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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

"Algebraic chromodynamics"      442n.20 473 479n.39
"Ghost" fermions      492n.48
"Mass gap" for constant quaternionic potential      126 527
"Quantization" of a classical theory      442n.20 533
$i_{2}$ , real $2\times 2$ representation of i, adjoint introduced      346
$i_{2}$ , real $2\times 2$ representation of i, defined      47
$i_{2}$ , real $2\times 2$ representation of i, transformation to bases which diagonalize      381
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra)      70 85 113 414 417 422—424 426—427 429 432 478 497 500
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra) and independent particle picture in Fock space      287
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra) and spin Hamiltonian      85—86
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra), asymptopia hypothesis      497—498 518—519 530
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra), conjugation      86 279
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra), spin matrices      84
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra), unitary products of left algebra units in second quantization      278
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion)      34 59 61 63 66 70—72 75—76 78 80—81 115 117 119 122 132 141 148 152—156 159—160 166—167 171—174 196 207—209 230 233 242—243 245 247—248 251 253—260 262 270—272 278 287 289 291 296 314—315 329n.12 332 334 335n.14 340 340n.16 351 359—361 369 373—387 390—395 397—398 401 404—405 407 409—410 478 479n.39 527—530
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) and matrix representations of symmetry generators      68 75—77 80—81 84 141n.6 238
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) and nonrelativistic reduction of Klein — Gordon equation      321—322
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) and zeroth-order basis rediagonalization in degenerate perturbation theory      142
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) with i replaced by $u_{T}$       116 286
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) with i replaced by general unit quaternion      93 227 254 283 322
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion), bases used to span quaternionic Fock space      271
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion), notation introduced      13
$\mathbf{R}$ , reduction of wave packet operation      520 522—524
$\mathbf{U}$ , unitary evolution operation      520 522—524
$\mathcal{CPT}$ theorem      216 216n.13 531
$\mathcal{CPT}$ theorem, apparent violation for $G\neq G^{'}$       482n.42 531
$\mathcal{CPT}$ theorem, restrictions on $\mathcal{T}$ violation model      216 216n.12
$\mathcal{CP}$ in time reversal violation model      216
$\mathcal{CP}$ violation, phenomenological      215—217 508—509 531
$\mathcal{CP}$ violation, phenomenological, Kobayashi — Maskawa form      216 518
$\mathcal{CP}$ violation, phenomenological, magnitude of      215—217
$\mathcal{CP}$ violation, phenomenological, milliweak      216
$\mathcal{CP}$ violation, phenomenological, superweak      216n.12
Abelian group      90—91 99—100
Abelian monopole      97n.8
Absolute value function      see "Modulus function"
Absorption      see "Scattering one-dimensional"
Action and Feynman path integral      110 353
Action as integral of Lagrangian density      375
Action, classical, not fundamental in quaternionic quantum mechanics      111—112
Action, invariance under $\mathcal{C}$ , $\mathcal{P}$ , $\mathcal{T}$       384—387
Action, principle for generalized dynamics      445—447
Action, real-valued, independent variations of      375
Adiabatic approximation      145—149 195n.2
Adiabatic approximation for zero energy state      148—149
Adiabatic approximation, effective expansion parameter for      147
Adiabatic switching      229
Adjoint defined for column vector      21—22
Adjoint defined for quaternion matrix      14
Adjoint of Grassmann quaternion      16
Adjoint of operator      22—23 51
Adjoint of product      15—16
Adjoint, relation between $D_{\mu}$ and $\bar{D_{\mu}}$       484
Adjoint, use of $\dagger$ as notation for      14 236 323 330
Albert's Theorem      7
Algebra of Dirac matrices and $i_{G}$ , $j_{G}$ , $k_{G}$       340 340n.16—17
Algebra over reals      5—7
Algebra, "charge" for complex generator matrices      436
Algebra, "charge" for quaternionic generator matrices      434—437 452
Algebra, "color charge"      480 480n.40
Algebra, absolute valued      7
Algebra, angular momentum      65 85 90n.4 395
Algebra, angular momentum, one-dimensional representation of      396 434—435 515
Algebra, anticommutator, of Dirac matrices      330 335
Algebra, basis elements      6
Algebra, Clifford      10n.9 114n.16 514n.11
Algebra, Clifford, of (3, 1) signature and quaternions      512
Algebra, commutator, of Dirac matrices and $i_{G}$       331 335
Algebra, complex      7
Algebra, complex, automorphism of      16n.16 30
Algebra, conformal, generators of      392
Algebra, conformal, nonzero energy quaternionic representations transformable to complex form      392
Algebra, division      8—9
Algebra, four-vector extension of quaternion      513 514n.11
Algebra, Grassmann      16
Algebra, Jordan      10—11
Algebra, Jordan, exceptional      11
Algebra, Jordan, infinite-dimensional      11
Algebra, Jordan, special      11
Algebra, left-acting      23—24 33 38—39 100n.10 see
Algebra, Lie, commutator      435—436
Algebra, Lie, generalized bracket      437 452
Algebra, multiquaternion      272n.1 479—480 511 530
Algebra, nonassociative      8 20 49—52 540
Algebra, noncommutative      7—8 11 20n.3 55 244 258
Algebra, octonion      7—8 11 49—52
Algebra, Poincare      361 388—398 429 489 530 see
Algebra, Poincare, nonzero energy quaternionic representations transformable to complex form      361—362 388—398 499 503 526 532
Algebra, Poincare, relation to locality      362 398
Algebra, quaternion      7 11—12 114
Algebra, quaternion, associativity of      12
Algebra, quaternion, automorphism of      16—17 30 515
Algebra, quaternion, complex $2\times 2$ matrix representation      495
Algebra, quaternion, real matrix representations      10n.9
Algebra, right-acting      24—25 34 38—39 100n.10 275 399 480 512 530
Algebra, simple      11n.12
Algebra, SU(2)      see "Angular momentum"
Algebra, supersymmetry      361
Analyticity properties      159 175
Analyticity properties of forward scattering amplitude      179—183 358 499 526
Analyticity properties of thermal Green's functions      291—293
Analyticity properties upper half plane, in decaying state theory      202—206
Angular momentum, algebra      see "Algebra"
Angular momentum, analog of left- and right-ordered Fourier transform      84n.12
Angular momentum, anti-self-adjoint operator for      64—66 80 118—119
Angular momentum, anti-self-adjoint operator for, action in coordinate representation      64
Angular momentum, anti-self-adjoint operator for, action of time reversal on      118—119
Angular momentum, anti-self-adjoint operator for, commutation relations      65
Angular momentum, anti-self-adjoint operator for, eigenstates and spectrum of      65
Angular momentum, anti-self-adjoint operator for, eigenstates, ray convention for      65
Angular momentum, anti-self-adjoint operator for, rotation group constructed with      65
Angular momentum, anti-self-adjoint operator for, self-adjoint squared operator      65 80
Angular momentum, complex linear operators      66—68 351 408n.4 512
Angular momentum, complex linear operators, action of time reversal on      118—119
Angular momentum, ladder operators      66—68
Angular momentum, orbital      85 502n.6
Angular momentum, orbital, scalars with respect to      85
Angular momentum, orbital, vectors with respect to      85
Angular momentum, partial wave analysis      166 175
Angular momentum, representation      65—66
Angular momentum, representation, left-acting algebra I, J, K in      65—66
Angular momentum, representation, matrices      68
Angular momentum, representation, matrices, right action      68 80—81 503
Angular momentum, self-adjoint operator for      66—68 351 512
Angular momentum, self-adjoint operator for, action in $\mathbb{C}(1, i)$ Hilbert subspace      66 166
Angular momentum, self-adjoint operator for, commutation relations      66
Angular momentum, spin      84
Angular momentum, spin, algebra of      85
Angular momentum, spin, nonrotational invariance of J      85 130
Angular momentum, spin, operators      85 119
Angular momentum, spin, rotational invariance of $J\sigma_{2}$       85 130
Angular momentum, total      85 90n.4
Angular momentum, total, action of time reversal on      119
Anomaly      387 387n.10 531
Anomaly, chiral      374 399 480 483 508
Anomaly, Witten      480
Anticommutation, assumption for symplectic components of fermions in charge conjugation analysis      384
Anticommutator in Foldy — Wouthuysen method      325
Anticommutator in Galilean analysis      93n.7 95
Anticommutator in uncertainty principle derivation      73
Anticommutator of $J_{\tilde{H}}$ , $K_{\tilde{H}}$ with $\tilde{H}_{0}$       139
Anticommutator of annihilation and creation operators      273 417 503
Anticommutator of Dirac matrices      330
Anticommutator of quasiparticle operators      280 285—286
Anticommutator, canonical, in operator gauge invariant theory as constraint      463 512
Anticommutator, canonical, in operator gauge invariant theory biunitary transform covariant      529
Anticommutator, notation defined      16
Associator      539—540
Asymptotic completeness      265 265n.8 527
Asymptotic particle spectrum      388 391 397

Asymptotic scattering states      4 66 128 188
Asymptotic scattering states in quantum chromodynamics      499n.3
Asymptotic scattering states, complex for suitable ray choice      63—64 159 160 172n.5 181 198 213 512
Asymptotic scattering states, dynamics of      113 497—499
Asymptotic scattering states, momentum operator definitions agree on      63—64
Asymptotic scattering states, structure, in multiparticle, multichannel scattering      254—262 see classification
Asymptotically $\mathbb{C}(1, i)$       497—498 500 530
Asymptotically $\mathbb{C}(1, i)$ , analogy with asymptotically flat      64n.5
Automorphism of number field, defined      29n.7
Axial-vector current      374 531
Baker — Campbell — Hausdorff formula      90n.4 102 105 110—111 111n.14 245 393n.14
Bargmann potentials      526
Bell inequalities      524n.19
Berry's phase      see "Phase geometric"
Bessel function of imaginary argument      137 137n.3 249n.4 311
Bessel function, spherical      167
big bang      532—533
Binding energies in atoms and nuclei      208
Birkhoff — von Neumann axioms      10—11
Bogoliubov transformation      533
Born approximation      174—175
Boson      238
Bound states      159—183 356
Bound states and potential component $V_{\beta}$       179n.7 526
Bound states and scattering theory      225—227 225n.1
Bound states, disappearance from spectrum      163—164
Bound states, energy shift      177
Bound states, stabilized by rest mass      164—165 171 177—179 208 258
Bound-state-associated scattering resonances (resolvent singularities)      127 159 163—164 175—179 183 356
Bracket, classical Poisson      528 536 539
Bracket, generalized or generalized Poisson      398—399 447—448 528 see "Operator-valued "Total
Bracket, generalized or generalized Poisson of general trace functional with total trace Hamiltonian      447
Bracket, generalized or generalized Poisson of two conserved functionals is conserved      448
Bracket, generalized or generalized Poisson, algebra of total trace generators under      437 452
Bracket, generalized or generalized Poisson, antisymmetric in arguments      447
Bracket, generalized or generalized Poisson, Jacobi identity for      447—448
Bracket, generalized or generalized Poisson, Jacobi identity invalid in octonionic Hilbert space      539—540
Bracket, generalized or generalized Poisson, Jacobi identity proved      535—540
Bracket, generalized or generalized Poisson, Leibnitz product rule for      448
BRST transformation      529 529n.25
c-number      33 33n.9 374 384 384n.7 399 442 475 481—482 532 536
Causality      182 499 see Kramers
Center of mass, coordinate defined      81 238 255—256
Center of mass, N-pair model      245—246
Center of mass, separation of motion      81 233
Center of mass, separation of motion, complex cluster      255—256 261
Center of mass, separation of motion, quaternionic cluster      256 261
Center of mass, separation of motion, three-body      240—242
Center of mass, separation of motion, two-body      239—240
Chaos      524n.19 526n.23
Charge conjugation invariance      381—382 384—385 482—483
Charge conjugation invariance, $\mathcal{C}$ defined as behavior under sign reversal of imaginary terms in symplectic components      384—385
Charge conjugation invariance, analysis assumes symplectic components of fermions anticommute      384
Charge conjugation invariance, notation C in quaternionic quantum mechanics, and C in complex quantum mechanics and phenomenology      217n.14
Charge conjugation invariance, real phase factors in      385 482—483
Charge conjugation invariance, relationship between complex and quaternionic definitions      482n.42 529
Charge conjugation invariance, second, quaternionic definition for G = G'      385
Charge conjugation operation, used to relate antiparticle to particle states      502
Chiral components of fermion      387
Chiral components of fermion, coupled by quaternionic gauge interactions      388 508
Chiral projections in Majorana representation, do not commute with covariant derivative $D_{\mu}$       387—388
Classical state      4—5
Classical system      4—5 519n.14 520
Classical system, Feynman formulation (versus quantum)      4—5
Classical system, Markov property of      5n.3
Cluster decomposition property      233 240 245—254 293—299 518—519 527 see
Cocycle (2-cocycle)      100 100n.9 see projective
Coherent state      5 111n.14 528
Color degree of freedom      503 532
Commutativity of complex phase shifts      516—517 517n.12
Commutator in Foldy — Wouthuysen method      325
Commutator in Galilean invariance analysis      90n.4
Commutator in Schur's Lemma derivation      103
Commutator in uncertainty principle derivation      73
Commutator in virial theorem derivation      108
Commutator of annihilation and creation operators      273
Commutator of annihilation and creation operators in Heisenberg picture      211—212
Commutator of annihilation and creation operators in interaction picture      211
Commutator of annihilation and creation operators in Schroedinger picture      210 411—412 416 422
Commutator of conformal boost and dilatation with energy      392
Commutator of conserved observables with Hamiltonian      269n.11
Commutator of covariant derivatives      364—365
Commutator of covariant derivatives with chiral projectors      388 531
Commutator of fermionic supersymmetry generators with energy      392
Commutator of permutation operator with identical particle Hamiltonian      238
Commutator of quasiparticle operators      280 286 424
Commutator of S-matrix with free particle Hamiltonian      228
Commutator of S-matrix with translation generator      267
Commutator of symmetry generators with Hamiltonian      74 238
Commutator, algebra of group generators      434—436
Commutator, algebra of Poincare generators      389—390 398
Commutator, canonical, in operator gauge invariant theory, as constraint      459 512
Commutator, canonical, in operator gauge invariant theory, biunitary transform covariant      529
Commutator, canonical, in operator gauge invariant theory, nonlocal form and Bell inequalities      524n.19
Commutator, conditions for independent particle behavior      243—245
Commutator, conditions for reduction of Hamiltonian modulus to complex self-adjoint form      125
Commutator, corrections to quaternionic path integral      110—111
Commutator, notation defined      13
Commutator, parafermion-like, of quasiparticles      504
Complete orthonormal set in complex inner product      43—45 272n.1 409—410
Complete orthonormal set in quaternionic inner product      42—43 409—410
Complete orthonormal set in real inner product      45
Complete orthonormal set, relationship of complex to quaternionic      40—44 111n.14 230 212n.1 407—410 439—440
Completeness relation      26 28 180 248 538
Completeness relation and unitarity deficiency      226—227 265
Completeness relation for energy eigenstates, applied to optical potential      128—129
Completeness relation for full scattering states and bound states      226 265
Completeness relation for in and out scattering states      223—224
Completeness relation for Klein — Gordon equation      309—310
Completeness relation, fails in octonionic Hilbert space      49—50 51n.20
Complex analyticity      18 179—183
Complex cluster      255 287 532—533
Complex conjugate or conjugation      7 13 16n.16 43 47n.19
Complex conjugate or conjugation and Frobenius — Schur classification      437
Complex conjugate or conjugation as time reversal operator      47n.19 49 112 174
Complex conjugate or conjugation, notation * used for      13 43
Complex form, transformation of quaternionic matrices to      360—361 396
Complex free particle wave equations      397
Complex function theory      526
Complex Hilbert space      see "Hilbert space i)$"/>
Complex number      3n.1 289n.8 see
Complex quantum field theory      see "Quantum field theory"
Complex quantum mechanics      3 19 20n.3 24 26—27 29—30 33—34 37 46—49 53 58 69 75—77 89 98 100n.9 107—108 110—111 111n.14 161 163—165 167 173 179 183 197 217n.14 233 238 255n.6 358 405
Complex quantum mechanics and indefinite inner product for Klein — Gordon equation      306
Complex quantum mechanics as effective dynamics for observed physics      498—499
Complex quantum mechanics of one degree of freedom      442n.21
Complex quantum mechanics, bound states in      168
Complex quantum mechanics, canonical commutators in, are constraints in generalized quantum dynamics      399 455—475 512
Complex quantum mechanics, classical limit of      528
Complex quantum mechanics, cluster decomposition property in      250 296 298
Complex quantum mechanics, conserved observables in      269n.11
Complex quantum mechanics, conversion of anti-self-adjoint to self-adjoint operators in      76
Complex quantum mechanics, decaying state problem in      203
Complex quantum mechanics, Ehrenfest and virial theorems      352—353
Complex quantum mechanics, embedding of real quantum mechanics in      47—49
Complex quantum mechanics, existence of multilinear tensor product and reduction to independent one-body problems      242—243 271 523
Complex quantum mechanics, existence of multilinear tensor product and reduction to independent one-body problems, second quantized treatment      283—284 286—287
Complex quantum mechanics, external potential problem obtained from multiparticle      233
Complex quantum mechanics, Feynman path integral in      110 353—354 489 531
Complex quantum mechanics, Foldy — Wouthuysen reduction of relativistic equation yields      303 322 342
Complex quantum mechanics, Galilean invariance in      89 94
Complex quantum mechanics, generalized quantum dynamics with total trace Lagrangian applied to      442 455—475
Complex quantum mechanics, harmonic oscillator in      123
Complex quantum mechanics, identical particles in      238 270—271
Complex quantum mechanics, left-acting operator I commutes with all operators in      34 456 456n.27
Complex quantum mechanics, multilinear tensor product for      243—244
Complex quantum mechanics, nonlinear corrections to      442n.21 524—525 530
Complex quantum mechanics, nonlinear corrections to, parameterization of      525 525n.20
Complex quantum mechanics, nonlinear corrections to, possible link to hierarchy problem      525
Complex quantum mechanics, perturbation theory for      131

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