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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

Galilean invariance or transformation, active, multiparticle case      234—237
Galilean invariance or transformation, passive      90n.3
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$       315 362—374 see
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , anti-self-adjoint operator      441
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , anti-self-adjoint operator, has operator formally real components      442
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , axial gauge for      488—489
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor      363
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor in terms of potential real components      382
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor is quaternion imaginary      364
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor, associated ray structure      368
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor, Bianchi identities      364 485
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor, gauge variation of      364—365
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , field-strength tensor, homogeneous gauge transformation rule      363
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , formally real component of operator can be nonzero      427
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , gauge transformation of      316 362—363
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , gauge transformation of, covariant wave equation under      363
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , gauge variation of      364—365
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation      365—368
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, four-component wave function      367
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, Fourier expansions      366
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, Hamiltonian      367
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, inner product time independent      367
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, Maxwellian field equations      366
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, momentum space inner product      365
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , linearized approximation, normalization N(p) for field strengths      366 366n.2
Gauge potential $B_{\mu}$ or $B^{'}_{\mu}$ , taken quaternion imaginary      317 362 364
Gauge potential field equations      364
Gauge potential field equations and conserved real currents $\mathcal{K}_{\nu},\ldots$       370—372
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$       364—365 368 371
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$ , are quaternion imaginary      364 368 371
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$ , constructed from Klein — Gordon solution      368 377
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$ , constructed from two Dirac solutions      371—372 378
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$ , covariantly conserved      364 368—369 371—372
Gauge potential field equations, source currents $\mathcal{J}_{\nu}$ and $\mathcal{J}^{'}_{\nu}$ , gauge transformation of      364 368 371
Gauge potential field equations, source currents: specializations and alternatives, constructed from Dirac solution      372
Gauge potential field equations, source currents: specializations and alternatives, constructed from Klein — Gordon solution      369
Gauge potential field equations, source currents: specializations and alternatives, using $\gamma^{5}_{M}$       374
Gauge potential field equations, source currents: specializations and alternatives, using covariantly constant unit e      374n.4
Gauge potential field equations, source currents: specializations and alternatives, using preferred quaternion unit i      369 372—374 374n.4
Gauge potential Lagrangian density and couplings G, G'      376
Gauge potential Lagrangian density and couplings G, G' in generalized quantum dynamics, C requires G = G'      482 531
Gauge potential Lagrangian density and couplings G, G', expressed in terms of real components of potentials      382
Gaussian Integrals      489—496 531
Gaussian integrals and supermatrix formalism      491—493 549—551
Gaussian integrals, complex specialization      489 531
Gaussian integrals, formulas derived      541—551
Gaussian integrals, formulas stated      490—494
Gaussian integrals, integration measures for      490
Gaussian integrals, simplify when numbers of bosonic and fermionic integrations equal      492—494 532
Gaussian integrals, unitary invariance of measure      541—542
General relativity      497 533 see
General relativity, analogies with quaternionic quantum mechanics      64n.5 511—516
General relativity, bitensor quantities      453n.24
General relativity, cosmological constant      see "Cosmological constant problem"
General relativity, energy-momentum in      64n.5 500—501
General relativity, energy-momentum tensor      483—484 533
General relativity, gravitationally defined energy      500—501
General relativity, identification of observables      453n.24 512 531
Generalized quantum dynamics      399 441—489 499 533 see generalized" "Operator-valued "Total
Generalized quantum dynamics and generalized Heisenberg picture quantum mechanics      448—449
Generalized quantum dynamics and quantum measurement theory      516 523—524
Generalized quantum dynamics, action principle for      445—446
Generalized quantum dynamics, analogies with classical mechanics      535n.1
Generalized quantum dynamics, analogies with classical mechanics, canonical momentum      446
Generalized quantum dynamics, constrained      455 455n.26 487—489 529
Generalized quantum dynamics, Feynman path integral possible for      531
Generalized quantum dynamics, for complex quantum mechanics      399 455—475
Generalized quantum dynamics, for complex quantum mechanics, biunitary operator gauging in      469—472
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate      456—461
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, canonical commutator a constraint      459 512
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, Lagrangian for coordinate q      456—457
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, Lagrangian for gauge potential $B_{0}$       457—458 458n.29 476
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, total trace dynamics      458—461
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, unitary dynamics      461
Generalized quantum dynamics, for complex quantum mechanics, bosonic self-adjoint Galilean coordinate, Weyl ordering and trace ordering agree      460—461
Generalized quantum dynamics, for complex quantum mechanics, free fermionic coordinate      461—464
Generalized quantum dynamics, for complex quantum mechanics, free fermionic coordinate, canonical anticommutator a constraint      463 512
Generalized quantum dynamics, for complex quantum mechanics, free fermionic coordinate, Lagrangian for $\psi$       461—462
Generalized quantum dynamics, for complex quantum mechanics, free fermionic coordinate, total trace dynamics      462—464
Generalized quantum dynamics, for complex quantum mechanics, free fermionic coordinate, unitary dynamics      464
Generalized quantum dynamics, for complex quantum mechanics, more than one coordinate      464—469
Generalized quantum dynamics, for complex quantum mechanics, multiple coordinates: constraint gives sum of bosonic commutators minus fermionic anticommutators      464—466 469 532
Generalized quantum dynamics, for complex quantum mechanics, operator gauge invariant extension of Yang — Mills action      472—475
Generalized quantum dynamics, for complex quantum mechanics, scalar field theory      466—469
Generalized quantum dynamics, for quaternionic quantum mechanics      475—489
Generalized quantum dynamics, for quaternionic quantum mechanics, chiral fermions excluded by biunitary gauge invariance      480 508 531
Generalized quantum dynamics, for quaternionic quantum mechanics, chiral symmetry of zero mass fermionic theory      481 508 531
Generalized quantum dynamics, for quaternionic quantum mechanics, classical gravitational coupling      483—484
Generalized quantum dynamics, for quaternionic quantum mechanics, discrete (C, P, T) symmetries      481—483
Generalized quantum dynamics, for quaternionic quantum mechanics, discrete (C, P, T) symmetries, C requires G = G'      482 508 531
Generalized quantum dynamics, for quaternionic quantum mechanics, discrete (C, P, T) symmetries, relation to $\mathcal{C}$ , $\mathcal{P}$ , $\mathcal{T}$ for complex case      482n.42
Generalized quantum dynamics, for quaternionic quantum mechanics, fermion coupled to axial current      483 531
Generalized quantum dynamics, for quaternionic quantum mechanics, fermion field pair with biunitary gauging      477—478 508—509
Generalized quantum dynamics, for quaternionic quantum mechanics, fermion field pair with biunitary gauging, fermionic current partial conservation      484—486
Generalized quantum dynamics, for quaternionic quantum mechanics, fermion field pair with biunitary gauging, total trace Hamiltonian formulation      486—489 530
Generalized quantum dynamics, for quaternionic quantum mechanics, fermion pair with a left or right complex gauging      478
Generalized quantum dynamics, for quaternionic quantum mechanics, Lagrangians and properties summarized      475—478 481—483
Generalized quantum dynamics, for quaternionic quantum mechanics, Majorana subscript M omitted      475
Generalized quantum dynamics, for quaternionic quantum mechanics, rules for converting classical to operator equations      475
Generalized quantum dynamics, for quaternionic quantum mechanics, scalar field with biunitary gauging      475—477
Generalized quantum dynamics, for quaternionic quantum mechanics, scalar field with biunitary gauging, specialized to self-adjoint scalar      477
Generalized quantum dynamics, for quaternionic quantum mechanics, scalar field with biunitary gauging, total trace Hamiltonian formulation      486—489
Generalized quantum dynamics, noncommuting dynamical variables      444
Generalized quantum dynamics, noncommuting dynamical variables, canonical momenta      446
Generalized quantum dynamics, nonunitary dynamics may be chaotic      524n.19 530
Generalized quantum dynamics, one- or two-dimensional models      530
Generalized quantum dynamics, operator equations of motion      446
Generalized quantum dynamics, perturbation expansion for      454 531
Generalized quantum dynamics, role of "canonical quantization" in      399 489 529
Generalized quantum dynamics, solvable models      530
Generalized quantum dynamics, unitary dynamics      453—454 530
Generalized quantum dynamics, unitary dynamics, allows Schroedinger picture      453—454
Generalized quantum dynamics, unitary dynamics, always valid in complex case      454 530
Generalized quantum dynamics, unitary dynamics, cotransforming states in      454
Geometric phase      see "Phase"
Geometry, noncommutative      442n.20
Golden rule, for transition probability per unit time      173 196—198 201 205—206
Goldstone theorem      396—397
Gram — Schmidt procedure      28
Grand unified theories      497—498 498n.2
Grand unified theories, scale of, and physics with a new kinematical structure      497—498 525
Grassmann elements      15—16 490
Grassmann elements, use to combine noncommuting exponents      111n.14
Grassmann integration      490n.47
Grassmann quaternion      15—16 490
Grassmann quaternion, product conjugation rule of      16 491 543
Gravity, classical      see "Equivalence principle" "General
Gravity, induced      532 see
Gravity, quantum      532—533 see 533
Green's function      169—170 175-176 183n.12 218—222 230
Green's function, advanced and retarded      219
Green's function, boundary conditions for      219
Green's function, formal integration of Schroedinger equation with      219
Green's function, Hermiticity properties      219
Green's function, integral equations relating full scattering state to in and out states      221—222
Green's function, Klein — Gordon equation      311—312
Green's function, multichannel      263—264
Green's function, thermal      270 289—293
Ground state      46n.17
Group of symmetry generators      74 434—437 452
Group of symmetry generators, matrix reps. $\mathbb{C}(1, i)$ for nonzero energy      75—76 238 391—392
Group, $SU(2) \times SU(2)$       501 509 511 528
Group, $SU(3) \times SU(2) \times U(1)$       497
Group, abelian      90—91 234 501
Group, compact      75 75n.10
Group, compact, as internal symmetry      433—434
Group, compact, complex generator algebra      435—436
Group, compact, quaternionic generator algebra      434—437 452
Group, compact, quaternionic irreducible and Frobenius — Schur classification      437—439
Group, compact, quaternionic irreducible representations of      399 433—441 527 530

Group, compact, quaternionic irreducible, one-dimensional of SU(2)      396 399 436 440—441 515 527 530 532 see
Group, complex representations of      75—76 75n.10 437—439
Group, complex representations of, class-1 twofold reducible      437
Group, complex representations of, reducibility over quaternions      75n.10 437—439
Group, complex representations of, relation to embedding of dynamics      439—440
Group, conformal      see "Algebra"
Group, Frobenius — Schur classification of      75n.10 437
Group, Frobenius — Schur classification of, and complex conjugate rep.      437
Group, Galilean      90—91 90n.4 100 103 234
Group, induced representation theory of      515 530
Group, Lorentz, $2\times 2$ complex matrix reps.      514n.11 515
Group, Lorentz, generators      389
Group, Lorentz, proper, orthochronous subgroup      389 389n.11
Group, Lorentz, reps. all complex transformable      395—397 514n.11 529
Group, Lorentz, rotation subgroup example      395—396
Group, non-abelian      501 see
Group, noncompact      396
Group, permutation      see "Symmetric"
Group, Poincare      76 90n.4 103 361—362 388—398 433—434
Group, Poincare, complex irreducible reps., spinorial induced by quaternionic      515 530
Group, Poincare, complex irreducible reps., Wigner analysis      391 394n.15
Group, Poincare, generators, "boost" $\tilde{K}_{\ell}$       390—391 513 515
Group, Poincare, generators, angular momentum $\tilde{J}_{\ell}$       85 390—391
Group, Poincare, generators, commutator algebra of      389—390 398
Group, Poincare, generators, energy-momentum four-vector $\tilde{p}_{\mu}$       389—390 499—500
Group, Poincare, generators, generalized bracket algebra of      451 489 530
Group, Poincare, generators, independence from internal symmetries      433—434
Group, Poincare, generators, spin-0      388—389
Group, Poincare, generators, spin-1/2      388—390
Group, Poincare, generators, total trace      451
Group, Poincare, nonzero energy quaternionic representations complex transformable      361—362 391 434 499 529 532
Group, Poincare, nonzero energy quaternionic representations complex transformable, conformal extension      392
Group, Poincare, nonzero energy quaternionic representations complex transformable, implications for field theory      398 434
Group, Poincare, nonzero energy quaternionic representations complex transformable, implications for free wave equations      397
Group, Poincare, nonzero energy quaternionic representations complex transformable, multicentral projective extension      392—394
Group, Poincare, nonzero energy quaternionic representations complex transformable, relation to locality      362 397—398
Group, Poincare, nonzero energy quaternionic representations complex transformable, standard basis used      390—394
Group, Poincare, nonzero energy quaternionic representations complex transformable, supersymmetric extension      361 392 499 532
Group, projective representation      90n.4 99—106 100n.9 392—393 531—532
Group, projective representation, central case and Schur's Lemma      103—105 527
Group, projective representation, complex case      102
Group, projective representation, contrasted with vector representation      392
Group, projective representation, generator algebra for      393 393n.14
Group, projective representation, multicentral case      90n.4 102—103 392—393 527
Group, projective representation, nonmulticentral      531—532
Group, projective representation, operator form of phase      101—103 392—393
Group, projective representation, phase space translation as example      105 392
Group, projective representation, phase spectral representation      101
Group, projective representation, quaternion automorphism on phase      100n.10 101 106
Group, projective representation, state dependence of phase      100—102 106
Group, projective representation, translation generator example      102
Group, ray representation      see "Projective representation"
Group, representation law      99
Group, rotation      65 75—76 395—396
Group, rotation, one-dimensional representation      396 399 515
Group, SO(3)      17 111n.14 434 497n.1 515
Group, SO(4)      434 528
Group, SU(2)      65 111n.14 479n.39 501 515 527
Group, SU(2), half-integer reps.      440
Group, SU(2), integer reps.      440
Group, SU(2), one-dimensional quaternionic rep.      396 434 440—441 527 530
Group, SU(2), one-dimensional quaternionic rep. and quaternionic field theory      434 532
Group, SU(2), one-dimensional quaternionic rep., induces complex spinorial rep.      515
Group, SU(2), one-dimensional quaternionic rep., one fermion coordinate as example      434—435
Group, SU(n)      473
Group, symmetric, and identical particles      76 233 237—238 270—271
Group, U(1)      473
Group, U(2)      479n.39 480
Group, U(n)      473
Group, unitary representation and Schur's Lemma      103—105
Guersey counterexample to octonion completeness      50
Hamilton, discovery of quaternions      7n.5
Hamiltonian      19 36—40 45—49 53 68—70 76 94 113 307n.2 431 499—500
Hamiltonian in multiparticle system      59—61 80—81 83 see
Hamiltonian in real quantum mechanics      47—48
Hamiltonian in single-particle system      89—95
Hamiltonian, $2\times 2$ matrix form      40—41 43—44 408
Hamiltonian, $2\times 2$ matrix form, self-adjoint      41 44 408
Hamiltonian, anti-self-adjoint and inner product conservation      51
Hamiltonian, anti-self-adjoint reduced to complex self-adjoint form      124—131
Hamiltonian, anti-self-adjointness conditions for      39—41 86 94—95 134 236—237 282 408
Hamiltonian, classical      528
Hamiltonian, complex self-adjoint      46 76 208 409
Hamiltonian, complex specializaton of quaternionic      117
Hamiltonian, coordinate representation      37—40
Hamiltonian, coordinate representation for charged scalar field      415
Hamiltonian, coordinate representation for delta function potential model      159
Hamiltonian, coordinate representation for Dirac equation      329 331
Hamiltonian, coordinate representation for Dirac free fermion field      417
Hamiltonian, coordinate representation for forced harmonic oscillator      209—210
Hamiltonian, coordinate representation for Hermitian scalar field      410
Hamiltonian, coordinate representation for identical particles      237—238
Hamiltonian, coordinate representation for N-pair model      246
Hamiltonian, coordinate representation for one-dimensional potential      183—184
Hamiltonian, coordinate representation for quaternionic harmonic oscillator      123
Hamiltonian, coordinate representation for quaternionic scalar field      422
Hamiltonian, coordinate representation for supersymmetric quantum mechanics      358—359
Hamiltonian, coordinate representation for three-dimensional potential      171
Hamiltonian, dependent on external parameters      133 145—149
Hamiltonian, effective constructed from dimension-6 operators      519—520 525
Hamiltonian, effective for three quasiparticle composites      509
Hamiltonian, eigenstates      see "Energy eigenstates"
Hamiltonian, Fock space      281—187
Hamiltonian, Fock space for one fermion coordinate      434—435
Hamiltonian, forced harmonic oscillator, Heisenberg picture      212
Hamiltonian, forced harmonic oscillator, interaction picture      211
Hamiltonian, forced harmonic oscillator, Schroedinger picture      210
Hamiltonian, form invariance under change of ray representative      96
Hamiltonian, free particle      113 137 249
Hamiltonian, fundamental in quaternionic quantum mechanics      112
Hamiltonian, gauge      460 460n.31
Hamiltonian, Heisenberg picture form      70 208 211—213
Hamiltonian, Hermitian in complex mechanics      46 76 208 409
Hamiltonian, interaction term      113 194 219
Hamiltonian, kinetic part of      87 106 108 111 128 137 172 196 219 255
Hamiltonian, kinetic part of, matrix element of      110
Hamiltonian, kinetic part of, rest mass in      164 177—178 208 247 255 258 260
Hamiltonian, kinetic part of, sign reversal in $\beta$ -symplectic      160
Hamiltonian, modulus, and phase of      60 83 113
Hamiltonian, modulus, and phase of, both commute with conserved operators      269n.11
Hamiltonian, modulus, and phase of, first-order perturbation theory for      134—139
Hamiltonian, modulus, and virial theorem      108—109
Hamiltonian, modulus, reduction to complex self-adjoint form      124—126
Hamiltonian, modulus, variational principle for      144—145
Hamiltonian, momentum representation for Dirac equation      333
Hamiltonian, momentum representation for Dirac free fermion field      418 432
Hamiltonian, momentum representation for Hermitian scalar field      411
Hamiltonian, momentum representation for Klein — Gordon equation      308 310—311 313—315
Hamiltonian, momentum representation for linearized gauge field strength      367
Hamiltonian, necessity for anti-self-adjoint form      40 98
Hamiltonian, one-parameter family in first-order perturbation theory      133
Hamiltonian, perturbation      131 194 201
Hamiltonian, perturbation, anti-self-adjointness conditions      134 195
Hamiltonian, perturbation, compact notation for matrix elements      132 195
Hamiltonian, perturbed and unperturbed      131
Hamiltonian, potential energy part of      108 111 255
Hamiltonian, real-valued      239—240 255—256
Hamiltonian, representation of symmetries of      74—76
Hamiltonian, representation-independent form      98 112 410
Hamiltonian, restrictions on from translational, rotational, and Galilean invariance      89—95 234—237
Hamiltonian, rotationally invariant      64 66—67 80 84—86 89—95
Hamiltonian, self-adjoint, for two-component semi-relativistic equation      351
Hamiltonian, simplification by choice of ray representative      95—99
Hamiltonian, simplified by omitting vector potential      98 106 109
Hamiltonian, spectral representation      60 195 197 213
Hamiltonian, spin      84—86
Hamiltonian, spin, optical potential reduction for      130—131
Hamiltonian, spin, symplectic decomposition for      85—86
Hamiltonian, spin, time reversal invariance restrictions      120 286
Hamiltonian, symmetry generators which anticommute with      75n.9 112—117 231—232

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