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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

Complex quantum mechanics, projective representation in      102
Complex quantum mechanics, ray structure of and phenomenology of nonlinear quantum mechanics      525
Complex quantum mechanics, reduction of quaternionic Schroedinger equation to      124—131
Complex quantum mechanics, regime of validity      498
Complex quantum mechanics, relationship to quaternionic quantum mechanics      40—44 111n.14 230 272n.1 407—410 439—440
Complex quantum mechanics, relationship to quaternionic quantum mechanics and class-1 complex representations      439—440
Complex quantum mechanics, relationship to real quantum mechanics      44—45
Complex quantum mechanics, scattering theory in      174 218 230
Complex quantum mechanics, scattering theory in, multichannel      262 266
Complex quantum mechanics, Schroedinger equation in      46 49
Complex quantum mechanics, Schroedinger equation in, as classical dynamical system      442n.21
Complex quantum mechanics, Schroedinger equation in, tests for nonlinear corrections      517 524—525
Complex quantum mechanics, Schur's Lemma in      104n.12
Complex quantum mechanics, shift of energy origin in      46
Complex quantum mechanics, Slater determinant in      271
Complex quantum mechanics, thermal Green's functions in      293 293n.12
Complex quantum mechanics, time reversal transformation in      47n.19 112—113 116—118 122
Complex quantum mechanics, uncertainty principle in      72
Complex quantum mechanics, WKB approximation in      158 158n.13
Complex subalgebra of quaternion      13 see i)$"/>"
Composite models of leptons/quarks      501—511 501n.4 519—520
Compton wavelength      312
Condensed matter physics      519—520 527
Conjugate or conjugation, complex      7 13 16n.16 112 174
Conjugate or conjugation, complex, notation * used for      13 43 86
Conjugate or conjugation, octonion      8
Conjugate or conjugation, quaternion      7—8 13—14 16 112
Conjugate or conjugation, quaternion of product      14 16 112 491 543
Conjugate or conjugation, quaternion, notation $\bar{}$ used for      13
Continuity conditions for wave function      161—162
Coordinate operator      26 59 76 89 234
Coordinate operator, eigenstates and eigenvalues      26 82
Coordinate operator, eigenstates and eigenvalues, notation for      25n.5
Coordinate representation, Galilean transformation operator in      91 234
Coordinate representation, Hamiltonian      38 40—44 89 95 98 112 see
Coordinate representation, left-acting algebra I, J, K in      38
Coordinate representation, momentum space translation generator      100
Coordinate representation, notation for operator and eigenvalue      25n.5 275n.3
Coordinate representation, wave function      37—38 40—44 55—57 95 112 126—127 271 304 311—312
Coordinate representation, wave function, symplectic components of      56—57 71
Coordinate representation, wave function, time reversal restrictions on      116 121—122
Coordinate, center of mass      81 238 241 245—246 255—256
Coordinate, relative, and Schroedinger equation in      239 241 246
Copenhagen interpretation      520 522
Correspondence principle      6 8—9
Correspondence principle, two-level      498
Cosmological constant problem      500—501 532
Cosmological constant problem and zero energy exception      532
Cosmological constant problem, supersymmetry and      500
Covariant derivative, for complex reductions $\mathcal{D}_{\mu}$       382—383 528
Covariant derivative, for complex reductions $\mathcal{D}_{\mu}$ , auxiliary matrices $\mathcal{M}_{\mu}$ , and $\mathcal{M}^{'}_{\mu}$       382
Covariant derivative, for complex reductions $\mathcal{D}_{\mu}$ , charge conjugation of      384—385
Covariant derivative, for complex reductions $\mathcal{D}_{\mu}$ , noncommutation with chiral projectors      388 531
Covariant derivative, for complex reductions $\mathcal{D}_{\mu}$ , time reversal of      386—387
Covariant derivative, nonrelativistic, anti-self-adjointness of      92
Covariant derivative, nonrelativistic, parallel transports unit quaternion e      92
Covariant derivative, relativistic, $\bar{\mathcal{D}_{\mu}}$       484—486
Covariant derivative, relativistic, $\bar{\mathcal{D}_{\mu}}$ , relation by adjoint to $\mathcal{D}_{\mu}$       484
Covariant derivative, relativistic, $\hat{\mathcal{D}}_{\mu}$ and $\hat{\mathcal{D}}^{'}_{\mu}$       317 363 368—369 476—478
Covariant derivative, relativistic, $\hat{\mathcal{D}}_{\mu}$ and $\hat{\mathcal{D}}^{'}_{\mu}$ , applied to complex quantum systems      456—460 462
Covariant derivative, relativistic, $\hat{\mathcal{D}}_{\mu}$ and $\hat{\mathcal{D}}^{'}_{\mu}$ , obey product rule for derivative      317
Covariant derivative, relativistic, $\mathcal{D}_{\mu}$       315 368 475—478 484
Covariant derivative, relativistic, $\mathcal{D}_{\mu}$ , does not obey product rule      315n.4 317
Covariant derivative, relativistic, $\mathcal{D}_{\mu}$ , gauge transformation of      316
Covariant derivative, relativistic, $\mathcal{D}_{\mu}$ , noncommutation with chiral projectors      388 508 531
Covariant derivative, relativistic, commutator identity      364—365
Covariant derivative, relativistic, for curved space-time $\nabla_{\lambda}$       483
Covariant derivative, relativistic, intertwining identities relating      317 352 368—369 371 476
Covariant derivative, relativistic, intertwining identities, trace of      318 376
Cyclic property of trace      12 15 290 318 376 443 443n.22 529
Cyclic property of trace, requires associativity      540
d'Alembertian      304 316
Dagger ( $\dagger$ ), indicates sections not dependent on nonrelativistic kinematics      87 99n.
Dagger ( $\dagger$ ), notation for adjoint      14 16 94 236
Decaying states      179n.8 201—208 213—217 526 see time-dependent"
Degenerate set of Hamiltonian eigenstates      74—75 201
Degenerate subspace or manifold      28 81 201
Density matrix      53 68—69 245n.3
Density matrix and clustering in multiparticle systems      233 250—252 295—299
Density matrix, defined      68
Density matrix, thermal equilibrium functional form      288
Density matrix, time evolution equation for      69
Derivation      see "Leibnitz product rule"
Derivative, first variational with respect to operator      445
Derivative, first variational with respect to operator, Leibnitz product rule for      448
Derivative, first variational with respect to operator, related to matrix element derivatives      539
Derivative, first variational with respect to operator, second and higher not defined      445
Determinant      191
Determinant, quaternionic, for self-adjoint matrices      490 494—496
Determinant, quaternionic, for self-adjoint matrices, properties summarized      495—496
Dimensional regularization      486 486n.44
Dirac bra-ket notation      5 19 21—22
Dirac delta function      39 198n.3 271
Dirac delta function as decaying state initial value      202
Dirac delta function, potential model for scattering      159—165
Dirac delta function, quaternionic generalization      493—494
Dirac delta function, singularity in canonical commutator      529
Dirac equation      87 303 329—345 348—350 362 399 see
Dirac equation, free particle case      329—337 492n.48
Dirac equation, free particle case in Dirac, Weyl, and generic complex (C) representation, $i_{G} = i$       331 333
Dirac equation, free particle case in Dirac, Weyl, and generic complex (C) representation, spinor orthogonality and spin sums      428n.13 430n.16
Dirac equation, free particle case in general (G) representation      329
Dirac equation, free particle case in general (G) representation, charge conjugation matrix      419n.10
Dirac equation, free particle case in general (G) representation, Dirac matrices $\alpha^{\ell}_{G}$ , $\beta_{G}$ , and $i_{G}$       331
Dirac equation, free particle case in general (G) representation, Hamiltonian for      331
Dirac equation, free particle case in general (G) representation, Poincare generators for      389 433 433n.17
Dirac equation, free particle case in Majorana (M) representation      329—330 419—421
Dirac equation, free particle case in Majorana (M) representation for adjoint wave function      330
Dirac equation, free particle case in Majorana (M) representation, $\gamma^{\mu}$ matrix reality and/or symmetry      369—370 378—379 433 477 480
Dirac equation, free particle case in Majorana (M) representation, $\gamma_{5}$ imaginary      387—388
Dirac equation, free particle case in Majorana (M) representation, Dirac anticommutator algebra      330
Dirac equation, free particle case in Majorana (M) representation, Dirac matrices $\alpha^{\ell}_{M}$ and $\beta_{M}$       330
Dirac equation, free particle case in Majorana (M) representation, Hamiltonian for      329
Dirac equation, free particle case in Majorana (M) representation, Poincare generators      433
Dirac equation, free particle case in Majorana (M) representation, subscript M omitted      475
Dirac equation, free particle case in Majorana (M) representation, time reversal matrix $A_{M}$       386 420
Dirac equation, free particle case, energy eigenstates and form of "anti-particle" states      333—334 417 429 432
Dirac equation, free particle case, four-component Lorentz spinor wave function      329
Dirac equation, free particle case, four-component Lorentz spinor wave function, bar denotes quaternion conjugation      330 335n.14
Dirac equation, free particle case, four-component Lorentz spinor wave function, real and symplectic component form      329
Dirac equation, free particle case, four-component Lorentz spinor wave function, spinor index summation convention      329
Dirac equation, free particle case, inner product      330—331
Dirac equation, free particle case, inner product, completeness relation      331—332
Dirac equation, free particle case, inner product, properties of      330—332
Dirac equation, free particle case, Lorentz transformation properties      334—337 389n.11
Dirac equation, free particle case, Lorentz transformation properties, $\gamma^{\mu}$ matrices and wave equation      334—335
Dirac equation, free particle case, Lorentz transformation properties, adjoint $\gamma^{\mu}$ matrices and wave equation      335
Dirac equation, free particle case, Lorentz transformation properties, algebra of $\gamma^{\mu}$ matrices and $i_{G}$       335 390
Dirac equation, free particle case, Lorentz transformation properties, bilinear covariants      337
Dirac equation, free particle case, Lorentz transformation properties, improper (space reflection) and $\gamma^{5}_{G}$       337
Dirac equation, free particle case, Lorentz transformation properties, matrix action on spinor      336—337
Dirac equation, free particle case, Lorentz transformation properties, sixteen independent $4\times 4$ matrices      337
Dirac equation, free particle case, momentum space, four-component wave function      332n.13 333
Dirac equation, free particle case, momentum space, Hamiltonian in complex representation      333
Dirac equation, free particle case, transformation from Majorana to general representation      331—332
Dirac equation, free particle case, two-component spinor wave function and complex inner product form of      329n.12 396n.17
Dirac equation, interacting case      338—345 348—350
Dirac equation, interacting case and source term for gauge potential      362 369—374
Dirac equation, interacting case in complex representation      340—345
Dirac equation, interacting case in complex representation, for one-dimensional potential step      400—407
Dirac equation, interacting case in complex representation, Hamiltonian for      341
Dirac equation, interacting case in complex representation, matrix $\gamma$       341—342
Dirac equation, interacting case in general representation      339—340
Dirac equation, interacting case in general representation, algebra of $\gamma^{\mu}$ matrices and $i_{G}$ , $j_{G}$ , $k_{G}$       340 340n.16—17
Dirac equation, interacting case in general representation, Hamiltonian for      339
Dirac equation, interacting case in general representation, state norm properties      339n.15
Dirac equation, interacting case in Majorana representation      338—339 348—350 369—374 378—381
Dirac equation, interacting case in Majorana representation, four equivalent covariant forms      369—370
Dirac equation, interacting case in Majorana representation, Hamiltonian for      338
Dirac equation, interacting case, general gauge principle for      338—339 434 501 528

Dirac equation, interacting case, general gauge principle for, covariant derivative      338 see
Dirac equation, interacting case, general gauge principle for, gauge potentials and transformation      316 338
Dirac equation, interacting case, general gauge principle for, specializations      338—342 348—350
Dirac equation, interacting case, inner product and properties      338
Dirac equation, interacting case, real currents $\mathcal{K}_{\nu},\ldots$       370—372
Dirac equation, nonrelativistic reduction      86—87 164 342—345
Dirac equation, nonrelativistic reduction, "semirelativistic" Foldy — Wouthuysen reduction      348—350
Dirac equation, nonrelativistic reduction, $\beta$ component of nonrelativistic potential arises as spin effect      303 345
Dirac equation, nonrelativistic reduction, Foldy — Wouthuysen method      342—345
Dirac equation, nonrelativistic reduction, Foldy — Wouthuysen method, conditions for validity      344 345
Dirac equation, nonrelativistic reduction, Foldy — Wouthuysen method, even and odd Hamiltonian terms defined      343
Dirac equation, nonrelativistic reduction, Foldy — Wouthuysen method, leading-order reduction      344—345
Dirac formulation of quantum mechanics      10—11
Dirac monopole      97n.8 528
Dispersion of operator      47n.19 71—74
Dispersion of operator, defined      71
Dispersion of operator, symplectic components of      72
Dispersion relations      179 see "Complex analyticity"
Dispersion relations, Kramers — Kronig      182 499
Dispersion relations, partial wave      183n.12 526
Dot as notation for time differentiation      89—90 93 146 153
Dynamical systems theory      533
Dynamics      see also "Hamiltonian" "Schroedinger
Dynamics for decaying state problem      201—202
Dynamics of complex quantum mechanics      44
Dynamics of densities and expectations      106—109
Dynamics of general quantum mechanics      50—51
Dynamics of octonionic quantum mechanics      51—52 539—540
Dynamics of quaternionic quantum mechanics      36—41 68—70
Dynamics of quaternionic quantum mechanics, finite-time transformation      37 68 70 208—209 228—232 453 523
Dynamics of quaternionic quantum mechanics, finite-time transformation and Feynman path integral      109—110
Dynamics of quaternionic quantum mechanics, finite-time transformation and quantum measurement paradox      523—524
Dynamics of quaternionic quantum mechanics, finite-time transformation, generator and Frobenius — Schur class      439—440
Dynamics of universe, or closed system      521—522
Dynamics, classical Hamiltonian      528
Dynamics, fundamental      533
Dynamics, generalized quantum      399 see
Dynamics, methods for      194
Ehrenfest theorem      106—108 352—353
Ehrenfest theorem, breakdown of      107—108
Eikonal approximation      156—58 see
Emch line of reasoning      34 76
Energy conservation in multiparticle scattering      269
Energy eigenstates      43n.16 45
Energy eigenstates for constant quaternionic potential      125—126
Energy eigenstates for free Dirac equation      333—334
Energy eigenstates for free Klein — Gordon equation      313—314
Energy eigenstates for perturbed and unperturbed systems      131
Energy eigenstates for quaternionic and related complex systems      43
Energy eigenstates for two-component semirelativistic equation      351
Energy eigenstates in complex quantum mechanics      46
Energy eigenstates in complex quantum mechanics and time reversal invariance      116 122
Energy eigenstates of instantaneous Hamiltonian      133 145
Energy eigenstates of instantaneous Hamiltonian as expansion basis      146
Energy eigenstates, absence of in real quantum mechanics      45 47
Energy eigenstates, canonical or standard form for      43 45 74 84 218 225 267 398 407 500
Energy eigenstates, consequences of time reversal for      115—117 120—122
Energy eigenstates, decay theory for      201—208
Energy eigenstates, degenerate      131
Energy eigenstates, dependent on external parameters      133 145—149
Energy eigenstates, expansion of state in terms of      45 146
Energy eigenstates, exponential time dependence      45 97 237
Energy eigenstates, nondegenerate      115 131 146
Energy eigenvalues      43n.16
Energy eigenvalues for perturbed and unperturbed systems      131
Energy eigenvalues for quaternionic and related complex systems      43—44
Energy eigenvalues of ground state      46n.17
Energy eigenvalues, dependent on external parameters      133 145—149
Energy eigenvalues, nonnegative in quaternionic quantum mechanics      45—46
Energy eigenvalues, zero point significant in quaternionic quantum mechanics      45—46 134 196 500—501
Energy scale for quaternionic physics      213 217 498 498n.2 525
Energy, bounded from below      500
Energy, gravitationally defined      500—501
Energy, peculiarities of in quaternionic quantum mechanics      268 269n.11
Energy, zero state as exception      75n.9 121 131 142—143 146 148—149 391—392 394 396—397 503 509—510 526—527
Energy, zero state as exception and cosmological constant problem      532
Energy, zero state as exception in Poincare analysis      395—397 499 526
Energy, zero state as exception, quaternionic representations allowed      141n.6
Energy, zero state as exception, relevance to Goldstone theorem      396—397
entropy      46n.17
Equivalence principle      511
Equivalence principle, extended      514—515 533 see
Euler — Lagrange equations for classical quaternionic wave equations      374—380
Euler — Lagrange equations, giving operator equations of motion in generalized dynamics      446
expectation value      28 47n.19 71—73
Expectation value, expressed in terms of total trace functional      448
Expectation value, notation for      68
Expectation value, thermal      288
Expectation value, time evolution of      68—70 106—109
Experimental signatures for quaternionic quantum mechanics      174 193 497 516—520 527
Experimental signatures for quaternionic quantum mechanics and tests for nonlinearities in quantum mechanics      524—525
Experimental signatures for quaternionic quantum mechanics and tests for nonlinearities in quantum mechanics, parameterization of      525 525n.20
Experimental signatures for quaternionic quantum mechanics and tests for nonlinearities in quantum mechanics, possible link to hierarchy problem      525
Experimental signatures for quaternionic quantum mechanics, multiparticle effects      518—520
Experimental signatures for quaternionic quantum mechanics, neutron-optical experiments      516—518 525n.21
Experimental signatures for quaternionic quantum mechanics, nonasymptotic situations      519—520
Experimental signatures for quaternionic quantum mechanics, scattering phase shifts      516—518
Experimental signatures for quaternionic quantum mechanics, time reversal violating effects      518
Families of leptons and quarks      501—511
Families of leptons and quarks in Harari — Shupe proposal      502—503 532
Fermion      238
Fermion, grading operator $(-1)^{F}$       442—443 442n.21 443n.22 461 481 535
Feynman formulation of classical versus quantum system      4—5
Feynman path integral      109—112 527 531
Feynman path integral in complex quantum mechanics      110 353—354 489 531
Feynman path integral, commutator contributions to      110—111
Feynman path integral, not expressible in terms of an action      112
Feynman path integral, perturbative and Gaussian integrals      489
Feynman remark on quantum measurement theory      522n.17
Fiber-bundle terminology (connection and curvature)      363n.1
Field (as defined in mathematics)      see "Number field"
First quantized case      399 see
Fock space      270—274 277 509 see
Foldy — Wouthuysen method      see "Dirac equation nonrelativistic "Klein nonrelativistic
Forced harmonic oscillator, Heisenberg picture      211—213 413—415
Forced harmonic oscillator, Heisenberg picture, equations of motion      212
Forced harmonic oscillator, interaction picture      210—211
Forced harmonic oscillator, Schroedinger picture      209—210
Formally real components of operator      23 38—40 98 120 139n.5 414 417 423—426 442 530—531
Formally real components of operator, subtleties in Heisenberg picture      425—426 426n.11 442
Formally real components of scalar potential      98
Formally real components of state vector      24—25 38—39 89n.2 275 530
Formula      see "Identity"
Four-current      see also "Gauge potential field equations"
Four-current, $J^{'}_{\nu}$       318—319
Four-current, $\mathcal{J}^{' \mathcal{A}}_{\nu}$       372—373
Four-current, $\mathcal{J}^{\mathcal{A}}_{\nu}$       369 373
Four-current, $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$       364—365 368 371
Four-derivative      304
Four-vector, inner product for      304
Fourier expansion      57 202 303 307 311—312 332—334
Fourier expansion of thermal Green's functions      291—293
Fourier expansion, left vs. right ordering in      57 84n.12 312n.3
Fourier expansion, normalization factor N(p)      307 411
Fourier expansion, notation $\int\limits_{+}$ for half space integral      58 307 366
Fourier expansion, notation $\sum\limits_{\ves{p}, +}$ for half space sum      412
Fourier expansion, real sine and cosine basis      57—58 307 332 366 412
Frobenius — Schur classification      75n.10 437
Functional for variational principles      144—145
Functional, "energy" for Klein — Gordon equation      307n.2
Functional, nonlocal in quaternionic fields, as asymptotic Poincare generators      398
Functional, positive definite      145
Functional, stationary for Schroedinger solutions      144—145
Galilean invariance or transformation      4 38n.15 64 87—95 100 327—328 345 442n.21 456 460 see
Galilean invariance or transformation, Abelian group of      90—91 90n.4 234
Galilean invariance or transformation, Abelian group of, projective representation      90n.4
Galilean invariance or transformation, action on coordinates and velocities      90 234
Galilean invariance or transformation, active      90n.3 98—99
Galilean invariance or transformation, active, assumptions made      235 527
Galilean invariance or transformation, active, multicentrality assumption in      90n.4 103 527

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