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Landsman N.P. — Mathematical topics between classical and quantum mechanics
Landsman N.P. — Mathematical topics between classical and quantum mechanics

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Название: Mathematical topics between classical and quantum mechanics

Автор: Landsman N.P.

Аннотация:

This monograph draws on two traditions: the algebraic formulation of quantum mechanics and quantum field theory, and the geometric theory of classical mechanics. These are combined in a unified treatment of the theory of Poisson algebras of observables and pure state spaces with a transition probability. The theory of quantization and the classical limit is discussed from this perspective. A prototype of quantization comes from the analogy between the C*- algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid. The parallel between reduction of symplectic manifolds in classical mechanics and induced representations of groups and C*- algebras in quantum mechanics plays an equally important role. Examples from physics include constrained quantization, curved spaces, magnetic monopoles, gauge theories, massless particles, and $theta$- vacua. The book should be accessible to mathematicians with some prior knowledge of classical and quantum mechanics, to mathematical physicists and to theoretical physicists who have some background in functional analysis.


Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 529

Добавлена в каталог: 18.07.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$C^*$-algebra      2 39
$C^*$-algebra of a Lie group      204
$C^*$-algebra of a Lie groupoid      285
$C^*$-algebra of compact operators      56
$C^*$-dynamical system      279
$R^*$-algebra      434
$\alpha$-density      273
$\Gamma$-representation      197
$\mathfrak g$-action      179
$\pi_\chi$-positive      362
$\sigma$-weak topology      59
$\theta$-angle      32 36^
*-algebra      39
*-algebra of bounded operators      39
*-algebra of finite-rank operators      56
1-cocycle on G      185
2-cocycle on $\mathfrak g^*$      181
AC-lattices      100
Action, $C^*$-algebra      23 291
Action, *-algebra      279
Action, algebroid      23 293
Action, groupoid      22 271
Action, Poisson algebra      298
Adjoint action      184
Adjoint bundle      235
Adjointable operator      28 356
Affine connection      154
Affine function      51
Affine geometry      154
Affinely parametrized geodesic      454
Aharonov — Bohm effect      36 430
Algebra      37
Amenable      205
Anchor      23 292
Anisotropic form      94
Annihilation operator      136
Antiunitary operator      441
Approximate unit      48
Associated bundle      17 226
Associated vector bundle      227
Associator identity      2 38
atom      92
Atomic lattice      92
Atomistic lattice      98
Automorphic action      139
Axioms for the pure state space of a $C^*$-algebra      104
Banach algebra      39
Bargmann — Fock space      449
Base of a bundle      224
Base of a groupoid      22 269
Based, gauge transformations      416
Based, loops      416
Basis of a transition probability space      7 81
Berezin quantization      10 114
Berezin quantization on coadjoint orbits      219
Berezin quantization on flat space      133
Berezin transform      447
Berezin — Toeplitz quantization      448
Bergman kernel      135 448
Bi-invariant metric      212
Bicommutant      58
Birkhoff's exchange axiom      442
Bochner integral      448
Bosonic Fock space      34 135
Bounded symmetric domain      447
Bundle      224
Bundle automorphism      235
Bundle of a-densities      273
c-extension      188
c-representation      197
c-unitary dual      205
Cameron — Martin formula, general      411
Cameron — Martin theorem, general      412
Canonical commutation relations      129
Canonical commutation relations in Weyl form      407
Canonical symplectic form      68
Casimir element      258
CCR algebra      407
Center of a lattice      94
Center of a Poisson algebra      318
Center of a von Neumann algebra      60
Central extension      181 188
Chain      100
CHARACTER      215
Christoffel symbol      158
Classical action      151
Classical constrained systems      313
Classical covariance condition      299
Classical dual pair      320
Classical imprimitivity theorem      26 322
Classical limit      8
Classical limit on a Riemannian manifold      173
Classical limit on flat space      148
Classical propagator      151
Classical system of imprimitivity      24 299
Classical transitive imprimitivity theorem      27 332
Closed projection      442
Co-equivariant momentum map      185
Coadjoint action      15 127 184
Coadjoint orbit      194
Coadjoint orbit from covariant reduction      396
Coadjoint orbit of a nilpotent group      341
Coadjoint orbit of a semidirect product      343
Coadjoint orbit of the Poincare group      393
Cogeodesic flow      160
Coherent pure state quantization      122
Coherent states      122
Coisotropic representation      245
Coisotropic submanifold      25 315
Commutant      58
Commutation relations on a Riemannian manifold      170
Commutative C*-algebra      40
Compact Lie groups      215
Complete element of a Poisson algebra      67
Complete lattice      92
Complete Riemannian manifold      159
Completely integrable      69
Completely positive map      117
Completeness of a quantization      109
Complex connection      415
Complexification      418
Component of a transition probability space      80
Condition of the frontier      473
Conjugate representation      207
Conjugate space      364
Connection      18 227
Connection coefficients      154
Connection, 1-form      228
Connection, G-invariant      247
Constrained quantization      386
Constrained quantization, and vacuum angles      427
Constrained quantization, author's method      387
Constrained quantization, BRST method      477
Constrained quantization, Dirac's method      389
Constrained systems, classical      313
Constrained systems, quantum      386
Constraint      314
Continuous field of $C^*$-algebras      110
Continuous field of states      112
Continuous functional calculus      44
Continuous quantization      112
Contravariant symbol      447
Convex hull      62
Convolution on a Lie group      202
Convolution on a Lie groupoid      275
Cotangent bundle reduction      231
Covariance      411
Covariance condition      280
Covariant Berezin quantization      384
Covariant derivative      158 229
Covariant localization in configuration space      384
Covariant localization in phase space      384
Covariant momentum      239
Covariant representation      33
Covariant representations      399
Covariant symbol      447
Covering property      99
Creation operator      136
Crossed product      279
Crossed product, *-algebra      279
Crossed product, Poisson algebra      298
Curvature      228
Cut locus      159
Cyclic representation      53
Cyclic vector      53
Dimension of a transition probability space      81
Dirac condition      109
Dirac method      389
Dirac monopole      264
Dirac quantization condition      269
Direct integral      284
Distance in a Riemannian manifold      157
Distribution on a manifold      69
Distributive lattice      92
Division ring      441
Dominant weight      217
Double commutant theorem      58
Dual bundle      225
Egorov's theorem      452
Enveloping algebra      199
Equation of geodesic deviation      160
Equivalent multipliers      189
Equivalent projective representations of a Lie algebra      198
Equivalent projective representations of a Lie group      197
Equivalent pure states      65
Equivalent quantizations      109
Equivalent representations of a $C^*$-algebra      52
Equivalent representations of a Poisson algebra      76
Equivariant momentum map      15 185
Ergodic measure      382
Euclidean group      348
Exchange property      442
Exponential Hilbert space      135
Exponential mapping      155
Exponential vector      135
Extreme boundary      61
Extreme point      61
Face      95
Factor      441
Faithful state      54
Fell induction      363
Fermi representation      480
Fiber      224
Fiber product      224
First cohomology group of $\mathfrak g$      180
First cohomology group of G      186
First-class constraint      314
Fock representation      407
Frobenius reciprocity theorem      209
Frobenius theorem      69
Fubmi-Study symplectic form      75
Full dual pair      471
Full Hilbert $C^*$-module      30 364
Fundamental vector field      227
G-invariant connection      247
Gaerding subspace      459
Gauge, algebroid      293
Gauge, group      33 34 235 405
Gauge, groupoid      22 272
Gauge, invariance      403
Gauge-covariant      239 256
Gauss law      418
Gaussian measure      411
Gelfand transform in a $C^*$-algebra      3 62
Gelfand transform in a commutative Banach algebra      43
Generalized distribution      69
Generalized system of imprimitivity      383
Generator      183
Geodesic      155
Geodesic flow      155
Geodesic motion      13
Geodesically complete      155
Geodesically convex      158
Geometric quantization      446
GNS-construction      3 52
Groenewold-van Hove theorem      449
Group action      183
Group C-algebra      15 204
Groupoid      21 269
Groupoid $C^*$-algebra      288
Groupoid action      300
h-connection      246
Half-densities on Lie groupoids      273
Hall coherent states      35 426
Hamiltonian      15
Hamiltonian $\mathfrak g$-action      180
Hamiltonian curve      67
Hamiltonian flow      4 67
Hamiltonian group action      184
Hamiltonian on a Riemannian manifold      173
Hamiltonian vector field      3 66
Hamiltonian, Riemannian geometry      159
Heat equation      425
Heisenberg group      11 126
Helicity      32 395
Hermitian form      93
Hermitian functional      49
Highest weight vector      218
Hilbert $C^*$-module      28 354
Hilbert $\mathfrak B$-module      354
Hilbert bundle      225
Hilbert space of half-densities      273
Hilbert subspace      124
Hilbert — Schmidt norm      56
Hopf fibration      264
Horizontal 1-forms      231
Horizontal cotangent space      231
Horizontal curve      154
Horizontal lift      154 227
Horizontal subbundle      227
Horizontal subspace      154 227
Horizontal vectors      227
Howe dual pair      471
Husimi function      452
Ideal      47
Ideal, left      47
Ideal, right      47
Implemented automorphism      259
Imprimitivity theorem, classical      26 322
Imprimitivity theorem, for gauge groupoids      378
Imprimitivity theorem, quantum      30 368
inclusion      269
Incomplete vector field      67
Induced representation      19 253
Induction      25
Induction, Fell      363
Induction, in stages      375
Induction, in stages, Mackey      378
Induction, in stages, quantum Marsden — Wemstein      377
Induction, in stages, Rieffel      375
Induction, Mackey      208 256
Induction, Rieffel      28 29 358
Inequivalent quantizations      36 430
Infinitesimal isometry      172
Integrable, Poisson manifold      470
Integrable, representation of $C^\infty(\mathfrak g^*{\_}, \mathbb R)$      183
Integrable, representation of $\mathfrak g$      199
Integral coadjoint orbit      215
Invariant measure      283
inversion      269
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