Quantum Mechanics of Non-Hamiltonian and Dissipative Systems (eBook)

Cover 1
Preface 6
Contents 8
A Very Few Preliminaries 16
1. Potential and conservative systems 16
2. Hamiltonian and non-Hamiltonian classical systems 17
3. Examples of non-Hamiltonian systems 17
4. Non-Hamiltonian and dissipative classical systems 18
5. Non-Hamiltonian and dissipative quantum systems 19
6. Quantization of non-Hamiltonian and dissipative systems 20
Part I: Quantum Kinematics 24
Chapter 1. Quantum Kinematics of Bounded Observables 26
1.1. Observables and states 26
1.2. Pre-Hilbert and Hilbert spaces 26
1.3. Separable Hilbert space 31
1.4. Definition and examples of operators 33
1.5. Quantum kinematical postulates 35
1.6. Dual Hilbert space 36
1.7. Dirac's notations 38
1.8. Matrix representation of operator 39
Chapter 2. Quantum Kinematics of Unbounded Observables 42
2.1. Deficiencies of Hilbert spaces 42
2.2. Spaces of test functions 43
2.3. Spaces of generalized functions 46
2.4. Rigged Hilbert space 48
2.5. Linear operators on a rigged Hilbert space 51
2.6. Coordinate representation 55
2.7. X-representation 58
Chapter 3. Mathematical Structures in Quantum Kinematics 62
3.1. Mathematical structures 62
3.2. Order structures 64
3.3. Topological structures 65
3.4. Algebraic structures 67
3.5. Examples of algebraic structures 72
3.6. Mathematical structures in kinematics 81
Chapter 4. Spaces of Quantum Observables 84
4.1. Space of bounded operators 84
4.2. Space of finite-rank operators 86
4.3. Space of compact operators 87
4.4. Space of trace-class operators 89
4.5. Space of Hilbert-Schmidt operators 93
4.6. Properties of operators from K1 (H) and K2 (H) 94
4.7. Set of density operators 96
4.8. Operator Hilbert space and Liouville space 98
4.9. Correlation functions 102
4.10. Basis for Liouville space 103
4.11. Rigged Liouville space 106
Chapter 5. Algebras of Quantum Observables 110
5.1. Linear algebra 110
5.2. Associative algebra 111
5.3. Lie algebra 111
5.4. Jordan algebra 113
5.5. Involutive, normed and Banach algebras 115
5.6. C*-algebra 118
5.7. W*-algebra 121
5.8. JB-algebra 126
5.9. Hilbert algebra 127
Chapter 6. Mathematical Structures on State Sets 130
6.1. State as functional on operator algebra 130
6.2. State on C*-algebra 132
6.3. Representations C*-algebra and states 138
6.4. Gelfand-Naimark-Segal construction 139
6.5. State on W*-algebra 143
Chapter 7. Mathematical Structures in Classical Kinematics 144
7.1. Symplectic structure 144
7.2. Poisson manifold and Lie-Jordan algebra 145
7.3. Classical states 148
7.4. Classical observables and C*-algebra 151
Chapter 8. Quantization in Kinematics 154
8.1. Quantization and its properties 154
8.2. Heisenberg algebra 162
8.3. Weyl system and Weyl algebra 164
8.4. Weyl and Wigner operator bases 167
8.5. Differential operators and symbols 172
8.6. Weyl quantization mapping 174
8.7. Kernel and symbol of Weyl ordered operator 176
8.8. Weyl symbols and Wigner representation 177
8.9. Inverse of quantization map 180
8.10. Symbols of operators and Weyl quantization 181
8.11. Generalization of Weyl quantization 189
Chapter 9. Spectral Representation of Observable 196
9.1. Spectrum of quantum observable 196
9.2. Algebra of operator functions 202
9.3. Spectral projection and spectral decomposition 204
9.4. Symmetrical and self-adjoint operators 207
9.5. Resolution of the identity 209
9.6. Spectral theorem 211
9.7. Spectral operator through ket-bra operator 214
9.8. Function of self-adjoint operator 216
9.9. Commutative and permutable operators 218
9.10. Spectral representation 220
9.11. Complete system of commuting observables 223
Part II: Quantum Dynamics 226
Chapter 10. Superoperators and its Properties 228
10.1. Mathematical structures in quantum dynamics 228
10.2. Definition of superoperator 232
10.3. Left and right superoperators 235
10.4. Superoperator kernel 238
10.5. Closed and resolvent superoperators 241
10.6. Superoperator of derivation 242
10.7. Hamiltonian superoperator 246
10.8. Integration of quantum observables 248
Chapter 11. Superoperator Algebras and Spaces 252
11.1. Linear spaces and algebras of superoperators 252
11.2. Superoperator algebra for Lie operator algebra 255
11.3. Superoperator algebra for Jordan operator algebra 256
11.4. Superoperator algebra for Lie-Jordan operator algebra 258
11.5. Superoperator C*-algebra and double centralisers 259
11.6. Superoperator W*-algebra 262
Chapter 12. Superoperator Functions 266
12.1. Function of left and right superoperators 266
12.2. Inverse superoperator function 268
12.3. Superoperator function and Fourier transform 269
12.4 Exponential superoperator function 270
12.5. Superoperator Heisenberg algebra 272
12.6. Superoperator Weyl system 273
12.7. Algebra of Weyl superoperators 274
12.8. Superoperator functions and ordering 276
12.9. Weyl ordered superoperator 278
Chapter 13. Semi-Groups of Superoperators 282
13.1. Groups of superoperators 282
13.2. Semi-groups of superoperators 284
13.3. Generating superoperators of semi-groups 288
13.4. Contractive semi-groups and its generators 290
13.5. Positive semi-groups 294
Chapter 14. Differential Equations for Quantum Observables 300
14.1. Quantum dynamics and operator differential equations 300
14.2. Definition of operator differential equations 301
14.3. Equations with constant bounded superoperators 303
14.4. Chronological multiplication 304
14.5. Equations with variable bounded superoperators 306
14.6. Operator equations with constant unbounded superoperators 309
14.7. Generating superoperator and its resolvent 310
14.8. Equations in operator Hilbert spaces 313
14.9. Equations in coordinate representation 316
14.10. Example of operator differential equation 317
Chapter 15. Quantum Dynamical Semi-Group 320
15.1. Dynamical semi-groups 320
15.2. Semi-scalar product and dynamical semi-groups 322
15.3. Dynamical semi-groups and orthogonal projections 324
15.4. Dynamical semi-groups for observables 326
15.5. Quantum dynamical semi-groups on W*-algebras 328
15.6. Completely positive superoperators 330
15.7. Bipositive superoperators 334
15.8. Completely dissipative superoperators 335
15.9. Lindblad equation 338
15.10. Example of Lindblad equation 343
15.11. Gorini-Kossakowski-Sudarshan equation 346
15.12. Two-level non-Hamiltonian quantum system 348
Chapter 16. Classical Non-Hamiltonian Dynamics 352
16.1. Introduction to classical dynamics 352
16.2. Systems on symplectic manifold 355
16.3. Systems on Poisson manifold 361
16.4. Properties of locally Hamiltonian systems 364
16.5. Quantum Hamiltonian and non-Hamiltonian systems 367
16.6. Hamiltonian and Liouvillian pictures 369
Chapter 17. Quantization of Dynamical Structure 376
17.1. Quantization in kinematics and dynamics 376
17.2. Quantization map for equations of motion 378
17.3. Quantization of Lorenz-type systems 385
17.4. Quantization of Poisson bracket 386
17.5. Discontinuous functions and nonassociative operators 392
Chapter 18. Quantum Dynamics of States 396
18.1. Evolution equation for normalized operator 396
18.2. Quantization for Hamiltonian picture 398
18.3. Expectation values for non-Hamiltonian systems 399
18.4. Adjoint and inverse superoperators 404
18.5. Adjoint Lie-Jordan superoperator functions 407
18.6. Weyl multiplication and Weyl scalar product 412
18.7. Weyl expectation value and Weyl correlators 415
18.8. Evolution of state in the Schrödinger picture 419
Chapter 19. Dynamical Deformation of Algebras of Observables 424
19.1. Evolution as a map 424
19.2. Rule of term-by-term differentiation 427
19.3. Time evolution of binary operations 429
19.4. Bilinear superoperators 432
19.5. Cohomology groups of bilinear superoperators 434
19.6. Deformation of operator algebras 437
19.7. Phase-space metric for classical non-Hamiltonian system 442
Chapter 20. Fractional Quantum Dynamics 448
20.1. Fractional power of superoperator 448
20.2. Fractional Lindblad equation and fractional semi-group 450
20.3. Quantization of fractional derivatives 459
20.4. Quantization of Weierstrass nondifferentiable function 463
Chapter 21. Stationary States of Non-Hamiltonian Systems 468
21.1. Pure stationary states 468
21.2. Stationary states of non-Hamiltonian systems 470
21.3. Non-Hamiltonian systems with oscillator stationary states 471
21.4. Dynamical bifurcations and catastrophes 474
21.5. Fold catastrophe 476
Chapter 22. Quantum Dynamical Methods 478
22.1. Resolvent method for non-Hamiltonian systems 478
22.2. Wigner function method for non-Hamiltonian systems 481
22.3. Integrals of motion of non-Hamiltonian systems 488
Chapter 23. Path Integral for Non-Hamiltonian Systems 490
23.1. Non-Hamiltonian evolution of mixed states 490
23.2. Path integral for quantum operations 492
23.3. Path integral for completely positive quantum operations 495
Chapter 24. Non-Hamiltonian Systems as Quantum Computers 502
24.1. Quantum state and qubit 502
24.2. Finite-dimensional Liouville space and superoperators 504
24.3. Generalized computational basis and ququats 506
24.4. Quantum four-valued logic gates 510
24.5. Classical four-valued logic gates 524
24.6. To universal set of quantum four-valued logic gates 527
Bibliography 536
Subject Index 548