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  .............................. 10
2.4 State space V() ....................................... 11
2.4.1 Hilbert space ................................... 14
2.5 Operators  () ........................................ 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  and  ................ 40
3.4.1 Momentum operator ............................. 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) ........................... 334
15.3.2 State vector | Ψ2(τ) ........................... 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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