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Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory . 94 6.4 Green’s Functions for λφ4 Theory ................. 97 6.5 The 2 2ScatteringAmplitudefromLSZ . 102 → 6.6 Appendix: ProofofEq.6.4.6 . 104 6.7 ProblemsSet6 ........................... 105 7 The Dirac Equation 107 7.1 Relativistic Wave Equations: Reprise . 107 7.2 TheDiracEquation. 108 7.3 Free Particle Solutions I: Interpretation . 109 7.4 FreeParticleSolutionsII:Spin. 112 7.5 Normalisation,GammaMatrices. 114 7.6 LorentzCovariance . 115 7.7 Parity ................................ 119 7.8 BilinearCovariants . 119 7.9 ChargeConjugation. 120 7.10Neutrinos .............................. 121 8 The Free Dirac Field 125 8.1 CanonicalQuantisation. 125 8.2 PathIntegralQuantisation. 127 8.3 TheFeynman PropagatorfortheDiracField. 129 8.4 Appendix: Grassmann Variables . 131 9 The Free Electromagnetic Field 133 9.1 The Classical Electromagnetic Field . 133 9.2 CanonicalQuantisation. 136 9.3 PathIntegralQuantisation. 138 CONTENTS 3 10 Quantum Electrodynamics 141 10.1 FeynmanRulesofQED . 141 10.2 Electron–MuonScattering . 143 10.3 Electron–ElectronScattering. 145 10.4 Electron–Positron Annihilation . 146 10.4.1 e+e− e+e− ........................ 146 → 10.4.2 e+e− µ+µ− and e+e− hadrons. 146 10.5 ComptonScattering→ . .→ . 147 4 CONTENTS Chapter 1 Preliminaries 1.1 Review of Classical Mechanics of Finite Sys- tem Consider a finite classical system of particles whose generalised coordinates are qi where i =1,...,N. For example N =3n for n particles in three dimensions. Suppose the Lagrangian given by L = T V , T and V being the kinetic and − potential energy, is: L = L(qi, q˙i) (1.1.1) whereq ˙i denotes the time derivative of qi. For each set of paths qi(t) connecting (qi1, t1) to (qi2, t2) the action is defined by: t2 S[qi(t)] = Ldt (1.1.2) Zt1 Note that S is a functional, i.e. a number whose value depends on a set of functions qi(t). The classical paths are those which minimise (or in general extremise) the numerical value of S, subject to the boundary conditions δqi =0 at the initial and final times t1 and t2. This is the well-known action principle (which can be derived from Feynman’s path integral approach to quantum mechanics – see later) and it leads to the N second order (in time) differential Lagrange equations of motion: d ∂L ∂L =0 (1.1.3) dt ∂q˙i − ∂qi In the Lagrange formalism the independent variables are the coordinates qi and the velocitiesq ˙i. The generalised momenta pi are derived quantities defined by: ∂L pi (1.1.4) ≡ ∂q˙i In the Hamiltonian formalism the variablesq ˙i are traded in for pi, and instead of working with a Lagrangian L = L(qi, q˙i) one works with a Hamiltonian 5 6 CHAPTER 1. PRELIMINARIES H = H(qi,pi) defined by the so-called Legendre transformation: H(q ,p ) p q˙ L(q , q˙ ) (1.1.5) i i ≡ i i − i i Xi where it is understood that theq ˙i are to be written as functions of qi and pi. From Eqs.1.1.3-1.1.5 one readily obtains the Hamilton equations of motion: ∂H ∂H p˙i = , q˙i = (1.1.6) − ∂qi ∂pi which are 2N first order (in time) differential equations. Just as (for a conser- vative force) L = T V , so H = T + V , so the Hamiltonian is just the total − energy in this case. Example 1: The Harmonic Oscillator First consider a single oscillator in oscillating in one dimension (e.g. mass on a spring) The Lagrangian is: 1 1 L = T V = mq˙2 ω2q2 (1.1.7) − 2 − 2 The generalised momentum is: ∂L p = = mq˙ (1.1.8) ∂q˙ The Hamiltonian is: p2 ω2 H = pq˙ L = + q2 (1.1.9) − 2m 2 Example 2: Coupled Harmonic Oscillators Now consider a linear chain of coupled oscillators, say corresponding to a vibrating line of atoms. Let the coordinate qi denote the displacement of the i th atom in the line from its equilibrium position, where we assume that all displacements− correspond to longitudinal vibrations along the length of the line of atoms. To make the system finite, let us form a closed loop of N atoms such that qi+N = qi. The Lagrangian is: N 1 Ω2 Ω2 L = q˙ 2 (q q )2 0 (q )2 (1.1.10) 2 i − 2 i − i+1 − 2 i Xi=1 where we have taken the atoms to have unit mass, and have included a nearest neighbour coupling plus a second independent frequency Ω0. The Lagrange equations of motion are: q¨ = Ω2(2q q q ) Ω2q (1.1.11) i − i − i−1 − i+1 − 0 i 1.2. REVIEW OF NON-RELATIVISTIC QUANTUM MECHANICS 7 The generalised momenta are: ∂L pi = =q ˙i (1.1.12) ∂q˙i The Hamiltonian is: N 1 Ω2 Ω2 H = p 2 + (q q )2 + 0 (q )2 (1.1.13) 2 i 2 i − i+1 2 i Xi=1 Normal Modes The oscillators above can be uncoupled by taking a discretised Fourier transform of the form N/2 q˜ q = k eij2πk/N (1.1.14) j N 1/2 k=X−N/2 N/2 p˜ p = k e−ij2πk/N (1.1.15) j N 1/2 k=X−N/2 which satisfies the periodic boundary conditions. Note that i2 = 1 and we have labelled the oscillators by j to avoid confusion. − Since the qj are real theq ˜k andp ˜k are complex and satisfy ⋆ ⋆ q˜−k =q ˜k, p˜−k =p ˜k. (1.1.16) Using the result: N ij2π(k−k′)/N e = δkk′ N jX=1 the Hamiltonian becomes, 1 N/2 H = p˜ p˜⋆ + ω2q˜ q˜⋆ (1.1.17) 2 k k k k k k=X−N/2 which corresponds to a sum of uncoupled harmonic oscillators of frequencies given by 2 2 2 2 ωk = Ω 4 sin (πk/N) + Ω0 (1.1.18) 1.2 Review of Non-Relativistic Quantum Me- chanics In quantum mechanics, 1 physical states are represented by vectors ψ > in a vector space (Hilbert space). Physical observables are represented by| linear 1We shall set h/(2π)= c = 1 throughout. 8 CHAPTER 1. PRELIMINARIES Hermitian operatorsω ˆ and the result of a measurement of some observableω ˆ is one of its (real) eigenvalues ω where ωˆ ω>= ω ω> (1.2.1) | | A very useful result is the completeness relation: ∞ I = ω ><ω Z−∞ | | which follows from the fact that the eigenvectors of an Hermitian operator form a complete orthonormal basis, where I is the unit operator. If the system is in some state ψ > and some observableω ˆ is measured then the system will be forced into one| of the eigenstates ψ > ω >. The result of the measurement | →| will be one of the eigenvalues ω which is not classically predictable, but which occurs with a probability P (ω) given by <ω ψ> 2 P (ω)= | | | (1.2.2) < ψ ψ > | The average or expectation value is therefore, < ψ ωˆ ψ > < ωˆ >= | | (1.2.3) < ψ ψ > | Now let us assume we are given a classical Hamiltonian in some particular basis of coordinates H = H(qi,pi). According to quantum mechanics we must replace the number-valued variables qi and pi by Hermitian operators which satisfy the commutation relations: [ˆqi, pˆj]= iIδij, [ˆqi, qˆj]=[ˆpi, pˆj]=0 (1.2.4) where I is the unit operator, 0 is the null operator, [A, B]= AB BA and δij is the Kronecker delta symbol defined in the usual way − δij =1 (i = j) δ =0 (i = j) ij 6 Each coordinate has its own complete set of eigenvectors and real eigenvalues qˆ q >= q q >, pˆ p >= p p > i| i i| i i| i i| i The position and momentum space wavefunctions are thus ψ(q , q ,...)=< q < q . ψ > 1 2 1| 2| | and ψ(p ,p ,...)=<p <p . ψ > 1 2 1| 2| | 1.2. REVIEW OF NON-RELATIVISTIC QUANTUM MECHANICS 9 In one dimension we have simply, ψ(q)=< q ψ >, ψ(p)=<p ψ > | | The way the time dependence of a quantum system is described is a matter of taste and convenience.
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