Geometric Formulations of Quantum Theory| Geometric Formulations of Quantum Theory |
In General > s.a. geometric quantization;
relativistic quantum mechanics; Superposition;
symplectic structure.
* Idea: Quantum states are
rays in Hilbert space, and one casts the main postulates of the theory in
terms of two geometric structures on phase space, a symplectic structure
and a Riemannian metric; Not to be confused with ideas on a geometric origin
of quantum mechanics.
* Formalisms: The Rivier-Margenau-Hill
and Born-Jordan-Shankara phase space ones are equivalent to the standard operator one.
@ Books: Giachetta et al 05 [geometry + algebraic
topology, see gq/04];
Bengtsson & Życzkowski 06;
Giachetta et al 10.
@ General references:
Castro FP(92) [Weyl geometry];
Varadarajan IJTP(93) [rev];
Klauder & Maraner AP(97)qp/96;
Ono PLA(97) [S1 bundle];
Wheeler ht/97-GRF;
Faraggi & Matone PLA(98)ht,
PLB(98)ht,
PLB(99)ht/98,
IJMPA(00)ht/98;
Iliev JPA(98)qp,
JPA(01)qp/98,
JPA(01)qp/98,
JPA(01)qp/98 [fiber bundles];
Brody & Hughston PRS(98),
JGP(01)qp/99;
Ziegler & Fuchssteiner qp/02;
Patwardhan qp/02 [on general spaces];
Bracken IJTP(03) [operator version of Poisson brackets];
Cariñena et al TMP(07)mp;
Clemente-Gallado & Marmo IJGMP(08);
Novello et al IJGMP(11)-a0901 [modification of euclidean structure of space];
Ercolessi & Morandi IJGMP(12);
Heydari a1503 [and applications];
Ciaglia et al in(19)-a1903 [rev];
Schwarz Sigma(20)-a1906;
> s.a. quantum mechanics [geometric aspects].
@ Metric on phase space, space of states:
Alicki & Klauder JPA(96);
Klauder qp/96,
LNP(99)qp/98;
Klauder qp/01;
Mehrafarin TMP(06) [on state space];
Hübschmann in(06)mp [holomorphic quantization on stratified Kähler spaces, overview];
Facchi et al PLA(10) [and Fisher information];
D'Amico et al PRL(12)-a1102 [metric on Fock space];
Cattaruzza et al AP(19)-a1811;
Schwarz a2102 [in terms of Jordan algebras];
> s.a. quantum mechanics.
@ For pilot-wave approaches:
Koch a0901 [geometrical dual];
Hurley & Vandyck IJGMP(13);
Tavernelli AP(16)-a1510 [de Broglie-Bohm theory, curvature induced by the quantum potential].
@ Formulations: Kryukov FP(06) [Hilbert manifolds and functional relativity];
Bertram IJTP(08)-a0801 [Jordan geometry];
Grigorescu JGSP(11)-a0905 [and configuration-space discretization];
Reginatto JPCS(14)-a1312 [from information geometry];
Herczeg & Waldron PLB(18)-a1709,
Herczeg et al a1805 [contact geometry];
Almalki & Kisil a1903 [coherent state transform and geometric dynamics];
Schwarz a1906;
Beggs & Majid a1912
[Schrödinger's equation from quantum geodesics].
@ Special topics: Sardanashvily qp/00 [evolution as parallel transport];
Marmo & Volkert PS(10)-a1006 [transformations and dynamics, separability and entanglement];
Karamatskou & Kleinert a1102 [quantum Maupertuis principle];
Molitor IJGMP(12) [statistical basis];
Clemente-Gallardo & Marmo IJGMP(15)-a1505 [Klein's program and groups of transformations];
Artacho & O'Regan PRB(17)-a1608 [with varying external parameters];
Elgressy & Horwitz a1704
[stability of trajectories as expectation values of quantum operators];
> s.a. Born-Jordan quantization; Ehrenfest
Dynamics; mixed states; quantum states.
Related Geometric Aspects > s.a. clifford algebra;
deformation quantization; euclidean geometry.
@ General references: Komar GRG(76);
Geroch ln;
Kibble CMP(79);
Bernard & Choquet-Bruhat 88;
Rzewuski RPMP(88);
Cirelli et al JMP(90),
JMP(90);
Collas PLA(90);
Dubrovin et al MPLA(90) [Kähler structures];
Anandan FP(91);
Dimakis & Müller-Hoissen JPA(92) [non-commutative symplectic geometry];
Schilling PhD(96);
Ashtekar & Schilling gq/97;
Isidro JPA(02)qp/01;
Cariñena et al a0707-ln [and quantum-classical transition];
Varadarajan 07;
Cariñena et al AIP(09)-a1209 [geometrical description of algebraic structures];
Grabowski et al JPA(18)-a1711 [quantum dynamics in infinite dimension, using Tulczyjew triples];
> s.a. formulations; geometric quantization.
@ Geometry of parameter space: Álvarez-Jiménez et al AdP-a1909 [the quantum metric tensor and its classical counterpart];
> s.a. geometric phase.
@ Mathematical references: Todorov BulgJP(12)-a1206 [including geometric and deformation quantization].
@ Obstructions: Gotay & Grundling RPMP(97)qp/96,
PAMS(00)dg/97,
et al JNS(96)dg;
Gotay in(00)mp/98;
> s.a. geometric quantization.
@ Boundary formulations:
Krtouš in(02)gq/03;
Oeckl FP(13)-a1212 [positive formalism];
> s.a. approaches and formulations
of quantum field theory.
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