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Quantum Field Theory I by Prof. Michael G. Schmidt 24 October 2007 Processed and LATEX-ed by Olivier Tieleman Supported by Adisorn Adulpravitchai and Jenny Wagner This is a first version of the lecture notes for professor Michael Schmidt’s course on quantum field theory as taught in the 2006/2007 winter semester. Comments and error reports are welcome; please send them to: o.tieleman at students.uu.nl 2 Contents 1 Introduction 5 1.1 Particle field duality ....................... 5 1.2 Short repetition of QM ...................... 5 1.2.1 Mechanics ........................ 6 1.2.2 QM States ........................ 6 1.2.3 Observables ........................ 7 1.2.4 Position and momentum ................. 7 1.2.5 Hamilton operator .................... 8 1.3 The need for QFT ........................ 9 1.4 History .............................. 12 1.5 Harmonic oscillator, coherent states .............. 12 1.5.1 Classical mechanics .................... 12 1.5.2 Quantization ....................... 13 1.5.3 Coherent states ...................... 14 1.6 The closed oscillator chain .................... 18 1.6.1 The classical system ................... 18 1.6.2 Quantization ....................... 19 1.6.3 Continuum limit ..................... 21 1.6.4 Ground state energy ................... 23 1.7 Summary ............................. 24 1.8 Literature ............................. 24 2 Free EM field, Klein-Gordon eq, quantization 27 2.1 The free electromagnetic field .................. 27 2.1.1 Classical Maxwell theory ................ 27 2.1.2 Quantization ....................... 30 2.1.3 Application: emission, absorption ............ 31 2.1.4 Coherent states ...................... 32 2.2 Klein-Gordon equation ...................... 32 2.2.1 Lagrange formalism for field equations ......... 32 2.2.2 Wave equation, Klein-Gordon equation ......... 35 2.2.3 Quantization ....................... 36 2.2.4 The Maxwell equations in Lagrange formalism .... 37 3 4 CONTENTS 3 The Schr¨odingerequation in the language of QFT 41 3.1 Second Quantization ....................... 41 3.1.1 Schr¨odinger equation ................... 41 3.1.2 Lagrange formalism for the Schr¨odinger equation ... 42 3.1.3 Canonical quantization ................. 43 3.2 Multiparticle Schr¨odinger equation ............... 45 3.2.1 Bosonic multiparticle space ............... 45 3.2.2 Interactions ........................ 47 3.2.3 Fermions ......................... 48 4 Quantizing covariant field equations 51 4.1 Lorentz transformations ..................... 51 4.2 Klein-Gordon equation for spin 0 bosons ............ 52 4.2.1 Klein-Gordon equation .................. 52 4.2.2 Solving the Klein-Gordon equation in 4-momentum space ........................... 53 4.2.3 Quantization: canonical formalism ........... 55 4.2.4 Charge conjugation .................... 58 4.3 Microcausality .......................... 59 4.4 Nonrelativistic limit ....................... 60 5 Interacting fields, S-matrix, LSZ 63 5.1 Non-linear field equations .................... 63 5.1.1 Relation to observation ................. 64 5.2 Interaction of particles in QFT ................. 64 5.2.1 The Yang-Feldman equations .............. 65 5.3 Green’s functions ......................... 66 5.4 Spectral representation of commutator vev ........... 69 5.5 LSZ reduction formalism ..................... 69 5.5.1 Generating functional .................. 71 6 Invariant Perturbation Theory 73 6.1 Dyson expansion, Gell-Mann-Low formula ........... 73 6.2 Interaction picture ........................ 76 6.2.1 Transformation ...................... 76 6.2.2 Scattering theory ..................... 78 6.2.3 Background field ..................... 80 6.3 Haag’s theorem .......................... 82 7 Feynman rules, cross section 83 7.1 Wick theorem ........................... 83 7.1.1 Φ4-theory ......................... 85 7.2 Feynman graphs ......................... 86 7.2.1 Feynman rules in x-space ................ 86 CONTENTS 5 7.2.2 Feynman rules in momentum space ........... 90 7.2.3 Feynman rules for complex scalars ........... 91 7.3 Functional relations ....................... 92 7.4 Back to S-matrix elements .................... 93 7.4.1 2-point functions ..................... 93 7.4.2 General Wightman functions .............. 95 7.5 From S-matrix to cross section ................. 96 7.5.1 Derivation ......................... 96 7.5.2 Scattering amplitudes and cross sections: an example 98 7.5.3 The optical theorem ................... 100 8 Path integral formulation of QFT 103 8.1 Path integrals in QM ....................... 103 8.1.1 Vacuum expectation values ............... 107 8.2 Path integrals in QFT ...................... 108 8.2.1 Framework ........................ 108 8.2.2 QFT path integral calculations ............. 110 8.2.3 Perturbation theory ................... 112 9 Lorentz group 115 9.1 Classification of Lorentz transformations ............ 115 9.2 Poincar´egroup .......................... 119 10 Dirac equation 123 10.1 Spinor representation of the Lorentz group ........... 123 10.2 Spinor fields, Dirac equation ................... 125 10.3 Representation matrices in spinor space ............ 126 10.3.1 Rotations ......................... 126 10.3.2 Lorentz transformations ................. 127 10.3.3 Boosts ........................... 128 10.4 Field equation ........................... 130 10.4.1 Derivation ......................... 130 10.4.2 Choosing the γ-matrices ................. 131 10.4.3 Relativistic covariance of the Dirac equation ...... 132 10.5 Complete solution of the Dirac equation ............ 133 10.6 Lagrangian formalism ...................... 135 10.6.1 Quantization ....................... 136 10.6.2 Charge conjugation .................... 137 11 Fermion Feynman Rules 139 11.1 Bilinear Covariants ........................ 139 11.2 LSZ reduction ........................... 140 11.3 Feynman rules .......................... 141 11.3.1 The Dirac propagator .................. 141 6 CONTENTS 11.4 Simple example in Yukawa theory ................ 144 12 QM interpretation of Dirac equation 149 12.1 Interpretation as Schr¨odinger equation ............. 149 12.2 Non-relativistic limit ....................... 152 12.2.1 Pauli equation ...................... 152 12.2.2 Foldy-Wouthousen transformation ........... 154 12.2.3 F.-W. representation with constant field ........ 155 12.3 The hydrogen atom ........................ 156 Chapter 1 Introduction 1.1 Particle field duality From classical physics we know both particles and fields/waves. These are two different concepts with different characteristics. But some experiments show, that there exists a duality between both. 1. electromagnetic waves waves ↔ photons (photoelectric effect, E = hν = ~ω) fields are quantized, consisting of particles called photons. 2. particles (e.g. electrons) may exhibit interference phenomena, like waves. Thus, particles must be described by a wavefunction ψ. How- ever, this has a probabilistic interpretation, it is not like an electro- magnetic field. The latter leads to quantum mechanics (QM), the former to quantum field theory (QFT). QM is nonrelativistic, and describes systems with fixed particle number. The quantization of the electromagnetic field requires quantum field theory, but is based on the same principles as quantum me- chanics. 1.2 Short repetition of QM QM cannot be derived from mechanics; rather, mechanics should follow from QM. But in obtaining appropriate Hamiltonians in QM, the correspondence principle, which substitutes quantities from mechanics by quantum mechan- ical operators, plays a key role. 7 8 CHAPTER 1. INTRODUCTION 1.2.1 Mechanics In the Lagrangian formulation of mechanics, we substitute the equations of motion by an extremal postulate for an action functional Z t S[L] = dtL (1.1) t0 of the Lagrange function L(qi, q˙i), where the qi are the (finitely many) gener- alized coordinates in the specific problem. Postulating δS = 0 for variations in the qi(t) and keeping the endpoints fixed, we obtain the Lagrange equa- tions ∂L d ∂L − = 0 (1.2) ∂qi dt ∂q˙i Here, ∂L/∂q˙i = pi are the generalized canonical momenta. The Hamiltonian H(pi, qi) is the Legendre transform of L: X H = q˙ipi − L (1.3) i and the Hamiltonian equations in phase space follow: ∂H ∂H p˙i = − , q˙i = (1.4) ∂qi ∂pi The Poisson bracket of two functions f(pi, qi), g(pi, qi) in phase space is defined as X ∂f ∂g ∂f ∂g {f(p , q ), g(p , q )} = − (1.5) i i i i P oisson ∂p ∂q ∂q ∂p i i i i i We have {pi, qj}P oisson = δij (1.6) and the Hamiltonian equations can be generalized to d f(p , q ) = {H, f} (1.7) dt i i P oisson ˙ In case of explicitly time-dependent f we have f(pi, qi) = {H, f}P oisson + ∂f/∂t. For these formal aspects see e.g. F. Scheck, “Mechanik”. Continuum mechanics can be obtained by taking the number of coordi- nates N to infinity, as will be seen in a specific example in QM. 1.2.2 QM States States are described by Hilbert space (ket) vectors |ψi ∈ H (or by the density matrix ρ; see below) with the following properties: 1.2. SHORT REPETITION OF QM 9 1. representation space: ψ(~x) = h~x|ψi are functions in the L2 Hilbert space H. These are coordinates in the h~x|-basis. 2. probabilistic interpretation: • |ψ(~x)|2 d3x is the probability to find the particle in the state |ψi in volume element d3x. • hϕ|ψi = R d3x ϕ∗(~x) ψ(~x) = R d3p ϕ∗(~p) ψ(~p) is called the inner product of ψ and ϕ • | hϕ|ψi |2 is the probability to find the state |ψi in |ϕi, and vice versa. 1.2.3 Observables Observables are described by self-adjoint linear operators A
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