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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

Probability and quantum measurement paradox      523 524n.19
Probability, amplitude      5—10 22 227 303
Probability, classical      4—6 8—9
Probability, conditional      5n.3
Probability, current      106—107 185—186 189 189n.13 352 402—405
Probability, density      106—107 160 185 352
Probability, density, boundedness of      160 172 183n.12
Probability, local conservation law      106—107 352 403—405
Probability, local conservation law for one-dimensional transfer matrices      185—186 189
Probability, quantum mechanical      5—6 8—9 22
Probability, quantum mechanical in complex quantum mechanics      42 44 306
Probability, quantum mechanical in quaternionic quantum mechanics      42 306
Probability, quantum mechanical in real quantum mechanics      44
Probability, total in $\beta$ -symplectic components      206—207
Probability, transition per unit time      173 196—198 201
Probability, transition to $\beta$ -symplectic final state      198 206—207 526
Probability, transition, expressed in terms of total trace      448—449
Probability, transition, preserved by symmetries      29 90 112
Projection operators      10—11
Projective geometry      10—11 29n.7
Projective representation      see "Group projective
Quantum chromodynamics      509
Quantum chromodynamics, suggested analogy with      499n.3
Quantum field theory, complex      315 362 381—388 516 529—530
Quantum field theory, complex embedded in quaternionic, general conditions for embedding      407—410
Quantum field theory, complex embedded in quaternionic, trivial embedding      409—410 424
Quantum field theory, complex embedded in quaternionic, trivial embedding, action of left-acting algebra      411—421
Quantum field theory, complex embedded in quaternionic, trivial embedding, charged scalar field      415—417
Quantum field theory, complex embedded in quaternionic, trivial embedding, Dirac free fermion field      417—421
Quantum field theory, complex embedded in quaternionic, trivial embedding, Hermitian scalar field      410—415
Quantum field theory, complex embedded in quaternionic, trivial embedding, Majorana free fermion fields      421 471
Quantum field theory, complex embedded in quaternionic, trivial embedding, must choose left-acting algebra      410—413
Quantum field theory, complex embedded in quaternionic, trivial embedding, no-go results for free fermions      419—421 427
Quantum field theory, complex, $\mathcal{C}$ , $\mathcal{P}$ , $\mathcal{T}$ symmetries in      381—382 384—387 528
Quantum field theory, complex, canonical commutation relations      411 422
Quantum field theory, complex, effective      498—500
Quantum field theory, complex, embedded in quaternionic Hilbert space      407—421 see complex
Quantum field theory, complex, Feynman path integral and Gaussian integrals      489 531
Quantum field theory, complex, generalized quantum dynamics with total trace Lagrangian applied to      442 455—475 see
Quantum field theory, complex, group generator commutator algebra      435—436
Quantum field theory, complex, preon models based on      501 501n.4
Quantum field theory, notation      see also "Fourier expansion"
Quantum field theory, notation, $\int\limits_{\nu}d^{3}x$ for integration over box $\nu$ of volume $(2\pi)^{3}$       411
Quantum field theory, notation, anti-self-adjoint canonical momentum $\tilde{\pi}$       412
Quantum field theory, notation, energy eigenvalue $\omega_{p}$       410—411
Quantum field theory, notation, normalization factor N(p)      411
Quantum field theory, notation, self-adjoint canonical momentum $\pi$       410
Quantum field theory, quaternionic      217n.14 399—496 516
Quantum field theory, quaternionic, absence of kinematic time reversal violation in      217n.14
Quantum field theory, quaternionic, conjectured asymptotic to complex      398 497—498
Quantum field theory, quaternionic, constructed as nontrivial embedding of complex      399 407—410 421—422 432 439—440
Quantum field theory, quaternionic, constructions valid in zero and one particle sectors only      429 432 436
Quantum field theory, quaternionic, free fields formed as superpositions of $\mathbb{C}(1, I)$ quantum fields      421—432
Quantum field theory, quaternionic, free fields formed as superpositions of $\mathbb{C}(1, I)$ quantum fields, fermion fields      427—432 529—530
Quantum field theory, quaternionic, free fields formed as superpositions of $\mathbb{C}(1, I)$ quantum fields, scalar fields      421—427
Quantum field theory, quaternionic, functional integral approach      492n.48
Quantum field theory, quaternionic, generalized quantum dynamics with total trace Lagrangian applied to      442 475—489 508 see
Quantum field theory, quaternionic, group generator Lie algebra      434—436 452
Quantum field theory, quaternionic, minimal fermionic theory with maximal gauge group      399—400 508
Quantum field theory, quaternionic, nontrivial, likely to be obtained by gauging one-dimensional quaternionic irreducible representation of SU(2)      434 440—441 532
Quantum field theory, quaternionic, Poincare generators not represented by a commutator algebra      398 451
Quantum field theory, quaternionic, role in physics      497—498
Quantum field theory, quaternionic, unitary versus asymptotically unitary evolution      524n.18
Quantum field theory, quaternionic, wave fields left-acting operators      399
Quantum measurement paradox      516 520—524 530
Quantum measurement paradox and internal consistency of quantum mechanics      523
Quantum measurement paradox and probabilistic aspect of quantum mechanics      523
Quantum measurement paradox, Bell inequalities, hidden variables and      524n.19
Quantum measurement paradox, decoherence, quantum histories, and      522
Quantum measurement paradox, Feynman's comment on      522n.17
Quantum measurement paradox, ideal measurement and      521 523
Quantum measurement paradox, implications of generalized quantum dynamics for      523—524
Quantum measurement paradox, implications of quaternionic quantum mechanics with unitary evolution operator for      523—524
Quantum measurement paradox, Penrose operations $\mathbf{R}$ and $\mathbf{U}$ in      520 522—524
Quantum measurement paradox, von Neumann chain or recursion in      522 523
Quantum measurement theory      497 516 520—525 520n.15
Quantum measurement theory and nonlinear corrections to complex quantum mechanics      524—525
Quantum mechanical interference      6 8
Quantum mechanical state      5—6
Quantum mechanical system      5
Quantum mechanical system, Feynman formulation (versus classical)      5
Quantum mechanics, axiomatic foundation      10—11
Quantum mechanics, complex      see "Complex"
Quantum mechanics, Dirac formulation      10—11
Quantum mechanics, Jordan formulation      10—11 448n.23
Quantum mechanics, nonlinear modifications of      524—525 530
Quantum mechanics, nonlinear modifications of, parameterization of      525 525n.20
Quantum mechanics, nonlinear modifications of, possible tie to hierarchy problem      525
Quantum mechanics, real      11—12 30 46—49
Quantum mechanics, real, embedding in complex      47—49
Quantum mechanics, real, Hamiltonian in      47—48
Quantum mechanics, real, Mackey analysis of      47 47n.19
Quantum mechanics, real, no energy eigenstates in      47
Quantum mechanics, real, relationship to complex      44—45 47—49 47n.19 407n.3
Quantum mechanics, real, Schroedinger equation in      47—49
Quantum mechanics, real, Stueckelberg analysis of      47 47n.19
Quark masses      see "Mass lepton
Quark model, enumeration of states in      505
Quasiparticle transformation      283—287 528 see
Quaternion or quaternionic, (3, 1) signature of      512
Quaternion or quaternionic, algebra      see "Algebra"
Quaternion or quaternionic, analyticity      18
Quaternion or quaternionic, anti-self-adjoint operator      see "Operator"
Quaternion or quaternionic, arithmetic      11—18
Quaternion or quaternionic, automorphism class      32 83
Quaternion or quaternionic, automorphism transformation      16—17 32 100—101 100n.10 254 296 299 320 434 482n.42 519
Quaternion or quaternionic, cluster      256
Quaternion or quaternionic, cluster energies and additive energy conservation      268—269
Quaternion or quaternionic, column vector      21
Quaternion or quaternionic, column vector, adjoint $\ddagger$ defined for      21—22
Quaternion or quaternionic, complex $2\times 2$ matrix representation      495
Quaternion or quaternionic, complex subalgebra of      13 see i)$"/>"
Quaternion or quaternionic, complexified      9 12 429n.14
Quaternion or quaternionic, conjugate      7—8 13 16 19n.1 512
Quaternion or quaternionic, conjugate of product      14 16 112
Quaternion or quaternionic, conjugate, not a time reversal operation      112
Quaternion or quaternionic, conjugate, notation $\bar{}$ used for      13 512
Quaternion or quaternionic, contravariant      513
Quaternion or quaternionic, covariant derivative      see "Covariant derivative"
Quaternion or quaternionic, defined      7 11—12
Quaternion or quaternionic, delta function      493—494
Quaternion or quaternionic, determinant, for self-adjoint matrices      400 490 494—496
Quaternion or quaternionic, determinant, for self-adjoint matrices, properties summarized      494—496
Quaternion or quaternionic, four-vector extension of      513—515
Quaternion or quaternionic, Gaussian integrals      400 489—496 531 see
Quaternion or quaternionic, Gaussian integrals, derived      541—551
Quaternion or quaternionic, Grassmann      15—16
Quaternion or quaternionic, harmonic oscillator      122—123
Quaternion or quaternionic, harmonic oscillator, coupled second-order equations for      123
Quaternion or quaternionic, harmonic oscillator, forced      194 209—213
Quaternion or quaternionic, harmonic oscillator, fourth-order equation for      123
Quaternion or quaternionic, Hilbert space      see "Hilbert space"
Quaternion or quaternionic, imaginary      12 14 37 40 73 148
Quaternion or quaternionic, imaginary part of      12 14
Quaternion or quaternionic, level of physical structure      497
Quaternion or quaternionic, matrix      14—15 see "Operator"
Quaternion or quaternionic, matrix, adjoint $\ddagger$ defined for      14
Quaternion or quaternionic, matrix, finite versus infinite-dimensional      15
Quaternion or quaternionic, matrix, reps., and octonionic quantum mechanics      66
Quaternion or quaternionic, matrix, trace Tr defined for      15
Quaternion or quaternionic, matrix, transpose $^{T}$ defined for      15
Quaternion or quaternionic, norm or modulus      14
Quaternion or quaternionic, nth roots of      14n.14
Quaternion or quaternionic, operators      15 19n.1 20 see
Quaternion or quaternionic, polar form      14 152
Quaternion or quaternionic, real      12 73
Quaternion or quaternionic, real part of      12
Quaternion or quaternionic, scalars      19n.1 20
Quaternion or quaternionic, Schwartz inequality      21 299
Quaternion or quaternionic, self-adjoint operator      see "Operator"

Quaternion or quaternionic, symplectic representation of      13
Quaternion or quaternionic, trace      12—13
Quaternion or quaternionic, unitary operator      see "Operator"
Quaternion or quaternionic, units or unit imaginary      11—12 17 32 58 91 234 236
Quaternion or quaternionic, units or unit imaginary, automorphism transformation of      17
Questions      see "Open questions"
Radial wave equation      166—171 175—177
Radius, larger $r_{>}$ and smaller $r_{<}$ defined      170
Ray representative      22 28—30 32—33 35 37 42 45—46 51n.20 54—55 63 65 68—69 74 80—84 95—101 115 118 120—122 144 148—150 159 161 174 230 233 236 255n.6 256 267—268 303 314 333 500
Ray representative, complex versus quaternionic change of ( $\omega$ versus $\xi$ )      272 320
Ray representative, indicated in state label      82 90n.4 100
Ray representative, invariance and phenomenology of nonlinear quantum mechanics      525
Ray representative, standard choice of      see "Energy eigenstates canonical
Ray representative, time-dependent change of      150—156
Ray representative, transformation, restricted      153—155
Rayleigh — Ritz variational principle      145 see "Variational
Rayleigh — Schroedinger perturbation theory      see "Perturbation theory stationary
Real momentum space basis      57—58
Reciprocity relation      215
Reciprocity relation, failure      174—175
Relativistic notation      304—305 320n.6
Relativistic notation, summation convention for Greek indices      304
Relativistic quaternionic field theory      see "Quantum field theory quaternionic"
Relativistic wave equation      58 87 87n.1 164 217n.14 303—388 362—388 399 501 see "Gauge "Klein "Lagrangian
Relativistic wave equation, limitations of      400 407
Reraying      see "Ray representative"
Resolvent      127 180—181
Resolvent, spectral representation for      180—182
Rest masses      159 247 250
Rest masses and additive energy conservation law      268—269 268n.10 287
Rest masses and generalized classification of asymptotic scattering states      260
Rest masses and scattering problem      164—165
Rest masses and stability of bound states      164—165 171 177—179 208 258
Riccati equation      153—155 158 527
Riemann sum      152
Riemann zeta function zeros      526n.23
Rotation generators and invariance      see "Angular momentum" "Group"
Rotation group SO(3)      17
Rotationally invariant system      see "Hamiltonian" "Wave
S-matrix      4 34 113 159 175 192 227—230 see multichannel "Scattering"
S-matrix in quantum chromodynamics      499n.3
S-matrix, always complex in quaternionic quantum mechanics      4 159 171—174 198 208 218 230 233 262 356 398 499—500 509 517
S-matrix, always complex in quaternionic quantum mechanics, multichannel generalization      266—267
S-matrix, channel, for multichannel scattering      266—261
S-matrix, commutes with free particle Hamiltonian      228
S-matrix, defined via Moeller wave operators      228
S-matrix, element defined      230 267
S-matrix, factorization in complex case      244
S-matrix, interaction picture expression      229
S-matrix, intertwining property in multichannel case      266—267
S-matrix, produces only energy-conserving transitions      230 267
S-matrix, related to transition matrix      173
S-matrix, symmetries of      218 231—232
S-matrix, symmetries of, unitary anticonimuting with $\tilde{H}$       231—232
S-matrix, symmetries of, unitary commuting with $\tilde{H}$       231
S-matrix, time reversal symmetry of      113 117 192
S-matrix, time reversal violating      174—175
S-matrix, unitarity of      163 174—175 192n.15 204—207 228 266
scalar      see "Hilbert space"
Scalar potential      40 59n.3 94—95 98 108 239
Scalar product      see "Inner product"
Scattering      see also "Multiparticle mutichannel "S-matrix"
Scattering, amplitude      179 see analyticity 179—183 499 526
Scattering, asymptotic state space      see "Asymptotic scattering states"
Scattering, exterior region for      160 166
Scattering, external potential      63 76 159—193 499
Scattering, external potential, time reversal violation in      159 174—175 213
Scattering, formal theory of      76 113 172 499 527
Scattering, formal theory of, multichannel time-dependent      218 262—269 527
Scattering, formal theory of, relationship to complex      230
Scattering, formal theory of, single-channel time-dependent      218—232
Scattering, free particle in and out states      see also "States free
Scattering, free particle in and out states, defined      220
Scattering, free particle in and out states, integral equations for      221—222
Scattering, full scattering states      see "States full
Scattering, fundamental question answered by S-matrix      228 266
Scattering, incident or incoming wave      162 173 177n.6 180n.10 186 197 400 403
Scattering, matrix      see "S-matrix"
Scattering, Moeller wave operators for      see "Moeller wave operators"
Scattering, multichannel      61n.4 64 218 262—269
Scattering, one-dimensional with absorption      190n.14 193n.17 518
Scattering, one-dimensional, compound barrier      188 189n.13 192 517—518
Scattering, one-dimensional, delta function potential      159—165 174 354—357
Scattering, one-dimensional, delta function potential, analyticity properties      182—183 182n.11 355 358
Scattering, one-dimensional, Dirac equation with step      400—407
Scattering, one-dimensional, general      163 183—193 517—518
Scattering, one-dimensional, left-right reflection symmetric      518
Scattering, one-dimensional, multiple reflection sum      192—193
Scattering, one-dimensional, reflected and transmitted waves      162
Scattering, one-dimensional, reflection and transmission coefficients      163 182 186—188 190—193 402—404
Scattering, one-dimensional, reflection and transmission coefficients, relations between left and right      188—192
Scattering, one-dimensional, reflection and transmission coefficients, unitarity sum rule for      163 187—188 190 192n.15 403—405
Scattering, one-dimensional, square well      188 401n.1
Scattering, outgoing wave      162 177n.6 180 403—404
Scattering, outgoing wave, Lippmann — Schwinger solution      173 179—180 180n.10 201
Scattering, phase shift      168 176
Scattering, three-dimensional      164—175 357—358
Scattering, three-dimensional, general      170—175
Scattering, three-dimensional, spherically symmetric      165—171 175—179
Scattering, time-dependent perturbation theory for      196—201
Schroedinger equation, for quaternionic quantum mechanics for three-body problem      240—242
Schroedinger equation, for quaternionic quantum mechanics, asymptotic, in multiparticle scattering      254—255
Schroedinger equation, for quaternionic quantum mechanics, Feynman path integral as alternative      109
Schroedinger equation, for quaternionic quantum mechanics, generic case not time reversal invariant      114 510—511
Schroedinger equation, for quaternionic quantum mechanics, one-dimensional and delta function potential model      159—165
Schroedinger equation, for quaternionic quantum mechanics, one-dimensional and WKB approximation      156—158 527
Schroedinger equation, for quaternionic quantum mechanics, radial      see "Radial wave equation"
Schroedinger equation, for quaternionic quantum mechanics, relative coordinate, in multiparticle system      239 242 245
Schroedinger equation, for quaternionic quantum mechanics, simplification by choice of ray representative      95—99 303
Schroedinger equation, for quaternionic quantum mechanics, time-dependent      36 38 40—41 44—49 68 99 106 146 218 281—282 325 409
Schroedinger equation, for quaternionic quantum mechanics, time-dependent and time reversal      112—113 129—130 174 510—511
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, canonical form for      96—99 233 237
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, for cotransforming states      454
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, for Fock space      281—282
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, optical potential used in      128—130 172 499
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, perturbation theory for      194—196
Schroedinger equation, for quaternionic quantum mechanics, time-dependent, residual ray rep. freedom      97
Schroedinger equation, for quaternionic quantum mechanics, time-independent      43 123 239 255—256
Schroedinger equation, for quaternionic quantum mechanics, time-independent, bound states in $\beta$ -symplectic component      161 334
Schroedinger equation, for quaternionic quantum mechanics, time-independent, canonical form for      96—99 145 159—160 194—195 237
Schroedinger equation, for quaternionic quantum mechanics, time-independent, optical potential for      127—128 130—131 181—182 294—295 499
Schroedinger equation, for quaternionic quantum mechanics, time-independent, reduction to complex      126—128 130—131 160
Schroedinger equation, for quaternionic quantum mechanics, two-component complex form      40—41 272n.1 407—409
Schroedinger picture      69 208 210 289 399 413
Schroedinger picture for Fock space construction      271
Schroedinger picture for forced harmonic oscillator      209—210
Schroedinger picture for unitary case of generalized quantum dynamics      454
Schroedinger picture, state vector and operators related to interaction picture      208
Schroedinger's cat      see "Quantum measurement paradox" "Quantum
Schur's lemma      99 103—105
Schwartz inequality      21 73 137
Second quantization      270—283 519 see
Seesaw mechanism      507—508 501n.1
Semirelativistic wave equation      see "Wave equation two-component
Shell model      509
Simultaneous diagonalization      see "Operator mutually
Slavnov — Taylor identities      529
Solid-state physics      see "Condensed matter physics"
Source current      see "Dirac equation" "Four-current" "Gauge "Klein
Space translation generator      see "Momentum"
Space-time translation, generators      499—501
Space-time translation, invariance      388
Special relativity      53n.1 see "Relativistic
Spherical functions (Bessel, Neumann, and Hankel)      167
Spherical harmonics      65 80
Spin      53 58 84—86 119—122 see
Spin, Hamiltonian      53 84—86 111n.14 345
Spin, Hamiltonian, optical potential for      130—131
Spin, Hamiltonian, time reversal restrictions      120—122
Spin-0 relativistic wave equation      see "Klein — Gordon equation"

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