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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

Multiparticle systems, center of mass separation, quaternionic cluster      256 261
Multiparticle systems, center of mass separation, three-body      241—242
Multiparticle systems, center of mass separation, two-body      239—240
Multiparticle systems, classification of asymptotic states      233 254—262 532—533
Multiparticle systems, classification of asymptotic states, assumptions made      254
Multiparticle systems, classification of asymptotic states, asymptotic Schroedinger equation      254—258
Multiparticle systems, classification of asymptotic states, complex cluster      255—256 287 518—519 532—533
Multiparticle systems, classification of asymptotic states, extension to positive cluster energies      260 287
Multiparticle systems, classification of asymptotic states, notation for partitioning in clusters      254
Multiparticle systems, classification of asymptotic states, null cluster      260
Multiparticle systems, classification of asymptotic states, quaternionic cluster      256 258 261 532—533
Multiparticle systems, classification of asymptotic states, wave function structure      256—262
Multiparticle systems, cluster decomposition property      233 240 245—254 293—299
Multiparticle systems, cluster decomposition property in finite subsystem of infinite system      253—254 293—299
Multiparticle systems, cluster decomposition property, breakdown      250 270 299 518—519 527
Multiparticle systems, cluster decomposition property, mean field approximation used to study      270 296—299
Multiparticle systems, cluster decomposition property, optical potential used to study      270 293—299
Multiparticle systems, energy, additive conservation law for      233 268—269 287
Multiparticle systems, energy, additive conservation law for, cluster energy conditions for      268 268n.10 287
Multiparticle systems, evolution operator factorization, complex case      244
Multiparticle systems, evolution operator factorization, failure in quaternionic case      245
Multiparticle systems, external potential problem obtained from      233
Multiparticle systems, Fock space      270—274 277 see second
Multiparticle systems, Fock space, class $\mathcal{C}$ of complex bases      270—272 528
Multiparticle systems, Fock space, class $\mathcal{C}^{'}$       283 528
Multiparticle systems, Fock space, class $\mathcal{R}$ of real bases      275n.4
Multiparticle systems, Fock space, defined      273
Multiparticle systems, Fock space, dynamics      281—282
Multiparticle systems, Fock space, identity operator      274 277
Multiparticle systems, Fock space, inner product      271—272 281
Multiparticle systems, Fock space, N-particle Hilbert space component      273
Multiparticle systems, Fock space, vacuum state      273
Multiparticle systems, Galilean analysis      234—237
Multiparticle systems, Galilean analysis, assumptions      234—235 527
Multiparticle systems, Galilean analysis, summary      236
Multiparticle systems, general reduction for $\tilde{H} =\tilde{H}_{(1)}+iH_{(2)}$       239—240
Multiparticle systems, Hamiltonian for      59—61 80—81 83 262
Multiparticle systems, Hamiltonian for, modeled as sum of one-body terms      242—243 270 283 509
Multiparticle systems, Hamiltonian for, noncommutativity and      244 283—285
Multiparticle systems, Hamiltonian for, quasiparticle transformation for      270 283—287
Multiparticle systems, Hamiltonian for, second quantized      270
Multiparticle systems, identical particles in      237—238
Multiparticle systems, independent particle behavior in complex specialization      245
Multiparticle systems, methods for      233—269
Multiparticle systems, momentum, in additive conservation law      233 267—268
Multiparticle systems, momentum, in anti-self-adjoint operator      59—60 80 238
Multiparticle systems, momentum, in anti-self-adjoint operator, action on asymptotic scattering states      261
Multiparticle systems, momentum, in anti-self-adjoint operator, commutes with S-matrix      267
Multiparticle systems, momentum, in anti-self-adjoint operator, eigenstates      61 76 255—256
Multiparticle systems, momentum, in self-adjoint operator      59—61
Multiparticle systems, momentum, in self-adjoint operator for individual particle      61
Multiparticle systems, momentum, in self-adjoint operator, action on asymptotic scattering states      261—262
Multiparticle systems, momentum, in total, as sum of cluster momenta      261
Multiparticle systems, N-pair model      245—254
Multiparticle systems, N-pair model, simplification in large N limit      250—251
Multiparticle systems, permutation, boson and fermion representations      238 270—271
Multiparticle systems, permutation, operator for coordinates      237—238
Multiparticle systems, permutation, order of P      271
Multiparticle systems, permutation, symmetry representations complex      238
Multiparticle systems, perturbation around $\mathbb{C}(1, i)$ limit      245—254
Multiparticle systems, perturbation around $\mathbb{C}(1, i)$ limit, zeroth-order approximation      247
Multiparticle systems, quasiparticle operators      280 283—287 503—511 528
Multiparticle systems, quasiparticle operators, annihilate/create one-particle states      270 284 510
Multiparticle systems, quasiparticle operators, factor ordering in inversion formulas      284
Multiparticle systems, quasiparticle operators, noncanonical commutator/anticommutator      280 431 504 510
Multiparticle systems, quasiparticle operators, obey nonstandard exclusion principle      270 285
Multiparticle systems, quasiparticle operators, properties of      504—505 510
Multiparticle systems, quasiparticle operators, restrictions from time reversal      285—286
Multiparticle systems, scattering      61n.4 64 see multichannel
Multiparticle systems, Schroedinger equation for relative coordinate wave function      239 245—246
Multiparticle systems, Schroedinger equation for three-body problem      240—242
Multiparticle systems, Schroedinger equation, x-rep. projected from Fock space      282
Multiparticle systems, second quantization in $\lambda$ -representation      270—287 519 see "Fock "Quasiparticle
Multiparticle systems, second quantization in $\lambda$ -representation, annihilation/creation operators      273 278—279 503
Multiparticle systems, second quantization in $\lambda$ -representation, annihilation/creation operators, commutator/anticommutator notation      273
Multiparticle systems, second quantization in $\lambda$ -representation, complete basis for Fock space      273—274 282n.6
Multiparticle systems, second quantization in $\lambda$ -representation, Hamiltonian for      281—282
Multiparticle systems, second quantization in $\lambda$ -representation, Hamiltonian for, n-body operator terms      282
Multiparticle systems, second quantization in $\lambda$ -representation, Hamiltonian for, number conserving and nonconserving      281—282 528
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra      270 272n.1 274—278 503
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra and change of representation      279
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra and Hamiltonian structure      282
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra, complex conjugation *      279 419 430
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra, properties derived      275—278
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra, properties stated      274—275
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra, quaternion conjugation $\bar{}$       282
Multiparticle systems, second quantization in $\lambda$ -representation, left-acting algebra, trace over defined      280 431
Multiparticle systems, second quantization in $\lambda$ -representation, notation used for operator and eigenvalue      275n.3 411n.6
Multiparticle systems, second quantization in $\lambda$ -representation, occupation number labels      274
Multiparticle systems, second quantization in $\lambda$ -representation, particle number in cluster p of arrangement channel a      287
Multiparticle systems, second quantization in $\lambda$ -representation, particle number, additive conservation law      287—288
Multiparticle systems, second quantization in $\lambda$ -representation, particle number, operator      281 285
Multiparticle systems, second quantization in $\lambda$ -representation, transformation to $\sigma$ -representation      277—279 277n.5
Multiparticle systems, simplification by reraying      233 236—237
Multiparticle systems, statistical mechanics      287—293 527
Multiparticle systems, statistical mechanics, dilute regime      287
Multiparticle systems, statistical mechanics, equilibrium density matrix      288
Multiparticle systems, statistical mechanics, thermal averages      270
Multiparticle systems, statistical mechanics, thermal Green's functions      270 289—293
Multiparticle systems, symmetrization for identical particles      233 237—238 270
Multiparticle systems, tensor product      233 240—245
Multiparticle systems, tensor product, existence of complex multilinear      243—244
Multiparticle systems, tensor product, nonexistence of quaternion multilinear      233 244—245 250 258 280
Multiparticle systems, translation invariant      237—239 267
Multiparticle systems, translation invariant, three-body problem      240—242
Multiparticle systems, translation invariant, two-body problem      239—240
Multiparticle, multichannel scattering      see also "Scattering" "S-matrix"
Multiparticle, multichannel scattering, S-matrix for $\mathbb{C}(1, i)$ with standard ray choice      233 266—267 509 517
Multiparticle, multichannel scattering, S-matrix for definition and properties      266—267
Multiparticle, multichannel scattering, time-dependent formal theory of      233 262—269
Multiparticle, multichannel scattering, time-dependent formal theory of, a-state and integral equations      263—264
Multiparticle, multichannel scattering, time-dependent formal theory of, arrangement channel a defined      262
Multiparticle, multichannel scattering, time-dependent formal theory of, arrangement channel Green's functions      263
Multiparticle, multichannel scattering, time-dependent formal theory of, full Hamiltonian Green's functions      263
Multiparticle, multichannel scattering, time-dependent formal theory of, Hilbert spaces for      263
Multiparticle, multichannel scattering, time-dependent formal theory of, Moeller wave operators for channel a      264—267
Negative energy solutions, Dirac equation      333—334
Negative energy solutions, Klein — Gordon equation      314
Nernst theorem      46n.17
Neutron-optical experiments      see "Experimental signatures for quaternionic quantum mechanics"
Noether theorem, total trace version      450—452
Noether theorem, total trace version, fermionic currents      484—486
Non-Abelian monopole      97n.8 528
Nonlinear modifications in quantum mechanics      524—525 see
Nonrelativistic kinematics      87 87n.1 89 164
Norm      see "Quaternion" "Hilbert
Normalization for bound state      161 168
Normalization for Fourier expansion      307 411
Normalization, box      28
Number field, automorphism      29n.7 30
Number field, defined      9—10
Number field, generalized Wigner theorem for      29—31
Number field, rational      7n.6
Number field, rational, p-adic norm for      7n.6
Number field, topological characterization of      10n.9
Observables      27 35 58 108
Observables, conserved, conditions for      269n.11
Observables, identification in operator gauge invariant theories      452—453 453n.24 512 531
Octonion      see "Algebra"
Octonionic quantum mechanics      9n.8 11 20 49—52 539—540
Octonionic quantum mechanics, attempted Schroedinger equation for      50—51
Octonionic quantum mechanics, failure of completeness in      49—50
Octonionic quantum mechanics, failure of cyclic trace property in      540
Octonionic quantum mechanics, failure of unitarity in      50—52
One quantum criterion      519 519n.14
Open questions      497
Open questions, involving quaternionic analogs of first quantized complex quantum mechanics topics      526—528
Open questions, involving quaternionic analogs of second quantization, relativistic quantum mechanics, and quantum field theory      528—533
Operator      see also "Quaternion"
Operator, "time"      526n.22

Operator, adjoint      22
Operator, annihilation for harmonic oscillator      209
Operator, annihilation, abstract      270
Operator, anti-Hermitian      see "Quaternion anti-self-adjoint"
Operator, antiunitary      29—30 112 118—119 214
Operator, bosonic      289n.9 443
Operator, colinear and counitary      22 30
Operator, complex anti-self-adjoint, spectral theory      80
Operator, complex antilinear      43 see
Operator, complex linear      22 47n.19 49 53 61—63 66—68 118—119 351 408n.4
Operator, complex linear, defined in general case      61
Operator, complex self-adjoint, spectral theory      80 142 204
Operator, creation for harmonic oscillator      209
Operator, creation, abstract      270
Operator, dimension-6, and phenomenology of tests for quaternionic and nonlinear quantum mechanics      519—520 525
Operator, elliptic      127 172n.4
Operator, fermion grading $(-1)^{F}$       442—443 442n.20 443n.22 461
Operator, fermionic      289n.9 443
Operator, gauge transformation      see "Operator-valued gauge transformation"
Operator, hermitian      see "Quaternion self-adjoint"
Operator, left-acting algebra      23—24 33—34
Operator, left-acting algebra in angular momentum representation      65—66
Operator, left-acting algebra in coordinate representation      38
Operator, matrix elements, reality or complexity properties      34
Operator, mutually commuting set of self- and anti-self-adjoint, all Hermitian      77
Operator, mutually commuting set of self- and anti-self-adjoint, spectral theory of      53 76—83
Operator, noncompact      443n.22
Operator, nonlocal      83 127
Operator, normal      28n.6
Operator, product expansion      216
Operator, quaternion anti-self-adjoint      19 23 28n.6 31—36 53 76—83 103—104 118—119 448n.23
Operator, quaternion anti-self-adjoint, expectation value of      35
Operator, quaternion anti-self-adjoint, modulus (or magnitude) and phase of      33 35 77
Operator, quaternion anti-self-adjoint, not trivially made self-adjoint      77 448n.23 490—491
Operator, quaternion anti-self-adjoint, spectral theory of      29—36 76—83 91 98 124 134 136 494
Operator, quaternion anti-self-adjoint, spin      85
Operator, quaternion anti-self-adjoint, unitary inversion operator for      35 139 286
Operator, quaternion linear      22—23 27 61 66 408n.4
Operator, quaternion self-adjoint      19 23 27—29 62—63 69—71 76—83 103—104 108 397 448n.23
Operator, quaternion self-adjoint, expectation value of      28
Operator, quaternion self-adjoint, spectral theory of      27—29 76—83 103—104 494
Operator, quaternion self-adjoint, thermal expectation      288
Operator, quaternion unitary      30—31 36—37 41 54 65 68 70 74 90 99 101 103—105 109 113 208—209 228—232 432 440 523
Operator, quaternion unitary, spectral theory of      35—36 91
Operator, real anti-self-adjoint      47—48
Operator, real linear      48
Operator, real self-adjoint      47n.19 48
Operator, real skew-symmetric, canonical form for      47
Operator, unit      see "Hilbert space unit
Operator, unitary      29—31 112
Operator-valued gauge transformation      399 442 442n.20—21 449—455 501 see generalized" "Generalized "Total
Operator-valued gauge transformation and complex quantum mechanics      455—475
Operator-valued gauge transformation, biunitary      431—432 449—450 483 529
Operator-valued gauge transformation, identification of invariant observables under      452—453 453n.24 512 531
Operator-valued gauge transformation, identification of invariant observables under, cotransforming states introduced      453
Operator-valued gauge transformation, summarized for quaternionic field models      475—478 483
Operator-valued gauge transformation, total trace Lagrangian invariant under      449—450 508
Operator-valued gauge transformation, unitary      431—432 449 529 see quaternion
Optical potential      114—115 127—128 177—179 183n.12 198n.4 218 499
Optical potential and cluster decomposition property      293—299
Optical potential and time reversal violation      114—115 129—130 174—175 215
Optical potential and time-dependent Schroedinger equation      128—130
Optical potential with spin      130—131
Optical potential, bound-state-associated resonances and singularities of      175—179
Optical potential, conjugate used in equation for $f_{\beta}$       128
Optical potential, Galilean invariance      527
Optical potential, isolated singularity in      175
Optical potential, properties of      127—128
Optical potential, spin-0 obtained from spin-1/2      131
Optical potential, total $V_{tot}$ , applied      200—201 203—204 215—216
Optical potential, total $V_{tot}$ , defined      172
Optical theorem      see "S-matrix unitarity
Parastatistics      238n.2 504—505 527
Parity, invariance      76 381—382 384 481—483
Parity, notation P in quaternionic quantum mechanics and $\mathcal{P}$ in complex quantum mechanics and phenomenology      217n.74
Parity, real phase factors in      384 481 483
Parity, relationship between complex and quaternionic definitions      482n.42
Path ordering operation $P_{\ell}$       156 185
Pauli spin matrices      84 111n.14 184—185 322 330 479n.39
Pauli spin matrices, used to represent quaternions      495 527
Pauli spinors      401
Pauli spinors, origin of      515—516
Permutation operator      237—238
Permutation operator, square not assumed unity      238n.2
Perturbation theory, stationary state or time-independent      124 131—143 168—170
Perturbation theory, stationary state or time-independent for asymptotic bound on $<I_{\tilde{H}}-I_{\tilde{H}_{0}}|x>$       138—139
Perturbation theory, stationary state or time-independent for multiparticle change from $\mathbb{C}(1, i)$       245—254
Perturbation theory, stationary state or time-independent for zero energy states      142—143
Perturbation theory, stationary state or time-independent, degenerate      140—143
Perturbation theory, stationary state or time-independent, dependence on origin of energy scale      134
Perturbation theory, stationary state or time-independent, first-order energy      133 177
Perturbation theory, stationary state or time-independent, first-order Hamiltonian modulus, phase      134—139
Perturbation theory, stationary state or time-independent, first-order left-acting algebra      134—139
Perturbation theory, stationary state or time-independent, first-order wave function      132—134
Perturbation theory, stationary state or time-independent, higher-order wave function      133—134 143
Perturbation theory, stationary state or time-independent, nondegenerate unperturbed energies      131—134 140
Perturbation theory, stationary state or time-independent, second-order energy and wave function      139—140
Perturbation theory, time-dependent      194—217
Perturbation theory, time-dependent in decaying state theory      201—208 213—217 499 526
Perturbation theory, time-dependent in decaying state theory, initial condition      201—202
Perturbation theory, time-dependent in decaying state theory, initial state      201
Perturbation theory, time-dependent in decaying state theory, mass and decay matrices      203—204
Perturbation theory, time-dependent in decaying state theory, upper half plane analyticity      202—206
Perturbation theory, time-dependent in decaying state theory, Weisskopf — Wigner approximation      202n.7 204—206
Perturbation theory, time-dependent in scattering theory      196—201
Perturbation theory, time-dependent in scattering theory, basic equation for      195 209
Perturbation theory, time-dependent in scattering theory, interaction picture      208—211
Perturbation theory, time-dependent in scattering theory, notation used for      194n.1
Perturbing Hamiltonian, compact notation for matrix elements      132 134 195
Phase of $\beta$ -symplectic potential      114n.17 188 191
Phase, approximation methods involving      145—158
Phase, dynamical (versus geometric)      148
Phase, geometric, adiabatic      145—149 527
Phase, geometric, analog in generalized dynamics      531
Phase, geometric, complex for nonzero energy      148
Phase, geometric, complex for nonzero energy, integral over closed orbit      148
Phase, geometric, complex for nonzero energy, reraying of      148
Phase, geometric, nonadiabatic      150—156
Phase, geometric, nonadiabatic and time-ordered integral properties      151—152
Phase, geometric, nonadiabatic, invariant angle associated with      152
Phase, geometric, nonadiabatic, quaternionic distinct from complex      153—156
Phase, geometric, nonadiabatic, quaternionic reraying of      152—156
Phase, geometric, nonadiabatic, Riccati equation and      153—155 527
Phase, geometric, nonadiabatic, trace of closed orbit integral      150—152
Phase, geometric, quaternionic for zero energy      149
Phase, geometric, quaternionic for zero energy, reraying of      149
Phase, geometric, quaternionic for zero energy, trace of closed orbit integral      149 152
Phase, role in generalized quantum dynamics      531
Phase, shift, in scattering      168 176
Phase, shift, in scattering, compound, and experimental tests      516—518
Planck scale (or mass)      497—498 500
Planck's constant      530 see
Potential      see also "Hamiltonian" "Scalar "Scattering" "Schroedinger "Vector
Potential, anti-self-adjointness implies real part vanishes for single-component wave function      40 236
Potential, delta function      159
Potential, left-right symmetric in one-dimensional scattering      518
Potential, local      397—398
Potential, spherically symmetric      165—171 175—179
Potential, taken as a quaternionic constant      59n.3 113—114 123 125 137n.2
Potential, taken as a quaternionic constant, energy eigenstates for      125—126
Potential, time-independent and time reversal operator      112—119
Potential, translation invariance restrictions on      237
Pregeometry      511—512 511n.10 516 532
Preons      501—503 532 see
Preons, chiral symmetry and      508n.8
Preons, fundamental doublet assumed      502
Preons, fundamental doublet assumed, "rishons" or "quips"      502n.5
Prime as notation for x differentiation      184 358
Principal value P      56 127—128 177 203
Probability and Markovian property      5 5n.3

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