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

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

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Название: Quaternionic Quantum Mechanics and Quantum Fields

Автор: Adler S.L.

Аннотация:

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

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Рубрика: Механика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1995

Количество страниц: 586

Добавлена в каталог: 10.12.2009

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Предметный указатель

Hamiltonian, symmetry generators which commute with      74—76 231 238
Hamiltonian, system, constrained in generalized quantum dynamics      455 487 529
Hamiltonian, system, constrained, invariant relations in      465n.35 469 529
Hamiltonian, system, constrained, primary and secondary constraints in      487
Hamiltonian, system, constrained, standard theory of      455n.26
Hamiltonian, time reversal invariant      112—122 198n.5
Hamiltonian, time-dependent      69—70
Hamiltonian, time-independent      68 70 109 194 196 201 219
Hamiltonian, total trace      399 see
Hamiltonian, translation invariant      59—62 80 83 89—95 237—239 255 267
Hamiltonian, trial      93 235
Harari — Shupe proposal      501—511 532
Harari — Shupe proposal, candidate dynamics      508—509
Harari — Shupe proposal, enumeration of states      505—508 532
Harari — Shupe proposal, enumeration of states, spin-1/2 mixed symmetry states      506—507
Harari — Shupe proposal, enumeration of states, spin-3/2 mixed symmetry states      506 508
Harari — Shupe proposal, families constructed in      501—503
Harari — Shupe proposal, naive counting rule for      502 502n.6
Harari — Shupe proposal, postulated rules for three-quasiparticle composites      503—505
Harari — Shupe proposal, role of residual forces      511
Harari — Shupe proposal, shell model dynamics      509—511
Harari — Shupe proposal, shell model dynamics, ground state wave function structure      510—511
harmonic oscillator      122—123
Hartree approximation      see "Mean field approximation"
Heaviside step function      170 289
Heisenberg picture      53 68—70 89 89n.2 194 208 234 289 397 399 413—415
Heisenberg picture and formally real field components      425—426 426n.11
Heisenberg picture for quaternionic forced harmonic oscillator      211—213
Heisenberg picture, generalized dynamics formulation      448—449
Heisenberg picture, H omitted as subscript for      89
Heisenberg picture, left-acting algebra time dependent      89n.2 234n.1 425—426
Heisenberg picture, operator and state vector defined      69—70
Heisenberg picture, operator and state vector defined, time development of      69 89 234
Heisenberg uncertainty principle      see "Uncertainty principle"
Helicity      391 391n.13
Hidden variables      524n.19
Hierarchy problem      see "Standard model"
Higgs mechanism      497n.1 508
Hilbert module      10n.10
Hilbert space for multichannel scattering      263
Hilbert space for one fermion degree of freedom      434—435
Hilbert space, $\mathbb{C}(1, i)$ subspace of, as arena for standard model physics      498—499
Hilbert space, $\mathbb{C}(1, i)$ subspace of, stable under quaternionic perturbations      207—208 499
Hilbert space, $\mathbb{C}(1, i)$ subspace, of quaternionic      63 66
Hilbert space, closed ray orbit in      150—156
Hilbert space, column vector      21 199
Hilbert space, complete      20 see
Hilbert space, complex      10—11 499 533 538
Hilbert space, four-dimensional, mutually commuting left algebra bases in      480
Hilbert space, generalized Wigner theorem      29—31 112
Hilbert space, halving of dimensions by complex into quaternionic embedding      407—410
Hilbert space, halving of dimensions by real into complex embedding      49
Hilbert space, indefinite metric      20n.3
Hilbert space, norm      20—21
Hilbert space, norm for symplectic components      71
Hilbert space, octonionic      10 539
Hilbert space, parameterized family of states in      150
Hilbert space, quantum mechanics, inner product conservation in      50—51
Hilbert space, quantum mechanics, requires associative multiplication      49—52 539—540
Hilbert space, quaternionic      10—11 19—21 90
Hilbert space, quaternionic and fundamental physics      499—501 533
Hilbert space, quaternionic, bosonic and fermionic subspaces      442—443
Hilbert space, quaternionic, complex quantum fields embedded in      407—421 498 see
Hilbert space, real      10—11 48
Hilbert space, row vector      21
Hilbert space, scalars of      19n.1 20 23 29
Hilbert space, scalars of, right-acting      24—25 34 38—39 100n.10 275 399 479 512 530
Hilbert space, separable      20
Hilbert space, unit (identity) operator of      34 73—74 see Fock
Hilbert space, unit rays of      22 29—30
Hilbert space, vectors of      20 22 29—30
Identity or formula for $<x|e^{s{\vec{\bar{p}}}^{2}/2m}|x^{''}>$       110 137
Identity or formula for $e^{\lambda A}Be^{-\lambda A}$       92n.6
Identity or formula for $\frac{\partial}{\partial t}e^{\Lambda(t)}$       458n.29
Identity or formula for $\gamma^{\mu}$ matrices $\gamma^{\mu}\frac{1}{2}[\gamma_{\mu}, \gamma_{\nu}]$       485
Identity or formula for $\int\limits^{\infty}_{0}ds\ e^{-sA}(AB+BA)e^{-sA}$       136
Identity or formula for $\int\limits^{\infty}_{0}ds\ e^{-sA}(AB+BC)e^{-sC}$       136n.1
Identity or formula for $\int\limits^{\infty}_{0}\frac{ds}{s^{3/2}}e^{-(As+B/s)}$       249—250 249n.4
Identity or formula for $\int\limits^{\infty}_{0}\frac{ds}{s^{3}}e^{-(As+B/s)}$       137 137n.3
Identity or formula for exponentiating 1/A, A>0      135—136 248
Identity or formula for quasiparticle operators      504—505
Identity or formula, Anandan — Aharonov, for <f(t)|f(t + dt)>      150n.9
Identity or formula, Baker — Campbell — Hausdorff      110—111
Identity or formula, Bianchi, for gauge field strength      485
Identity or formula, commutator of covariant derivatives      364—365
Identity or formula, Dirac spinor orthogonality and spin sums      428n.13 430n.16
Identity or formula, Feynman, for 1/(ab)      138
Identity or formula, golden rule approximation      205—206
Identity or formula, Jacobi      see "Jacobi identity"
Identity or formula, principal value, for $(E + i\varepsilon - E_{\ell})^{-1}$       203
Identity or formula, reproducing, for $\int\frac{d\phi d\chi}{4\pi^{2}}f(\kappa+\bar{\phi}i\phi+\bar{\chi}\chi)$       494n.49 532
Identity or formula, Trotter product, for unitary operators      109
Imprimitivity theorem      515
Inner product (scalar product), associative multiplication needed for conservation      50—51
Inner product (scalar product), complex $\mathbb{C}(1, i)$       26—27 41—45 49 181 212n.1 408—409 408n.4
Inner product (scalar product), imaginary      49
Inner product (scalar product), invariance transformation of      27 41
Inner product (scalar product), quaternionic      20—22 25—27 42—43 271—272 281 408n.4 409
Inner product (scalar product), quaternionic, reality or complexity properties      34
Inner product (scalar product), real      26—27 44 49
Inner product (scalar product), real linear      272n.1
Inner product (scalar product), symplectic      42 408n.4 408
Instanton      528
Integrable systems      526
Integration measures      171
Interaction picture      194 208—211 415
Interaction picture for quaternionic forced harmonic oscillator      210—211
Interaction picture, dynamics of states in      208—209
Interaction picture, evolution operator      228—232 228n.4
Interaction picture, expression for Moeller wave operator and S-matrix      229
Interaction picture, expression for Moeller wave operator and S-matrix symmetries      231—232
Interaction picture, operator, for generalized quantum dynamics      454 531
Interaction picture, state vector and operator, related to Schroedinger picture      208
Intertwining identities      see "Covariant derivative" "Moeller "S-matrix"
Inverse scattering theory      526
Isotropy, spatial      500
Jacobi identity for commutator      70 363
Jacobi identity for generalized Poisson bracket      447—448 535—540 see generalized"
Jacobi identity for generalized Poisson bracket, fails in octonionic Hilbert space      539—540
Jauch theorem counterexample      34n.10
Jordan algebra      10—11 see
Jordan formulation of quantum mechanics      10—11 448n.23
Jost function      183n.12 526
Junction conditions for wave function      160 162 164
Klein paradox      334 399—407 see interacting
Klein paradox, complex quantum mechanics limit      405—407
Klein paradox, conclusions from      407
Klein paradox, two interpretations      405—407
Klein — Gordon equation      87 303—328 362 see
Klein — Gordon equation, free particle case      303—315
Klein — Gordon equation, free particle case, coordinate space inner product      305—306
Klein — Gordon equation, free particle case, coordinate space inner product, completeness relation      309—310
Klein — Gordon equation, free particle case, coordinate space inner product, connected to charge structure      306
Klein — Gordon equation, free particle case, coordinate space inner product, indefinite      305—306 306n.1 309
Klein — Gordon equation, free particle case, coordinate space inner product, is Lorentz scalar      305
Klein — Gordon equation, free particle case, coordinate space inner product, relation to four-current $J^{'}_{\nu}$       305
Klein — Gordon equation, free particle case, coordinate space inner product, rewritten in momentum space      310—311
Klein — Gordon equation, free particle case, coordinate space inner product, time independence      305 311
Klein — Gordon equation, free particle case, energy eigenstates and form of "antiparticle" states      314
Klein — Gordon equation, free particle case, four-component wave function      308
Klein — Gordon equation, free particle case, four-component wave function, Hamiltonian for      308 310—311
Klein — Gordon equation, free particle case, Lorentz scalar wave function      304
Klein — Gordon equation, free particle case, momentum space inner product      307—315
Klein — Gordon equation, free particle case, momentum space inner product, completeness relation      309
Klein — Gordon equation, free particle case, momentum space inner product, definite      309
Klein — Gordon equation, free particle case, momentum space inner product, expectation in      309
Klein — Gordon equation, free particle case, momentum space inner product, is Lorentz scalar      307n.2
Klein — Gordon equation, free particle case, momentum space inner product, rewritten in coordinate space      310—312
Klein — Gordon equation, free particle case, momentum space inner product, time independence      309 312—313
Klein — Gordon equation, free particle case, normalization integral      307n.2

Klein — Gordon equation, free particle case, Poincare generators for      389 433
Klein — Gordon equation, free particle case, relativistic notation      304—305 see convention "Relativistic
Klein — Gordon equation, free particle case, two-component wave function      312—314
Klein — Gordon equation, free particle case, two-component wave function, Hamiltonian for      313—315
Klein — Gordon equation, interacting case      315—328
Klein — Gordon equation, interacting case and source term for gauge potential      362
Klein — Gordon equation, interacting case, coordinate space inner product      318—319 323
Klein — Gordon equation, interacting case, coordinate space inner product, real and complex projections      319
Klein — Gordon equation, interacting case, coordinate space inner product, relation to four-current $J^{'}_{\nu}$       318
Klein — Gordon equation, interacting case, gauge covariance of      316
Klein — Gordon equation, interacting case, general gauge principle      315—316 368 434 501 528
Klein — Gordon equation, interacting case, general gauge principle, covariant derivative      315 see
Klein — Gordon equation, interacting case, general gauge principle, gauge potentials and transformation      315—316
Klein — Gordon equation, interacting case, general gauge principle, Lagrangian density and coupling g      375
Klein — Gordon equation, interacting case, general gauge principle, specializations      319—321 320n.6 363 369 434 477—478
Klein — Gordon equation, nonrelativistic reduction      321—328
Klein — Gordon equation, nonrelativistic reduction and Galilean invariance      327—328
Klein — Gordon equation, nonrelativistic reduction, "semirelativistic" Foldy — Wouthuysen reduction      345—348
Klein — Gordon equation, nonrelativistic reduction, doesn't have form studied in Part II      327
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method      322—327
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, auxiliary quantities for      322
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, conditions for validity      326n.9 345
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, even and odd Hamiltonian terms defined      324—325
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, Hamiltonian self-adjointness condition      323
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, leading-order reduction      327—328
Klein — Gordon equation, nonrelativistic reduction, Foldy — Wouthuysen method, must pick out $\mathbb{C}(1, e)$ subalgebra      321—322
Kramers degeneracy      217n.14
Kronecker delta      271 485
Lagrangian      307n.2 399 415n.7 see "Lagrangian "Total
Lagrangian density      362 374n.4 374—388
Lagrangian density for interacting Klein — Gordon equation general gauging      375—377
Lagrangian density for interacting Klein — Gordon equation specialized gaugings      377
Lagrangian density for pair of interacting Dirac equations      377—378
Lagrangian density for single interacting Dirac equation      379—380
Lagrangian density, complex quantum field models      381—384
Lagrangian density, complex quantum field models, invariance under $\mathcal{C}$ , $\mathcal{P}$ , $\mathcal{T}$       384—387 528
Lagrangian density, obtained by complex restriction of pair Lagrangian      380—381
Lagrangian density, trick of replacing $\mathcal{L}$ by $tr\mathcal{L}$       376 378—380
Lattice of propositions      12
Lee — friedrichs model      526n.22
Left-acting operator algebra      23—24 38 55—56 58—61 63 82 85 89n.2 98 101—102 109 125 134—139 199 248 410—421 441—442 481—482 512 528 see left-acting"
Left-acting operator algebra and Fock space construction      270 274—283 410—421
Left-acting operator algebra and formally real operator components      426
Left-acting operator algebra, asymptotic limiting of $I_{\tilde{H}}$       138—139 261
Left-acting operator algebra, multiple expansions over      478—479 511 530
Left-acting operator algebra, time dependence in Heisenberg picture      89n.2 234n.1 425—426
Left-acting operator algebra, trace over defined      280
Left-right symmetric theory      508 508n.9
Legendre transformation      446
Leibnitz Product Rule      315n.4 536—537
Leibnitz product rule for total trace functional and generalized bracket      448
Lemma 1, applied      84 391—392 394
Lemma 1, stated      76
Lemma 2, applied      80—81 83 390 392 401 500
Lemma 2, Corollary 1, applied      82
Lemma 2, Corollary 1, stated      81
Lemma 2, Corollary 2, applied      83
Lemma 2, Corollary 2, stated      82—83
Lemma 2, Corollary 3, applied      351
Lemma 2, Corollary 3, stated      83
Lemma 2, stated      77
Lemma 3, stated      375
Lemma 4, applied      376—381
Lemma 4, stated      395 395n.16
Lemma 5, discussed      533
Lemma 5, stated      514
Levinson's theorem      183n.12
Lippmann — Schwinger equation      173
Lippmann — Schwinger equation, outgoing wave scattering solution      173 179—180 180n.10 201
Locality, and basis in which S-matrix and Poincare representations are complex      398
Lorentz frame, preferred singled out by Lemma 5      533
Lorentz invariance      388
Lorentz invariance and curved space-time generalization      483—484
Lorentz scalar charge integral, conditions for      305
Lorentz scalar wave function for Klein — Gordon equation      304
Lorentz transformation, homogeneous      336 389n.11 513—515
Lorentz transformation, inhomogeneous      305
Many-body problem      see "Multiparticle systems"
Markovian property      5 5n.3
Mass scale for quaternionic physics      213 217 498 498n.2 525
Mass scale, electroweak      214n.10
Mass scale, hadronic      214n.10
Mass zero, fermion theories      374 387—388 508 531
Mass zero, higher spin composites, no-go theorems for      503
Mass, lepton and quark      507—508 511
Mass, lepton and quark, origin of      531
Mass, reduced      239
Mass, total      239 241
Matrix      see also "Operator" "Quaternion"
Matrix, $2\times 2$ null and unit      330 386
Matrix, auxiliary $\mathcal{M}_{\mu}$ and $\mathcal{M}^{'}_{\mu}$       382
Matrix, auxiliary $\mathcal{M}_{\mu}$ and $\mathcal{M}^{'}_{\mu}$ , charge conjugation of      384—385
Matrix, auxiliary $\mathcal{M}_{\mu}$ and $\mathcal{M}^{'}_{\mu}$ , time reversal of      386—387
Matrix, complex anti-self-adjoint and self-adjoint      80
Matrix, complex unitary      80 314 437
Matrix, decay      203—204
Matrix, mass      203—204
Matrix, quaternion anti-self-adjoint      19 29—35 143 490—491 494—495 546—549
Matrix, quaternion self-adjoint      19 27—29 490—491 494—496 542—545
Matrix, quaternion unitary      29 33 35—36 78 494—495 541—542 546
Matrix, supermatrix generalization, applied to Gaussian integrals      491—494 549—551
Mean field approximation      296—299 509 531
Mean field approximation, notation for subsystem expectations      298
Metric tensor      304
Metric, convention for      53n.1 304 512
Metric, curved space-time      483 513—515
Metric, Minkowski      512—514
Microscopic units      3n.1
Modulus function      5—9 14 52 52n.22
Moeller wave operators      218 222—232 244 264—267
Moeller wave operators, defined      222 224 264
Moeller wave operators, integrals for      223
Moeller wave operators, interaction picture expression      228—229
Moeller wave operators, intertwining action between free and full Hamiltonian      224 265
Moeller wave operators, isometric property      224 230
Moeller wave operators, limiting expressions for      223 264
Moeller wave operators, multichannel, for arrangement channel a      264—267
Moeller wave operators, not unitary when there are bound states      225—226
Moeller wave operators, range of      265
Moeller wave operators, S-matrix defined in terms of      228
Moeller wave operators, S-matrix symmetries and      231—232
Moeller wave operators, unitarity deficiency action on      226 265
Momentum, anti-self-adjoint operator for      53—64 66 98 209 500
Momentum, anti-self-adjoint operator for, action in coordinate representation      54
Momentum, anti-self-adjoint operator for, action of time reversal on      118—119
Momentum, anti-self-adjoint operator for, in multiparticle system      59—60 80—81 238 267
Momentum, anti-self-adjoint operator for, spectral representation for      56 500
Momentum, anti-self-adjoint operator for, translation group constructed with      54 59—60 63 99
Momentum, canonical      410—415 446 459 463 487—489
Momentum, complex linear definition of      62—63 351 512
Momentum, complex linear definition of, action of time reversal on      118—119
Momentum, conservation in multiparticle scattering      268
Momentum, representation      53
Momentum, representation, left-acting algebra I, $J_{p}$ , $K_{p}$ in      56
Momentum, representation, left-acting algebra I, J, K in      56 413
Momentum, representation, wave function      55—57 271 307 310—311
Momentum, self-adjoint operator for      53 58—64 66 107—108 126 351 499 512
Momentum, self-adjoint operator for, action in $\mathbb{C}(1, i)$ Hilbert subspace      63—64
Momentum, self-adjoint operator for, action on asymptotic scattering states      81 139
Momentum, self-adjoint operator for, commutator with coordinates      61
Momentum, self-adjoint operator for, in complex quantum mechanics      76
Momentum, self-adjoint operator for, in multiparticle systems      59—61
Momentum, self-adjoint operator for, in system with constant potential      126
Momentum, self-adjoint operator for, translation group constructed with      59
Monopole, 't Hooft — Polyakov      528
Monopole, Dirac      97n.8 528
Multiparticle systems and experimental signatures for quaternionic quantum mechanics      518—520
Multiparticle systems, center of mass separation      233
Multiparticle systems, center of mass separation, complex cluster      255—256 261
Multiparticle systems, center of mass separation, N-pair model      245—246

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