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Quantum Field Theory – 230A QUANTUM FIELD THEORY – 230A Eric D’Hoker Department of Physics and Astronomy University of California, Los Angeles, CA 90095 2004, October 3 Contents 1 Introduction 3 1.1 Relativity and quantum mechanics . ... 4 1.2 WhyQuantumFieldTheory?......................... 6 1.3 Further conceptual changes required by relativity . .......... 6 1.4 Some History and present significance of QFT . ..... 8 2 Quantum Mechanics – A synopsis 10 2.1 ClassicalMechanics.............................. 10 2.2 Principles of Quantum Mechanics . ... 12 2.3 Quantum Systems from classical mechanics . ...... 13 2.4 Quantum systems associated with Lie algebras . ....... 15 2.5 Appendix I : Lie groups, Lie algebras, representations . ........... 16 2.6 Functionalintegralformulation . ..... 19 3 Principles of Relativistic Quantum Field Theory 22 3.1 The Lorentz and Poincar´egroups and algebras . ....... 23 3.2 ParityandTimereversal . 25 3.3 Finite-dimensional Representations of the Lorentz Algebra ......... 26 3.4 Unitary Representations of the Poincar´eAlgebra . .......... 29 3.5 The basic principles of quantum field theory . ...... 32 3.6 Transformation of Local Fields under the Lorentz Group . ......... 34 3.7 DiscreteSymmetries .............................. 39 3.7.1 Parity .................................. 39 3.7.2 Timereversal .............................. 39 3.7.3 ChargeConjugation .......................... 40 3.8 Significance of fields in terms of particle contents . ......... 40 3.9 Classical Poincar´einvariant field theories . .......... 41 4 Free Field Theory 43 4.1 TheScalarField ................................ 43 4.2 The Operator Product Expansion & Composite Operators . ....... 51 4.3 TheGaugeorSpin1Field........................... 54 4.4 The Casimir Effect on parallel plates . .... 59 4.5 Bose-Einstein and Fermi-Dirac statistics . ........ 62 4.6 TheSpin1/2Field ............................... 65 4.6.1 The Weyl basis and Free Weyl spinors . 66 4.7 Thespinstatisticstheorem. ... 71 4.7.1 Parastatistics? ............................. 71 4.7.2 Commutation versus anti-commutation relations . ....... 72 5 Interacting Field Theories – Gauge Theories 73 5.1 InteractingLagrangians . .. 73 5.2 AbelianGaugeInvariance . 74 5.3 Non-AbelianGaugeInvariance. ... 74 5.4 Componentformulation . .. .. 76 5.5 Bianchiidentities ............................... 77 5.6 Gaugeinvariantcombinations . ... 77 5.7 Classical Action & Field equations . .... 78 5.8 Lagrangians including fermions and scalars . ........ 79 5.9 Examples .................................... 80 6 The S-matrix and LSZ Reduction formulas 81 6.0.1 InandOutStatesandFields . 82 6.0.2 TheS-matrix .............................. 83 6.0.3 Scatteringcrosssections . 83 6.1 Relating the interacting and in- and out-fields . ........ 85 6.2 The K¨allen-Lehmann spectral representation . ......... 87 6.3 TheDiracField................................. 89 6.4 TheLSZReductionFormalism . 90 6.4.1 In- and Out-Operators in terms of the free field . .... 90 6.4.2 Asymptotics of the interacting field . ... 91 6.4.3 Thereductionformulas. 92 6.4.4 TheDiracField............................. 93 6.4.5 ThePhotonField............................ 94 2 1 Introduction Quantum Field Theory (abbreviated QFT) deals with the quantization of fields. A familiar example of a field is provided by the electromagnetic field. Classical electromagnetism describes the dynamics of electric charges and currents, as well as electro-magnetic waves, such as radio waves and light, in terms of Maxwell’s equations. At the atomic level, however, the quantum nature of atoms as well as the quantum nature of electromagnetic radiation must be taken into account. Quantum mechanically, electromagnetic waves turn out to be composed of quanta of light, whose individual behavior is closer to that of a particle than to a wave. Remarkably, the quantization of the electromagnetic field is in terms of the quanta of this field, which are particles, also called photons. In QFT, this field particle correspondence is used both ways, and it is one of the key assumptions of QFT that to every elementary particle, there corresponds a field. Thus, the electron will have its own field, and so will every quark. Quantum Field Theory provides an elaborate general formalism for the field–particle correspondence. The advantage of QFT will be that it can naturally account for the cre- ation and annihilation of particles, which ordinary quantum mechanics of the Schr¨odinger equation could not describe. The fact that the number of particles in a system can change over time is a very important phenomenon, which takes place continuously in everyone’s daily surroundings, but whose significance may not have been previously noticed. In classical mechanics, the number of particles in a closed system is conserved, i.e. the total number of particles is unchanged in time. To each pointlike particle, one associates a set of position and momentum coordinates, the time evolution of which is governed by the dynamics of the system. Quantum mechanics may be formulated in two stages. 1. The principles of quantum mechanics, such as the definitions of states, observables, are general and do not make assumptions on whether the number of particles in the system is conserved during time evolution. 2. The specific dynamics of the quantum system, described by the Hamiltonian, may or may not assume particle number conservation. In introductory quantum mechanics, dynamics is usually associated with non-relativistic mechanical systems (augmented with spin degrees of freedom) and therefore assumes a fixed number of particles. In many important quantum systems, however, the number of particles is not conserved. A familiar and ubiquitous example is that of electromagnetic radiation. An excited atom may decay into its ground state by emitting a single quantum of light or photon. The photon was not “inside” the excited atom prior to the emission; it was “created” by the excited atom during its transition to the grounds state. This is well illustrated as 3 follows. An atom in a state of sufficiently high excitation may decay to its ground state in a single step by emitting a single photon. However, it may also emit a first photon to a lower excited state which in turn re-emits one or more photons in order to decay to the ground state (see Figure 1, (a) and (b)). Thus, given initial and final states, the number of photons emitted may vary, lending further support to the fact that no photons are “inside” the excited state to begin with. Other systems where particle number is not conserved involve phonons and spin waves in condensed matter problems. Phonons are the quanta associated with vibrational modes of a crystal or fluid, while spin waves are associated with fluctuating spins. The number of particles is also not conserved in nuclear processes like fusion and fission. 1.1 Relativity and quantum mechanics Special relativity invariably implies that the number of particles is not conserved. Indeed, one of the key results of special relativity is the fact that mass is a form of energy. A particle at rest with mass m has a rest energy given by the famous formula E = mc2 (1.1) The formula also implies that, given enough energy, one can create particles out of just energy – kinetic energy for example. This mechanism is at work in fire and light bulbs, where energy is being provided from chemical combustion or electrical input to excite atoms which then emit light in the form of photons. The mechanism is also being used in particle accelerators to produce new particles through the collision of two incoming particles. In Figure 1 the example of a photon scattering off an electron is illustrated. In (c), a photon of low energy ( m c2) is being scattered elastically which results simply e in a deflection of the photon and a recoil of the electron. In (d), a photon of high energy ( m c2) is being scattered inelastically, resulting not only in a deflection of the photon e and a recoil of the electron, but also in the production of new particles. The particle data table also provides numerous examples of particles that are unstable and decay. In each of these processes, the number of particles is not conserved. To list just a few, n p+ + e− +¯ν → e π0 γ + γ → π+ µ+ + ν → µ µ+ e+ + ν +¯ν → e µ As already mentioned, nuclear processes such as fusion and fission are further examples of systems in which the number of particles is not conserved. 4 photon photon photon initial initial final final (a) (b) scattered photon scattered photon photon photon new particles recoiled electron recoiled electron (c) (d) Figure 1: Production of particles : (a) Emission of one photon, (b) emission of two photons between the same initial and final states; (c) low energy elastic scattering, (d) high energy inelastic scattering of a photon off a recoiling electron. 5 1.2 Why Quantum Field Theory ? Quantum Field Theory is a formulation of a quantum system in which the number of particles does not have to be conserved but may vary freely. QFT does not require a change in the principles of either quantum mechanics or relativity. QFT requires a different formulation of the dynamics of the particles involved in the system. Clearly, such a description must go well beyond the usual Schr¨odinger equation, whose very formulation requires that the number of particles in a system be fixed. Quantum field theory may be formulated for non-relativistic systems in which the number of particles is not conserved, (recall spin waves, phonons, spinons etc). Here, however, we shall concen- trate
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