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Quantum Field Theory I

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Quantum Field Theory I Physics 523 & 524: Quantum Field Theory I Kevin Cahill Department of Physics and Astronomy University of New Mexico, Albuquerque, NM 87131 Contents 1 Quantum fields and special relativity page 1 1.1 States 1 1.2 Creation operators 1 1.3 How fields transform 3 1.4 Translations 7 1.5 Boosts 8 1.6 Rotations 9 1.7 Spin-zero fields 9 1.8 Conserved charges 12 1.9 Parity, charge conjugation, and time reversal 13 1.10 Vector fields 15 1.11 Vector field for spin-zero particles 18 1.12 Vector field for spin-one particles 18 1.13 Lorentz group 23 1.14 Gamma matrices and Clifford algebras 25 1.15 Dirac's gamma matrices 27 1.16 Dirac fields 28 1.17 Expansion of massive and massless Dirac fields 41 1.18 Spinors and angular momentum 44 1.19 Charge conjugation 49 1.20 Parity 51 2 Feynman diagrams 53 2.1 Time-dependent perturbation theory 53 2.2 Dyson's expansion of the S matrix 54 2.3 The Feynman propagator for scalar fields 56 2.4 Application to a cubic scalar field theory 60 2.5 Feynman's propagator for fields with spin 62 ii Contents 2.6 Feynman's propagator for spin-one-half fields 63 2.7 Application to a theory of fermions and bosons 66 2.8 Feynman propagator for spin-one fields 71 2.9 The Feynman rules 73 2.10 Fermion-antifermion scattering 74 3 Action 76 3.1 Lagrangians and hamiltonians 76 3.2 Canonical variables 76 3.3 Principle of stationary action in field theory 78 3.4 Symmetries and conserved quantities in field theory 79 3.5 Poisson Brackets 84 3.6 Dirac Brackets 85 3.7 Dirac Brackets and QED 86 4 Quantum Electrodynamics 93 4.1 Global U( 1) Symmetry 93 4.2 Abelian gauge invariance 94 4.3 Coulomb-gauge quantization 96 4.4 QED in the interaction picture 97 4.5 Photon propagator 98 4.6 Feynman's rules for QED 100 4.7 Electron-positron scattering 101 4.8 Trace identities 104 4.9 Electron-positron to muon-antimuon 105 4.10 Electron-muon scattering 111 4.11 One-loop QED 112 4.12 Magnetic moment of the electron 124 4.13 Charge form factor of electron 132 5 Nonabelian gauge theory 134 5.1 Yang and Mills invent local nonabelian symmetry 134 5.2 SU(3) 135 6 Standard model 138 6.1 Quantum chromodynamics 138 6.2 Abelian Higgs mechanism 138 6.3 Higgs's mechanism 140 6.4 Quark and lepton interactions 146 6.5 Quark and Lepton Masses 148 6.6 CKM matrix 153 6.7 Lepton Masses 156 6.8 Before and after symmetry breaking 157 Contents iii 6.9 The Seesaw Mechanism 161 6.10 Neutrino Oscillations 163 7 Path integrals 165 7.1 Path integrals and Richard Feynman 165 7.2 Gaussian integrals and Trotter's formula 165 7.3 Path integrals in quantum mechanics 166 7.4 Path integrals for quadratic actions 170 7.5 Path integrals in statistical mechanics 175 7.6 Boltzmann path integrals for quadratic actions 180 7.7 Mean values of time-ordered products 183 7.8 Quantum field theory 185 7.9 Finite-temperature field theory 189 7.10 Perturbation theory 192 7.11 Application to quantum electrodynamics 196 7.12 Fermionic path integrals 200 7.13 Application to nonabelian gauge theories 208 7.14 Faddeev-Popov trick 208 7.15 Ghosts 211 7.16 Integrating over the momenta 212 8 Landau theory of phase transitions 214 8.1 First- and second-order phase transitions 214 9 Effective field theories and gravity 217 9.1 Effective field theories 217 9.2 Is gravity an effective field theory? 218 9.3 Quantization of Fields in Curved Space 221 9.4 Accelerated coordinate systems 228 9.5 Scalar field in an accelerating frame 229 9.6 Maximally symmetric spaces 233 9.7 Conformal algebra 236 9.8 Conformal algebra in flat space 237 9.8.1 Angles and analytic functions 241 9.9 Maxwell's action is conformally invariant for d = 4 247 9.10 Massless scalar field theory is conformally invariant if d = 2 248 9.11 Christoffel symbols as nonabelian gauge fields 249 9.12 Spin connection 254 10 Field Theory on a Lattice 256 10.1 Scalar Fields 256 10.2 Finite-temperature field theory 257 iv Contents 10.3 The Propagator 261 10.4 Pure Gauge Theory 261 10.5 Pure Gauge Theory on a Lattice 264 References 266 Index 267 1 Quantum fields and special relativity 1.1 States A Lorentz transformation Λ is implemented by a unitary operator U(Λ) which replaces the state jp; σi of a massive particle of momentum p and spin σ along the z-axis by the state s 0 (Λp) X (j) U(Λ)jp; σi = D (W (Λ; p)) jΛp; s0i (1.1) p0 s0σ s0 where W (Λ; p) is a Wigner rotation W (Λ; p) = L−1(Λp)ΛL(p) (1.2) and L(p) is a standard Lorentz transformation that takes (m;~0) to p. 1.2 Creation operators The vacuum is invariant under Lorentz transformations and translations U(Λ; a)j0i = j0i: (1.3) A creation operator ay(p; σ) makes the state jp; σi from the vacuum state j0i jpσi = ay(p; σ)j0i: (1.4) The creation and annihilation operators obey either the commutation rela- tion y 0 0 y 0 0 y 0 0 (3) 0 [a(p; s); a (p ; s )]− = a(p; s) a (p ; s ) − a (p ; s ) a(p; s) = δss0 δ (p − p ) (1.5) 2 Quantum fields and special relativity or the anticommutation relation y 0 0 y 0 0 y 0 0 (3) 0 [a(p; s); a (p ; s )]+ = a(p; s) a (p ; s ) + a (p ; s ) a(p; s) = δss0 δ (p − p ): (1.6) The two kinds of relations are written together as y 0 0 y 0 0 y 0 0 (3) 0 [a(p; s); a (p ; s )]∓ = a(p; s) a (p ; s ) ∓ a (p ; s ) a(p; s) = δss0 δ (p − p ): (1.7) A bracket [A; B] with no signed subscript is interpreted as a commutator. Equations (1.1 & 1.4) give s 0 (Λp) X (j) U(Λ)ay(p; σ)j0i = D (W (Λ; p)) ay(Λp; s0)j0i: (1.8) p0 s0σ s0 And (1.3) gives s 0 (Λp) X (j) U(Λ)ay(p; σ)U −1(Λ)j0i = D (W (Λ; p)) ay(Λp; s0)j0i: (1.9) p0 s0σ s0 SW in chapter 4 concludes that s 0 (Λp) X (j) U(Λ)ay(p; σ)U −1(Λ) = D (W (Λ; p)) ay(Λp; s0): (1.10) p0 s0σ s0 If U(Λ; b) follows Λ by a translation by b, then s 0 (Λp) X (j) U(Λ; b)ay(p; σ)U −1(Λ; b) = e−i(Λp)·a D (W (Λ; p)) ay(Λp; s0) p0 s0σ s0 s 0 (Λp) X y(j) = e−i(Λp)·a D (W −1(Λ; p)) ay(Λp; s0) p0 s0σ s0 s 0 (Λp) X ∗(j) = e−i(Λp)·a D (W −1(Λ; p)) ay(Λp; s0) p0 σs0 s0 (1.11) 1.3 How fields transform 3 The adjoint of this equation is s 0 (Λp) X ∗(j) U(Λ; b)a(p; σ)U −1(Λ; b) = ei(Λp)·a D (W (Λ; p)) a(Λp; s0) p0 s0σ s0 s 0 (Λp) X y(j) = ei(Λp)·a D (W (Λ; p)) a(Λp; s0) p0 σs0 s0 s 0 (Λp) X (j) = ei(Λp)·a D (W −1(Λ; p)) a(Λp; s0): p0 σs0 s0 (1.12) These equations (1.11 & 1.12) are (5.1.11 & 5.1.12) of SW. 1.3 How fields transform The \positive frequency" part of a field is a linear combination of annihila- tion operators Z + X 3 ` (x) = d p u`(x; p; σ) a(p; σ): (1.13) σ The \negative frequency" part of a field is a linear combination of creation operators of the antiparticles Z − X 3 y ` (x) = d p v`(x; p; σ) b (p; σ): (1.14) σ To have the fields (1.13 & 1.14) transform properly under Poincar´etrans- formations X U(Λ; a) +(x)U −1(Λ; a) = D (Λ−1) +(Λx + a) ` ``¯ `¯ `¯ Z X −1 X 3 = D``¯(Λ ) d p u`¯(Λx + a; p; σ) a(p; σ) `¯ σ X U(Λ; a) −(x)U −1(Λ; a) = D (Λ−1) −(Λx + a) ` ``¯ `¯ `¯ Z X −1 X 3 y = D``¯(Λ ) d p v`¯(Λx + a; p; σ) b (p; σ) `¯ σ (1.15) the spinors u`(x; p; σ) and v`(x; p; σ) must obey certain rules which we'll now determine. 4 Quantum fields and special relativity First (1.12 & 1.13) give Z + −1 X 3 −1 U(Λ; a) ` (x)U (Λ; a) = U(Λ; a) d p u`(x; p; σ) a(p; σ)U (Λ; a) σ Z X 3 −1 = d p u`(x; p; σ) U(Λ; a)a(p; σ)U (Λ; a) σ (1.16) s Z 0 X (Λp) X (j) = d3p u (x; p; σ) ei(Λp)·a D (W −1(Λ; p)) a(Λp; s0): ` p0 σs0 σ s0 Now we use the identity d3p d3(Λp) = (1.17) p0 (Λp)0 to turn (1.16) into Z + −1 X 3 i(Λp)·a U(Λ; a) ` (x)U (Λ; a) = d (Λp) u`(x; p; σ) e σ s 0 p X (j) × D (W −1(Λ; p)) a(Λp; s0): (1.18) (Λp)0 σs0 s0 Similarly (1.11, 1.14, & 1.17) give Z − −1 X 3 y −1 U(Λ; a) ` (x)U (Λ; a) = U(Λ; a) d p v`(x; p; σ) b (p; σ)U (Λ; a) σ Z X 3 y −1 = d p v`(x; p; σ) U(Λ; a)b (p; σ)U (Λ; a) (1.19) σ s Z 0 X (Λp) X ∗(j) = d3p v (x; p; σ) e−i(Λp)·a D (W −1(Λ; p)) by(Λp; s0) ` p0 σs0 σ s0 s Z 0 X p X ∗(j) = d3(Λp) v (x; p; σ) e−i(Λp)·a D (W −1(Λ; p)) by(Λp; s0): ` (Λp)0 σs0 σ s0 So to get the fields to transform as in (1.15), equations (1.18 & 1.19) say 1.3 How fields transform 5 that we need X X X Z D (Λ−1) +(Λx + a) = D (Λ−1) d3p u (Λx + a; p; σ) a(p; σ) ``¯ `¯ ``¯ `¯ `¯ `¯ σ Z X −1 X 3 = D``¯(Λ ) d (Λp) u`¯(Λx + a;Λp; σ) a(Λp; σ) `¯ σ Z X 3 i(Λp)·a = d (Λp) u`(x; p; σ) e (1.20) σ s 0 p X (j) × D (W −1(Λ; p)) a(Λp; s0) (Λp)0 σs0 s0 Z X 3 0 i(Λp)·a = d (Λp) u`(x; p; s ) e s0 s 0 p X (j) × D (W −1(Λ; p)) a(Λp; σ) (Λp)0 s0σ σ and X X X Z D (Λ−1) −(Λx + a) = D (Λ−1) d3p v (Λx + a; p; σ) by(p; σ) ``¯ `¯ ``¯ `¯ `¯ `¯ σ Z X −1 X 3 y = D``¯(Λ ) d (Λp) v`¯(Λx + a;Λp; σ) b (Λp; σ) `¯ σ Z X 3 −i(Λp)·a = d (Λp) v`(x; p; σ) e (1.21) σ s 0 p X ∗(j) × D (W −1(Λ; p)) by(Λp; s0) (Λp)0 σs0 s0 Z X 3 0 −i(Λp)·a = d (Λp) v`(x; p; s ) e s0 s 0 p X ∗(j) × D (W −1(Λ; p)) by(Λp; σ): (Λp)0 s0σ σ Equating coefficients of the red annihilation and blue creation operators, we find that the fields will transform properly if the spinors u and v satisfy the 6 Quantum fields and special relativity rules s 0 X −1 p X (j) −1 0 i(Λp)·a D ¯(Λ ) u¯(Λx + a;Λp; σ) = D (W (Λ; p))u (x; p; s ) e `` ` (Λp)0 s0σ ` `¯ s0 (1.22) s 0 X −1 p X ∗(j) −1 0 −i(Λp)·a D ¯(Λ )v¯(Λx + a;Λp; σ) = D (W (Λ; p))v (x; p; s )e `` ` (Λp)0 s0σ ` `¯ s0 (1.23) which differ from SW's by an interchange of the subscripts σ; s0 on the rotation matrices D(j).
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