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

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Ќазвание: Quaternionic Quantum Mechanics and Quantum Fields

јвтор: Adler S.L.


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.

язык: en

–убрика: ћеханика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1995

 оличество страниц: 586

ƒобавлена в каталог: 10.12.2009

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ѕредметный 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 422Ч424 426Ч427 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      85Ч86
$\mathbb{C}(1, I)$ (complex subalgebra of left-acting algebra), asymptopia hypothesis      497Ч498 518Ч519 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 70Ч72 75Ч76 78 80Ч81 115 117 119 122 132 141 148 152Ч156 159Ч160 166Ч167 171Ч174 196 207Ч209 230 233 242Ч243 245 247Ч248 251 253Ч260 262 270Ч272 278 287 289 291 296 314Ч315 329n.12 332 334 335n.14 340 340n.16 351 359Ч361 369 373Ч387 390Ч395 397Ч398 401 404Ч405 407 409Ч410 478 479n.39 527Ч530
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) and matrix representations of symmetry generators      68 75Ч77 80Ч81 84 141n.6 238
$\mathbb{C}(1, i)$ (complex subalgebra of quaternion) and nonrelativistic reduction of Klein Ч Gordon equation      321Ч322
$\mathbb{C}(1, i)$ (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 522Ч524
$\mathbf{U}$, unitary evolution operation      520 522Ч524
$\mathcal{CPT}$ 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      215Ч217 508Ч509 531
$\mathcal{CP}$ violation, phenomenological, Kobayashi Ч Maskawa form      216 518
$\mathcal{CP}$ violation, phenomenological, magnitude of      215Ч217
$\mathcal{CP}$ violation, phenomenological, milliweak      216
$\mathcal{CP}$ violation, phenomenological, superweak      216n.12
Abelian group      90Ч91 99Ч100
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      111Ч112
Action, invariance under $\mathcal{C}$, $\mathcal{P}$, $\mathcal{T}$      384Ч387
Action, principle for generalized dynamics      445Ч447
Action, real-valued, independent variations of      375
Adiabatic approximation      145Ч149 195n.2
Adiabatic approximation for zero energy state      148Ч149
Adiabatic approximation, effective expansion parameter for      147
Adiabatic switching      229
Adjoint defined for column vector      21Ч22
Adjoint defined for quaternion matrix      14
Adjoint of Grassmann quaternion      16
Adjoint of operator      22Ч23 51
Adjoint of product      15Ч16
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.16Ч17
Algebra over reals      5Ч7
Algebra, "charge" for complex generator matrices      436
Algebra, "charge" for quaternionic generator matrices      434Ч437 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 434Ч435 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      8Ч9
Algebra, four-vector extension of quaternion      513 514n.11
Algebra, Grassmann      16
Algebra, Jordan      10Ч11
Algebra, Jordan, exceptional      11
Algebra, Jordan, infinite-dimensional      11
Algebra, Jordan, special      11
Algebra, left-acting      23Ч24 33 38Ч39 100n.10 see
Algebra, Lie, commutator      435Ч436
Algebra, Lie, generalized bracket      437 452
Algebra, multiquaternion      272n.1 479Ч480 511 530
Algebra, nonassociative      8 20 49Ч52 540
Algebra, noncommutative      7Ч8 11 20n.3 55 244 258
Algebra, octonion      7Ч8 11 49Ч52
Algebra, Poincare      361 388Ч398 429 489 530 see
Algebra, Poincare, nonzero energy quaternionic representations transformable to complex form      361Ч362 388Ч398 499 503 526 532
Algebra, Poincare, relation to locality      362 398
Algebra, quaternion      7 11Ч12 114
Algebra, quaternion, associativity of      12
Algebra, quaternion, automorphism of      16Ч17 30 515
Algebra, quaternion, complex $2\times 2$ matrix representation      495
Algebra, quaternion, real matrix representations      10n.9
Algebra, right-acting      24Ч25 34 38Ч39 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      179Ч183 358 499 526
Analyticity properties of thermal Green's functions      291Ч293
Analyticity properties upper half plane, in decaying state theory      202Ч206
Angular momentum, algebra      see "Algebra"
Angular momentum, analog of left- and right-ordered Fourier transform      84n.12
Angular momentum, anti-self-adjoint operator for      64Ч66 80 118Ч119
Angular momentum, anti-self-adjoint operator for, action in coordinate representation      64
Angular momentum, anti-self-adjoint operator for, action of time reversal on      118Ч119
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      66Ч68 351 408n.4 512
Angular momentum, complex linear operators, action of time reversal on      118Ч119
Angular momentum, ladder operators      66Ч68
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      65Ч66
Angular momentum, representation, left-acting algebra I, J, K in      65Ч66
Angular momentum, representation, matrices      68
Angular momentum, representation, matrices, right action      68 80Ч81 503
Angular momentum, self-adjoint operator for      66Ч68 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 Ч 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 285Ч286
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      539Ч540
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      63Ч64 159 160 172n.5 181 198 213 512
Asymptotic scattering states, dynamics of      113 497Ч499
Asymptotic scattering states, momentum operator definitions agree on      63Ч64
Asymptotic scattering states, structure, in multiparticle, multichannel scattering      254Ч262 see classification
Asymptotically $\mathbb{C}(1, i)$      497Ч498 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 Ч Campbell Ч Hausdorff formula      90n.4 102 105 110Ч111 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      532Ч533
Binding energies in atoms and nuclei      208
Birkhoff Ч von Neumann axioms      10Ч11
Bogoliubov transformation      533
Born approximation      174Ч175
Boson      238
Bound states      159Ч183 356
Bound states and potential component $V_{\beta}$      179n.7 526
Bound states and scattering theory      225Ч227 225n.1
Bound states, disappearance from spectrum      163Ч164
Bound states, energy shift      177
Bound states, stabilized by rest mass      164Ч165 171 177Ч179 208 258
Bound-state-associated scattering resonances (resolvent singularities)      127 159 163Ч164 175Ч179 183 356
Bracket, classical Poisson      528 536 539
Bracket, generalized or generalized Poisson      398Ч399 447Ч448 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      447Ч448
Bracket, generalized or generalized Poisson, Jacobi identity invalid in octonionic Hilbert space      539Ч540
Bracket, generalized or generalized Poisson, Jacobi identity proved      535Ч540
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 481Ч482 532 536
Causality      182 499 see Kramers
Center of mass, coordinate defined      81 238 255Ч256
Center of mass, N-pair model      245Ч246
Center of mass, separation of motion      81 233
Center of mass, separation of motion, complex cluster      255Ч256 261
Center of mass, separation of motion, quaternionic cluster      256 261
Center of mass, separation of motion, three-body      240Ч242
Center of mass, separation of motion, two-body      239Ч240
Chaos      524n.19 526n.23
Charge conjugation invariance      381Ч382 384Ч385 482Ч483
Charge conjugation invariance, $\mathcal{C}$ defined as behavior under sign reversal of imaginary terms in symplectic components      384Ч385
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 482Ч483
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 $D_{\mu}$      387Ч388
Classical state      4Ч5
Classical system      4Ч5 519n.14 520
Classical system, Feynman formulation (versus quantum)      4Ч5
Classical system, Markov property of      5n.3
Cluster decomposition property      233 240 245Ч254 293Ч299 518Ч519 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      516Ч517 517n.12
Commutator in Foldy Ч 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      211Ч212
Commutator of annihilation and creation operators in interaction picture      211
Commutator of annihilation and creation operators in Schroedinger picture      210 411Ч412 416 422
Commutator of conformal boost and dilatation with energy      392
Commutator of conserved observables with Hamiltonian      269n.11
Commutator of covariant derivatives      364Ч365
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      434Ч436
Commutator, algebra of Poincare generators      389Ч390 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      243Ч245
Commutator, conditions for reduction of Hamiltonian modulus to complex self-adjoint form      125
Commutator, corrections to quaternionic path integral      110Ч111
Commutator, notation defined      13
Commutator, parafermion-like, of quasiparticles      504
Complete orthonormal set in complex inner product      43Ч45 272n.1 409Ч410
Complete orthonormal set in quaternionic inner product      42Ч43 409Ч410
Complete orthonormal set in real inner product      45
Complete orthonormal set, relationship of complex to quaternionic      40Ч44 111n.14 230 212n.1 407Ч410 439Ч440
Completeness relation      26 28 180 248 538
Completeness relation and unitarity deficiency      226Ч227 265
Completeness relation for energy eigenstates, applied to optical potential      128Ч129
Completeness relation for full scattering states and bound states      226 265
Completeness relation for in and out scattering states      223Ч224
Completeness relation for Klein Ч Gordon equation      309Ч310
Completeness relation, fails in octonionic Hilbert space      49Ч50 51n.20
Complex analyticity      18 179Ч183
Complex cluster      255 287 532Ч533
Complex conjugate or conjugation      7 13 16n.16 43 47n.19
Complex conjugate or conjugation and Frobenius Ч 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      360Ч361 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 26Ч27 29Ч30 33Ч34 37 46Ч49 53 58 69 75Ч77 89 98 100n.9 107Ч108 110Ч111 111n.14 161 163Ч165 167 173 179 183 197 217n.14 233 238 255n.6 358 405
Complex quantum mechanics and indefinite inner product for Klein Ч Gordon equation      306
Complex quantum mechanics as effective dynamics for observed physics      498Ч499
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 455Ч475 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      352Ч353
Complex quantum mechanics, embedding of real quantum mechanics in      47Ч49
Complex quantum mechanics, existence of multilinear tensor product and reduction to independent one-body problems      242Ч243 271 523
Complex quantum mechanics, existence of multilinear tensor product and reduction to independent one-body problems, second quantized treatment      283Ч284 286Ч287
Complex quantum mechanics, external potential problem obtained from multiparticle      233
Complex quantum mechanics, Feynman path integral in      110 353Ч354 489 531
Complex quantum mechanics, Foldy Ч 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 455Ч475
Complex quantum mechanics, harmonic oscillator in      123
Complex quantum mechanics, identical particles in      238 270Ч271
Complex quantum mechanics, left-acting operator I commutes with all operators in      34 456 456n.27
Complex quantum mechanics, multilinear tensor product for      243Ч244
Complex quantum mechanics, nonlinear corrections to      442n.21 524Ч525 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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