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Adler S.L. — Quaternionic Quantum Mechanics and Quantum Fields

Adler S.L. — Quaternionic Quantum Mechanics and Quantum Fields

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Название: Quaternionic Quantum Mechanics and Quantum Fields

Автор: Adler S.L.

Аннотация:

It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex Hilbert space. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Authored by a world-renowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum mechanics for the purpose of determining whether quaternionic Hilbert space is the appropriate arena for the long sought-after unification of the standard model forces with gravitation. Significant results from earlier literature, together with many new results obtained by the author, are integrated to give a coherent picture of the subject. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic Hilbert space. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still challenge the field. This well-written treatise is a very significant contribution to theoretical physics. It will be eagerly read by a wide range of physicists.

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Рубрика: Механика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1995

Количество страниц: 586

Добавлена в каталог: 10.12.2009

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Предметный указатель

Spin-1 relativistic wave equation      see "Gauge potential"
Spin-1/2 relativistic wave equation      see "Dirac equation"
SPIRES keywords      7n.5
SQUID (Superconducting Quantum Interference Device)      519 522
Stability of atoms and nuclei      see "Rest masses and
Standard model      3 213—214 216—217 475 497 497n.1 498n.2 501 504 516 520 533
Standard model as an asymptotic dynamics      497
Standard model, hierarchy problem of      516 525
Standard model, running couplings of      498
Standard model, time reversal violation phenomenology      216—217 518
States (in Hilbert space)      61 118 see
States (in Hilbert space), bound and unitarity deficiency      226
States (in Hilbert space), correspond to unit rays      29
States (in Hilbert space), density matrix for      68
States (in Hilbert space), described by vectors      20—22
States (in Hilbert space), dispersion of operator in      71
States (in Hilbert space), energy eigenstate expansion of      45 195
States (in Hilbert space), energy eigenstates introduced      43
States (in Hilbert space), expectation defined in terms of      68 71n.8
States (in Hilbert space), free particle in and out, completeness      223—224
States (in Hilbert space), free particle in and out, defined      220
States (in Hilbert space), free particle in and out, normalization      225
States (in Hilbert space), free particle in and out, related to full scattering state      221—222
States (in Hilbert space), full scattering      218
States (in Hilbert space), full scattering, normalization      225—226
States (in Hilbert space), full scattering, not complete when bound states      225—227 225n.1
States (in Hilbert space), initial system in decay      201
States (in Hilbert space), stationary      see also "Energy eigenstates"
States (in Hilbert space), stationary, methods for approximating      124—145 194
States (in Hilbert space), time development of      36—37
States (in Hilbert space), wave function defined from      38
Stationary state perturbation theory      see "Perturbation theory"
Statistical mechanics      287—293 527 532 see
Statistics, Bose      238
Statistics, Fermi      238
Statistics, fractional      238n.2 527
Statistics, para-      238n.2 527
Statistics, spin connection with      481 531
Stone's theorem      31
String theory      3 498
Sturm — Liouville system boundary conditions      168 168n.2
Subasymptotic states momentum not well defined on      64
Subsystem density matrix      252—253
Subsystem density matrix, cluster decomposition property      253—254 295—299
Subsystem density matrix, cluster decomposition property, abbreviated notations used in      295n.13 296
Superconductivity, BCS theory of      511
Supermatrix formalism, applied to quaternionic Gaussian integrals      491—494 549—551
Superposition      5 8—10 521
Supersymmetric or supersymmetry and fermionic current in quaternionic field theory      484—486
Supersymmetric or supersymmetry, case of Gaussian integral formulas      492—494
Supersymmetric or supersymmetry, extensions of Poincare algebra      361 499 532
Supersymmetric or supersymmetry, hints of in operator constraints and Gaussian integrals      465 492—493 492n.48 532
Supersymmetric or supersymmetry, quantum mechanics (Witten model)      358—359 see two-component
Supersymmetric or supersymmetry, total trace Lagrangian theories      532
Symmetry transformation or generators      29—31 34 36 53—70 74—76 99 112 238 388 397
Symmetry transformation or generators and anti-self-adjoint generators      29—31 53—68 388 433—434
Symmetry transformation or generators in complex quantum mechanics      76 433
Symmetry transformation or generators in complex quantum mechanics, Poincare and internal symmetry      433
Symmetry transformation or generators in generalized quantum dynamics      451—452
Symmetry transformation or generators of complex versions of relativistic wave equations      362 384—387
Symmetry transformation or generators, complex classification applies in quaternionic case      433—434
Symmetry transformation or generators, generators which anticommute with Hamiltonian      75n.9
Symmetry transformation or generators, most general composition law for      100 see projective
Symmetry transformation or generators, Wigner analysis of group representations      53 74—76
Symmetry transformation or generators, Wigner analysis of group representations, leads to complex representations      75—76 141n.6 166 238
Symplectic component representation      18—19 26—27 40—42 56—57 71—73 85—86 93—94 93n.7 114n.17 123 126 132 140—141 143 146—148 153—173 184 201—207 218 244 255n.5 259—260 272n.1 329 333—334 341—345 354—358 391 395 482n.42 491 541—542
Symplectic component representation and embedded fermion fields      427—431
Symplectic component representation and running wave solutions      160 171—172 218
Symplectic component representation, $\beta$ component perturbations      246 248 270 295—296
Symplectic component representation, bound states in $\beta$ components      161 334
Symplectic component representation, coupled complex equations for      126 160 166 196
Symplectic component representation, defined and notation introduced      13
Symplectic component representation, exponential decay of $\beta$ component      160 164—165 167 178 188 192n.16 517
Symplectic component representation, positive and negative energy states in      314
Symplectic component representation, sign reversal for kinetic energy of $\beta$ component      160
Symplectic component representation, use in complex Lagrangians      381—384
Symplectic component representation, use in Foldy — Wouthuysen method      325—327
Symplectic group      13n.13
T-matrix      see "Transition matrix"
Tensor, product problem      240—245 250 271
Tensor, three-index antisymmetric      11—12
Thermo field dynamics      527
Three-body problem      240—242
Time development or evolution      see "Dynamics"
Time reversal violation      4 159 174—175 531
Time reversal violation and tests for quaternionic effects      518
Time reversal violation in elementary particle physics      194 213—217 387 518 see violation"
Time reversal violation, conditions for vanishing      114—115 174 191 215
Time reversal, bosonic states with eigenvalues 1 and -1 related      215n.11
Time reversal, notation T in quaternionic quantum mechanics and $\mathcal{T}$ in complex quantum mechanics and phenomenology      214—217 217n.14
Time reversal, relationship between complex and quaternionic definitions      482n.42
Time reversal, symmetry transformation or invariance      29 35 75n.9
Time reversal, symmetry transformation or invariance in complex Lagrangian field models      381—382 385—387
Time reversal, symmetry transformation or invariance in quaternionic field theories      481—483
Time reversal, symmetry transformation or invariance in spin zero systems      112—119 122 510—511
Time reversal, symmetry transformation or invariance in systems with spin      119—122 510—511
Time reversal, symmetry transformation or invariance of quaternionic harmonic oscillator      122—123
Time reversal, symmetry transformation or invariance, implemented by complex conjugation      47n.19 49 112 214 386 386n.9
Time reversal, symmetry transformation or invariance, not implementable by conjugation      112
Time reversal, symmetry transformation or invariance, real phase factors in      386 481 483
Time reversal, unitary operator $U_{T}$ for invariance      112—118
Time reversal, unitary operator $U_{T}$ for invariance and complex antiunitary operator $\mathcal{T}$       118—119
Time reversal, unitary operator $U_{T}$ for invariance and constant phase of $H_{\beta}$       114n.17 188 191
Time reversal, unitary operator $U_{T}$ for invariance and linear dependence of $V_{2}$ and $V_{3}$       114—115 191
Time reversal, unitary operator $U_{T}$ for invariance and right algebra element $u_{T}$       114
Time reversal, unitary operator $U_{T}$ for invariance in generalized quantum dynamics      481—483
Time reversal, unitary operator $U_{T}$ for invariance, action on energy eigenstates      115—117 121—122
Time reversal, unitary operator $U_{T}$ for invariance, action on momentum/angular momentum      114 118—119
Time reversal, unitary operator $U_{T}$ for invariance, action on operators      116—117
Time reversal, unitary operator $U_{T}$ for invariance, action on S-matrix      113 117 192 231—232
Time reversal, unitary operator $U_{T}$ for invariance, dependent on structure of Hamiltonian      114—115
Time reversal, unitary operator $U_{T}$ for invariance, extended for systems with spin      119
Time reversal, unitary operator $U_{T}$ for invariance, necessary condition for      112—113
Time reversal, unitary operator $U_{T}$ for invariance, not universal      113—114
Time reversal, unitary operator $U_{T}$ for invariance, sufficient conditions for      113—115 120
Time translation generator      see "Hamiltonian"
Time-ordering operators $T_{\ell}$ , $T_{r}$ , applied      70 149 151—152 209 229 231—232 353
Time-ordering operators $T_{\ell}$ , $T_{r}$ , defined      37
Total trace dynamics      see "Generalized quantum dynamics"
Total trace functional or Trace functional      398—399 442—489 499 see generalized" "Generalized "Operator-valued
Total trace functional or Trace functional, action principle for      444—446
Total trace functional or Trace functional, energy-momentum tensor      483—484 533
Total trace functional or Trace functional, expectation and transition probabilities expressed in terms of      448—449
Total trace functional or Trace functional, Hamiltonian      399 446 499 523

Total trace functional or Trace functional, Hamiltonian, constrained      455 529
Total trace functional or Trace functional, Hamiltonian, nonunitary dynamics may be chaotic      524n.19
Total trace functional or Trace functional, Hamiltonian, operator equations of motion      446
Total trace functional or Trace functional, Hamiltonian, transformation to Schroedinger picture      454
Total trace functional or Trace functional, Hamiltonian, unitary dynamics a special case      453—454 524 524n.18
Total trace functional or Trace functional, Lagrangian      399 444 523 528—529
Total trace functional or Trace functional, Lagrangian and complex quantum mechanics      455—475
Total trace functional or Trace functional, Lagrangian and constrained systems      455 455n.26 529
Total trace functional or Trace functional, Lagrangian for quaternionic field models      475—478 483
Total trace functional or Trace functional, Lagrangian, gauge-fixing conditions      455
Total trace functional or Trace functional, Lagrangian, operator equations of motion      446
Total trace functional or Trace functional, Lagrangian, operator-valued gauge invariance of      449—450
Total trace functional or Trace functional, Noether theorem generalized to      450—452
Total trace functional or Trace functional, Noether theorem generalized to, and generator algebra closure      452
Total trace functional or Trace functional, Noether theorem generalized to, for linear transformations      451—452
Total trace functional or Trace functional, Noether theorem generalized to, Poincare generators      451
Total trace functional or Trace functional, operator derivative and      445
Total trace functional or Trace functional, operator derivative and, higher derivatives not defined      445
Total trace functional or Trace functional, operator derivative and, Leibnitz product rule for      448
Total trace functional or Trace functional, operator derivative and, relation to matrix element derivatives      539
Total trace functional or Trace functional, operator Euler — Lagrange equations for      446
Total trace functional or Trace functional, operator Euler — Lagrange equations for, as operator constraints      455
Total trace functional or Trace functional, operator Euler — Lagrange equations for, operator gauge covariant      450
Trace operation, tr defined for octonion      51n.20
Trace operation, tr defined for quaternion      12
Trace operation, Tr defined for quaternion operators      15 443
Trace operation, Tr defined for quaternion operators and Witten index      443n.22
Trace operation, Tr defined for quaternion operators, cyclic property      15 290 443
Trace operation, Tr defined for quaternion operators, properties of      443—444 443n.22 459 459n.30 463 529
Trace operation, tr defined for quaternion, cyclic property      12 318 376
Trace operation, tr defined for quaternion, over left-acting algebra      280 431
Transformation function      see "Probability amplitude" "Wave
Transition matrix      173 196—201 see "Scattering amplitude"
Transition matrix and outgoing wave function      201
Transition matrix, coupled equations for      197
Transition matrix, operator form for      200
Transition matrix, related to S-matrix      173
Translation group      see "Momentum"
Translation invariant multiparticle system      see "Hamiltonian in
Translation invariant system      see "Hamiltonian"
Transpose of column vector      21
Transpose of product      15
Transpose, defined for quaternion matrix      15
Transpose, use of $^{T}$ as notation for      15 39 41 191 309 323 330 367 408n.4 477 490
Trotter product formula      109
Two-body problem      239—240
Uncertainty principle      47n.19 53 70—74
Unification of forces      3
Unitarity of S-matrix      228
Unitarity, deficiency and Moeller wave operators      226 265
Unitarity, failure in octonionic Hilbert space quantum mechanics      51—52 52n.22
Unitarity, sum rule      163 174 204—207
Unitary operator      see "Operator quaternion
Vacuum spontaneous symmetry breaking      75n.9 387 387n.10 396—397 508 526
Vacuum state      273
Vacuum state, doublet      429n.14
Variational principles      144—145
Variational principles for mean field approximation      297
Variational principles for smallest eigenvalue of Hamiltonian modulus      145
Variational principles, Rayleigh — Ritz analog      145
Vector potential      93—98 239—240
Vector potential, "string" for monopole      97n.8
Vector potential, quaternion imaginary      94
Vector-like theory      508 511
Vectors in coordinate space, conventions for three-vectors and four-vectors      53n.1 320n.6
Vectors in Hilbert space      see "Hilbert space vectors
Velocity operator      93 235
Vierbein      513
Virial theorem      108—109 352—353
Wave equation, effective      see "Optical potential" "Quantum complex"
Wave equation, effective, relativistic      303—388 397 see "Gauge "Klein "Lagrangian
Wave equation, effective, two-component semirelativistic      58 66n.6 87n.1 111n.14 165n.1 179n.8 348 350—361 429n.14
Wave equation, effective, two-component semirelativistic and supersymmetric quantum mechanics      358—359
Wave equation, effective, two-component semirelativistic and Witten model      358—359
Wave equation, effective, two-component semirelativistic, Ehrenfest and virial theorems      352—353
Wave equation, effective, two-component semirelativistic, energy eigenstates      351
Wave equation, effective, two-component semirelativistic, Feynman path integral      353—354
Wave equation, effective, two-component semirelativistic, probability current conservation      352
Wave equation, effective, two-component semirelativistic, scattering theory and bound states      354—358
Wave equation, effective, two-component semirelativistic, self-adjoint generators      351
Wave equation, effective, two-component semirelativistic, transformation to complex form      360—361 441
Wave function      19—20 26 see "Momentum representation"
Wave function for energy eigenstate (giving time-independent Schroedinger equation)      97
Wave function for identical particles      238 270—271
Wave function, asymptotic scattering      see "Asymptotic scattering states"
Wave function, bound state normalization      161 168
Wave function, boundedness      160 172 183n.12 186
Wave function, complex, related to real      49
Wave function, continuity conditions for      161—162 400 403
Wave function, eikonal form for      156
Wave function, four-component quaternionic      184 308 367
Wave function, four-component quaternionic, spinor      329 332n.13
Wave function, free particle      180 219—221
Wave function, junction conditions      160 162 164 355
Wave function, n-component      39
Wave function, phase, methods based on      145—158
Wave function, relative coordinate, in multiparticle system      239 241 245
Wave function, rotational invariance analysis for      68 80 179
Wave function, single-component      40 58 64 84 87 91n.5 98 240
Wave function, two-component complex      40 272n.1 358 382 407—409 528
Wave function, two-component quaternionic      84 312—314
Wave function, two-component quaternionic, spinor      329n.12
Wave function, two-component semirelativistic      58 see
Wave packet      227 227n.2 266—267 405—407
Wave packet, group velocity for      406
Weisskopf — Wigner approximation      202n.7 204—206
Weyl ordering      460—461
Wigner analysis of symmetries of Hamiltonian      53
Wigner theorem for unit ray mappings      29—31 36 112
Witten index      443n.22
Witten model for supersymmetric quantum mechanics      358—359 see two-component
WKB approximation      156—158 527
WKB approximation, connection formulas not known in quaternionic case      158n.13 527
WKB approximation, equations for real components of eikonal integrand      157—158
WKB approximation, Riccati equation and      158 527
WKB approximation, Riccati equation and, algebraic equation in complex limit      158
WKB approximation, slow variation assumption      156
wronskian      169—170 183n.12
Yang — Mills gauge potential and field      363 442 460n.31 488 488n.45 527
Yang — Mills gauge potential and field, identification of observables for      453n.24
Yang — Mills gauge potential and field, identification of observables for, path-ordered integrals used      453n.24
Yang — Mills gauge potential and field, operator gauge-invariant extension      472—475
Yang — Mills gauge potential and field, supersymmetric      484
Yang — Mills gauge potential and field, ubiquitous appearance in standard model physics      501
Zero energy states      see "Energy zero

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