ(y1)...ψ¯ (z1)...|O >=

n ∞ (i) R R iS0[A,ψψ¯ ] ∑0 n! dw1 ...dwn [dAµ (x)dψ(x)dψ¯ (x)]e Lint(w1)...Lint(wn) Aµ1 (x1)...ψ(y1)...ψ¯ (z1)... (i)n R ∞ R iS0[A,ψψ¯ ] ∑0 n! dw1 ...dwn [dAµ (x)dψ(x)dψ¯ (x)]e Lint(w1)...Lint(wn) (8.32)

What remains, at each order n is a Gaussian integral of a polynomial in the ﬁelds. Anther way of writing the generating functional which is sometimes valuable is to use the explicit forms of the generating functionals for the non-interacting ﬁelds and correct it for interactions. The expression is

h i Z 1 Z[J] = exp iS 1 δ , 1 ∂ ,− 1 δ exp − dxdy Jµ (x)∆ (x,y)Jν (y) + η¯ (x)g(x,y)η(y) int i δJ i ∂η¯ i δη 2 µν ∞ in h in Z 1 1 δ 1 ∂ 1 δ µ ν ¯ = ∑ Sint i δJ , i ∂η¯ ,− i δη exp − dxdy J (x)∆µν (x,y)J (y) + η(x)g(x,y)η(y) n=1 n! 2 (8.33)

In this formula, the perturbation theory is gotten by taking functional derivatives which is generally easy and systematic. We will ﬁnd several applications fo this formula in the following sections. Finally the counter-terms that are in the interaction Lagrangian density are to be determined by imposing various conditions on multi-point functions which, to be maintained, must have the renormalization constants corrected, order-by-order in perturbation theory. They are thus dependent of the coupling constant e, in fact it turns out that they only depend on e2 and

∞ ∞ 2n (2n) 2n (2n) δZ2 = ∑ e δZ2 δZ3 = ∑ e δZ3 (8.34) n=1 n=1 ∞ δm = ∑ e2nδm(2n) (8.35) n=1 (8.36)

(2n) (2n) (2n) 2 where δZ2 ,δZ3 ,δm are independent of e .

8.3 Wick’s Theorem In order to evaluate the Gaussian integrals in the summation on the right-hand-side of equation (8.32), we can use formulae which we derived in the previous chapters for dealing with similar integrals in the study of free photons and non-interacting Dirac ﬁelds. In this way, we obtain the functional version of Wick’s 8.4 Feynman diagrams 141

theorem, which is usually presented in the context of the interaction picture and time-dependent perturbation theory. Our master formula is

R iS0[A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e Aµ1 (x1)...ψa1 (y1)...ψ¯b1 (z1)... R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0

deg x x g y z = ∑ (−1) ∏ ∆µiµ j ( i, j) ∏ a jbk ( j, k) (8.37a) pairings photon pairs ψψ¯ pairs

where deg is the number of exchanges of positions that is needed to change the order of the Fermion integration variables ψ and ψ¯ from the ordering in the integrand to the ordering in the pairing. In the pairings, photons are only paired with photons and ψ’s are always paired with ψ¯ ’s. Each photon is labeled by its position coordinate xi and its vector index µi. Each Fermion has a position and Dirac index y j,a j or zk,bk.

These become indices on the Green functions ∆µiµ j (xi,x j) and ga jbk (y j,zk) which are associated with each pair. Clearly, the pairings can only be implemented if there are an even number of Aµ ’s and the same number of ψ’s as ψ¯ ’s in the integrand. If the number of Aµ ’s is odd or if the numbers of ψ’s and ψ¯ ’s are not equal, the contribution vanishes. An example of an application of the formula (8.37a) is

R iS0[A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e Aµ1 (x1)Aµ2 (x2)ψa1 (y1)ψa2 (y2)ψ¯b1 (z1)ψ¯b2 (z2) R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0

= −∆µ1µ2 (x1,x2)ga1b1 (y1,z1)ga2b2 (y2,z2) + ∆µ1µ2 (x1,x2)ga1b2 (y1,z2)ga2b1 (y2,z1)

We see on the right-hand-side of this equation the two possible pairings which give a non-zero result. They differ by a sign since the differ by one in the number of Fermion interchanges needed to put the operator product into the order where the ψ-ψ¯ pairs are adjacent. Another example, containing an interaction is

R iS [A,ψ,ψ¯ ] Z [dAµ (x)dψ(x)dψ¯ (x)]e 0 ψ¯ (w)A/(w)ψ(w) Aµ (x)ψa(y)ψ¯b(z) − ie dw R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0 Z ν = −ie dw gac(y,w)γcdgdb(w,z) ∆νµ (w,x)

We see in this example that the Green functions g and ∆ connect the external points x,y,z to the point w where the interaction occurs. The indices of the gamma-matrices in the interaction are matched with the indices on the Green functions.

8.4 Feynman diagrams The perturbative expansion in quantum field theory has a useful diagrammatic representation where the diagrams are called Feynman diagrams. These diagrams were a great advance in the formulation of field theory in that they make lower order perturbation theory quite easy and intuitive. For the impatient reader, we shall summarize the rules for drawing Feynman diagrams in a later section. In the following, we will give a pedagogical introduction. In a Feynman diagram, the Green function gab(y,z) is represented by a oriented solid line, as depicted in figure 8.1. The photon Green function ∆µν (x,y) is represented by an un-oriented wiggly line as depicted in figure 8.2. The interaction vertex is an intersection of a photon line and an electron line as depicted in figure 8.3. For now, let us ignore the counter-terms which must also appear in the interaction. We can easily include them later on when they are needed. By using the symbols for the Green functions and interaction 142 Chapter 8. Quantum Electrodynamics

Figure 8.1: The Fermion Green function is represented by an oriented solid line.

Figure 8.2: The photon Green function is represented by an un-oriented wiggly line. .

Figure 8.3: The vertex is a point which absorbs and emits an electron line at a spacetime point w which is connected to a photon line. .

vertex, we can associate a Feynman diagram with any of the pairings that is produced by the application of Wick’s theorem.

In fact, we can go beyond that the simple representation of the results of using Wick’s theorem by diagrams and replace both the perturbation theory formula (8.32) and Wick’s theorem by the appropriate rules for drawing all of the Feynman diagrams which those formulae would produce. The rules for drawing diagrams are called the Feynman rules.

Before we introduce that technique, let us consider, as an example, the leading contribution to the three-point function < O|T Aµ (x)ψa(y)ψ¯b(z)|O >. The ﬁrst step is to apply the perturbation theory fomula 8.4 Feynman diagrams 143

(8.32) to For this correlation function to be non-zero, the interaction is required. The leading order is

< O|T ψa(y)ψ¯b(z)|O >=

R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0 Aµ (x)ψa(y)ψ¯b(z) (8.38) R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0

R iS [A,ψ,ψ¯ ] Z [dAµ (x)dψ(x)dψ¯ (x)]e 0 ψ¯ (w)A/(w)ψ(w) Aµ (x)ψa(y)ψ¯b(z) − ie dw (8.39) R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0

R iS [A,ψ,ψ¯ ] Z [dAµ (x)dψ(x)dψ¯ (x)]e 0 Aµ (x)ψa(y)ψ¯b(z) + ie dw × . R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0

R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0 ψ¯ (w)A/(w)ψ(w) × + ... (8.40) R iS [A,ψ,ψ¯ ] [dAµ (x)dψ(x)dψ¯ (x)]e 0 Z ν = −ie dw gac(y,w)γcdgdb(w,z) ∆νµ (w,x) + ... (8.41)

In the above formula, the line labeled (8.38) is the zero’th order contribution, the line labeled (8.39) comes from inserting one power of the interaction into the correlation function and line (8.40) is the correction to the denominator with one power of the interaction. The ellipses denote higher orders in e. The second step is to use Wick’s theorem to evaluate the Gaussian functional integrals. The first, zero’th order term, line (8.38), vanishes, as does the third term, line (8.40), with the correction to the denominator. Only the term in the line labeled (8.39) survives and the result is the last line (8.41) of the equation above. The contribution corresponds to the Feynman diagram in figure ??. Feynman rules for drawing this diagram would have read like the following. 1. First of all, we place the external vertices x, µ;y,a;z,b at the boundary of the graph, as is depicted in figure ??. We place the desired number of internal vertices in the centre of the graph. The number of vertices depends on the order of perturbation theory. Here, the desired order is e, so we use one insertion of −ieR dwψ¯ (w)A/(w)ψ(w). The vertices with counter-terms do not contribute at this order, since they begin at order e2. The external Dirac field lines for ψ’s are labeled with incoming arrows and the external lines with ψ¯ ’s have outgoing arrows. 2. Then, we connect the lines in all possible ways that are compatible with the orientation of the Dirac field lines to make the Feynman diagrams. The result of this procedure is the one diagram depicted in figure ??. In this case, there are no minus signs from re-ordering of Fermions. The diagram should have an overall factor of −ie. Plugging in the Green functions for the lines in the diagram, we obtain the last line, labeled (8.41) in the equation above. 3. Note that, in this diagrammatic approach, the denominator was never mentioned. In fact, all contributions of corrections to the denominator will cancel with certain corrections to the numerator, those which have sub-diagrams which are not connected to any of the external lines. This is a result of Goldstone’s theorem, which we will discuss in some detail shortly. The upshot of the theorem is that we never have to consider corrections to the denominator and we also discard all of the contributions which have disconnected sub-diagrams. In this example, there were no such contributions. We will encounter them in the next example. As another example, we shall consider the computation of the corrections in perturbation theory, to 2 the two-point function of the Dirac field, < O|T ψa(y)ψ¯b(z)|O >. We will concentrate on the order e contributions. We begin the diagrammatic approach by writing the external points y,z at the edges of the diagram. Then, since the computation is of the order e2, we draw two vertices in the interior of the graph. We label the vertices by their coordinates w1,w2. This starting point is depicted in figure 8.4. We then connect the lines in all of the possible ways that are compatible with their orientation. The result is shown in figure 8.5. Note that there are four distinct diagrams. Once we have generated the diagrams, there will be several important simplifications. Goldstone’s theorem allows us to discard the diagrams which have parts which are disconnected from any external points. These are the third and fourth diagrams in figure 8.5. We are left with the first two diagrams in figure 8.5. 144 Chapter 8. Quantum Electrodynamics

Figure 8.4: We begin to draw the Feynman diagrams by putting the external points, x and y at the edge of the graph and the vertices in the center. .

Figure 8.5: The Feynman diagrams which are obtained by connecting the lines in ﬁgure 8.4 in all possible ways. .

The second diagram contains a subgraph which is connected to the rest of the graph by only one photon line. Such a subgraph is called a tadpole. Vanishing of tadpoles is a special case of Furry’s theorem, which we will also discuss in more detail later. Furry’s theorem states that any subgraph with no external electron lines and any odd number of external photon lines must vanish. This leaves us with the first diagram in figure 8.5 as the contribution to the two-point function of the electron. Once we have it, we can find the explicit mathematical formula for the correction by associating the lines with Green’s functions and the vertices with Dirac matrices. One of the details that we shall need is the overall coefficient of the contribution. In the (−ie)2 perturbation theory formula which had two vertices, the factor in front would have been 2! . The factor of 1 2! is canceled by the fact that there are two contributions to the set of all pairings in Wick’s theorem which produce the Feynman diagram in figure (8.5). One can think of these as simply being the choice of two vertices to pair the incoming Fermion line with. After that, there is no more multiplicity, so the net factor in (−ie)2 2 front is 2 × 2! = (−ie) and the contribution, written in terms of Green functions, is Z 2 µ ν < O|T ψa(y)ψ¯b(z)|O >= gab(y,z) − e dw1dw2 [g(y,w1)γ g(w1,w2)γ g(w2,z)]ab ∆µν (w1,w2) Z h i +ie2 dw g(y,w)(−iδZ(2)∂/ − iδm(2))g(w,z) + ... 2 ab (8.42) where we have included the zero’th order contribution and the contribution of the counter-terms. The ellipses denote orders higher than e2. We leave the derivation of the term with counter-terms as an exercise for the reader.

Now, let us consider the corrections to the photon two-point function. The Feynman diagrams will have two external photon lines and two internal vertices with the starting point depicted in figure 8.6. Discarding the tadpoles and diagrams with disconnected parts, we immediately arrive at the diagram in figure 8.7. The 1 factor of 2! cancels as it did for the Dirac field two-point-function, however the re-ordering of the fermions produces a minus sign. It will turn out that this is quite general, Feynman diagrams have factors of (−1) for each Fermion loop, and this contribution to the photon has one Fermion loop. Explicitly, the photon 8.5 Connected Correlations and Goldstone’s theorem 145

Figure 8.6: We begin ﬁnding the Feynman diagrams which correct the two-point function of the photon by drawing two external photon lines and two vertices. .

Figure 8.7: The Feynman diagram which contributes to the two-point correlation function of the photon ﬁeld at order e2 .

two-point function is

< O|T Aµ1 (x1)Aµ2 (x2)|O >= ∆µν (x1,x2) Z 2 ρ σ + e dw1dw2∆µρ (x1,w1)Tr[γ g(w1,w2)γ g(w2,w1)]∆σν (w2,x2) Z 2 (2) ρσ 2 ρ σ + ie dw∆µρ (x1,w)δZ1 (η ∂ − ∂ ∂ )∆σν (w,x2) + ... (8.43)

8.5 Connected Correlations and Goldstone’s theorem The role of the denominator in equation (8.32) is to cancel those contributions in the numerator which have parts that are not connected by Green functions to any of the external points x1,...,y1,...,z1,.... This theorem allows the convenience that, when we are computing the perturbative contribution at a given order, we can ignore the denominator and we can also ignore pairings in the numerator whenever there are any parts which are disconnected from any of the external points. In this section, we will ﬁnd a simple proof of this theorem in the context of quantum electrodynamics. However, it is much more general than the example that we will discuss. It applies to the perturbative expansion of any quantum ﬁeld theory.

8.5.1 Connected correlation functions In the above, we have been concerned with Feynman diagrams where there is a sub-diagram which is not connected to any of the external lines. Such a sub-diagram is called a vacuum bubble and, as a consequence of Goldstone’s theorem, they cancel and we need not consider Feynman diagrams which contain vacuum bubbles when we compute correlation functions. In this section, we shall ﬁnd a proof that vacuum bubbles cancel. We will do this by ﬁnding another interesting and useful fact, that the functional given by the logarithm of the generating functional

W[J,η,η¯ ] ≡ lnZ[J,η,η¯ ]

is itself a generating functional for connected Feynman diagrams. A disconnected Feynman diagram is defined as one which can be split into two sub-diagrams in such a way that no vertex in one of the sub-diagrams is connected by an internal line with any vertex of the other sub-diagram. The sub-diagrams could each be attached to external lines of the full diagram. If one of them is not attached to any external lines, it is a vacuum bubble. The contributions to multi-point functions can, and generally do have disconnected diagrams. A connected Feynman diagram is defined as one which is not disconnected. Alternatively, a connected Feynman diagram is defined as a Feynman diagram which has the property that one can trace a path between any two vertices, internal or external in the entire diagram by continually 146 Chapter 8. Quantum Electrodynamics following internal and external lines which are in the diagram. A diagram is said to be disconnected if it is not connected. For example, in free field theory, if we set the electrodynamic coupling constant e to zero,

R 1 µ ν dydx[− J (y)∆µν (y,z)J (z)−η¯ (y)g(y,z)η(z)] Ze=0[J,η,η¯ ] = e 2 and, taking the logarithm, Z 1 W [J,η,η¯ ] = dydx − Jµ (y)∆ (y,z)Jν (z) − η¯ (y)g(y,z)η(z) e=0 2 µν This formula tells us that, in free ﬁeld theory, the only connected correlation functions are two-point functions. Thus, the existence of connected functions beyond two-point functions can only occur when there are interactions. We have already seen this when we computed the leading order contribution to the three-point function < O|Aµ (x)ψa(y)ψ¯b(z)|O > and noticed that the leading term is of order e. First, we need to ﬁnd a proof that W[J,η,η¯ ] generates connected correlation functions. Let us assume that we have a perturbative expansion of W[J,η,η¯ ], ∞ n W[J,η,η¯ ] = ∑ e Wn[J,η,η¯ ] n=0 and that we know that, up to some order, k, each of the terms W0[J,η,η¯ ],...,Wk[J,η,η¯ ] generate only R 1 connected correlation functions. This is certainly true for W0 = − 2 J∆J − η¯ gη which only contains connected two-point functions. The generating functional for connected correlation functions has a general expression given in equation (8.33) which we copy here for the reader’s convenience

iS [ δ , δ , δ ] R 1 µ ν int δJ δη¯ δη − dydx[ 2 J (y)∆µν (y,z)J (z)+η¯ (y)g(y,z)η(z)] W[J,η,η¯ ] e e e = δ δ δ R 1 µ ν iSint[ , , ] − dydx[ J (y)∆µν (y,z)J (z)+η¯ (y)g(y,z)η(z)] e δJ δη¯ δη e 2 Jηη¯ =0 d Now, consider taking the derivative deW[J,η,η¯ ] and using the expression for W[J,η,η¯ ] from the equation above, to get d W[J,η,η¯ ] de d δ δ δ d = e−W[J,η,η¯ ] i S , , eW[J,η,η¯ ]+ < O|i S [A,ψ,ψ¯ ]|O > de int δJ δη¯ δη de int which we can rewrite as d d δ δW δ δW δ δW d W = i S + , + , + − < O|i S [A,ψ,ψ¯ ]|O > de de int δJ δJ δη¯ δη¯ δη δη de int where we have used the identities δ δ δW δ δ δW δ δ δW e−W eW = + , e−W eW = + , e−W eW = + δJ δJ δJ δη¯ δη¯ δη¯ δη δη δη and the remaining derivatives operate on whatever functionals appear to the right of them. Now, let us make use of the explicit expression for the interaction terms, We can expand the right-hand-side as d W[J,η,η¯ ] = de ! Z δ 3W δ 3W − dw µ − γab µ µ J (w)δηa(w)δη¯b(w) J (w)δηa(w)δη¯b(w) Jηη¯ =0 Z δW δ 2W δW δW δW − dw µ + γab µ µ δηa(w) J (w)δη¯b(w) δηa(w) δJ (w) δη¯b(w) δ 2W δW δW δ 2W + + + µ µ counterterms δηa(w)δJ (w) δη¯b(w) δJ (w) δηa(w)δη¯b(w) 8.6 Fourier transform 147

where we leave the form of the contribution of the counter-terms as an exercise for the reader. Next, let us consider a Taylor expansion of both sides of the above equation in the parameter e. Consider the order ek term in such a Taylor expansion. It determines Wk+1 once all of W0,W1,...,Wk are known. More-over, we are assuming that W0,W1,...,Wk all generate connected diagrams and the right-hand-side depends only on these, explicitly,

(k + 1)Wk+1[J,η,η¯ ] = ! Z δ 3W δ 3W − dw µ k − k γab µ µ J (w)δηa(w)δη¯b(w) J (w)δηa(w)δη¯b(w) Jηη¯ =0

Z k δW δ 2W δW δW δW − dw µ p k−p + p q r γab ∑ µ ∑ δp+q+r,k µ p=0 δηa(w) J (w)δη¯b(w) pqr δηa(w) δJ (w) δη¯b(w) ! k δ 2W δW n δW δ 2W + q k−q + q k−q + − ∑ µ ∑ µ counter terms (8.44) q=0 δηa(w)δJ (w) δη¯b(w) q=0 δJ (w) δηa(w)δη¯b(w)

Because functional derivatives of W0,W1,...,Wk are all connected and the right-hand-side contains contributions which connects these connected parts to the point w, which is subsequently integrated the right -hand-side is a summation over connected functionals. Therefore Wk+1 must be a sum of connected diagrams. Then, given that W0 is connected, the fact that all Wn are connected follows by mathematical induction.

8.5.2 Goldstone’s Theorem Now that we have established that W[J,η,η¯ ] is a generating functional for connected correlation functions, it is a simple consequence that the correlation functions that are generated by Z[J,η,η¯ ] = eW[J,η,η¯ ] cannot have vacuum bubbles. Functional derivatives of Z[J,η,η¯ ] are given by polynomials in functional derivatives of W[J,η,η¯ ] and in such an expression, all of the parts of all of the terms are connected to some external line, they can contain no vacuum bubbles. From this, we conclude that the vacuum bubbles that are generated by perturbation theory must cancel. Moreover, the only contributions that corrections to the functional integral in the denominator of the generating functional can contain are those with vacuum bubbles, since such corrections cannot be connected to any external line. Thus, the absence of vacuum bubbles in correlation functions allows us to ignore them whenever they occur in our use of Wick’s theorem and, simultaneously, to ignore perturbative contributions to the denominator. This is a great simpliﬁcation of many computations.

8.6 Fourier transform Due to space-time translation invariance, all of the Green functions that we consider are functions of the differences of the coordinates. This make the Fourier transform an essential tool in evaluating the results of perturbation theory. Remember that we had the expression

4 Z d k µ µ g(x,y) = eikµ (x −y )g(k) (2π)4 i/k − m g(k) = = −[i/k + m]−1 µ 2 kµ k + m − iε 4 Z d k µ µ ∆ (x,y) = eikµ (x −y )∆ (k) µν (2π)4 µν η 1 k k (k) = −i µν − − µ ν ∆µν ν 1 ν 2 kν k − iε ξ (kν k − iε) 148 Chapter 8. Quantum Electrodynamics

Figure 8.8: The orientation of momentum assignments to external lines is depicted. We will take all of the external momenta as incoming. The fact that they must sum to zero is a result of the conservation of momentum. The momentum flow in an electron line is along its orientation. Thus, the outgoing electron lines carry negative momenta. Photon lines must also be assigned a direction of momentum flow. For external photons lines, as depicted here, the momentum is taken as flowing inward.

Figure 8.9: An example of a Feynman diagram with momentum assignments to external and internal lines. The total momentum is conserved at each of the four vertices. The total momentum µ of the diagram is also conserved, q1 + q2 + `1 + `2 = 0. This leaves one momentum arbitraty, k1 . 4 R d k1 The analytic expression for the diagram must be integrated over this loop momentum (2π)4 . 8.6 Fourier transform 149

Figure 8.10: The Feynman diagram which contributes to the two-point correlation function of the electron ﬁeld at order e2 in momentum space..

Figure 8.11: The Feynman diagram which contributes to the two-point correlation function of the photon field at order e2 in momentum space.. for the Dirac field and photon Green functions. The Fourier transforms of the two-point correlation functions of the interacting quantum field theory are defined by

4 Z d k µ µ < O|T ψ(x)ψ¯ (y)|O > = eikµ (x −y )S(k) (8.45) (2π)4 4 Z d k µ µ < O|T A (x)A (y)|O > = eikµ (x −y )D (k) (8.46) µ ν (2π)4 µν

Then, when we take the Fourier transforms of both sides of equations (8.42) and (8.43), we have

 4 Z d p (2) S(k) = g(k) + e2g(k) − ∆ (p)γλ1 g(k − p)γλ2 + Z i/k + δm(2) g(k) + ... (8.47) (2π)4 λ1λ2 2

Dµν (k) = ∆µν (k)+ Z d4 p + e2∆ (k) Tr[g(p − k)γρ g(p)γσ ] − iδZ(2)(k2ηρσ − kρ kσ ) ∆ (k) µρ (2π)4 3 σν + ... (8.48)

We observe that the momentum space equations (8.47) and (8.48) are simpler than their coordinate space counterparts, equations (8.42) and (8.43). There is only one integral over four dimensional momentum space which remains to be done in order to find the order e2 corrections given in equations (8.47) and (8.48). (2) (2) (2) We will discuss how this integral is done and how the constants δZ2 ,δm ,δZ3 are chosen later in this chapter. It is useful to draw the Feynman diagram in a way that can be used to find the momentum space contribution directly. The goal is to compute a correlation function. We begin by defining the Fourier transform of a generic correlation function,

< O|T Aµ1 (x1)...ψa1 (y1)...ψ¯b1 (z1)...|O >= Z d4 p d4q d4` = 1 ... 1 ... 1 ...eip1x1+...+iq1y1+...i`1z1+...Γ˜ (p ,...,q ,...,` ,...) (8.49) (2π)4 (2π)4 (2π)4 µ1...a1...b1... 1 1 1

Translation invariance of the quantum ﬁeld theory implies that the correlation functions depend only on the differences of the coordinates. This leads to conservation of momentum in their Fourier transform, in that the sum of all of the momenta must vanish, p1 + ... + q1 + ... + `1 + ... = 0. This conservation of momentum 150 Chapter 8. Quantum Electrodynamics

must be enforced by a Dirac delta function. It is useful to extract this delta function from the full correlation function as ˜ Γµ1...a1...b1...(p1,...,q1,...,`1,...) 4 = (2π) δ(p1 + ... + q1 + ... + `1 + ...)Γµ1...a1...b1...(p1,...,q1,...,`1,...) (8.50)

From now on, whenever we write Γµ1...a1...b1...(p1,...,q1,...,`1,...) we will assume that it only makes sense when the sum of its momentum arguments is zero. Examples are the photon and electron two-point functions

Dµ1µ2 (p) = Γµ1µ2 (p,−p) (8.51) Sab(p) = Γab(p,−p) (8.52)

respectively. The momentum space Feynman rules will calculate the momentum-space correlation functions

Γµ1...a1...b1...(p1,...,q1,...,`1,...). To do this, we follow the instructions for finding the Feynman diagrams which contribute to a given correlation function to some order in perturbation theory. Once we have a diagram, we must label the four-momenta of all of the lines in the diagram. Electron lines are oriented and the direction of momentum flow in an electron line will be taken as the direction of its orientation. Photon lines are normally not oriented, but they must be assigned a direction of momentum flow. The momenta of the external photon lines p1,..., must each taken as entering the diagram. The incoming photon lines with index µ1, µ2,... are labeled with momenta p1, p2,... flowing from the external points into the graph. The electron lines are oriented as incoming to the graph, having Dirac indices a1,a2,... and momenta q1,q2,... entering the diagram. The outgoing electron lines are oriented outgoing from the diagram. Those with indices b1,b2,... are labeled with momenta −`1,−`2,... oriented as outgoing from the diagram. The conventions for assigning momenta to external lines are depicted in figure 8.8. Momenta must be assigned to internal lines of a Feynman diagram in such a way that the direction of momentum flow in each line is indicated. The other constraint is that the total momentum is conserved at each vertex. This is the only constraint on momenta and it can leave a certain number of momenta undetermined. The number of indetermined 4-momenta is equal to the number of loops in the diagram. For example, consider the diagram in figure 8.9. Momentum is conserved at each vertex. The momentum space Feynman diagrams for our examples of order e2 corrections to the electron and photon two-point functions are depicted in figures 8.10 and 8.11.

8.7 Furry’s theorem Furry’s theorem states that an electron loop emitting an odd number of photons must vanish. It is useful since it allows us to eliminate some possibilities for Feynman diagrams. The vanishing of the tadpole, which was a Fermion loop emitting one photon loop is a simple example. To prove this theorem, we consider an electron loop emitting n photons,

Z d4 p Trg(p)γλ1 g(p + q )γλ2 g(p + q + q )...g(p + q + ... + q )γλn (8.53) (2π)4 1 1 2 1 n−1

We remember charge conjugation transformations which are implemented by a unitary matrix C with the property C †C = 1 , C †γ µ C = (γ µ )∗ = (γ µ )†t = (γ0γ µ γ0)t The superscript t stands for transpose. If we apply it to the momentum space Green function,

C †g(k)C = −(γ0g(−k)γ0)t

Then, inserting CC † between each matrix product in the trace in the integrand in (8.53) yields

Z d4 p Trg(p)γλ1 g(p + q )γλ2 g(p + q + q )...g(p + q + ... + q )γλn = (2π)4 1 1 2 1 n−1 Z 4 d p † † λ † † † λn † TrCC g(p)CC γ 1 CC ...CC g(p + q + ... + qn− )CC γ CC = (2π)4 1 1 8.8 One-particle irreducible correlation functions 151

Z d4 p = (−1)n Tr[γ0γ0g(−p)γ0γ0γλ1 γ0γ0 ...γ0γ0g(p − q − ... − q )γ0γ0γλn γ0γ0]t = (2π)4 1 n−1 Z d4 p = (−1)n Tr[g(−p)γλ1 gt (−p − q )γλ2 g(−p − q − q )...g(p − q − ... − q )γλn ]t = (2π)4 1 1 2 1 n−1 Z d4 p = (−1)n Tr[γλn g(−p − q − ... − q )...g(−p − q − q )γλ2 gt (−p − q )γλ1 g(−p)] (2π)4 1 n−1 1 2 1

Now, changing variables, p → −p − q1 − ... − qn recovers Z d4 p (−1)n Trg(p)γλn g(p + q )γλn−1 g(p + q + q )...g(p + q + ... + q )γλ1 (2π)4 1 1 2 1 n−1 This ﬁnal expression is the loop with reversed orientation and it has a minus sign when n is odd. In applying the Feynman rules, both orientations of the loop will occur in a given set of Feynman diagrams. They must therefore cancel when n is odd. The utility of Furry’s theorem is that when we are doing a perturbative computation of a correlation function, we can drop all closed Fermion loops which contain an odd number of vertices.

8.8 One-particle irreducible correlation functions Equations (8.47) and (8.48) have been written in a suggestive form which we could present as

S(k) = g(k) − g(k)Σ(k)g(k) + ... (8.54)

λ1λ2 Dµν (k) = ∆µν (k) − ∆µλ1 (k)Π (k)∆λ2ν (k) + ... (8.55) 4 Z d p (2) Σ(k) = e2 ∆ (p)γλ1 g(k − p)γλ2 − Z i/k − δm(2) + ... (8.56) (2π)4 λ1λ2 2 Z d4 p Πρσ (k) = e2 − Tr[g(p − k)γρ g(p)γσ ] + iδZ(2)(k2ηρσ − kρ kσ ) + ... (8.57) (2π)4 3 In fact all contributions to the electron and photon two point functions have to begin and end with electron or photon Green functions. We deﬁne the one-particle irreducible two-point function Σ(k) as the set of all Feynman diagrams with two ports, one where an incoming g(k) could be attached and one where an outgoing g(k) could be attached and which could not be severed into two disconnected diagrams by cutting a single electron line. Then, it is clear that the full electron two-point function S(k) can be reconstructed from the one-particle irreducible one Σ(k) by simply writing the geometric sum

S(k) = g(k) − g(k)Σ(k)g(k) + g(k)Σ(k)g(k)Σ(k)g(k) + ... (8.58)

We could do the same for the photon where the one-particle irreducible photon two-point function Πµν (k) is equal to the sum of all Feynman diagram with two ports which can be connected to two photons and which cannot be severed into two disconnected graphs by cutting a single photon line. It can also never be severed by cutting an electron line so, it is truly “one-particle-irreducible”. The photon two point function can be reconstructed once we know the one-particle irreducible function as

D(k) = ∆(k) − ∆(k)Π(k)∆(k) + ∆(k)Π(k)∆(k)Π(k)∆(k) + ... (8.59)

Equations (8.58) and (8.59) are geometric sums which can be written in closed form as

S−1(k) = g−1(k) + Σ(k) (8.60) D−1(k) = ∆−1(k) + Π(k) (8.61)

The IPI-irreducible functions Σ(k) and Π(k) are sometimes called the “Dirac ﬁeld self-energy” and the “photon self-energy”, respectively. Also, the photon self-energy is given by the one-particle-irreducible current-current correlation function

µν µ ν Π (x,y) =< O|T j (x)j (y)|O >IPI (8.62) where jµ (x) = −eψ¯ (x)γ µ ψ(x) is the electromagnetic current. 152 Chapter 8. Quantum Electrodynamics

8.9 Some calculations In this section, we will complete the computation of the order e2 corrections to the two-point functions of the photon ﬁeld and the Dirac ﬁeld.

8.9.1 The photon two-point function

Let us consider the two-point function, of the photon whose Fourier transform we denoted as Dµν k. The inverse of this function is given by

−1 −1 Dµν (k) = ∆µν (k) + Πµν (k) Here −1 2 ∆µν (k) = i k ηµν − kµ kν + iξkµ kν

is the inverse of the free field Green function. The function Πµν (k) is sometimes called the “vacuum polarization” or the “photon self-energy”. We will study it in perturbation theory to order e2. The photon self-energy, at the order to which we shall compute it, is divergent and it will require a regularization of the divergent parts in order to define them precisely. We will use a regularization which respects gauge invariance and current conservation. With this regularization, we will find by explicit computation that the vacuum polarization has the form 2 2 Πµν (k) = i k ηµν − kµ kν Π(k ) (8.63)

2 The appearance of the transverse projection operator, k ηµν − kµ kν , can be expected for two reasons. First µ of all, we can formulate a general argument to show that k Πµν (k) = 0. This is basically because Πµν (k) is made from current-current correlation functions and the current is conserved. Secondly, we expect that it must be made from a Lorentz covariant function of kµ and the only covariant quantities with two Lorentz 2 2 indices are δµν times a function of k and kµ kν times a function of k . These two facts tell us that the tensor structure must be as shown in equation (8.63). In the following, we shall ﬁnd that this is indeed the case by doing an explicit calculation. The photon self-energy, to the leading order in perturbation theory, is given by the integral

4 Z d p (2) h i Πλ1λ2 (k) = −e2 Trg(p − k)γλ1 g(p)γλ2 + iδZ k2ηλ1λ2 − kλ1 kλ2 (8.64) (2π)4 3 First of all, we note that, since g(p) ∼ 1/p for large p, the integral is infinite, the divergence coming from the large momentum domain. In order to make sense of it, we must re-define the quantum field theory in such a way that the integral is finite and also in such a way that we can recover the quantum field theory that we are interested in, including the divergent Feynman integrals in some limit. This is called “regularization”. There are many choices for such a regularization. The one which we shall choose is called “dimensional regularization”. It assumes that the dimension of space-time is a parameter, 2ω, rather than four. Generally, one assumes that there is still, always, one time dimension and 2ω − 1 space dimensions. We do the integral in 2ω space-time dimensions where ω is adjusted to be small enough that the integration makes sense. We then promote the result to a function of a complex variable ω and we analytically continue it to the vicinity 1 of 2ω ∼ 4 in the complex ω-plane. The original divergences of the integral will show up as poles ∼ 4−2ω at four dimensions. We will then find a procedure for dealing with the divergent contributions. That procedure is called “renormalization”. When the space-time dimension is exactly four, e2 is a dimensionless parameter. However, away from four dimensions, e2 takes on a scaling dimension. For various reasons, we shall find it advantageous to keep e2 dimensionless by multiplying it by a constant which makes up the dimensions that it needs. For this, we replace e2 by e2µ4−2ω where µ has the dimensions of mass, or momentum and, for now, is otherwise arbitrary. With dimensional regularization, equation (8.64) becomes

2ω Z d p (2) h i Πλ1λ2 (k) = −e2µ4−2ω Trg(p − k)γλ1 g(p)γλ2 + iδZ k2ηλ1λ2 − kλ1 kλ2 (2π)2ω 3 2ω Z d p −i(/p − /k) + m −i/p + m (2) h i Πλ1λ2 (k) = −e2µ4−2ω Tr γλ1 γλ2 + iδZ k2ηλ1λ2 − kλ1 kλ2 (2π)2ω (p − k)2 + m2 − iε p2 + m2 − iε 3 (8.65) 8.9 Some calculations 153 where we have inserted the explicit expressions for g(p) and, now, the Lorentz indices λ1,λ2 take on 2ω values. Now, we shall take the trace over the Dirac gamma-matrices. The formulas for the traces of these matrices is discussed in section 8.9.3 below. We will have to make an assumption about how the dimension of the gamma-matrices changes as we go from four space-time dimensions to 2ω dimensions. We will assume that in 2ω dimensions, they are 4κ × 4κ where, near four dimensions, κ = 1 + (2 − ω)κ(1) + .... Upon taking the traces over gamma-matrices, equation (8.65) becomes

Z d2ω p −(p − k)λ1 pλ2 − pλ1 (p − k)λ2 + ηλ1λ2 p · (p − k) + ηλ1λ2 m2 Πλ1λ2 (k) = −4κe2µ4−2ω (2π)2ω [(p − k)2 + m2 − iε][p2 + m2 − iε] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k η − k k (8.66)

In order to make the integration easier, we shall also do a “Wick rotation” which replaces p0 → ip0, q0 → iq0. The propagator in the integrand, for example,

" # 1 1 1 1 = − + p2 + m2 − iε p 2 2 p 2 2 p 2 2 2 ~k + m p0 + ~k + m − iε p0 − ~k + m − iε

0 p 2 2 p 2 2 Has poles at p = − ~k + m +iε and p0 = ~k + m −iε, that is, in the second and in the fourth quadrant of the complex p0 plane. It is analytic in the first and third quadrants of the complex p0-plane.It is easy to see that this is so for any of the propagators in the integral. Wick rotation is a replacement of the integration contour for p0, from the one in the integral which follows the entire real axis in the complex p0-plane from −∞ to +∞, by one which follows the imaginary axis from −i∞ to +i∞ plus a compensating closed contour which follows the real axis from −∞ to ∞, the infinite quarter-circle in the first quadrant from ∞toi∞, the imaginary axis from i∞ to −i∞, then a quarter circle at infinity in the third quadrant from −i∞ to −∞. Since the entire integrand is analytic in the region inside the closed, compensating contour, by Cauchy’s theorem for line integrals in the complex plane, the contribution from the compensating contour vanishes. The result is an integration of p0 which follows the imaginary axis in the complex plane. This is equivalent to simply replacing p0 by ip0 everywhere in the integrand. Once that is done, we can also analytically continue in the variables q0 which allows us to replace q0 by iq0 wherever it appears. We can also change the sign of Π00 and multiply Π0a by i in order to render that vacuum polarization as if we were simply computing it in Euclidean space. We do the integrals and find the result and, afterwards, we analytically continue back. to Minkowski space. After the Wick rotation, the vacuum polarization is

Z d2ω p −(p − k)λ1 pλ2 − pλ1 (p − k)λ2 + δ λ1λ2 p · (p − k) + δ λ1λ2 m2 Πλ1λ2 (k) = −4iκe2µ4−2ω (2π)2ω [(p − k)2 + m2][p2 + m2] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k δ − k k (8.67)

Next, we shall use Feynman parameters to combine the two factors in the denominator. The general Feynman parameter formula is reviewed in section 8.9.4. Applied to our integral, the formula yields

Πλ1λ2 (k) = Z 1 Z d2ω p −2pλ1 pλ2 + 2α(1 − α)kλ1 kλ2 + δ λ1λ2 [p2 − α(1 − α)k2 + m2] = − i e2 4−2ω d 4 κ µ α 2ω 2 2 2 2 0 (2π) [p + m + α(1 − α)k ] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k δ − k k where, to simplify the denominator, we have translated the variable, p → p + αk. and we have dropped all terms in the numerator which are odd in k. We use symmetry to replace 2pλ1 pλ2 where it appears in the 154 Chapter 8. Quantum Electrodynamics

1 λ1λ2 2 numerator of the integrand by ω δ p

Πλ1λ2 (k) = Z 1 Z d2ω p 2α(1 − α)kλ1 kλ2 − δ λ1λ2 ( 1 − 1)p2 + δ λ1λ2 [−α(1 − α)k2 + m2] = − i e2 4−2ω d ω 4 κ µ α 2ω 2 2 2 2 0 (2π) [p + m + α(1 − α)k ] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k δ − k k Z 1 Z d2ω p 2α(1 − α)kλ1 kλ2 − (2 − 1 )δ λ1λ2 α(1 − α)k2 + 1 m2 = − i e2 4−2ω d ω ω 4 κ µ α 2ω 2 2 2 2 0 (2π) [p + m + α(1 − α)k ] Z 1 Z d2ω p −δ λ1λ2 ( 1 − 1) − i e2 4−2ω d ω 4 κ µ α 2ω 2 2 2 0 (2π) [p + m + α(1 − α)k ] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k δ − k k 4iκe2µ4−2ω Γ[2 − ω] Z 1 2α(1 − α)kλ1 kλ2 − (2 − 1 )δ λ1λ2 α(1 − α)k2 + 1 m2 = − d ω ω ω α 2 2 2−ω (4π) 0 [m + α(1 − α)k ] 4iκe2µ4−2ω Γ[1 − ω] Z 1 −δ λ1λ2 ( 1 − 1) − d ω ω α 2 2 1−ω (4π) 0 [m + α(1 − α)k ] h i 8κe2µ4−2ω Γ[2 − ω] Z 1 α(1 − α) = i λ1λ2 k2 − kλ1 kλ2 d δ ω α 2 2 2−ω (4π) 0 [m + α(1 − α)k ] h i (2) 2 λ1λ2 λ1 λ2 + iδZ3 k δ − k k

In the second-last step above, we have done the integral over p using a formula (8.78b) for dimensionally regularized integrals which is quoted and derived in section 8.9.5. We have also used the property of the Euler gamma function (1 − ω)Γ[1 − ω] = Γ[2 − ω]. From the above expression, we identify the vacuum polarization function in 2ω space-time dimensions,

8e2κµ4−2ω Γ[2 − ω] Z 1 α(1 − α) (k2) = d + Z(2) Π ω α 2 2 2−ω δ 3 (8.68) (4π) 0 [m + α(1 − α)k ] This is our result for the dimensionally regularized vacuum polarization which contributes to the self-energy of the photon. It is still ultraviolet divergent. The divergence resides in the singularity of the Euler gamma function as we put the dimension 2ω to four. We have also not determined the renormalization constant (2) δZ3 . The Euler gamma-function has an expansion near four dimensions

1 π2 γ2 Γ[2 − ω] = − γ + + (2 − ω) + ... 2 − ω 12 2 and " # e2 1 e2 Z 1 m2 + α(1 − α)k2 (k2) = − d ( − ) + Z(2) Π 2 2 αα 1 α ln (1) δ 3 6π 2 − ω 2π 0 4πeγ+κ µ2

We have added the contribution of the counter-term, which is necessary to cancel the singularity at ω → 2. It ∼ 1 Z should be determined so that the pole in the photon two-point-function is k2 . To do this, we choose δ 3 so that

e2 1 e2 m2 δZ(2) = − + ln 3 6π2 2 − ω 2π2 4πeγ+κ(1) µ2 Then 2 Z 1 2 2 e k Π(k ) = − 2 dαα(1 − α) ln 1 + α(1 − α) 2 2π 0 m 8.9 Some calculations 155

Figure 8.12: The Feynman diagram which contributes to the two-point correlation function of the photon ﬁeld at order e2 with tthe counter-term diagram included...

Now, we can do an analytic continuation back to Minkowski space by putting k0 → −ik0. The result is

2 Z 1 2 2 e k − iε Π(k ) = − 2 dαα(1 − α) ln 1 + α(1 − α) 2 (8.69a) 2π 0 m 2 Z 1 2 2 2 e k − iε Πµν (k ) = −i(ηµν k − kµ kν ) 2 dαα(1 − α) ln 1 + α(1 − α) 2 (8.69b) 2π 0 m

where k2 can now be time-like, that is, k2 < 0. The logarithm on in the integrand has a cut singularity on the real axis which stretches along the real axis from k2 = −4m2 to k2 = −∞. We have retained the iε in order to deﬁne the integrand at the singularity. Although the last integral that remains to be done in (8.69b) is elementary, we ﬁnd it illuminating to leave this result in integral form.

8.9.2 The Dirac ﬁeld two-point function

We found in the discussion above that the Feynman integral corresponding to the Feynman diagram in ﬁgure 8.14 is Z d4 p Σ (k) = e2 γ µ g (k − p)γν ∆ (p) − δZ(2)[i/k] − δm(2)δ (8.70) ab (2π)4 ac cd db µν 2 ab ab or, explicitly, with the Green functions Z d4 p i/k − i/p − m −iη (k) = e2 µ ν µν − Z(2)i/k − m(2) Σ 4 γ 2 2 γ ν δ 2 δ (8.71) (2π) (k − p) + m − iε pν p − iε where we have chosen to use the Feynman gauge, ξ = 1. We will attempt to do this integral. First of all, we observe that the integral will be divergent and it must be regularized. We will use dimensional regularization. We will also do a Wick rotation. The “Wick rotation” replaces k0 by ik0 and p0 by ip0. The result is Z d2ω p i/k − i/p − m Σ(k) = e2µ4−2ω γ µ γ − δZ(2)i/k − δm(2) (2π)2ω [(k − p)2 + m2][p2] µ 2 Z d2ω p −(2ω − 2)i(/k − /p) − 2ωm = e2µ4−2ω − δZ(2)i/k − δm(2) (2π)2ω [(k − p)2 + m2][p2] 2 156 Chapter 8. Quantum Electrodynamics

Figure 8.13: The Feynman diagrams which contributes to the one-particle irreducible two-point correlation function of the Dirac ﬁeld at order e2, including the counter-terms.

Figure 8.14: The Feynman diagram which contributes to the one-particle irreducible two-point correlation function of the electron at order e2 . 8.9 Some calculations 157

µ ν ν µ where we have used γ γ γµ = (2ω − 2)γ and γ γµ = 2ω in 2ω dimensions. The space-time metric is now Euclidean, that is, p2, rather than standing for ~p2 −(p0)2 now stands for ~p2 +(p0)2. Also, in /p or /k, what was 0 0 0 0 0 0 0 /p =~γ · p−γ p is replaced by ~γ ·~p+γˆ p where γˆ = −iγ .The euclidean space Dirac matrices γˆµ = (~γ,γˆ ) obey the euclidean space algebra γˆµ ,γˆν = 2δµν . We will undo this Wick rotation later. We will also drop the hats from the euclidean gamma matrices and simply remember that when we are in euclidean space we use euclidean matrices, when we are in Minkowski space, we use the Minkowski space gamma-matrices. We combine the denominators using the Feynman parameter formula to get

Z 1 Z d2ω p −(2ω − 2)i(/k − /p) − 2ωm (k) = e2 4−2ω d − Z(2)i/k − m(2) Σ µ α 2ω 2 2 2 2 δ 2 δ (8.72) 0 (2π) [(p − αk) + α(1 − α)k + αm ] Now, we change variables, p → p + αk and drop the term that is linear in p in the numerator as its integral must vanish due to euclidean rotation invariance of the rest of the integrand. We get

Z 1 Z d2ω p −(2ω − 2)i(1 − α)/k − 2ωm (k) = e2 4−2ω d − Z(2)i/k − m(2) Σ µ α 2ω 2 2 2 2 δ 2 δ (8.73) 0 (2π) [p + α(1 − α)k + αm ] We can use this dimensional regularization formula to do the integral over p, The result is Γ[2 − ω] Z 1 −(2ω − 2)i(1 − α)/k − 2ωm (k) = e2 4−2ω d − Z(2)i/k − m(2) Σ µ ω α 2 2 2−ω δ 2 δ (4π) 0 [α(1 − α)k + αm ] Then, we can study our integral near four dimensions. In an asymptotic expansion around four dimensions, we obtain e2 1 4πµ2e1−γ m4 k2 m2 Σ(k) = −i/k + ln + − 1 ln + 1 − (4π)2 2 − ω m2 k4 m2 k2  3  e2  1 4πµ2e 2 −γ m2 k2  (2) / (2) − m 2 + ln 2 − 2 + 1 ln 2 + 1 − δZ2 ik − δm 4π 2 − ω m k m 

1 The ultraviolet divergence of the Feynman integral is reﬂected in the appearance of the pole ∼ 2−ω at dimension four. We will determine the counter-terms so that both parts vanish when we put k2 → −m2,

e2 1 4πµ2e1−γ δZ(2) = + ln + 1 2 (4π)2 2 − ω m2  3  2 2 2 −γ (2) e  1 4πµ e  δm = −m 2 + ln 2 4π 2 − ω m 

The result is

e2 m4 k2 − iε m2 Σ(k) = −i/k − 1 ln + 1 − − 1 (4π)2 (k2 − iε)2 m2 k2 − iε e2 m2 k2 − iε + m + 1 ln + 1 (8.74a) 4π2 k2 − iε m2 e2 m4 k2 − iε m2 S−1(k) = −i/k 1 + − 1 ln + 1 − − 1 (4π)2 (k2 − iε)2 m2 k2 − iε e2 m2 k2 − iε − m 1 − + 1 ln + 1 (8.74b) 4π2 k2 − iε m2

We have done the inverse Wick rotation to go back to Minkowski space, that is we put k0 → −ik0 so that k2 now has the Minkowski signature, k2 =~k2 − (k0)2. k2 can now be negative and the functions that we have found have singularities in the region where k2 < 0. We have therefore included the iε everywhere that it should be placed in order to help to deﬁne these singularities. 158 Chapter 8. Quantum Electrodynamics

8.9.3 Traces of gamma matrices In computations with Dirac ﬁelds we are usually faced with the task of computing traces of gamma-matrices. Here, we shall summarize some facts about these traces. First of all, we observe that the trace of any odd number of gamma-matrices must vanish. To see this, consider Trγ µ1 ...γ µ2n+1 = Tr(γ5)2γ µ1 ...γ µ2n+1 = −Trγ5γ µ1 ...γ µ2n+1 γ5 = −Trγ µ1 ...γ µ2n+1 = 0 where we have used (γ5)2 = 1 and the fact that γ5 anti-commutes with all of the gamma-matrices. Then to take a trace of an even number of gamma matrices, we observe that Trγ µ1 γ µ2 ...γ µ2n = 2gµ1µ2 Trγ µ3 ...γ µ2n − Trγ µ2 γ µ1 γ µ3 ...γ µ2n µ and we continue this process until γ1 is on the far right of the trace. Then we use cyclicity of the trace to bring it back to the left where it began. Since the number of interchanges will have been odd, the result is the equation n Trγ µ1 γ µ2 ...γ µ2n = ∑(−1)i−1η µ1µi Trγ µ2 ...( i0th missing )...γ µ2n i=2 Then, we continue until the product is completely reduced. The result is a factor of four times a sum over all distinct pairings of the indices of Trγ µ1 γ µ2 ...γ µ2n = 4 ∑ (−1)# ∏ η µiµ j pairings pairs where # is the number of exchanges of neighbours that is needed, beginning with the original order of the gamma matrices, to put all of the pairs in a given pairing next to each other (where you never exchange two members of a pair). For example Trγ µ γν = 4η µν (8.75) Trγ µ γν γλ γρ = 4η µν ηλρ − 4η µλ ηνρ + 4η µρ ηλν (8.76)

We ahall also often use euclidean space gamma-matrices. For them, the Minkowski space metric ηµν which appears in the formulae above is simply replaced by the Euclidean space metric δµν .

8.9.4 Feynman Parameter Formula Feynman parameters are useful in doing Feynman integrals with multiple denominators. The general formula is

1 n ν1−1 νn−1 1 Γ[ν + ... + ν ] Z dα ...dαnδ (1 − ∑ αi)α ...α = 1 n 1 i=1 1 n (8.77) ν1 νn ν1+...+νn D1 ...Dn Γ[ν1]...Γ[νn] 0 [α1D1 + ... + αnDn] We prove this formula as follows. Begin with the Schwinger representation of the left-hand-side, which uses the integral representation of the Euler gamma function (8.79) 1 1 Z ∞ = d ν−1e−λD ν λλ D Γ[ν] 0 where the D-dependence on the left-hand-side is recovered by scaling the integration variable λ. Applying this formula leads to 1 n 1 Z ∞ dα = i e−αiDi ν1 νn ∏ 1−νi D1 ...Dn 1 Γ[νi] 0 α Z ∞ n Z ∞ d 1 αi −αiDi = dλ e δ(λ − αi) ∏ Γ[ν ] 1−νi ∑ 0 1 i 0 αi Z ∞ d n Z ∞ d λ 1 αi −λαiDi = e δ(1 − αi) 1−∑i νi ∏ 1−νi ∑ 0 λ 1 Γ[νi] 0 α 1 ν1−1 νn−1 Γ[ν + ... + ν ] Z dα ...dαnδ(1 − ∑αi)α ...α = 1 n 1 1 n ν +...+ν Γ[ν1]...Γ[νn] 0 [α1D1 + ... + αnDn] 1 n 8.9 Some calculations 159

R ∞ In the second line above, we have inserted 1 = 1 dλδ(λ − ∑αi) into the integrand. In the third line above, we have re-scaled all of the α0s by λ. In the fourth line above, we have integrated over λ to produce an Euler gamma-function. An application of this formula to an expression with two denominators, for example, is

1 Z 1 1 2 2 2 2 = dα 2 2 2 2 [(p − k) + m ][p + m ] 0 [(p − αk) + α(1 − α)k + m ]

This formula is useful since the eventual integration over p can be made more symmetric by translating the integration variable, p → p + αk.

8.9.5 Dimensional regularization integral In this subsection, we will derive the following integral formulae:

in Minkowski space (8.78a) Z d2ω p 1 Γ[A − ω] 1 = i (8.78b) (2π)2ω [p2 + m2 − iε]A (4π)ω Γ[A] [m2]A−ω in Euclidean space (8.78c) Z d2ω p 1 Γ[A − ω] 1 = (8.78d) (2π)2ω [p2 + m2]A (4π)ω Γ[A] [m2]A−ω

which is very useful in doing loop integrals with dimensional regularization. We can derive these formulas as follows. We will concentrate on the Minkowski space one (8.78b) We begin with the integral

Z d2ω p 1 dp 2ω−1 0 2 2 2 A (2π) [−p0 +~p + m − iε]

We first perform a “Wick rotation” of the integration of p0. This begins with the observation that, the integrand is analytic in the first and third quadrants in the complex p0-plane. We add to this contour the quarter-circles at the boundaries of the first and third quadrants and add and subtract the integration along the imaginary axis. Then, the integral over the closed loop consisting of the real axis, the quarter-circles at the boundaries of the first and third quadrants and the imaginary axis vanishes, as this contour encloses no poles. That remains is an integral from −∞ to ∞ along the imaginary axis. Now, wherever it appears, 2 2 2 2 2 p = p0 + p1 + p2 + p3 and the integration measure gets a factor of i.

2ω 2ω Z d p 1 i Z ∞ ds Z d p 2 2 i = e−s(p +m ) 2ω 2 2 A 1−A 2ω (2π) [p + m ] Γ[A] 0 s (2π) Z ∞ Z 2 ω i ds −sm2 d p −sp2 = 1−A e 2 e Γ[A] 0 s (2π) i Z ∞ ds 2 Γ[ω − A] 1 = e−sm = i ω 1+ω−A ω 2 A−ω (4π) Γ[A] 0 s (4π) Γ[A] [m ]

where we have used the integral representation of Euler’s gamma function,

Z ∞ ds −s Γ[x] = 1−x e (8.79) 0 s

Remember that, for n = 1,2,3,..., Γ[n] = (n−1)! and zΓ[z] = Γ[z+1]. Also, it has poles at x = 0,−1,−2,... 1 1 2 π2 and, Γ(z) = z +γ + 2 γ + 6 +... with γ = 0.57721566490153286060... the Euler-Mascheroni constant. 160 Chapter 8. Quantum Electrodynamics

8.10 Quantum corrections of the Coulomb potential The Coulomb interaction between electric charges will be modified by quantum effects in quantum electrodynamics. In order to analyze how this comes about, we can ask the question as to how the energy is modified if we introduce a time-independent classical charge distribution. We want to find the energy of the lowest energy state in the presence of the charge distribution compared to the energy of the vacuum state in the absence of the charge distribution. We couple a classical charge distribution to quantum electrodynamics by adding a term to the Hamiltonian. We will denote the Hamiltonian of quantum electrodynamics by H and R 3 0 0 the additional Hamiltonian would be H˜ = e d xA0(x)J (~x) where J (~x) is the classical charge distribution. We can answer this question in perturbation theory. The first order perturbation vanishes,

δE =< O|H˜ |O >= 0

Z ∞ Z 4 Z ∞ 2 2 d k i~k·(~x−~y)−ik0t 0 ~ V(|~x −~y|) = −ie dtD00(~x,t;~y,0) = −ie 4 dte D00(k ,k) −∞ (2π) −∞ 3 Z d k ~ = −ie2 eik·(~x−~y)D (0,~k) (2π)3 00

2 We could try this expression out for the free photon where D = ∆ in that case −ie2∆ (0,~k) = e and 00 00 00 ~k2 Z 3 2 2 d k i~k·(~x−~y) e e V0(|~x −~y|) = e = (2π)3 ~k2 4π|~x −~y| which is the classical Coulomb potential. The Coulomb law for the electric interaction of two charged particles is obtained from the time components of the photon two-point function

2 2 0 e −ie D00(k = 0,~k) = ~k2[1 − Π(~k2)] At distance scales much longer than the Compton wave-length of the electron, we can use the expansion in equation (8.69a) to get

2 2 0 e − ie D00(k = 0,~k) = ~k2[1 − Π(~k2)] e2 = 2 ~ 2 ~ 4 ~ 6 ~k2 1 − e 1 k − 1 k + 1 k + ... 2π2 30 m2 280 m4 1890 m6 " # ! e2 1 e2 ~k2 1 e2 2 1 e2 ~k4 = 1 + + − + ... ~k2 30 2π2 m2 900 2π2 280 2π2 m4 8.10 Quantum corrections of the Coulomb potential 161

The ﬁrst correction to the Coulomb potential is a delta function, that is, a contact interaction,

e2 e4 V(r) = + δ 3(~r) + ... + ... 4πr 60π2m2 which is called the Uehling term. As a perturbation, it affects only the s-wave atomic orbitals and it accounts for about -50 MHz of the 1000 MHz Lamb shift. Another interesting limit is the high-energy limit where momenta are much greater than the electron mass, where using (8.69b) gives us

e2 −ie2D (~k2) = (8.80) 00 h 2 ~ 2 i ~k2 1 − e ln k + ... 12π2 m2

In the very short distance limit, the logarithm in the denominator could be large and it could compensate for 2 ~ 2 the small size of the coupling constant in the product e ln k . The theory becomes non-perturbative there. 12π2 m2 What is more, there is a value of~k2 for which the expression in equation (8.81) has a pole This is called the “Landau pole” and it is evidence that the high energy cutoff cannot be entirely removed from quantum electrodynamics. In this dimensional regularization that we have used, it means that On the positive side, this pole appears at a phenomenally high energy which is beyond any accessible scale, far beyond the energies where we have any right to know that quantum electrodynamics should be a consistent theory. It is interesting to deﬁne a running coupling constant, e2(µ). This is a coupling constant which would be the strength of the Coulomb interaction at the wave-vector scale~k2 = µ2,

e2 e2( ) = µ h 2 i 1 − e2 ln µ + ... 12π2 m2 and, consequently,

e2(µ) −ie2D (~k2) = (8.81) 00 k2 he differential dependence of the running coupling constant on the scale is encoded in the beta function

d e3(µ) β = µ2 e(µ) = + ... dµ2 12π2 where the dots denote higher order corrections. In terms of the ﬁne structure constant, which in our natural units is deﬁned as α = e2/4π, the beta function is

d 2α2 β(α) = µ2 α = + ... dµ2 3π

Once the beta function is known, the comparison of the coupling constant at two differing scales is given by

2 Z e(µ2) µ2 de ln 2 = µ1 e(µ1) β(e)

The solution of the differential equation for the beta function is a comparison of the couplings at two different scales

2 1 1 1 µ1 2 − 2 = 2 ln 2 + ... e (µ2) e (µ1) 12π µ2

The beta function of electrodynamics is positive. This means that the Landau pole is unavoidable. 162 Chapter 8. Quantum Electrodynamics

8.11 Renormalization In the sections above, we have chosen the counterterms in such a way that the Dirac ﬁeld and the photon two-point functions have a pole and residue identical to the free ﬁeld two-point functions, that is, in

Z d4 p < O|T ψ(x)ψ¯ (y)|O >= eip(x−y)S(p) (8.82a) (2π)4 1 S(p) = 2 2 (8.82b) −i/p[1 + Σ1(p )] − m[1 + Σ2(p )] + iε Z d4k < O|T A (x)A (y)|O >= eik(x−y)D (k) (8.82c) µ ν (2π)4 µν −i k k D (k) = µ ν + µν ξ (k2 − iε)2 Z d4k k k −i 1 + eik(x−y) η − µ ν (8.82d) (2π)4 µν k2 − iε k2 − iε 1 + Π(k2) 2 2 2 2 2 Σ1(p → −m ) = 0, Σ2(p → −m ) = 0, Π(k → 0) = 0 (8.82e)

we have used the counterterms to ensure that the three renormalization conditions in equation (8.82e) are satisfied to order e2. This fixes the pole and the residue of the pole in each two-point-function to be identical to those for the non-interacting Dirac and photon fields, respectively,

S(p) = g(p) + ﬁnite as p2 → −m2 (8.83) k k D (k) = ∆ (k) + η − µ ν · ﬁnite as k2 → 0 (8.84) µν µν µν k2 − iε

This was achieved by adjusting all of the available counterterms to order e2. Moreover, we could continue to compute higher orders, order by order in perturbation theory, at each order adjusting the counterterms so that the two-point functions had the forms in equations (8.83) and (8.84). Once we have done this “renormalization”, anything else that we compute is a prediction of the theory. 3 Consider, for example, the order e contribution to the three-point function < O|T Aµ (w)ψ(x)ψ¯ (y)|O >. Its momentum space expression is given by Z dkdpdq ikx+ipy+iqz 4 < O|T A (x)ψa(y)ψ¯ (z)|O >= e Γ (k, p,q)(2π) δ(k + p + q) (8.85) µ b (2π)12 µab

and we deﬁne an irreducible three-point function Gµab(k, p,q) by the equation ν Γµab(k, p,q) = Dµ (k)Sac(p)Gνcd(k, p,q)Sdb(−q) (8.86) (Remember that the momentum q is directed inward, toward the vertex, thus the S(−q) for the ﬁnal leg.) Then, to order e3 in perturbation theory, it is given by

Z d2ω ` G (k, p,q) = −ieµ2−ω γ + ie3µ6−3ω γλ g(p + `)γ g(−q + `)γσ ∆ (`) µab µab (2π)2ω µ λσ 2−ω −ieµ Z2γµab + ... + ... (8.87) where we have anticipated that the remaining integral is divergent in four dimensions and we have implemented dimensional regularization. Let us ﬁrst study this integral by taking the limit of it where the photon wave-vector vanishes and the Dirac ﬁeld momenta obey their mass-shell conditions, Z 2ω 2−ω 3 6−3ω d ` λ σ Gµab(0, p,−p) 2 2 = −ieµ γµab + ie µ γ g(p + `)γµ g(p + `)γ ∆ (`) p =−m (2π)2ω λσ 2−ω −ieµ Z2γµab + ... (8.88) 8.11 Renormalization 163

/ −1 1 ∂ We remember that g(p + `) = −i/p − i` − m and we observe that g(p + `)γµ g(p + `) = i ∂ pµ g(p + `). This allows us to write the right-hand-side of the above equation as

∂ G ( , p,−p) = e 2−ω S−1(p) µab 0 p2=−m2 µ µ (8.89) ∂ p p2=−m2

where S(p) is the Dirac ﬁeld two-point function. What is more, the renormalization condition for this two-point function implies that its derivative, when we put the momentum on-shell, is given by the free ﬁeld limit

Gµab(0, p,−p) p2=−m2 = −ieγµab (8.90) which can be taken as defining e as the electric charge of the Dirac field. We emphasize that this is a result of the renormalization that we have done so far, the definition of e follows from our determining the counterterms so that the two-point functions have the forms in equations (8.83) and (8.84). Equation (8.89) that we found be explicit computation to order e3 is not an accident, indeed it is a result of symmetry. The results of symmetry can be encoded into some identities for correlation functions called the Ward-Takahashi identities. Equation (8.89) is one of those identities. We will derive it in the next subsection. What it tells us about renormalization is that if, order by order in perturbation theory, if we adjust the counterterms δZ2,δZ3,δm so that the two-point functions have the forms given by equations (8.83) and (8.84), the three-point function will also be renormalized and it will obey the equation (8.90).

8.11.1 The Ward-Takahashi identities To ﬁnd the Ward-Takahashi identities, we recall the form of the gauge-ﬁxed action,

Z Z ξ S = dx − 3 F (x)F µν (x) − (∂ Aµ (x))2 − iZ ψ¯ (x)∂ψ/ (x) 4 µν 2 µ 2

−i(m + δm)ψ¯ (x)ψ(x) − Z2eψ¯ (x)A/(x)ψ(x)

Under a gauge transformation, Z µ δ S + dx Aµ (x)J (x) + η¯ (x)ψ(x) + ψ¯ (x)η(x) Z 2 ν ν = dxχ(x) ξ∂ ∂ν A (x) − ∂ν J (x) + ieη¯ (x)ψ(x) − ieψ¯ (x)η(x)

The measure in the functional integral should be invariant under this change of variables. Therefore, upon doing this inﬁnitesimal gauge transformation to the integration variables in the functional integral, and noting that the result of the integral cannot depend on the fuction χ(x) in the change of variables, we obtain the identity for the generating functional

 1 δ δ δ ξ∂ 2∂ − ∂ Jν (x) + ieη¯ (x) − ieη(x) Z[A,η,η¯ ] = 0 (8.91a) µ i δJµ (x) ν δη¯ (x) δη(x)

This is a functional version of a collection of identities for correlation functions that are called the Ward- Takahashi identities. These identities have some interesting implications. For example, upon taking a functional derivative 1 δ and putting the sources to zero, we obtain an identity for the photon two-point function, i δJλ (y) 1 ∂ 2∂ < O|T Aν (x)A (y)|O >= −i ∂ δ(x − y) (8.92) ν λ ξ λ This tells us that the longitudinal part of the full photon two-point function is identical to the longitudinal part of the free photon two-point function ∆νλ (x,y). The upshot is that, if we do all of our computations with a regularization of the quantum ﬁeld theory which respects gauge and Lorentz invariance, the parameter 164 Chapter 8. Quantum Electrodynamics

ξ does not renormalize. Indeed, the dimensional regularization, which have used in our examples, is gauge 1 µ 2 and Lorentz invariant. This justifies our omitting a counter-term for the gauge fixing term −ξ 2 (∂µ A (x)) in the action. Taking more derivatives by J and setting the sources to zero finds, for n ≥ 2,

2 ν −∂ ∂ν < O|T A (x)Aλ1 (y1)...Aλn (yn)|O >= 0, n ≥ 2 which, together with symmetry of the time ordered product, tells us that four and higher-point photon correlation functions obey

µ1 k1 Γµ1...µr (k1,k2,...,kr) = 0, r ≥ 3 (8.93)

Since Γµ1...µr (k1,k2,...,kr) is a completely symmetric, (8.93) implies that its contraction on the appropriate index on any of its momenta vanishes. Taking two functional derivatives 1 δ −1 δ and setting the sources to zero, we obtain i δη¯a(y) i δηb(z)

2 ξ∂ ∂µ < O|T Aµ (x)ψa(y)ψ¯b(z)|O >= e[δ(x − y) − δ(x − z)] < O|T ψa(y)ψ¯b(z)|O >= 0 (8.94)

Then, we plug in equations (8.85) and (8.86) and we use the identity (8.92) to get

µ −1 −1 k Gµab(k, p,q) = ieS (k + p) − ieS (p) (8.95)

A derivative by k, limits k → 0, p2 → −m4 and use of the renormalization conditions yields the identity quoted in equation (8.90).

8.12 Summary of this Chapter The gauge-fixed renormalized Lagrangian density of quantum electrodynamics, a relativistic quantum field theory with interacting Dirac fields and photons is

1 ξ L = −iψ¯ [∂/ + m]ψ − F F µν − (∂ Aµ )2 − eψ¯ A/ψ 4 µν 2 µ δZ − iδZ ψ¯ ∂ψ/ − iδmψψ¯ − 3 F F µν − δZ eψ¯ A/ψ 2 4 µν 2 where m is the Dirac Fermion mass, e is the Dirac Fermion charge

∞ 2 n δm = ∑(e ) δmn 1 ∞ ∞ 2 n 2 n δZ2 = ∑(e ) δZ2,n , δZ3 = ∑(e ) δZ3,n 1 1 and the renormalization conditions are, for the renormalized photon and Dirac ﬁeld two-point correlation functions:

−1 2 2 S (k) = −[i/k + m] 1 + (O(k + m ) −1 −1 2 2 2 Dµν (k) = ∆µν (k) − i(k ηνν − kµ kν /k ) O(k ) We also ﬁnd that, once the renormalization conditions are obeyed, the irreducible vertex function obeys

Gµ (0, p,−p)|p2=−m2 = −ieγµ Our goal is to calculate the correlation function

< O|T Aµ1 (x1)...Aµ (xr)ψa1 (y1)...ψas (ys)ψ¯b1 (z1)...ψ¯bt (zt )|O >

2 as an asymptotic expansion in e . The points x1,...,xr,y1,...,yr,z1 ...,zt which are arguments of the ﬁelds in the correlation function are called “external vertices”. We will call the points at which interactions are located, w1,...,wm the “internal vertices”. To compute the n’th order of perturbation theory, we follow the rules: 8.12 Summary of this Chapter 165

1. A Dirac field green function is represented by an oriented solid line. An example is depicted in figure 8.1. 2. A photon green function is represented by an un-oriented wiggly line. An example is depicted in figure 8.2. 3. A vertex is denoted by a location where a Dirac field line intersects a single photon line. An example is depicted in figure 8.3. 4. We draw and label the internal vertices in the centre of the diagram. They are the points labeled by w1,...,wm. Each internal vertex will both absorb and emit a solid line and it will be an endpoint of a wiggly line. 5. We will then generate a number of drawings. In each drawing, we depict one of the possible pairings of the points in the diagram by connecting the paired points with their appropriate lines. Only certain pairing are allowed. Pairs must consist of either two points which absorb and emit wiggly lines or a point which emits a solid line with a point which absorbs a solid line. All other pairings are forbidden. The line connecting a pair of points which both emit wiggly lines must be a wiggly line. The line which connects a pair of points which emit and absorb a solid line must be a solid line with orientation beginning at the point which emits a solid line and ending at the point which absorbs a solid line. Each distinct pairing will generate its own Feynman diagram. Each diagram which we generate will be a contribution to the correlation function which we are computing. This procedure gives us a systematic enumeration of the diagrams. 6. There are a few simplifications. (a) Diagrams which have a sub-diagram which is not connected to any external legs is called a diagram with a “vacuum bubble”. Diagrams with vacuum bubbles cancel the contributions of the denominator in the functional integral formula for the correlation function. Thus we can ignore vacuum bubbles and we can ignore the denominator (Goldstone’s theorem). (b) Any closed solid line (Dirac field loop) which emits an odd number of photons can be discarded (Furry’s theorem). (c) All diagrams with the same topology will have the same contribution. If these can be identified, their multiplicity should be recorded as a factor in front of the eventual contribution of the 1 diagram. In quantum electrodynamics, this multiplicity always cancels the n! factor which occurs in the n’the order of perturbation theory. In some other field theories it could sometimes cancel that factor only partially. 7. Then, once we have the topologically distinct diagrams, we must form the analytic expression that each one corresponds to. For this, we will go to momentum space where we are then evaluating perturbative contributions to

Γµ1...µra1...asb1...bt (p1,..., pr,q1,...,qs,`1,...,`t )

In this correlation function, all momenta p1,..., pr,q1,...,qs,`1,...,`t are taken as incoming to the Feynman diagrams irrespective of the orientation of the incoming lines. To each topologically distinct Feynman diagram, we (a) Assign momenta to all of the external and internal lines in the diagram. For all lines, both solid and wiggly, there is a definite direction to the momentum flow. Normally, for solid lines, it will be in the direction of their orientation. The momenta going into the diagram from each external vertex and the index are equal to the momentum and index for the corresponding entry in Γ. These momenta are chosen as they are depicted in figure 8.8. The momenta of internal lines are constrained so that each of the vertices conserves momentum. (b) Each line in the Feynman diagram corresponds to a Green function. Each solid line is an Dirac field Green function [i/k − m] g (k) = −[i/k + m − iε]−1 = ab ab ab k2 + m2 − iε where k is the momentum of the line flowing from its initial to final points, its initial point has Dirac index a and its final point has Dirac index b. Each wiggly line is a photon Green function

 η 1 k k (k) = −i µν − − µ ν ∆µν ν 1 ν 2 kν k − iε ξ (kν k − iε) 166 Chapter 8. Quantum Electrodynamics

where k is the momentum of the line, one endpoint has Lorentz index µ and the other endpoint has Lorentz index ν. (c) The contribution has an overall factor of (−ie)n if we are computing the n’th order and n is the number of internal vertices. (d) The vertices contribute factors of γλ1 ...γλn , one for each internal vertex. A Dirac field line c1d1 cndn ending at the vertex is represented by a Green function which contracts its second Dirac index with the c-type index of the vertex. An Dirac field line beginning at the vertex is represented by a Green function which contracts its first index with the d-type index of the vertex. (e) Conservation of momentum at all of the vertices constrains the momenta of internal lines. The number of independent momenta is equal to the total number of internal lines minus the number of vertices. These remaining momenta must be integrated. They are called “loop momenta” as they are associated with internal loops of the Feynman diagram. For each of them, we write the 4 integration R d k . (2π)4 (f) The remaining integrations must be done. This is usually accomplished by various analytic and numerical techniques. (g) Then, we must also find the diagrams with counterterms. These are composed of all of the lower order contributions to the same correlation functions where we i. replace a Dirac field Green function

2q 2q g(k) → g(k)[i/kδZ2,qe + δM2,qe ]g(k)

ii. replace a photon Green function

2q ∆(k) → ∆(k)[−δZ3,qe ]∆(k)

iii. We replace a vertex

µ 2q µ γ → δZ1,qe γ

We do this in all possible ways with all possible counterterms so that the order of e2 totals n. We will then use the renormalization conditions (??)-(??) to determine the counterterms. 9. Formal developments

9.1 In-ﬁelds, the Haag expansion and the S-matrix The ﬁeld equations of quantum electrodynamics

2 2 − ∂ Aµ (x) = eψ¯ (x)γµ ψ(x) + δZ2eψ¯ (x)γµ ψ(x) + δZ3∂ Aµ (x) ∂/ + m ψ(x) = ieA/(x)ψ(x) + ieδZ2A/(x)ψ(x) − δZ2∂ψ/ (x) − δmψ(x) h ←− i ←− ψ¯ (x) − ∂/ + m = −ieψ¯ (x)A/(x) − ieδZ2ψ¯ (x)A/(x) + δZ2ψ¯ (x) ∂/ − δmψ¯ (x)

could be rewritten as integral equations,

Z in 2 Aµ (x) = Aµ (x) + dy∆R(x,y) eψ¯ (y)γµ ψ(y) + δZ2eψ¯ (y)γµ ψ(y) + δZ3∂ Aµ (y) (9.1) Z in ψ(x) = ψ (x) + dyGR(x,y) ieA/(y)ψ(y) + ieδZ2A/(y)ψ(y) − δZ2∂ψ/ (y) − δmψ(y) (9.2) Z ←− in h i ψ¯ (x) = ψ¯ (x) + dy −ieψ¯ (y)A/(y) − ieδZ2ψ¯ (y)A/(y) + δZ2ψ¯ (y) ∂/ − δmψ¯ (y) G¯R(y,x) (9.3)

where we have used retarded Green functions

2 0 0 − ∂ ∆R(x,y) = δ(x − y) , ∆R(x,y) = 0 when x < y 0 0 ∂/ + m GR(x,y) = δ(x − y) , GR(x,y) = 0 when x < y h ←− i G¯R(y,x) − ∂/ + m = δ(x − y)

and we call the solutions of the homogeneous (free) wave equations the in-ﬁelds. They must obey the free wave equations

2 in ∂ Aµ (x) = 0 ∂/ + mψin(x) = 0 h ←− i ψ¯ in(x) − ∂/ + m = 0 168 Chapter 9. Formal developments

The Heisenberg representation ﬁelds approach the in-ﬁelds as their time arguments approach −∞,

in Aµ (x) = weak lim Aµ (x) x0→−∞ in ψa (x) = weak lim ψa(x) x0→−∞ in ψ¯b (x) = weak lim ψ¯b(x) x0→−∞

where “weak lim” means the limit of the expectation values of the operators in properly normalized quantum states. The in-fields are free fields whose Fock space we can easily construct. Moreover, the gauge fixing µ µ in constraint, ∂ Aµ (x) = 0 implies the same for the in-field ∂ Aµ (x) = 0. The Heisenberg representation fields have the equal-time commutation relations ∂ 0 0 1 Aµ (x), 0 Aν (y) δ(x − y ) = iηµν δ(x − y) ∂y 1 + δZ3 n † o 0 0 1 ψa(x),ψb (y) δ(x − y ) = δabδ(x − y) 1 + δZ2 The role of the counter-terms in the integral representation of the field equations (9.1)-(9.3) is to cancel contributions to the interaction terms which satisfy the free field equations (and would therefore be singular when integrated against the Green functions). As well, they are chosen to cancel singularities due to ultraviolet divergences that are encountered in the products of the composite operators in the interaction. What is more, we have chosen the counter-terms in such a way that the pole and residue in the two-point functions are identical to those in the free field two-point functions. This will eventually tell us that the in-fields have the commutation relations,

 ∂ Ain(x), Ain(y) δ(x0 − y0) = iη δ(x − y) µ ∂y0 ν µν n in in† o 0 0 ψa (x),ψb (y) δ(x − y ) = δabδ(x − y)

If we use perturbation theory and iterate equations (9.1)-(9.3), we obtain the Heisenberg representation fields as functionals of the in-fields. This representation of the fields is called the Haag expansion,

in Aµ (x) = Aµ (x)+ Z + dx dy dz ...F (x,x ,...,y ,...,z ,...) : Ain (x )...ψin (y )...ψ¯ in (z )... : ∑ 1 1 1 µν1...a1...b1... 1 1 1 ν1 1 a1 1 b1 1 (9.4) ψ(y) = ψin(y)+ Z + dx dy dz ...F (y,x ,...,y ,...,z ,...) : Ain (x )...ψin (y )...ψ¯ in (z )... : (9.5) ∑ 1 1 1 aν1...a1...b1... 1 1 1 ν1 1 a1 1 b1 1 ψ¯ (z) = ψ¯ in(z)+ Z + dx dy dz ...F (z,x ,...,y ,...,z ,...) : Ain (x )...ψin (y )...ψ¯ in (z )... : (9.6) ∑ 1 1 1 bν1...a1...b1... 1 1 1 ν1 1 a1 1 b1 1

where the summations are over all of the allowed in-field content beginning with quadratic terms. The dots : ... : denote normal ordering, that is putting all annihilation operators to the right of all creation operators. This is accompanied by the usual minus signs if the order of Fermionic operators are interchanged. The Haag expansion defines the Heisenberg representation fields as operators in the in-field Fock space.

9.2 Spectral Representation In this section, we will discuss some properties of correlation functions which hold beyond perturbation theory and which tell us about properties of the quantum ﬁeld theory. These properties are called the spectral representation. Since spectral representations are greatly complicated by the spin of particles, we begin with the simplest example of scalar operators. 9.2 Spectral Representation 169

9.2.1 Gauge invariant scalar operators There are some things that we can say about the two-point correlation function of any two operators that are independent of perturbation theory. One of them is a spectral representation1 which gives us a way of extracting information about the states of a quantum ﬁeld theory from the two-point correlation functions. Consider a gauge invariant scalar ﬁeld, O(x) and its two-point function

< O|T O(x)O(y)|O >

µν Examples of scalar operators in quantum electrodynamics are ψ¯ (x)ψ(x) and Fµν (x)F (x). Note that since the elementary fields are the vector Aµ (x) and the spinor ψ(x), a scalar operator must be a composite, that is, a local operator containing a product of two or more of the elementary fields. Similarly, a gauge invariant operator must also be a composite. Let us assume that the quantum field theory is translation invariant and that the corresponding Noether charge, which is the spatial integral of the energy momentum tensor Z Pµ = d3xT 0µ (x)

which generates space-time translations, ∂µ O(x) = −i Pµ ,O(x) which implies

µ µ O(x) = e−iPµ x O(0)eiPµ x

The total four-momentum Pµ is a Hermitian operator which has real eigenvalues. In any reasonable quantum ﬁeld theory, the spectrum of the energy P0 must be bounded from below in that its eigenvalues are greater than or equal to some real number. We assume that ground state |O > of the quantum ﬁeld theory is an eigenstate of Pµ , so that Pµ |O >= p(0)µ |O > and that it is the eigenstate of P0 which has the smallest possible eigenvalue, p(0)0. In the translation invariant vacuum state of quantum electrodynamics, we can add a constant to the Hamiltonian so that p(0)0 = 0. We also assume that the vacuum is a normalized state < O|O >= 1. Then, consider

µ µ < O|O(x)O(y)|O >= ∑ < O|O(0)|pˆ >< pˆ|O(0)|O > eipˆµ (x −y ) |pˆ> 4 Z d p µ µ ipµ (x −y ) 2 4 = 4 e ∑ | < pˆ|O(0)|O > | (2π) δ(p − pˆµ ) (2π) |pˆ> We deﬁne the spectral density

1This is sometimes called the Kallen-Lehman-Umezawa-Kamefuchi representation [uk][k][l]. 170 Chapter 9. Formal developments

We can use this expression to ﬁnd the time-ordered function,

Z d4 p < O|T O(x)O(y)|O >= G(p) (9.11) (2π)4 Z ∞ 2 2 −i G(p) = dm σ(m ) 2 2 (9.12) 0 p + m − iε This equation has profound consequences. It tells us that, as a complex function of a complex variable, −p2, the two-point function has singularities only on the positive real axis. It also determines the nature of the singularities, they must be either simple poles or cut singularities. The spectral function σ(m2). is proportional to the probability of ﬁnding a state with four-momentum of pµ in those states which are obtained by operating O(0) on the ground state, that is, in O(0)|O >. In equation (9.16) we have used Lorentz invariance of the theory which tells us that it can only depend on the momentum of those states in µ 0 the combination of the “invariant mass”, −pµ p . The θ(p ) is simply there to remind us that the ground state, with p0 = 0 is the lowest energy state of the theory, so all other states must have higher energy, that is, positive p0. If the state has just one particle, the invariant mass of the state must be ﬁxed at −p2 = m2 where m is the mass of the particle. Other states generically have continuum values of −p2. Nevertheless, as well as positivity of the energy p0, the total momentum should be time-like, −p2 ≥ 0. If the state O(0)|O > contains single particle state with mass m, the spectral function has a part which can be nonzero only when p2 = −m2. In that case, the spectral function contains a delta function,

2 2 2 2 ρ(µ ) = κδ(m − m0) + σ˜ (m )

where σ˜ (m2) comes from the sum over other states, which could be more single particle states (which would lead to more delta functions), and states with a continuum spectrum of m2. We conclude that studying two-point correlation functions and examining them for poles allows us to find the masses of the single-particle states of the quantum field theory. The masses are just given by the position of the poles in the Fourier transforms of these correlation functions. In the above arguments, demonstrated that this is so for all operators which are scalar fields. If the fields have indices that transform under Lorentz transformations, the argument is more complicated.

9.2.2 The Dirac ﬁeld In this section, we will derive and discuss the implications of the “spectral representation” of the two-point correlation function of the Dirac fermion and of the photon. The representation itself has the form

Z ∞ 2 2 2 −ρ1(µ )i/p + ρ2(µ ) G(p) = dµ 2 2 (9.13) 0 p + µ − iε

where ρ1 and ρ2 are spectral densities and G(p) is the Fourier transform of the two-point correlation function

Z 4 d p ip(x−y) < O|T ψa(x)ψ¯ (y)|O >= e G(p) (9.14) b (2π)4

Equation (9.13) contains some non-trivial information about what the two-point correlation function tells us about the quantum ﬁeld theory. The spectral functions are deﬁned by i 2 2 0 ¯ 4 3 −ρ1(−p )i/p + ρ2(−p ) θ(p ) ≡ ∑ < O|ψa(0)|n >< n|ψb(0)|O > δ (p − Pn) (2π) n

where ∑n |n >< n| is a sum over all states in the Hilbert space. The spectral function carries information about the states which can be created from the vacuum state by the electron ﬁeld operator for which < O|ψa(0)|n > 2 2 is non-zero. The intermediate states must have time-like total momenta, Pn < 0 which means that ρ1(−p ) 2 2 and ρ2(−p ) can be nonzero only when −p ≥ 0. We can show that the spectral function must obey the sum rule

Z ∞ 2 2 dµ ρ1(µ ) = 1 (9.15) 0 9.2 Spectral Representation 171 and, as well, the inequalities

2 2 2 ρ1(−p ) ≥ 0 , ρ1(−p ) ≥ m|ρ2(−p )|

From equation (9.13) we see that, if we view G(p) as a function of a complex variable −p2, G(p) is singular 2 2 2 wherever at least one of ρ1(−p ) or ρ2(−p ) have support (i.e. are non-zero), that is for all values of −Pn of the intermediate states in the correlation function for which < n|ψ¯b(0)|O > is non-zero. If that support is 2 2 discrete, in that sense that ρi ∼ δ(p +m ), have a pole singularity, otherwise, they will have a cut singularity. The cut lies on parts of the positive real axis in the complex −p2 plane. Let us consider the quantity < O|ψa(x)ψ¯b(y)|O >. Before we begin, let us remember what this quantity would be in free ﬁeld theory. There, remember that we had an expansion of the ﬁeld operator in terms of creation and annihilation operators

Z 3 d k h (+) i~k·~x−iE(k)t (−) i~k·~x+iE(k)t † i ψ(x) = 3 ψs (k)e as(k) + ψs (k)e bs (−k) (2π) 2 p where k0 = ~k2 + m2 ≡ E(k) and the positive and negative energy state wave-functions obey

0 (+) 0 (−) (−iE(k)γ + i~γ ·~k + m)ψs = 0 , (iE(k)γ + i~γ ·~k + m)ψs = 0

i i jk i i j ~ ~ The label s denotes the helicity state. With Σ = ε 4 γ ,γ , the helicity matrix is k · Σ and the positive and |~k| negative energy wave-functions are also eigenfunction of the helicity matrix with eigenvalues s = 2 and |~k| s = − 2 , |~k| |~k| ~k ·~Σ ψ(+) = ± ψ(+) , ~k ·~Σ ψ(−) = ± ψ(−) ± 2 ± ± 2 ± Being eigenstates of hermitian matrices, the helicity and the hamiltonian (in momentum space, the Dirac hamiltonian is the matrix h = −γ0~γ ·~k+iγ0m)), the eigenstates obey orthogonality and completeness relations

(+)† (+) (−)† (−) (+)† (−) ψs (k)ψs0 (k) = δss0 , ψs (k)ψs0 (k) = δss0 , ψs (k)ψs0 (k) = 0

0 (+) (+) m + iE(k)γ − i~γ ·~k ∑ ψs (k)ψ¯s (k) = s=± −2iE(k) 0 (−) (−) m − iE(k)γ − i~γ ·~k ∑ ψs (k)ψ¯s (k) = s=± 2iE(k) 4 Z d k µ ikµ (x−y) 2 2 0 < O|ψa(x)ψ¯ (y)|O >= i e [−i/k + m]δ(k + m )θ(k ) b (2π)3 4 Z d k µ ikµ (x−y) 2 2 0 < O|ψ¯ (y)ψa(x)|O >= −i e [−i/k + m]δ(k + m )θ(−k ) b (2π)3 where we have used the identity 1 δ(k2 + m2) = δ(k0 − E(k)) + δ(k0 + E(k) 2E(k) Now, let us go beyond free ﬁeld theory and ask what we can say about these quantities in the interacting ﬁeld theory, where we do not know the solution of the theory in terms of creation and annihilation operators. We can begin by representing the unit operator in the Hilbert space of many fermion-antifermion-photon states as a sum over a complete set of states, I = ∑n |n >< n|, between the operators,

< O|ψa(x)ψ¯b(y)|O >= ∑ < O|ψa(x)|n >< n|ψ¯b(y)|O > n The sum over n includes the vacuum state and all other states in the Hilbert space. We no longer know precisely what these states are, but we can assume that, if the quantum ﬁeld theory is a sensible quantum mechanical theory, such a set of states exists. All that we need is that it can be done in a way which 172 Chapter 9. Formal developments is consistent with Lorentz invariance. By conservation of charge, all of those states |n > which actually contribute to the sum, that is, those with non-zero matrix elements < n|ψ¯ |O >, must have the same fermion number as an anti-fermion. This excludes the vacuum, but it must have other states which should have 0 energies, Pn which are positive and greater than the vacuum energy (which to be Lorentz invariant, we set to zero). They should also have total momentum ~Pn. Another physical requirement is that their energy- 0 ~ momentum relation is time-like, Pn > |Pn|. µ µ Now, we use the translation invariance of the theory to recall that ψ(x) = e−iPµ x ψ(0)eiPµ x and −iP yµ iP yµ µ µ µ ψ¯ (y) = e µ ψ¯ (0)e µ . We assume that P |n >= Pn |n > and P |O >= 0 to write

µ iPnµ (x−y) < O|ψa(x)ψ¯b(y)|O >= ∑ < O|ψa(0)|n >< n|ψ¯b(0)|O > e n

We deﬁne the spectral density, σ(p), by i 0 ¯ 4 3 σab(p)θ(p ) ≡ ∑ < O|ψa(0)|n >< n|ψb(0)|O > δ (p − Pn) (2π) n The Heavyside step function simply indicates that the energies of the intermediate states are positive. Then

4 Z d p µ 0 ipµ (x−y) < O|ψa(x)ψ¯ (y)|O >= i σ (p)θ(p )e b (2π)3 ab

Lorentz invariance and some discrete symmetries of the Dirac theory tell us that σab(p), which is a 4 × 4 matrix, has two independent parts 2

2 2 σ(p) = −ρ1(−p )i/p + ρ2(−p ) (9.16)

It is non-zero for only those values of pµ which are the energies and momenta of the possible intermediate states. We then have the expression

Z d4 p Z ∞ ¯ ip(x−y) 2 2 2 0 2 2 < O|ψa(x)ψb(y)|O >= i 3 e dµ δ(−p + µ )θ(p ) −ρ1(µ )i/p + ρ2(µ ) ab (2π) 0

Z ∞ Z d4 p 2 2 / 2 ip(x−y) 2 2 0 = i dµ −iρ1(µ )∂ + ρ2(µ ) ab 3 e δ(−p + µ )θ(p ) 0 (2π) We can also derive Z ∞ Z d4 p ¯ 2 2 / 2 ip(x−y) 2 2 0 < O|ψb(y)ψa(x)|O >= −i dµ −iρ1(µ )∂ + ρ2(µ ) ab 3 e δ(−p + µ )θ(−p ) 0 (2π) 2We can expand the 4 × 4 matrix σ in the basis of 16 linearly independent Hermitian matrices made out of the 5 µ µ 5 i µ ν Dirac matrices, {I ,γ ,γ ,iγ γ , 4 [γ ,γ ]}. Any four-by-four matrix can be written as a linear superposition of these matrices with coefﬁcient which are complex numbers. To be Lorentz covariant, the coefﬁcients must be made from the i µ ν momentum pµ . Since pµ pν 4 [γ ,γ ] = 0, the remaining possibilities are

2 2 5 2 µ µ 5 σ(p) = σ0(p )I + σ1(p )γ + σ2(p )pµ γ + σ3 pµ γ γ Then, we observe that the Dirac theory has a parity symmetry. If we make the replacement

ψ˜ (t,x,y,z) = γ5γ1ψ(t,−x,y,z) , ψ˜¯ (t,x,y,z) = −ψ¯ (t,−x,y,z)γ5γ1 ,

(A˜0(t,x,y,z),A˜1(t,x,y,z),A˜2(t,x,y,z),A˜3(t,x,y,z)) =

(A0(t,−x,y,z),−A1(t,−x,y,z),A2(t,−x,y,z),A3(t,−x,y,z)) the action (and consequently the equations of motion) are invariant. This tells us that

σ(p0, p1, p2, p3) = −γ1γ5σ(p0,−p1, p2, p3)γ1γ5 and requiring this symmetry sets σ1 and σ3 to zero. What remains, with some renaming is the expression (9.16). 9.3 S-matrix and Reduction formula 173

9.3 S-matrix and Reduction formula

One of the precepts of scattering theory is the ability of the experimenter who is doing the scattering experiment to prepare in-coming states from particles which are isolated from each other and are not interacting yet. These particles are then sent on a trajectory where they approach each other, interact, scatter and then depart to be detected in a particle detector, again after they have become well-separated and are effectively non-interacting particles, except of course their interaction with the detector. Such experiments are the primary tool that physicists use to examine nature at the most fundamental level. The description of scattering by quantum field theory relies on decoupling of fields at large initial or final times. Imagine that we have studied the two-point function of a particular local operator, say the fermion field, ψ(x), and we have discovered that, in momentum space, it has an isolated pole in the positive real axis in the complex −p2-plane. Let us say that the residue of that pole is one, that is, that

Z d4 p i/p − m < O|T ψ(x)ψ¯ (y)|O >= eip(x−y) + ... (2π)4 p2 + m2 − iε

where the ellipses denote terms, possibly dependent on pµ , but which remain finite when we put −p2 → m2. Here, in principle, rather than the electron field, we could examine the two-point function of any local operator. The existence of this single-particle state implies that ψ(x) interpolates a free field in the large negative or positive time limit,

weak lim ψ(x) = ψin/out(x) (9.17) x0→−∞/+∞ (∂/ + m)ψin/out(x) = 0 (9.18)

This means that, when we operate with that operator on the vacuum, it creates a single particle, with the same quantum numbers as the operator possesses. The term “weak limit” means that the equation only holds true for matrix elements of the equation between discrete normalizable states in the Hilbert space where one takes the matrix element ﬁrst and then afterward we take the limit. The S matrix can be written in terms of in and out-ﬁelds as

←− ←− R h inµ 2 δ in δ i R δ (− / +m) ¯ in A (−∂ ) +ψ¯ (∂/+m) δψ ∂ ψ δJµ δψ¯ S = : e Z[J,η,η¯ ]e : Jηη¯ =0

9.3.1 Some intuition about asymptotic behaviour

Let us consider a photon inside a larger correlation function

< A|T Aµ (x)O1(y1)...O2(yn)|B >

We ﬁrst of all wish to project the photon ﬁeld onto a transverse wave-function of a free photon. The result is

3 ←−−! Z d x µ −ikµ x ∗µ ∂ ∂ i e ε (~k) − < A|T A (x)O (y )...O (yn)|B > q ∂x0 ∂x0 µ 1 1 2 (2π)32|~k|

and then we want to examine the difference between what this correlation function becomes as the time component of the photon field x0 goes to infinity and to minus infinity, out ~ in ~ < A|as (k)T O1(y1)...O2(yn)|B > − < A|T O1(y1)...O2(yn)as (k)|B > ←−− 3 ! Z d x µ ∂ ∂ −ikµ x ∗µ ~ = i lim − lim e εs (k) − < A|T Aµ (x)O1(y1)...O2(yn)|B > 0 0 q ∂x0 ∂x0 x →∞ x →−∞ (2π)22|~k|

where, in the last equation, we have used the fact that the wave-function obeys the free wave equation in order to write its second time deravative as a laplacian operating on it, then integrated by parts to get the full wave-operator operating on the arguments of the photon in the correlation function.

9.3.2 In and out-ﬁelds in QED

The weak large and small time limits of the Heisenberg picture operators ψ(x), ψ¯ (x) and Aµ (x) become free in0out in0out in0out ﬁelds, ψ (x), ψ¯ (x) and Aµ (x) which obey free wave equations

(∂/ + m)ψin/out(x) = 0 (9.19) ←− ψ¯ in/out(x)(− ∂/ + m) = 0 (9.20) 2 in/out − ∂ Aµ (x) = 0 (9.21) The expansion of the electron ﬁeld in terms of creation and annihilation operators begins with solutions of the Dirac equation, which, after a Fourier transform, has the form

0 (+) (−iE(k)γ + i~γ ·~k + m)ψs (k) = 0 (9.22) 0 (−) (iE(k)γ + i~γ ·~k + m)ψs (k) = 0 (9.23) p where E(k) = ~k2 + m2 and the superscript (+) or (−) denotes a positive and negative energy solution, respectively. The label s denotes the helicity. The Helicity operator is

kˆ ·~Σ

where kˆ =~k/|~k| and i h i Σi = − εi jk γ j,γk 8 The Helicity matrix commutes with the Dirac equation and the positive and negative energy wave-functions 1 1 are also eigenfunctions of helicity with eigenvalues s = 2 and s = − 2 , 1 1 kˆ ·~Σ ψ(+) = ± ψ(+) , kˆ ·~Σ ψ(−) = ± ψ(−) ± 2 ± ± 2 ± The spinors obey orthogonality and completeness relations

(+)† (+) (−)† (−) (+)† (−) ψs (k)ψs0 (k) = δss0 , ψs (k)ψs0 (k) = δss0 , ψs (k)ψs0 (k) = 0

0 (+) (+) m + iE(k)γ − i~γ ·~k ∑ ψs (k)ψ¯s (k) = 1 −2iE(k) s=± 2 9.3 S-matrix and Reduction formula 175

0 (−) (−) m − iE(k)γ + i~γ ·~k ∑ ψs (k)ψ¯s (k) = 1 2iE(k) s=± 2 The solution of the Dirac equation is

Z 3 d k h (+) i~k·~x−iE(k)t in/out ψin/out(x) = 3 ∑ ψs (k)e as (k)+ (2π) 2 1 s=± 2 (−) −i~k·~x+iE(k)t in/out† i +ψs (k)e bs (k) (9.24) where the creation and annihilation operators anti-commute and those with non-vanishing anti-commutators are

n in/out in/out† 0 o ~ ~ 0 as (k),as0 (k ) = δss0 δ(k − k ) (9.25) n in/out in/out† 0 o ~ ~ 0 bs (k),bs0 (k ) = δss0 δ(k − k ) (9.26)

Orthogonality of wave-functions tells us that

Z 3 in/out d x −ik·x+iE(k)t (+) 0 in/out as (k) = − 3 e ψ¯s (k)γ ψ (x) (2π) 2

Z 3 in/out† d x ik·x−iE(k)t in/out 0 (+) as (k) = − 3 e ψ¯ (x)γ ψs (k) (2π) 2 Z 3 in/out† d x ik·x+iE(k)t (−) 0 in/out bs (k) = − 3 e ψ¯s (k)γ ψ (x) (2π) 2 Z 3 in/out d x −ik·x−iE(k)t in/out 0 (−) bs (k) = − 3 e ψ¯ (x)γ ψs (k) (2π) 2 The photon in/out ﬁelds satisfy the relativistic wave equation. It has the expansion in creation and annihilation operators

3 Z d k h ~ Ain/out(x) = eik·~x−iωt αin/out(k) µ ∑ p 3 µ s (2π) 2ω(k) −i~k·~x+iωt in/out† i +e αµ (k) (9.27)

p where ω(k) = ~k2 and the creation and annihilation operators commute with the non-vanishing commutation relation being h in/out in/out† 0 i ~ ~ 0 αµ (k),αν (k ) = ηµν δ(k − k ) The creation and annihilation operators are obtained from the photon ﬁeld operator by Z in/out 3 ∗ −→ ←− in/out aµ (k) = i d x ψk (x) ∂ t − ∂ t Aµ (x) (9.28) Z in/out† 3 −→ ←− in/out aµ (k) = −i d x ψk(x) ∂ t − ∂ t Aµ (x) (9.29) where

e−i~k·~x+i|~k|t ψk(x) = q (9.30) (2π)32|~k| is the positive energy photon wave-function. 176 Chapter 9. Formal developments

The non-vanishing anti electron-commutation and commutation relations of the positron and photon in/out ﬁeld creation and annihilation operators are

n in/out in/out† 0 o ~ ~ 0 as (k),as0 (k ) = δss0 δ(k − k ) (9.31) n in/out in/out† 0 o ~ ~ 0 bs (k),bs0 (k ) = δss0 δ(k − k ) (9.32) h in/out in/out† 0 i ~ ~ 0 αµ (k),αν (k ) = ηµν δ(k − k ) (9.33)

The vacuum state is annihilated by all of the annihilation operators

in/out in/out in/out as (k)|O >= 0,bs (k)|O >= 0,αµ (k)|O >= 0 (9.34) Physical states are equivalence classes of Fock states which 1. are annihilated by the physical state condition

µ in/out k αµ (k)|phys >= 0 2. are equivalent, |ψ >≡ |ψ0 > if they differ by a state with zero norm, || |ψ > −|ψ0 > || = 0 A representative of each equivalence class can be chosen so that we begin with the vacuum |O >, which is a physical state, and we create all states containing photons using only the operators

in/out ∗µ in/out αs (k) ≡ εs (k)αµ (k)

µ 0 ~ where εs (k) are the two physical polarization vectors. They satisfy εs (k) = 0 and k ·~εs(k) = 0. They are ∗µ orthogonal εs (k)εs0µ (k) = δss0 and kik j εi(k)ε∗ j(k) = δ i j − ∑ s s 2 s ~k We then build the in or the out-ﬁeld Hilbert spaces by taking all superpositions of the basis vectors

{| >,ain†(k)| >,bin†(k)| >, in†(k)| >,ain†(k )bin†(k )| >,...} (9.35) O s O s O αs O s1 1 s2 2 O Both the in-fields and the out-fields can be regarded as complete in the sense that their Hilbert spaces are complete sets of states describing the theory. The in-fields and out-fields have identical properties and the spaces of states that are created by operating creation operators in their vacuum states are identical. In a certain sense, all Hilbert spaces are identical. They are infinite dimensional vector spaces with a countable basis. However, in this case the in and out-field Hilbert spaces, there is a canonical identification of the states in the two spaces which identifies states which have the same energy, momentum and electric charge. This would not be possible for free field Hilbert spaces if they were not identical. The canonical identification between the two is called the S-matrix. It is an operator S which is unitary

SS† = 1 , S−1 = S† (9.36) and which maps in-states to out states.

out † in out † in out † in ψ (x) = S ψ (x)S , ψ¯ (x) = S ψ¯ (x)S , Aµ (x) = S Aµ (x)S (9.37) Any state in the out-ﬁeld Hilbert space is a superposition of states in the in-ﬁeld Hilbert space

† |out n >= ∑Snm |in m > m To proceed, we need to make some assumptions about the behaviour of the quantum ﬁeld theory. We assume that the vacuum is stable. This means, we assume that the vacuum state |O > is also the vacuum state for the in-states and the out-states.

The vacuum state satisﬁes

S|O >= |O > (9.40)

The S-matrix summarizes the time evolution of the quantum ﬁeld theory that is needed to study scattering experiments. For example, the initial data for a scattering experiment is an in-state that is created by in-state creation operators, as a concrete example, a state with two in-coming electrons, a† (k )a† (k )| >. This ins1 1 ins2 2 O state evolves with time and eventually becomes the out-state a† (k )a† (k )| >. The out-state is a outs1 1 outs2 2 O superposition of in-states,

a† (k )a† (k )| >= S†a† (k )a† (k )| > (9.41) outs1 1 outs2 2 O ins1 1 ins2 2 O (The above equation follows from (9.37) and (9.40). ) We could ask, for example, what the probability amplitude for two electrons and nothing else to emerge from the scattering experiment. This amplitude would be

0 0 † † < O|a 0 (k )a 0 (k ) a (k1)a (k2)|O > ins1 1 ins2 2 outs1 outs2 0 0 † † † =< O|a 0 (k )a 0 (k ) S a (k1)a (k2)|O > (9.42) ins1 1 ins2 2 ins1 ins2 or

† 0 † 0 ∗ < O|ains1 (k1)ains2 (k2) S a 0 (k1)a 0 (k2) |O > ins1 ins2 (9.43)

Thus, to ﬁnd scattering matrix elements, we need to evaluate the overlap of out-states and in-states. What about one-particle states? We would expect that, having no other particles to interact with, a one-particle state would remain intact during its time evolution, that is, that a one-particle in-state evolves to a one particle out-state. Here, we are talking about stable particles. An absolutely stable particle, like the electron in quantum electrodynamics, cannot decay to other particles. An electron in isolation, and traveling with constant momentum, should remain in its state indeﬁnitely. Thus, a reasonable assumption is that one-particle in-states evolve to one-particle out-states which are identical to the one-particle in-state, that is,

† 0 ~ ~ 0 < O|aouts(k)ains0 (k )|O > = δss0 δ(k − k ) , (9.44) † 0 ~ ~ 0 < O|bouts(k)bins0 (k )|O > = δss0 δ(k − k ) , (9.45) † 0 ~ ~ 0 < O|αoutµ (k)αinν (k )|O > = δµν δ(k − k ) (9.46) These statements will have a profound inﬂuence on what we call in or out-states. They must be tuned so that they are stable. This tuning is intimately related to renormalization. Now, we shall consider some of the master identities of the LSZ formalism. Given an arbitrary local operator O(x) (or more generally a product of local operators at a set of points which we shall generically call x), the following are a set of remarkably useful identities:

Z out in 4 (+) / as (k)O(x) ∓ O(x)as (k) = − d yψ¯sk (y) ∂ y + m T [ψ(y)O(x)] (9.47)

Z h ←− i in† out† 4 / (+) O(x)as (k) ± as (k)O(x) = − d yT [O(x)ψ¯ (y)] − ∂ y + m ψsk (y) (9.48) 178 Chapter 9. Formal developments

Z h ←− i in out 4 / (−) O(x)bs (k) ± bs (k)O(x) = − d yT [O(x)ψ¯ (y)] − ∂ y + m ψsk (y) (9.49)

Z out† in† 4 (−) / bs (k)O(x) ∓ O(x)bs (k) = − d yψ¯sk (y) ∂ y + m T [ψ(y)O(x)] (9.50)

Z out in 4 ∗ 2 αµ (k)O(x) − O(x)αµ (k) = i d zψk (z) −∂z T Aµ (z)O(x) (9.51)

Z out† in† 4 2 αµ (k)O(x) − O(x)αµ (k) = −i d zψk(z) −∂z T Aµ (z)O(x) (9.52)

These identities should be understood as applying only to their matrix elements between normalizable states. For example, if in equations (9.51) and (9.52) we use O(x) = Aν (x), and then the vacuum expectation value, we get

out ∗ < O|αµ (k)Aν (x)|O >= ψk (x) (9.53)

in† < O|Aν (x)αµ (k)|O >= ψk(x) (9.54)

These equations tell us that, when the Heisenberg ﬁeld operator operates on the vacuum state, it creates a single-particle state, as well as other states, Z 3 ∗ in† Aν (x)|O >= d k ψk (x)αν (k)|O > + multi − particle states (9.55)

This is also true for the fermions, amongst all of the states that the Dirac ﬁeld creates when it operatrs on the vacuum, there is a single electron or single positron state, Z 3 (+) in† ψ¯b(x)|O >= ∑ d k [ψ¯ks (x)]bas (k)|O > + multi − particle states (9.56) s Z 3 (−) in† ψa(x)|O >= ∑ d k [ψks (x)]abs (k)|O > + multi − particle states (9.57) s

We can get another remarkable fact by taking the inner product of two states in (9.56), for example Z 3 (+) (+) < O|ψa(x)ψ¯b(y)|O >= −∑ d k [ψks (x)ψ¯ks (y)]ab + ... s Z 3 (−) (−) < O|ψ¯b(y)ψ¯a(x)|O >= −∑ d k [ψks (x)ψ¯ks (y)]ab + ... s

9.3.3 Proof of the LSZ identities In the following, we shall fashion a proof of the LSZ identities that are given in equations (9.47)-(9.52). Let us begin with equation (9.47). The left-hand-side can be rewritten as Z out in 3 (+) i~k·~y 0 −iE(k)y0 out as (k)O(x) ∓ O(x)as (k) = d yψ¯s (k)e γ e T ψ (y)O(x) Z 3 (+) i~k·~y 0 −iE(k)y0 h in i + d yψ¯s (k)e γ e T ψ (y)O(x) (9.58) 9.3 S-matrix and Reduction formula 179

On the right-hand-side of this equation, we have projected the in and out annihilation operators for the electron out of ψin(x) and ψout(x) by integrating them against the conjugate of the positive energy wave-function,

(+)† i~k·~y−iE(k)y0 (+) i~k·~y−iE(k)y0 0 ψs (k)e = −ψ¯s (k)e γ so that Z out/in 3 (+) i~k·~y−iE(k)y0 0 out/in as (k) = − d yψ¯s (k)e γ ψ (x) Note that the left-hand-sides of these last equations are independent of time. We are therefore allowed to adjust the time argument y0 inside the integral as we please. We have done this in equation (9.59) to replace ψ(y) by ψinout(y) in the appropriate places. Now, we observe that each of the two terms on the right-hand-side are independent of time, that it, inside the integrals, we can choose the time to be any value. We can do this in each one individually. In the first one we choose the time y0 near plus infinity. In the second one we choose y0 near minus infinity. Then, in the first term, we can replace ψout(y) by ψ(y) and in the second term we can replace ψin(y) by ψ(y). We will also use the time ordering to place the operator ψ(y) to the left of the operator O(x) in both terms. The plus or minus sign on the left-hand-side correspond to O(x) being a bosonic or fermionic operator. The result is

out in as (k)O(x) ∓ O(x)as (k) = Z 3 (+) i~k·~y 0 −iE(k)y0 lim d yψ¯s (k)e γ e T [ψ(y)O(x)] y0→∞ Z 3 (+) i~k·~y 0 −iE(k)y0 + lim d yψ¯s (k)e γ e T [ψ(y)O(x)] (9.59) y0→−∞ Next, we shall use the identity Z ∞ d lim f (t) − lim f (t) = dt f (t) t→∞ t→−∞ −∞ dt to re-write the expression (9.59) as Z out in 4 (+) i~k·~y 0 −iE(k)y0 as (k)O(x) ∓ O(x)as (k) = − d yψ¯s (k)e γ ∂y0 e T [ψ(y)O(x)] Z 4 (+) / = − d yψ¯sk (y) ∂ y + m T [ψ(y)O(x)] where we denote (+) (+) i~k·~y −iE(k)y0 ψ¯sk (y) ≡ ψ¯s (k)e e and, in ∓ it is minus if O(x1) is bosonic and plus if O(x1) is fermonic. In this expression we have used

−iE(k)y0 −iE(k)y0 ∂y0 e = e ∂y0 − iE(k) and the fact that (+) 0 −→ (+) 0−→ ←− ψ¯sk (y)γ ∂ y0 − iE(k) = ψ¯sk γ ∂ y0 − ∇ y ·~γ + m in order to form the Dirac operator. This brings us to Z out in 4 (+) / as (k)O(x) ∓ O(x)as (k) = − d yψ¯sk (y) ∂ y + m T [ψ(y)O(x)]

In a similar manner, we can show that Z h ←− i in† out† 4 / (+) O(x)as (k) ± as (k)O(x) = d yT [O(x)ψ¯ (y)] − ∂ y + m ψsk (y)

We have produced the two equations (9.47) and (9.48). The formulae (9.49) and (9.50) are found by an analogous procedure. Now, let us turn to equation (9.51). The creation and annihilation operators can be obtained by −−→ ←−−! Z ∂ ∂ αin(k) = i d3xψ∗(x) − Ain(x) µ k ∂x0 ∂x0 µ 180 Chapter 9. Formal developments −−→ ←−−! Z ∂ ∂ αin†(k) = −i d3xψ (x) − Ain(x) µ k ∂x0 ∂x0 µ Then, consider

out in αµ (k)O(z) − O(z)αµ (k) = ←→ Z ∂ = i d3xψ∗(x) (Aout(x)O(z) − O(z)Ain(x)) k ∂x0 µ µ Since the two terms on the right-hand-side are actually independent of time, we can write these as

out in αµ (k)O(z) − O(z)αµ (k) = ←→ Z 3 ∗ ∂ i lim − lim d xψk (x) 0 T Aµ (x)O(z) x0→∞ x0→−∞ ∂x

where T Aµ (x)O(z) is the time-ordered product. The time ordering sets the operators in the correct order when x0 is taken to plus or minus infinity. Writing the difference of infinite time limits as the definite integral of a derivative gives

out in αµ (k)O(z) − O(z)αµ (k) = ←→ Z ∂ Z ∂ i dx0 d3xψ∗(x) T A (x)O(z) ∂x0 k ∂x0 µ and taking the time derivatives yields

out in αµ (k)O(z) − O(z)αµ (k) = −→ ←− Z 2 2 ! 4 ∗ ∂ ∂ i d xψk (x) − T Aµ (x)O(z) ∂x02 ∂x02

which, using the fact that the wave-function obeys the wave equation and integrating by parts gives Z out in 4 ∗ 2 αµ (k)O(z) − O(z)αµ (k) = i d xψk (x) −∂x T Aµ (x)O(z) Z out† in† 4 2 αµ (k)O(z) − O(z)αµ (k) = −i d xψk(x) −∂x T Aµ (x)O(z)

In the second equation we obtain by a conjugation of the ﬁrst, and the obvious re-identiﬁcation of O(x). We have produced equations (9.51) and (9.52).

9.3.4 An Example: Electron-electron scattering We can use the master formulae (9.47)-(9.52) to study a few simple examples. Consider the amplitude which describes electron-electron scattering. There are two incoming electrons and two out-going electrons. The S-matrix element is

< |aout(k )aout(k )ain†(k )ain†(k )| > O s1 1 s2 2 s3 3 s4 4 O Z 4 (+) out in† in† = − d y ψ¯ (y ) ∂/ + m < O|a (k )ψ(y )a (k )a (k )|O > 2 s2k2 2 2 s1 1 2 s3 3 s4 4 + < |aout(k )ain (k )ain†(k )ain†(k )| > O s1 1 s2 2 s3 3 s4 4 O Z 4 (+) out in† in† = − d y ψ¯ (y ) ∂/ + m < O|a (k )ψ(y )a (k )a (k )|O > 2 s2k2 2 2 s1 1 2 s3 3 s4 4 + (~k −~k ) < |aout(k )ain†(k )| > − (~k −~k ) < |aout(k )ain†(k )| > δs2s3 δ 2 3 O s1 1 s4 4 O δs2s4 δ 2 4 O s1 1 s3 3 O Z 4 (+) out in† in† = − d y ψ¯ (y ) ∂/ + m < O|a (k )ψ(y )a (k )a (k )|O > 2 s2k2 2 2 s1 1 2 s3 3 s4 4 ~ ~ ~ ~ ~ ~ ~ ~ + δs2s3 δs1s4 δ(k1 − k4)δ(k2 − k3) − δs2s4 δs1s3 δ(k1 − k3)δ(k2 − k4) (9.60) 9.3 S-matrix and Reduction formula 181 so that, ﬁnally, we have

< |aout(k )aout(k )ain†(k )ain†(k )| >= O s1 1 s2 2 s3 3 s4 4 O ~ ~ ~ ~ ~ ~ ~ ~ = δs2s3 δs1s4 δ(k1 − k4)δ(k2 − k3) − δs2s4 δs1s3 δ(k1 − k3)δ(k2 − k4) Z + d4y d4y d4y d4y ψ¯ (+) (y )∂/ + mψ¯ (+) (y )∂/ + m· 1 2 3 4 s1k1 1 1 s2k2 2 2 h ←− i (+) h ←− i (+) · < O|T ψ(y )ψ(y )ψ¯ (y )ψ¯ (y )|O >c − ∂/ + m ψ (y ) − ∂/ + m ψ (y ) (9.61) 1 2 3 4 3 s3k3 3 4 s4k4 4

Where the last term contains the connected time-ordered four-point correlation function. What we have accomplished is to write the formula for an element of the S-matrix in terms of the time ordered correlation function. On the right-hand-side of this formula, we are required to “amputate” the fermion legs and attach the fermion wave-functions to the places where the legs have been removed.

Example: Photon-photon scattering

Now, in a scattering experiment, an in-state evolves to an out-state,

ain†(k )...ain†(k )| >→ aout†(k )...aout†(k )| > µ1 1 µn n O µ1 1 µn n O which is a superposition of in-states. The coefﬁcients in that superposition are elements of the S-matrix. Formally, aout†(k )...aout†(k )| >= S†ain†(k )...ain†(k )| > µ1 1 µn n O µ1 1 µn n O Z = d3 p ...d3 p ain†(p )...ain†(p )| > S†( p ,..., p ; k ,..., k ) ∑ ∑ 1 ` ν1 1 νn ` O ν1 1 ν` ` µ1 1 µn n ` ν1...ν` where the S-matrix element is

S†( p ,..., p ; k ,..., k ) =< |ain (p )...ain (p ) aout†(k )...aout†(k )| > ν1 1 ν` ` µ1 1 µn n O ν1 1 ν` ` µ1 1 µn n O or, upon taking the Hermitian conjugate,

S( k ,..., k ; p ,..., p ) =< | aout(k )...aout(k ) ain†(p )...ain†(p )| > µ1 1 µn n ν1 1 ν` ` O µ1 1 µn n ν1 1 ν` ` O We can use equation (9.51) in the S-matrix element, with O(z) = 1, to get

< | aout(k )...aout(k ) ain†(p )...ain†(p )| >= O µ1 1 µn n ν1 1 ν` ` O

< | aout(k )...aout (k )ain (k ) ain†(p )...ain†(p )| > + O µ1 1 µn−1 n−1 µn n ν1 1 ν` ` O Z i d4x ∗ (x )− 2 < | aout(k )...aout (k )A (x ) ain†(p )...ain†(p )| > nψkn n ∂xn O µ1 1 µn−1 n−1 µn n ν1 1 ν` ` O

` ~ = ∑ δ(kn −~pi)ηµnνi · i=1 · < | aout(k )...aout (k )(k ) ain†(p )...ain† (p )ain† (p )...ain†(p )| > + O µ1 1 µn−1 n−1 n ν1 1 νi−1 i−1 νi+1 i+1 ν` ` O Z i d4x ∗ (x )− 2 < | aout(k )...aout (k )A (x ) ain†(p )...ain†(p )| > nψkn n ∂xn O µ1 1 µn−1 n−1 µn n ν1 1 ν` ` O Iterating this argument gives

n Z ` Z i dx ∗ (x )− 2 i dy ∗ (y ) − 2 · ∏ iψki i ∂xi ∏ jψp j j ∂x j 1 1

· < O|T Aµ1 (xl1)...Aµn (xn)Aν1 (y1)...Aν` (ν`)|O > 182 Chapter 9. Formal developments

plus terms with some momentum delta functions and lower order correlation functions of A’s. We can use the generating functional to ﬁnd this contribution as

n Z δ ` Z δ dx ψ∗ (x )−∂ 2 dy ψ∗ (y ) −∂ 2 Z[J,η,η¯ ] ∏ i ki i xi µ ∏ j p j j x j ν j δJ i (xi) δJ (y j) 1 1 Jηη¯ =0 Now, we can observe that we can ﬁnd the above quantity as the matrix element in the same initial and ﬁnal states of the operator

1 Z δ n dxAinµ (x)− 2 Z[J, , ¯ ] : ∂ µ : η η n! δJ (x) Jηη¯ =0 where the bracket : ... : stands for normal ordering. The remaining terms with lower order correlation functions turn out to be taken care of by using the formula

R dxAinµ (x) − 2 δ ( ∂ ) Jµ (x) Sphotons = : e δ : Z[J,η,η¯ ] Jηη¯ =0

9.4 More generating functionals We have found the generating functional, Z[J,η,η¯ ] to be an important tool for encoding the data contained in the quantum ﬁeld theory which we have been studying. Functional derivatives of it by the sources yields the correlation functions. In this section, we are going to introduce two other types of generating functional, W[J,η,η¯ ], for connected correlations functions and Γ[< A >,< ψ >,< ψ¯ >] for connected one-particle irreducible correlations functions.

9.4.1 Connected correlation functions The perturbative computation of a general correlation function sums over all of the Feynman diagrams contributing to that correlation function, following a few simple rules which we have outlined in detail in the previous chapter. The connected correlation function is defined to be the sum of all connected Feynman diagrams which contribute to the correlation function. These diagrams are a subset of all of the diagrams which contribute, which can generally be classified as being either connected or disconnected. A connected Feynman diagram is defined as a Feynman diagram which has the property that one trace a path between any two vertices, internal or external in the entire diagram by continually following internal and external lines which are in the diagram. A diagram is said to be disconnected if it is not connected. Connected correlation function also have an equivalent algebraic definition which is independent of perturbation theory. This defines a multi-point function as a sum of all possible factorizations of the correlation function into connected parts. For example, the fermion four-point function can be written as a connected four-point function plus products of connected two-point functions,

< O|T ψa1 (x1)ψa2 (x2)ψ¯b1 (y1)ψ¯b2 (y2)|O >

=< O|T ψa1 (x1)ψa2 (x2)ψ¯b1 (y1)ψ¯b2 (y2)|O >C

+ < O|T ψa1 (x1)ψ¯b2 (y2)|O >C< O|T ψa2 (x2)ψ¯b1 (y1)|O >C

− < O|T ψa1 (x1)ψ¯b1 (y1)|O >C< O|T ψa2 (x2)ψ¯b2 (y2)|O >C where the subscript C denotes the connected correlation function. (Due to charge conservation, two-point func-

tions of complex fermions are automatically connected, < O|T ψa1 (x1)ψ¯b1 (y1)|O >C=< O|T ψa1 (x1)ψ¯b1 (y1)|O >.) It turns out that the inverse of this equation,

< O|T ψa1 (x1)ψa2 (x2)ψ¯b1 (y1)ψ¯b2 (y2)|O >C

=< O|T ψa1 (x1)ψa2 (x2)ψ¯b1 (y1)ψ¯b2 (y2)|O >

− < O|T ψa1 (x1)ψ¯b2 (y2)|O >< O|T ψa2 (x2)ψ¯b1 (y1)|O >

+ < O|T ψa1 (x1)ψ¯b1 (y1)|O >< O|T ψa2 (x2)ψ¯b2 (y2)|O > 9.4 More generating functionals 183 defines the connected correlation function. It turns out that this is equivalent to the perturbation theory definition. We will prove the latter statement in the following. There is a nice combinatorial formula for finding connected Feynman diagrams. If Z[J,η,η¯ ] is the generating functional for all possible correlation functions, the generating functional for connected correlation functions is simply related to it by the formula

W[J,η,η¯ ] = lnZ[J,η,η¯ ] (9.62) in that, the connected correlation functions will be given by

< O|T Aµ1 (x1)...ψa1 (y1)...ψ¯b1 (z1)...|O >C

1 δ 1 δ 1 δ = ...... W[J,η,η¯ ] (9.63) µ1 ¯ i δJ (x1) i δηa1 (y1) −i δηb1 (z1) J=η=η¯ =0

We can immediately see that this is so in the free ﬁeld theory limit, e → 0 where

R 1 µ ν dydx[− J (y)∆µν (y,z)J (z)−η¯ (y)g(y,z)η(z)] Ze=0[J,η,η¯ ] = e 2 (9.64) and

Z 1 W [J,η,η¯ ] = dydx − Jµ (y)∆ (y,z)Jν (z) − η¯ (y)g(y,z)η(z) (9.65) e=0 2 µν

This tells us that, in free ﬁeld theory, the only connected correlation functions are two-point functions. This we already know, of course, since Wick’s theorem tells us that, without interaction vertices, all correlations functions are written as products of two-point functions where the points are external vertices of the diagram. In that way. all correlation functions are written as sums of contributions where each contribution is factorized into two-point functions. Now, let us turn on the interactions. The generating functional for connected correlation functions is

eW[J,η,η¯ ] = R δ µ δ δ −e dw γ µ −R dydx 1 Jµ (y) (y,z)Jν (z)+ ¯ (y)g(y,z) (z) e δη(w) δJ (w) δη¯ (w) e [ 2 ∆µν η η ] (9.66) −eR dw δ µ δ δ R 1 µ ν (w) γ µ ¯ (w) − dydx[ J (y)∆µν (y,z)J (z)+η¯ (y)g(y,z)η(z)] e δη δJ (w) δη e 2 Jηη¯ =0

Now, consider

d W[J,η,η¯ ] de Z Z −W[J,η,η¯ ] δ µ δ δ W[J,η,η¯ ] = e −e dw γ e + i < ψ¯ A/ψ > ¯ δη(w) δJµ (w) δη¯ (w) Jηη=0

The last term in the above equation comes from the derivative of the denominator in (9.66). Its role is to d make de dW[0,0,0] = 0

d Z W[J,η,η¯ ] = i < ψ¯ A/ψ > − de Jηη¯ =0 Z δ δW δ δW δ δW dw + γ µ + + (9.67) δη(w) δη(w) δJµ (w) δJµ (w) δη¯ (w) δη¯ (w)

In the above equation, we have taken the functional derivatives of the exponential eW . The remaining functional derivatives act on whatever is to the right of them. If there is no functional to the right, they vanish. 184 Chapter 9. Formal developments

We can expand the right-hand-side as

d W[J,η,η¯ ] = de ! Z δ 3W δ 3W − dw µ − γab µ µ J (w)δηa(w)δη¯b(w) J (w)δηa(w)δη¯b(w) Jηη¯ =0 Z δW δ 2W δW δW δW − dw µ + γab µ µ δηa(w) J (w)δη¯b(w) δηa(w) δJ (w) δη¯b(w) δ 2W δW δW δ 2W + + µ µ (9.68) δηa(w)δJ (w) δη¯b(w) δJ (w) δηa(w)δη¯b(w)

If we consider the Taylor expansion of W[J,η,η¯ ] in e,

∞ n W[J,η,η¯ ] = ∑ e Wn[J,η,η¯ ] n=o

the above equation determines Wn+1 once all of W0,W1,...,Wn are known. Explicitly,

(n + 1)Wn+1[J,η,η¯ ] = ! Z δ 3W δ 3W − dw µ n − n γab µ µ J (w)δηa(w)δη¯b(w) J (w)δηa(w)δη¯b(w) Jηη¯ =0

Z n δW δ 2W δW δW δW − dw µ p n−p + p q r γab ∑ µ ∑ δn,p+q+r µ p=0 δηa(w) J (w)δη¯b(w) pqr δηa(w) δJ (w) δη¯b(w) ! n δ 2W δW n δW δ 2W + q n−q + q n−q ∑ µ ∑ µ (9.69) q=0 δηa(w)δJ (w) δη¯b(w) q=0 δJ (w) δηa(w)δη¯b(w)

Moreover, the right-hand-side contains contributions which are all connected to the point w, which is subsequently integrated. Therefore, Wn+1must be a sum of connected diagrams. Then, given that W0 is connected, the fact that all Wn are connected follows by mathematical induction.

9.4.2 One-particle irreducible correlation functions Amongst the Feynman diagrams which contribute to a connected correlation function, such as the one which we discussed in the previous section, are diagrams that are one-particle reducible. By definition, a one-particle reducible Feynman diagram is a diagram which can be separated into two disconnected parts by cutting one internal line, either a photon line or an electron line. A reducible diagram can always be made from irreducible diagrams by connecting the irreducible components with the appropriate single lines. What is more, in momentum space, in assembling a reducible diagram from irreducible components, not further momentum integrals need to be done. An irreducible correlation function is one whose perturbative computation involves only irreducible Feynman diagrams. Interestingly, such correlation functions have a definition beyond perturbation theory. We can find a generating functional for them. We begin with the classical fields that are defined as functional derivatives 1 δ < A (x) >= W[J,η,η¯ ] (9.70) µ i δJµ (x) 1 ∂ < ψa(x) >= W[J,η,η¯ ] (9.71) i ∂η¯a(x) 1 ∂ < ψ¯a(x) >= − W[J,η,η¯ ] (9.72) i ∂ηa(x)

and we consider the generating functional which is a Legendre transform of [J,η,η¯ ], Z µ Γ[< A >,< ψ >,< ψ¯ >] = W[J,η,η¯ ] − i < Aµ > J + η¯ < ψ > + < ψ¯ > η (9.73) 9.4 More generating functionals 185

Here, we are supposed to solve the equations (9.72) to ﬁnd the sources J, η and η¯ as functionals of the classical ﬁelds, < Aµ >, < η > and < η¯ < ψ > and < barψ >. Then, it is easy to show that

1 δ Γ[< A >,< ψ >,< ψ¯ >] = −Jµ (x) (9.74) i δ < Aµ (x) > <ψ>,<ψ¯ >

1 δ − Γ[< A >,< ψ >,< ψ¯ >] = −η¯a(x) (9.75) i δ < ψa(x) > .<ψ¯ >

1 δ Γ[< A >,< ψ >,< ψ¯ >] = −ηa(x) (9.76) i δ < ψ¯a(x) > ,<ψ>

To set the sources to zero, we are supposed to adjust the classical ﬁelds so that their irreducible one-point functions are zero. In quantum electrodynamics, there are generally at < A >=< ψ >=< ψ¯ >= 0, although in other ﬁeld theories, they need not be. Then, we have the generating functional for one-particle irreducible correlation functions,

1 δ < O|T Aµ1 (x1)...ψa1 (y1)...ψ¯b1 (z1)...|O >irreducible= ... i δ < Aµ1 (x1) >

1 δ 1 δ ...... Γ[< A >,< ψ >,< ψ¯ >] (9.77) ¯ i δ < ψa1 (y1) > −i δ < ψb1 (z1) > =<ψ>=<ψ¯ >=0

Now, let us consider the variation of the original action that is given by perturbing its quadratic parts

Z 1 δS = dxdy A (x)σ µν (x,y)A (y) + ψ¯ (x)τ (x,y)ψ (y) (9.78) 2 µ ν a ab b

µν Here σ (x,y) and τab(x,y) are kernels. It is easy, using the same arguments as we used in the previous section to see that the variation of the generating functional for connected correlation function varies as

Z 1 1 δ 1 δW 1 δ 1 δW δW = dxdy + σ µν (x,y) + 2 i δJµ (x) i δJµ (x) i δJν (y) i δJν (y) δ δW δ δW + + τab(x,y) + δηa(x) δηa(x) δη¯b(y) δη¯b(y) Z 1 δ 2W δW δW = dxdy σ µν (x,y) − − 2 δJµ (x)δJν (y) δJµ (x) δJν (y) δ 2W δW δW +τab(x,y) + (9.79) δηa(x)δη¯b(y) δηa(x) δη¯b(y)

Now, a functional derivative by σ or τ removes a photon or a fermion propagator. In the ﬁrst case, when we remove a photon propagator from W, the result is

δW 1 δ 2W δW δW = − − (9.80) δσ µν (x,y) 2 δJµ (x)δJν (y) δJµ (x) δJν (y)