| Preface | p. v |
| A Very Few Preliminaries | p. 1 |
| Quantum Kinematics | p. 9 |
| Quantum Kinematics of Bounded Observables | p. 11 |
| Observables and states | p. 11 |
| Pre-Hilbert and Hilbert spaces | p. 11 |
| Separable Hilbert space | p. 16 |
| Definition and examples of operators | p. 18 |
| Quantum kinematical postulates | p. 20 |
| Dual Hilbert space | p. 21 |
| Dirac's notations | p. 23 |
| Matrix representation of operator | p. 24 |
| Quantum Kinematics of Unbounded Observables | p. 27 |
| Deficiencies of Hilbert spaces | p. 27 |
| Spaces of test functions | p. 28 |
| Spaces of generalized functions | p. 31 |
| Rigged Hilbert space | p. 33 |
| Linear operators on a rigged Hilbert space | p. 36 |
| Coordinate representation | p. 40 |
| X-representation | p. 43 |
| Mathematical Structures in Quantum Kinematics | p. 47 |
| Mathematical structures | p. 47 |
| Order structures | p. 49 |
| Topological structures | p. 50 |
| Algebraic structures | p. 52 |
| Examples of algebraic structures | p. 57 |
| Mathematical structures in kinematics | p. 66 |
| Spaces of Quantum Observables | p. 69 |
| Space of bounded operators | p. 69 |
| Space of finite-rank operators | p. 71 |
| Space of compact operators | p. 72 |
| Space of trace-class operators | p. 74 |
| Space of Hilbert-Schmidt operators | p. 78 |
| Properties of operators from K[superscript 1](H) and K[superscript 2](H) | p. 79 |
| Set of density operators | p. 81 |
| Operator Hilbert space and Liouville space | p. 83 |
| Correlation functions | p. 87 |
| Basis for Liouville space | p. 88 |
| Rigged Liouville space | p. 91 |
| Algebras of Quantum Observables | p. 95 |
| Linear algebra | p. 95 |
| Associative algebra | p. 96 |
| Lie algebra | p. 96 |
| Jordan algebra | p. 98 |
| Involutive, normed and Banach algebras | p. 100 |
| C*-algebra | p. 103 |
| W*-algebra | p. 106 |
| athit J B-algebra | p. 111 |
| Hilbert algebra | p. 112 |
| Mathematical Structures on State Sets | p. 115 |
| State as functional on operator algebra | p. 115 |
| State on C*-algebra | p. 117 |
| Representations C*-algebra and states | p. 123 |
| Gelfand-Naimark-Segal construction | p. 124 |
| State on W*-algebra | p. 128 |
| Mathematical Structures in Classical Kinematics | p. 129 |
| Symplectic structure | p. 129 |
| Poisson manifold and Lie-Jordan algebra | p. 130 |
| Classical states | p. 133 |
| Classical observables and C*-algebra | p. 136 |
| Quantization in Kinematics | p. 139 |
| Quantization and its properties | p. 139 |
| Heisenberg algebra | p. 147 |
| Weyl system and Weyl algebra | p. 149 |
| Weyl and Wigner operator bases | p. 152 |
| Differential operators and symbols | p. 157 |
| Weyl quantization mapping | p. 159 |
| Kernel and symbol of Weyl ordered operator | p. 161 |
| Weyl symbols and Wigner representation | p. 162 |
| Inverse of quantization map | p. 165 |
| Symbols of operators and Weyl quantization | p. 166 |
| Generalization of Weyl quantization | p. 174 |
| Spectral Representation of Observable | p. 181 |
| Spectrum of quantum observable | p. 181 |
| Algebra of operator functions | p. 187 |
| Spectral projection and spectral decomposition | p. 189 |
| Symmetrical and self-adjoint operators | p. 192 |
| Resolution of the identity | p. 194 |
| Spectral theorem | p. 196 |
| Spectral operator through ket-bra operator | p. 199 |
| Function of self-adjoint operator | p. 201 |
| Commutative and permutable operators | p. 203 |
| Spectral representation | p. 205 |
| Complete system of commuting observables | p. 208 |
| Quantum Dynamics | p. 211 |
| Superoperators and its Properties | p. 213 |
| Mathematical structures in quantum dynamics | p. 213 |
| Definition of superoperator | p. 217 |
| Left and right superoperators | p. 220 |
| Superoperator kernel | p. 223 |
| Closed and resolvent superoperators | p. 226 |
| Superoperator of derivation | p. 227 |
| Hamiltonian superoperator | p. 231 |
| Integration of quantum observables | p. 233 |
| Superoperator Algebras and Spaces | p. 237 |
| Linear spaces and algebras of superoperators | p. 237 |
| Superoperator algebra for Lie operator algebra | p. 240 |
| Superoperator algebra for Jordan operator algebra | p. 241 |
| Superoperator algebra for Lie-Jordan operator algebra | p. 243 |
| Superoperator C*-algebra and double centralisers | p. 244 |
| Superoperator W*-algebra | p. 247 |
| Superoperator Functions | p. 251 |
| Function of left and right superoperators | p. 251 |
| Inverse superoperator function | p. 253 |
| Superoperator function and Fourier transform | p. 254 |
| Exponential superoperator function | p. 255 |
| Superoperator Heisenberg algebra | p. 257 |
| Superoperator Weyl system | p. 258 |
| Algebra of Weyl superoperators | p. 259 |
| Superoperator functions and ordering | p. 261 |
| Weyl ordered superoperator | p. 263 |
| Semi-Groups of Superoperators | p. 267 |
| Groups of superoperators | p. 267 |
| Semi-groups of superoperators | p. 269 |
| Generating superoperators of semi-groups | p. 273 |
| Contractive semi-groups and its generators | p. 275 |
| Positive semi-groups | p. 279 |
| Differential Equations for Quantum Observables | p. 285 |
| Quantum dynamics and operator differential equations | p. 285 |
| Definition of operator differential equations | p. 286 |
| Equations with constant bounded superoperators | p. 288 |
| Chronological multiplication | p. 289 |
| Equations with variable bounded superoperators | p. 291 |
| Operator equations with constant unbounded superoperators | p. 294 |
| Generating superoperator and its resolvent | p. 295 |
| Equations in operator Hilbert spaces | p. 298 |
| Equations in coordinate representation | p. 301 |
| Example of operator differential equation | p. 302 |
| Quantum Dynamical Semi-Group | p. 305 |
| Dynamical semi-groups | p. 305 |
| Semi-scalar product and dynamical semi-groups | p. 307 |
| Dynamical semi-groups and orthogonal projections | p. 309 |
| Dynamical semi-groups for observables | p. 311 |
| Quantum dynamical semi-groups on W*-algebras | p. 313 |
| Completely positive superoperators | p. 315 |
| Bipositive superoperators | p. 319 |
| Completely dissipative superoperators | p. 320 |
| Lindblad equation | p. 323 |
| Example of Lindblad equation | p. 328 |
| Gorini-Kossakowski-Sudarshan equation | p. 331 |
| Two-level non-Hamiltonian quantum system | p. 333 |
| Classical Non-Hamiltonian Dynamics | p. 337 |
| Introduction to classical dynamics | p. 337 |
| Systems on symplectic manifold | p. 340 |
| Systems on Poisson manifold | p. 346 |
| Properties of locally Hamiltonian systems | p. 349 |
| Quantum Hamiltonian and non-Hamiltonian systems | p. 352 |
| Hamiltonian and Liouvillian pictures | p. 354 |
| Quantization of Dynamical Structure | p. 361 |
| Quantization in kinematics and dynamics | p. 361 |
| Quantization map for equations of motion | p. 363 |
| Quantization of Lorenz-type systems | p. 370 |
| Quantization of Poisson bracket | p. 371 |
| Discontinuous functions and nonassociative operators | p. 377 |
| Quantum Dynamics of States | p. 381 |
| Evolution equation for normalized operator | p. 381 |
| Quantization for Hamiltonian picture | p. 383 |
| Expectation values for non-Hamiltonian systems | p. 384 |
| Adjoint and inverse superoperators | p. 389 |
| Adjoint Lie-Jordan superoperator functions | p. 392 |
| Weyl multiplication and Weyl scalar product | p. 397 |
| Weyl expectation value and Weyl correlators | p. 400 |
| Evolution of state in the Schrodinger picture | p. 404 |
| Dynamical Deformation of Algebras of Observables | p. 409 |
| Evolution as a map | p. 409 |
| Rule of term-by-term differentiation | p. 412 |
| Time evolution of binary operations | p. 414 |
| Bilinear superoperators | p. 417 |
| Cohomology groups of bilinear superoperators | p. 419 |
| Deformation of operator algebras | p. 422 |
| Phase-space metric for classical non-Hamiltonian system | p. 427 |
| Fractional Quantum Dynamics | p. 433 |
| Fractional power of superoperator | p. 433 |
| Fractional Lindblad equation and fractional semi-group | p. 435 |
| Quantization of fractional derivatives | p. 444 |
| Quantization of Weierstrass nondifferentiable function | p. 448 |
| Stationary States of Non-Hamiltonian Systems | p. 453 |
| Pure stationary states | p. 453 |
| Stationary states of non-Hamiltonian systems | p. 455 |
| Non-Hamiltonian systems with oscillator stationary states | p. 456 |
| Dynamical bifurcations and catastrophes | p. 459 |
| Fold catastrophe | p. 461 |
| Quantum Dynamical Methods | p. 463 |
| Resolvent method for non-Hamiltonian systems | p. 463 |
| Wigner function method for non-Hamiltonian systems | p. 466 |
| Integrals of motion of non-Hamiltonian systems | p. 473 |
| Path Integral for Non-Hamiltonian Systems | p. 475 |
| Non-Hamiltonian evolution of mixed states | p. 475 |
| Path integral for quantum operations | p. 477 |
| Path integral for completely positive quantum operations | p. 480 |
| Non-Hamiltonian Systems as Quantum Computers | p. 487 |
| Quantum state and qubit | p. 487 |
| Finite-dimensional Liouville space and superoperators | p. 489 |
| Generalized computational basis and ququats | p. 491 |
| Quantum four-valued logic gates | p. 495 |
| Classical four-valued logic gates | p. 509 |
| To universal set of quantum four-valued logic gates | p. 512 |
| Bibliography | p. 521 |
| Subject Index | p. 533 |
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