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Introduction to Quantum Field Theory and Quantum Statistics

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Introduction to Quantum Field Theory and Quantum Statistics 1 Introduction to Quantum Field Theory and Quantum Statistics Michael Bonitz Insitut f¨urTheoretische Physik und Astrophysik Kiel University February 9, 2015 preliminary lecture notes, not for distribution 2 Contents 1 Canonical Quantization 11 1.1 Minimal action principle . 11 1.1.1 Classical mechanics of a point particle . 16 1.1.2 Canonical momentum and Hamilton density of classical fields . 17 1.2 Conservation laws in classical field theory . 19 1.2.1 Translational invariance. Energy and momentum con- servation . 21 1.3 Field quantization . 24 1.4 Phonons . 26 1.4.1 Application of canonical field theory . 27 1.4.2 Expansion in terms of eigenfunctions . 28 1.4.3 Quantization of the displacement field . 30 1.5 Photons . 34 1.5.1 Maxwell's equations. Electromagnetic potentials. Field tensor . 34 1.5.2 Lagrange density of the free electromagnetic field . 38 1.5.3 Normal mode expansion of the electromagnetic field . 42 1.5.4 Quantization of the electromagnetic field . 44 1.6 EMF Quantization in Matter . 48 1.6.1 Lagrangian of a classical relativistic particle . 48 1.6.2 Relativistic particle coupled to the electromagnetic field . 50 1.6.3 Lagrangian of charged particles in an EM field . 51 1.6.4 Quantization of the electromagnetic field coupled to charges 54 1.6.5 Quantization of the EM field in a dielectric medium or plasma . 56 1.7 Quantization of the Schr¨odinger field . 58 1.8 Quantization of the Klein-Gordon field . 58 3 4 CONTENTS 1.9 Coupled equations for the Schr¨odingerand Maxwell fields . 58 1.10 Problems to Chapter 1 . 58 2 Second Quantization 59 2.1 Second quantization in phase space . 59 2.1.1 Classical dynamics in terms of point particles . 59 2.1.2 Point particles coupled via classical fields . 60 2.1.3 Classicle dynamics via particle and Maxwell fields . 61 2.2 Quantum mechanics and first quantization . 65 2.3 The ladder operators . 66 2.3.1 One-dimensional harmonic oscillator . 67 2.3.2 Generalization to several uncoupled oscillators . 70 2.4 Interacting Particles . 71 2.4.1 One-dimensional chain and its normal modes . 71 2.4.2 Quantization of the 1d chain . 75 2.4.3 Generalization to arbitrary interaction . 77 2.4.4 Quantization of the N-particle system . 80 2.5 Continuous systems . 81 2.5.1 Continuum limit of 1d chain . 81 2.5.2 Equation of motion of the 1d string . 82 2.6 Problems to chapter 2 . 85 3 Fermions and bosons 87 3.1 Spin statistics theorem . 87 3.2 N-particle wave functions . 89 3.2.1 Occupation number representation . 91 3.2.2 Fock space . 92 3.2.3 Many-fermion wave function . 92 3.2.4 Many-boson wave function . 93 3.2.5 Interacting bosons and fermions . 95 3.3 Second quantization for bosons . 97 3.3.1 Creation and annihilation operators for bosons . 97 3.4 Second quantization for fermions . 104 3.4.1 Creation and annihilation operators for fermions . 104 3.4.2 Matrix elements in Fock space . 110 3.4.3 Fock Matrix of the binary interaction . 115 3.5 Field operators . 118 3.5.1 Definition of field operators . 118 3.5.2 Representation of operators . 119 3.6 Momentum representation . 122 CONTENTS 5 3.6.1 Creation and annihilation operators in momentum space . 122 3.6.2 Representation of operators . 123 3.7 Problems to Chapter 3 . 126 4 Bosons and fermions in equilibrium 127 4.1 Density operator . 127 4.2 Path Integral Monte Carlo . 127 4.3 Configuration Path Integral . 127 4.3.1 Canonical ensemble . 128 4.3.2 Evaluation of Correlation energy contributions . 138 4.3.3 Electron gas at finite temperature . 138 4.4 Problems to Chapter 4 . 138 5 Dynamics of field operators 139 5.1 Equation of motion of the field operators . 139 5.2 General Operator Dynamics . 143 5.3 Extension to time-dependent hamiltonians . 144 5.4 Schr¨odingerdynamics of the field operators . 145 5.5 Dynamics of the density matrix operator . 147 5.6 Fluctuations and correlations . 151 5.6.1 Fluctuations and correlations . 151 5.6.2 Stochastic Mean Field Approximation . 154 5.6.3 Iterative Improvement of Stochastic Mean Field Approx- imation . 155 5.6.4 Ensemble average of the field operators . 155 5.6.5 Field operators and reduced density matrices . 156 5.6.6 BBGKY-hierarchy . 159 5.7 Problems to Chapter 5 . 159 6 Hubbard model 161 6.1 One-dimensional Hubbard chain . 161 6.2 Ensemble averages . 166 7 Nonequilibrium Green Functions 169 7.1 Introduction . 170 7.2 Nonequilibrium Green functions . 173 7.2.1 Keldysh Contour . 173 7.2.2 One-Particle Nonequilibrium Green Function . 182 7.2.3 Matrix Representation of the Green Function . 187 7.2.4 Langreth-Wilkins Rules . 191 6 CONTENTS 7.2.5 Proporties of the Nonequilibrium Green function . 195 7.3 Kadanoff-Baym-Ansatz . 202 7.4 Keldysh-Kadanoff-Baym Equations . 210 7.4.1 Derivation of the Martin-Schwinger hierarchy . 210 7.4.2 Selfenergy. Keldysh-Kadanoff-Baym equations . 218 7.4.3 Equilibrium Limit. Dyson Equation . 220 7.5 Many-Body Approximations . 221 7.5.1 Requirements for a Conserving Scheme . 222 7.5.2 Perturbation Expansions. Born approximation . 223 7.5.3 Vertex function . 228 7.5.4 Bethe-Salpeter equation . 232 7.5.5 Strong coupling. T-matrix selfenergy . 232 7.6 Single-time equations . 234 7.6.1 The Reconstruction Problem for the One-Particle Green Function . 235 7.6.2 The Generalized Kadanoff-Baym Ansatz . 239 8 Matter in strong fields 243 8.1 Free particle in a time-dependent field . 243 8.1.1 Classical approach . 243 8.1.2 Quantum-mechanical approach . 245 8.1.3 Alternative analytical methods . 252 8.1.4 Computational methods . 253 8.2 Atoms in electromagnetic fields . 253 8.2.1 Schr¨odinger equation for atom in EM field . 253 8.2.2 Stationary electrical fields. Stark effect. 254 8.2.3 Ionization processes . 255 8.3 Excitation and ionization processes in strong fields . 256 8.3.1 Classification of laser intensities . 256 8.3.2 Multiphoton processes in weak fields (γ 1) . 256 8.3.3 Strong field effects, γ 1 ................. 258 8.3.4 Strong field approximation. Keldysh theory. 258 8.4 Strong field phenomena . 260 8.4.1 Above threshold ionization . 260 8.4.2 Higher harmonics generation (HHG) . 260 8.5 Problems to Chapter 8 . 261 A Perturbation expansion for NEGF 263 A.1 Functional Derivative of a Contour-Ordered Operator Product . 263 CONTENTS 7 A.2 Selfenergy . 265 A Solutions to problems 269 A.1 Problems from chapter 2 . 269 A.2 Problems from chapter 3 . 269 A.3 Problems from chapter 4 . 271 A.4 Problems from chapter 5 . 271 A.5 Problems from chapter 6 . 272 Bibliography 293 Chapter 2 Introduction to second quantization 2.1 Second quantization in phase space 2.1.1 Classical dynamics in terms of point particles We consider systems of a large number N of identical particles which interact via pair potentials V and may be subject to an external field U. The system is described by the Hamilton function N N X p2 X X H(p; q) = i + U(r ) + V (r − r ) (2.1) 2m i i j i=1 i=1 1≤i<j≤N where p and q are 3N-dimensional vectors of all particle momenta and coordi- nates, p ≡ fp1; p2;::: pN g and q ≡ fr1; r2;::: rN g. Examples of the external potentials can be the electrostatic potential of a capacitor, the potential of an atomic nucleus or the potential an electron feels at a solid surface. The interaction potentials can arise from gravitational fields, from the Coulomb interaction of charged particles, the magnetic interaction of currents and so on. From the hamiltonian (2.1) the equations of motion follow by applying Hamilton's equations1, @H pi q_i = = ; (2.2) @pi m @H @U X @V p_i = − = − − ; (2.3) @qi @qi @qi j6=i 1Generalized equations of motion can, of course, also be obtained for non-Hamiltonian (dissipative) systems 59 60 CHAPTER 2. SECOND QUANTIZATION which is to be understood as two systems of 3N scalar equations for x1; y1; : : : zN and px1; py1; : : : pzN where @=@q ≡ f@=@r1; : : : @=@rN g, and the last equalities are obtained by inserting the hamiltonian (2.1). The system (2.3) is nothing but Newton's equations containing the forces arising from the gradient of the external potential and the gradient from all pair interactions involving the given particle, i.e. for any particle i = 1 :::N @U(r1;::: rN ) X @V (ri − rj) _pi = − − : (2.4) @ri @ri j6=i Consider, as an example, a system of identical charged particles with charge ei which may be subject to an external electrostatic potential φext and interact eiej with each other via the Coulomb potential Vc = . Then Newton's equa- jri−rj j tion (2.4) for particle i contains, on the r.h.s., the gradients of the potential U = eiφext and of the N − 1 Coulomb potentials involving all other particles. @eiφext(ri) X @ eiej _pi = − − ; (2.5) @ri @ri jri − rjj j6=i 2.1.2 Point particles coupled via classical fields An alternative way of writing Eq. (2.5) is to describe the particle interaction not by all pair interactions but to compute the total electric field, E(r; t), all particles produce in the whole space. The force particle \i" experiences is then just the Lorentz force, eiE, which is minus the gradient of the total electrostatic potential φ which is readily identified from the r.h.s.
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