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];
> s.a. quantum mechanics [geometric aspects].
@ Metric on phase space, state space: Alicki & Klauder JPA(96);
Klauder qp/96,
LNP(99)qp/98;
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]; > 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 a1709,
Herczeg et al a1805 [contact geometry].
@ 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; 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;
Klauder qp/01 [metric
on phase space]; 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 a1711 [quantum dynamics in infinite dimension, using Tulczyjew triples]; > s.a. formulations; geometric quantization.
@ 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 1 jun 2018