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]; Kryukov FP(06) [Hilbert manifolds and functional relativity]; Cariñena et al TMP(07)mp; Bertram IJTP(08)-a0801 [Jordan geometry]; Clemente-Gallado & Marmo IJGMP(08); Grigorescu JGSP(11)-a0905 [and configuration-space discretization]; Novello et al IJGMP(11)-a0901 [modification of euclidean structure of space]; Ercolessi & Morandi IJGMP(12); Reginatto JPCS(14)-a1312 [from information geometry]; 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].
@ 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]; > 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 pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 14 apr 2017