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ОбложкаBaaquie B.E. Path integrals and hamiltonians: principles and methods. - New York: Cambridge university press, 2014. - xvii, 417 p.: ill. - Bibliogr.: p.409-411. - Ind.: p.413-417. - ISBN 978-1-107-00979-0
Шифр: (И/В31-В11) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Preface ........................................................ xv
Acknowledgements ............................................ xviii

1  Synopsis ..................................................... 1

Part one. Fundamental principles ................................ 5
2  The mathematical structure of quantum mechanics .............. 7
   2.1  The Copenhagen quantum postulate ........................ 7
   2.2  The superstructure of quantum mechanics ................ 10
   2.3  Degree of freedom space fig.1 .............................. 10
   2.4  State space V(fig.1) ....................................... 11
        2.4.1  Hilbert space ................................... 14
   2.5  Operators fig.2(fig.1) ........................................ 14
   2.6  The process of measurement ............................. 18
   2.7  The Schrцdinger differential equation .................. 19
   2.8  Heisenberg operator approach ........................... 22
   2.9  Dirac-Feynman path integral formulation ................ 23
   2.10 Three formulations of quantum mechanics ................ 25
   2.11 Quantum entity ......................................... 26
   2.12 Summary ................................................ 27
3  Operators ................................................... 30
   3.1  Continuous degree of freedom ........................... 30
   3.2  Basis states for state space ........................... 35
   3.3  Hermitian operators .................................... 36
        3.3.1  Eigenfunctions; completeness .................... 37
        3.3.2  Hamiltonian for a periodic degree of freedom .... 39
   3.4  Position and momentum operators fig.4 and fig.3 ................ 40
        3.4.1  Momentum operator fig.3 ............................. 41
   3.5  Weyl operators ......................................... 43
   3.6  Quantum numbers; commuting operator .................... 46
   3.7  Heisenberg commutation equation ........................ 47
   3.8  Unitary representation of Heisenberg algebra ........... 48
   3.9  Density matrix: pure and mixed states .................. 50
   3.10 Self-adjoint operators ................................. 51
        3.10.1 Momentum operator on finite interval ............ 52
   3.11 Self-adjoint domain .................................... 54
        3.11.1 Real eigenvalues ................................ 54
   3.12  Hamiltonian's self-adjoint extension .................. 55
        3.12.1 Delta function potential ........................ 57
   3.13 Fermi pseudo-potential ................................. 59
   3.14 Summary ................................................ 60
4  The Feynman path integral ................................... 61
   4.1  Probability amplitude and time evolution ............... 61
   4.2  Evolution kernel ....................................... 63
   4.3  Superposition: indeterminate paths ..................... 65
   4.4  The Dirac-Feynman formula .............................. 67
   4.5  The Lagrangian ......................................... 69
        4.5.1 Infinite divisibility of quantum paths ........... 70
   4.6  The Feynman path integral .............................. 70
   4.7  Path integral for evolution kernel ..................... 73
   4.8  Composition rule for probability amplitudes ............ 76
   4.9  Summary ................................................ 79
5  Hamiltonian mechanics ....................................... 80
   5.1  Canonical equations .................................... 80
   5.2  Symmetries and conservation laws ....................... 82
   5.3  Euclidean Lagrangian and Hamiltonian ................... 84
   5.4  Phase space path integrals ............................. 85
   5.5  Poisson bracket ........................................ 87
   5.6  Commutation equations .................................. 88
   5.7  Dirac bracket and constrained quantization ............. 90
        5.7.1  Dirac bracket for two constraints ............... 91
   5.8  Free particle evolution kernel ......................... 93
   5.9  Hamiltonian and path integral .......................... 94
   5.10 Coherent states ........................................ 95
   5.11 Coherent state vector .................................. 96
   5.12 Completeness equation: over-complete ................... 98
   5.13 Operators; normal ordering ............................. 98
   5.14 Path integral for coherent states ...................... 99
        5.14.1 Simple harmonic oscillator ..................... 101
   5.15 Forced harmonic oscillator ............................ 102
   5.16 Summary ............................................... 103
6  Path integral quantization ................................. 105
   6.1  Hamiltonian from Lagrangian ........................... 106
   6.2  Path integral's classical limit ħ → 0.................. 109
        6.2.1  Nonclassical paths and free particle ........... 111
   6.3  Fermat's principle of least time ...................... 112
   6.4  Functional differentiation ............................ 115
        6.4.1  Chain rule ..................................... 115
   6.5  Equations of motion ................................... 116
   6.6  Correlation functions ................................. 117
   6.7  Heisenberg commutation equation ....................... 118
        6.7.1  Euclidean commutation equation ................. 121
   6.8  Summary ............................................... 122

Part two  Stochastic processes ................................ 123
7  Stochastic systems ......................................... 125
   7.1  Classical probability: objective reality .............. 127
        7.1.1  Joint, marginal and conditional probabilities .. 128
   7.2  Review of Gaussian integration ........................ 129
   7.3  Gaussian white noise .................................. 132
        7.3.1  Integrals of white noise ....................... 134
   7.4  Ito calculus .......................................... 136
        7.4.1  Stock price .................................... 137
   7.5  Wilson expansion ...................................... 138
   7.6  Linear Langevin equation .............................. 140
        7.6.1  Random paths ................................... 142
   7.7  Langevin equation with potential ...................... 143
        7.7.1  Correlation functions .......................... 144
   7.8  Nonlinear Langevin equation ........................... 145
   7.9  Stochastic quantization ............................... 148
        7.9.1  Linear Langevin path integral .................. 149
   7.10 Fokker-Planck Hamiltonian ............................. 151
   7.11 Pseudo-Hermitian Fokker-Planck Hamiltonian ............ 153
   7.12 Fokker-Planck path integral ........................... 156
   7.13 Summary ............................................... 158

Part three. Discrete degrees of freedom ....................... 159
8  Ising model ................................................ 161
   8.1  Ising degree of freedom and state space ............... 161
        8.1.1  Ising spin's state space V ..................... 163
        8.1.2  Bloch sphere ................................... 164
   8.2  Transfer matrix ....................................... 165
   8.3  Correlators ........................................... 167
        8.3.1  Periodic lattice ............................... 168
   8.4  Correlator for periodic boundary conditions ........... 169
        8.4.1  Correlator as vacuum expectation values ........ 171
   8.5  Ising model's path integral ........................... 171
        8.5.1  Ising partition function ....................... 172
        8.5.2  Path integral calculation of Cr ................ 173
   8.6  Spin decimation ....................................... 175
   8.7  Ising model on 2 ħ N lattice .......................... 176
   8.8  Summary ............................................... 179
9  Ising model: magnetic field ................................ 180
   9.1  Periodic Ising model in a magnetic field .............. 180
   9.2  Ising model's evolution kernel ........................ 182
   9.3  Magnetization ......................................... 183
        9.3.1  Correlator ..................................... 184
   9.4  Linear regression ..................................... 185
   9.5  Open chain Ising model in a magnetic field ............ 189
        9.5.1  Open chain magnetization ....................... 190
   9.6  Block spin renormalization ............................ 191
        9.6.1  Block spin renormalization: magnetic field ..... 195
   9.7  Summary ............................................... 196
10 Fermions ................................................... 198
   10.1 Fermionic variables ................................... 199
   10.2 Fermion integration ................................... 200
   10.3 Fermion Hilbert space ................................. 201
        10.3.1 Fermionic completeness equation ................ 203
        10.3.2 Fermionic momentum operator .................... 204
   10.4 Antifermion state space ............................... 204
   10.5 Fermion and antifermion Hilbert space ................. 206
   10.6 Real and complex fermions: Gaussian integration ....... 207
        10.6.1 Complex Gaussian fermion ....................... 209
   10.7 Fermionic operators ................................... 211
   10.8 Fermionic path integral ............................... 211
   10.9 Fermion-antifermion Hamiltonian ....................... 214
        10.9.1 Orthogonality and completeness ................. 216
   10.10 Fermion-antifermion Lagrangian ....................... 217
   10.11 Fermionic transition probability amplitude ........... 219
   10.12 Quark confinement .................................... 220
   10.13 Summary .............................................. 222

Part four. Quadratic path integrals ........................... 223
11 Simple harmonic oscillator ................................. 225
   11.1 Oscillator Hamiltonian ................................ 226
   11.2 The propagator ........................................ 226
        11.2.1 Finite time propagator ......................... 227
   11.3 Infinite time oscillator .............................. 230
   11.4 Harmonic oscillator's evolution kernel ................ 230
   11.5 Normalization ......................................... 233
   11.6 Generating functional for the oscillator .............. 234
        11.6.1 Classical solution with source ................. 234
        11.6.2 Source free classical solution ................. 236
   11.7 Harmonic oscillator's conditional probability ......... 239
   11.8 Free particle path integral ........................... 240
   11.9 Finite lattice path integral .......................... 241
        11.9.1 Coordinate and momentum basis .................. 243
   11.10 Lattice free energy .................................. 243
   11.11 Lattice propagator ................................... 245
   11.12 Lattice transfer matrix and propagator ............... 246
   11.13 Eigenfunctions from evolution kernel ................. 249
   11.14 Summary .............................................. 250
12 Gaussian path integrals .................................... 251
   12.1 Exponential operators ................................. 252
   12.2 Periodic path integral ................................ 253
   12.3 Oscillator normalization .............................. 254
   12.4 Evolution kernel for indeterminate final position ..... 256
   12.5 Free degree of freedom: constant external source ...... 260
   12.6 Evolution kernel for indeterminate positions .......... 261
   12.7 Simple harmonic oscillator: Fourier expansion ......... 264
   12.8 Evolution kernel for a magnetic field ................. 267
   12.9 Summary ............................................... 270

Part five. Action with acceleration ........................... 271
13 Acceleration Lagrangian .................................... 273
   13.1 Lagrangian ............................................ 273
   13.2 Quadratic potential: the classical solution ........... 275
   13.3 Propagator: path integral ............................. 277
   13.4 Dirac constraints and acceleration Hamiltonian ........ 280
   13.5 Phase space path integral and Hamiltonian operator .... 283
   13.6 Acceleration path integral ............................ 286
   13.7 Change of path integral boundary conditions ........... 289
   13.8 Evolution kernel ...................................... 291
   13.9 Summary ............................................... 293
14 Pseudo-Hermitian Euclidean Hamiltonian ..................... 294
   14.1 Pseudo-Hermitian Hamiltonian; similarity
        transformation ........................................ 295
   14.2 Equivalent Hermitian Hamiltonian HO ................... 297
   14.3 The matrix elements of ℮-tQ ........................... 298
   14.4 ℮-tQ and similarity transformations ................... 301
   14.5 Eigenfunctions of oscillator Hamiltonian HO ........... 304
   14.6 Eigenfunctions of H and Ht ............................ 305
        14.6.1 Dual energy eigenstates 3....................... 307
   14.7 Vacuum state; ℮Q/2 .................................... 309
   14.8 Vacuum state and classical action ..................... 312
   14.9 Excited states of H ................................... 313
        14.9.1 Energy ω1 eigenstate Ψ10(x, v) ................. 314
        14.9.2 Energy ω2 eigenstate Ψ01(x, v) ................. 315
   14.10 Complex ω1, ω2 ....................................... 317
   14.11 State space V of Euclidean Hamiltonian ............... 318
        14.11.1 Operators acting on V ......................... 320
        14.11.2 Heisenberg operator equations ................. 322
   14.12 Propagator: operators ................................ 323
   14.13 Propagator: state space .............................. 324
   14.14 Many degrees of freedom .............................. 327
   14.15 Summary .............................................. 329
15 Non-Hermitian Hamiltonian: Jordan blocks ................... 330
   15.1 Hamiltonian: equal frequency limit .................... 331
   15.2 Propagator and states for equal frequency ............. 331
   15.3 State vectors for equal frequency ..................... 334
        15.3.1 State vector | Ψ1(τ)fig.5 ........................... 334
        15.3.2 State vector | Ψ2(τ)fig.5 ........................... 335
   15.4 Completeness equation for 2 × 2 block ................. 336
   15.5 Equal frequency propagator ............................ 337
   15.6 Hamiltonian: Jordan block structure ................... 339
   15.7 2 × 2 Jordan block .................................... 340
        15.7.1 Hamiltonian .................................... 342
        15.7.2 Schrцdinger equation for Jordan block .......... 343
        15.7.3 Time evolution ................................. 344
   15.8 Jordan block propagator ............................... 344
   15.9 Summary ............................................... 347

Part six. Nonlinear path integrals ............................ 349
16 The quartic potential: instantons .......................... 351
   16.1 Semi-classical approximation .......................... 352
   16.2 A one-dimensional integral ............................ 353
   16.3 Instantons in quantum mechanics ....................... 355
   16.4 Instanten zero mode ................................... 362
   16.5 Instanten zero mode: Faddeev-Popov analysis ........... 364
        16.5.1 Instanten coefficient N ........................ 368
   16.6 Multi-instantons ...................................... 370
   16.7 Instanten transition amplitude ........................ 371
        16.7.1 Lowest energy states ........................... 372
   16.8 Instanten correlation function ........................ 373
   16.9 The dilute gas approximation .......................... 374
   16.10 Ising model and the double well potential ............ 376
   16.11 Nonlocal Ising model ................................. 377
   16.12 Spontaneous symmetry breaking ........................ 380
        16.12.1 Infinite well ................................. 381
        16.12.2 Double well ................................... 381
   16.13 Restoration of symmetry .............................. 381
   16.14 Multiple wells ....................................... 383
   16.15 Summary .............................................. 383
17 Compact degrees of freedom ................................. 385
   17.1 Degree of freedom: a circle ........................... 386
        17.1.1 Poisson summation formula ...................... 387
        17.1.2 The S1 Lagrangian .............................. 388
   17.2 Multiple classical solutions .......................... 388
        17.2.1 Large radius limit ............................. 391
   17.3 Degree of freedom: a sphere ........................... 391
   17.4 Lagrangian for the rigid rotor ........................ 393
   17.5 Cancellation of divergence ............................ 395
   17.6 Conformation of DNA ................................... 399
   17.7 DNA extension ......................................... 401
   17.8 DNA persistence length ................................ 403
   17.9 Summary ............................................... 405
Conclusions ................................................... 409
References Index .............................................. 413

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