Formulations of Quantum Theory  

In General > s.a. path integrals; representations; stochastic quantum mechanics; wigner function [phase space approach].
* Approaches: The main ones are canonical quantization (including geometric and group quantization, and for quantum field theory also covariant quantization), path integral, and stochastic quantization; They are known to be equivalent for some classes of systems.
@ Comparisons: Mansfield AP(87) [for strings]; Tuynman JMP(87); Fukutaka & Kashiwa AP(88); Panfil MPLA(89); Shiekh CJP(90); Landsman & Linden NPB(91); De Jonghe PLB(93); Rédei SHPMP(96) [von Neumann's point of view]; Styer et al AJP(02); Ali & Englis RVMP(05)mp/04 [rev]; Sternheimer LMP(05) [quantization as functor and deformation].
@ Star quantization: Alcalde JMP(90); Hakioglu & Dragt JPA(01)qp [Moyal-Lie]; > s.a. deformation.
@ Hydrodynamic formalism: Madelung ZP(26); Takabayasi PTP(53); Harvey PR(66); Jánossy FP(73), FP(74), FP(76); Sonego FP(91); Mita AJP(01); Chavoya-Aceves qp/02-wd; Holland AP(05)qp/04 [particle/wave pictures].
@ Algebraic: Sudarshan et al AIHP(88); Rieffel qp/97-in [operator algebra]; Slavnov qp/01, qp/04.
@ Categorification: Schlesinger JMP(99); Morton TAC(06)m.QA [combinatorial model for harmonic oscillator].
@ Related topics: Haag CMP(96); Olavo qp/96; Gray qp/97, Sutherland FP(98) [density formalism]; Klauder 00; Watson & Klauder JMP(00)qp [affine variables]; Corbett & Durt qp/02 [ito quantum real numbers]; Krtous in(02)gq/03 [boundary quantum mechanics]; Anderson & Wheeler IJGMP(06)ht/04 [biconformal spaces]; Singh gq/04 [without spacetime, non-commutative Hamilton-Jacobi]; Gozzi & Mauro qp/06-in [dimensional reduction of Koopman-von Neumann]; > s.a. hilbert space, operator theory.

Geometric Point of View > 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 & Zyczkowski 06 [r JPA(08)#1].
@ 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; 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 a0801 [Jordan geometry]; > s.a. quantum mechanics [geometric aspects].
@ Metric on phase space, state space: Alicki & Klauder JPA(96); Klauder qp/96, qp/98-in; Mehrafarin TMP(06) [on state space]; Hübschmann mp/06-in [holomorphic quantization on stratified Kähler spaces, overview]; > s.a. quantum mechanics.
@ Evolution as parallel transport: Sardanashvily qp/00; > s.a. quantum states.

Other Approaches and Concepts > s.a. hamilton-jacobi; hilbert space [rigged]; logic; quantum states and systems.
* Ambiguities: May arise because of different choices of Lagrangians, operator orderings, representations, complex structures...
@ Ambiguities, surprises: Redmount et al gq/99; Gieres RPP(00)qp/99; Cislo & Lopuszanski JMP(01)mp/00 [1+1 sho with different L's]; de Souza Dutra JPA(06)-a0705 [orderings and representations]; > s.a. duality, systems.
@ As an evolution problem: Yajima CMP(87) [solutions of initial value problem]; Busch & Lahti FP(89) [past and future of a system]; Gergely AP(02)ht/03 [Hamiltonian form]; > s.a. schrödinger equation.
@ Lagrangian: Dyson AJP(90), Hojman & Shepley JMP(91) [need]; Acatrinei JPA(04)ht/02 [examples without]; Sharan & Chingangbam qp/03 [as connection 1-form].
@ Linearity: Jordan PRA(06)qp/05; Holman qp/06 [assessment of arguments]; Jordan qp/07-in; Ercolessi et al IJMPA(07)-a0706 [alternative linear structures on TQ].
@ With gauge freedom: Wawrzycki CMP(04)mp/03, mp/03-in [covariance]; Isidro & de Gosson MPLA(07)qp/06 [Abelian gerbe over phase space]; > s.a. gauge.
@ Histories-based: Larsson ht/04 [histories phase space, harmonic oscillator and free scalar]; Sorkin JPA(07)qp/06 [quantum measure].
@ Stochastic: Comisar PR(65) [as Brownian motion]; de la Peña-Auerbach JMP(71) [with spin]; Guerra & Marra PRD(84); Garbaczewski PRD(86) [H atom]; Garbaczewski & Vigier PRA(92); Gillespie PRA(94) [argument against Markov process]; Garbaczewski & Olkiewicz PRA(95) [argument for; + comments]; > s.a. stochastic process and quantization.
@ From equations of motion: Ho et al PRL(07) [and model phase transition]; Kochan ht/07.
@ From classical ensemble: Parwani JPA(05) [using uncertainty measure]; Hegseth a0704 [using imperfect information]; > s.a. foundations, Rokhsar-Kivelson.
@ Topological quantization: Nettel et al a0801 [based on Maupertuis' formalism for classical mechanics]; > s.a. quantum oscillators.
@ 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.
@ Related topics: Zabey FP(75) [reconstruction theorems]; Gudder IJTP(92) [ito measurement and influence function]; Sudarshan PRA(94) [composite and unstable systems, scattering theory]; Ni qp/98 [i and non-commutativity]; Caticha FP(00)qp/98 [inner product and histories]; Coecke qp/05-ln [with picture calculus]; Sergi qp/05 [non-Hamiltonian]; Mohrhoff qp/06 [from stability of matter]; Hewitt-Horsman & Vedral NJP(07) [Deutsch-Hayden approach].
> Approaches: see canonical quantization; modified versions; quantum computing; quantum theory [books].
> Related concepts: see axioms; entropy; locality; matrix; Momentum; probability; topology; Unitarity.
> Techniques: see computational physics; green functions; Perturbation Methods; symmetry [including reduction].

Application to Other Fields
@ References: Aerts & Czachor JPA(04)qp/03 [quantitative linguistics].

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