Quantum fields with classical perturbations Jan Derezi´nski Dept. of Math. Methods in Phys., Faculty of Physics, University of Warsaw Hoza 74, 00-682 Warszawa, Poland email [email protected] September 13, 2018 Abstract The main purpose of these notes is a review of various models of Quan- tum Field Theory involving quadratic Lagrangians. We discuss scalar and 1 vector bosons, spin 2 fermions, both neutral and charged. Beside free the- ories, we study their interactions with classical perturbations, called, de- pending on the context, an external linear source, mass-like term, current or electromagnetic potential. The notes may serve as a first introduction to QFT. Contents 0 Introduction 7 1 Basic concepts 11 1.1 Minkowski space . 11 1.1.1 Coordinates in Minkowski space . 11 1.1.2 Causal structure . 12 1.1.3 Fourier transform . 12 arXiv:1307.1162v4 [math-ph] 5 May 2014 1.1.4 Lorentz and Poincar´egroups . 13 1.1.5 Double coverings of Lorentz and Poincar´egroups . 13 1.1.6 Complex Lorentz groups . 15 1.2 General concepts of quantum field theory . 16 1.2.1 Quantum mechanics . 16 1.2.2 Time reversal . 16 1.2.3 Relativistic quantum mechanics . 18 1.2.4 Haag-Kastler axioms for observable algebras . 19 1.2.5 Haag-Kastler axioms for field algebras . 20 1.2.6 Global symmetries . 21 1 1.2.7 Neutral quantum fields . 21 1.2.8 Wightman axioms for neutral fields . 22 1.2.9 Relationship between Haag-Kastler and Wightman axioms 23 1.2.10 Global symmetries in the Wightman formalism . 23 1.2.11 Charged fields . 24 1.2.12 Wightman axioms for neutral and charged fields . 25 1.2.13 U(1) symmetry . 26 1.2.14 Charge conjugation . 26 1.2.15 Parity invariance . 27 1.2.16 Time reversal invariance . 27 1.2.17 The CPT Theorem . 28 1.2.18 The CPT Theorem in a P and T -invariant theory . 28 1.2.19 N-point Wightman and Green's functions . 29 1.3 General scattering theory . 30 1.3.1 Time ordered exponential . 30 1.3.2 Heisenberg picture . 31 1.3.3 Time-dependent perturbations . 32 1.3.4 Time ordered Green's functions . 33 1.3.5 Adiabatic switching and the energy shift . 34 1.3.6 Adiabatic switching and Green's functions . 37 1.3.7 Adiabatic scatttering theory . 38 2 Neutral scalar bosons 39 2.1 Free neutral scalar bosons . 40 2.1.1 Special solutions and Green's functions . 40 2.1.2 Space of solutions . 42 2.1.3 Classical fields . 44 2.1.4 Poisson brackets . 45 2.1.5 Smeared fields . 45 2.1.6 Lagrangian formalism . 46 2.1.7 Stress-energy tensor . 47 2.1.8 Diagonalization of the equations of motion . 48 2.1.9 Plane waves . 49 2.1.10 Positive frequency space . 50 2.1.11 Quantization . 51 2.1.12 Quantization in terms of smeared fields . 53 2.1.13 Quantization in terms of C∗-algebras . 54 2.1.14 Two-point functions . 54 2.2 Neutral scalar bosons with a linear source . 55 2.2.1 Classical fields . 55 2.2.2 Lagrangian and Hamiltonian formalism . 56 2.2.3 Quantization . 57 2.2.4 Operator valued source . 58 2.2.5 Scattering operator . 59 2.2.6 Green's functions . 60 2.2.7 Path integral formulation . 61 2 2.2.8 Feynman rules . 62 2.2.9 Vacuum energy . 63 2.2.10 Problems with the scattering operator . 63 2.2.11 Energy shift and scattering theory for a stationary source 64 2.2.12 Travelling source . 65 2.2.13 Scattering cross-sections . 68 2.2.14 Inclusive cross-section . 68 2.3 Neutral scalar bosons with a mass-like perturbation . 70 2.3.1 Classical fields . 70 2.3.2 Lagrangian and Hamiltonian formalism . 71 2.3.3 Dynamics in the interaction picture . 71 2.3.4 Quantization . 72 2.3.5 Quantum Hamiltonian . 73 2.3.6 Path integral formulation . 74 2.3.7 Feynman rules . 74 2.3.8 Vacuum energy . 76 2.3.9 Pauli-Villars renormalization . 77 2.3.10 Renormalization of the vacuum energy . 78 2.3.11 Method of dispersion relations . 79 2.3.12 Wick rotation . 80 2.3.13 Dimensional renormalization . 81 2.3.14 Energy shift . 82 3 Massive photons 83 3.1 Free massive photons . 84 3.1.1 Space of solutions . 84 3.1.2 Classical 4-potentials . 84 3.1.3 Poisson brackets . 85 3.1.4 Smeared 4-potentials . 86 3.1.5 Lagrangian formalism and stress-energy tensor . 86 3.1.6 Diagonalization of the equations of motion . 88 3.1.7 Plane waves . 91 3.1.8 Positive frequency space . 91 3.1.9 Spin averaging . 92 3.1.10 Quantization . 92 3.2 Massive photons with an external 4-current . 94 3.2.1 Classical 4-potentials . 94 3.2.2 Lagrangian and Hamiltonian formalism . 95 3.2.3 Quantization . 96 3.2.4 Causal propagators . 98 3.2.5 Feynman rules . 99 3.2.6 Path integral formulation . 99 3.2.7 Energy shift . 101 3.3 Alternative approaches . 102 3.3.1 Classical 4-potentials without the Lorentz condition . 102 3.3.2 The Lorentz condition . 103 3 3.3.3 Diagonalization of the equations of motion . 104 3.3.4 Positive frequency space . 106 3.3.5 \First quantize, then reduce" . 107 3.3.6 Quantization without reduction on a positive definite Hilbert space . 107 3.3.7 The Gupta-Bleuler approach . 108 4 Massless photons 109 4.1 Free massless photons . 110 4.1.1 Space of solutions and the gauge invariance . 110 4.1.2 Classical 4-potentials . 112 4.1.3 Smeared 4-potentials . 113 4.1.4 Lagrangian formalism and the stress-energy tensor . 113 4.1.5 Diagonalization of the equations of motion . 114 4.1.6 Positive frequency space . 115 4.1.7 Spin averaging . 116 4.1.8 Quantization . 116 4.1.9 Quantization in terms of C∗-algebras . 118 4.2 Massless photons with an external 4-current . 118 4.2.1 Classical fields . 118 4.2.2 Lagrangian and Hamiltonian formalism . 120 4.2.3 Quantization . 121 4.2.4 Causal propagators . 122 4.2.5 Path integral formulation . 122 4.2.6 The m ! 0 limit . 123 4.2.7 Current produced by a travelling particle . 124 4.2.8 Energy shift . 125 4.3 Alternative approaches . 125 4.3.1 Manifestly Lorentz covariant formalism . 125 4.3.2 The Lorentz condition . 126 4.3.3 Positive frequency space . 127 4.3.4 \First quantize, then reduce" . 128 4.3.5 Quantization with a subsidiary condition . 128 4.3.6 The Gupta-Bleuler approach . 129 5 Charged scalar bosons 129 5.1 Free charged scalar bosons . 130 5.1.1 Classical fields . 130 5.1.2 Smeared fields . 131 5.1.3 Lagrangian formalism . 132 5.1.4 Classical 4-current . 133 5.1.5 Stress-energy tensor . 134 5.1.6 Diagonalization of the equations of motion . 134 5.1.7 Plane waves . 136 5.1.8 Positive and negative frequency subspace . 137 5.1.9 Quantization . 138 4 5.1.10 Quantum 4-current . 140 5.1.11 Quantization in terms of smeared fields . 142 5.2 Charged scalar bosons in an external 4-potential . 143 5.2.1 Classical fields . 143 5.2.2 Lagrangian and Hamiltonian formalism . 144 5.2.3 Classical discrete symmetries . 145 5.2.4 Quantization . ..