Formulations of Quantum Theory  

In General > s.a. representations; stochastic quantum mechanics; phase space approach and wigner functions.
* 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.
* Some alternatives: In the canonical formulation, one can use different types of Hilbert spaces or different representations of the observable algebra; And one can use non-canonical formulations of quantum theory; > see modified formulations.
@ 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)mar; Ali & Englis RVMP(05)mp/04 [rev]; Sternheimer LMP(05) [quantization as functor and deformation]; David a1211-ln; Heunen a1412 [classical reformulation].
@ Star quantization: Alcalde JMP(90); Hakioglu & Dragt JPA(01)qp [Moyal-Lie]; > s.a. deformation quantization.
@ Hydrodynamic formalism: Madelung ZP(26); Takabayasi PTP(53); Jánossy FP(73), FP(74), FP(76); Sonego FP(91); Mita AJP(01)apr; Chavoya-Aceves qp/02-wd; Holland AP(05)qp/04 [particle/wave methods]; Spera a0902; Hushwater a1005; Chiarelli a1205 [including system with local noise]; Kodama & Koide a1412 [and stochastic quantization]; Sanz FP(15)-a1501 [role of the phase]; Renziehausen & Barth a1806 [Bohmian mechanics]; > s.a. interference; interpretations; pilot wave.
@ Complex formulations: Nielsen & Ninomiya in hp/06, Nagao & Nielsen PTEP(13)-a1205 [future-included / not excluded]; > s.a. modified formulations [non-Hermitian]; quantum oscillators.
@ Other formulations: Ralston a1203 [without Planck's constant]; Girotti 13 [functional formulation]; Brunet a1305 [without the notion of state]; Wallace a1604 [without projection postulate or eigenvalue-eigenvector link]; Castellani IJQI-a1810 [history operator]; Gouba a1912-ln [recent trend].
@ Related topics: Haag CMP(96); Olavo qp/96; Gray qp/97, Sutherland FP(98) [density formalism]; Corbett & Durt qp/02, SHPMP(09) [in terms of non-standard, quantum real numbers]; Anderson & Wheeler IJGMP(06)ht/04 [biconformal spaces]; Singh gq/04 [without spacetime, non-commutative Hamilton-Jacobi]; Gozzi & Mauro AIP(06)qp [dimensional reduction of Koopman-von Neumann]; Coecke CP(10)-a0908 [alternative to Hilbert-space formalism]; Abramsky Syn-a0910, a0910 [physical systems as Chu spaces]; Aerts & Sozzo LNCS(12)-a1204 [Quantum Model Theory (QMod)]; Weinberg PRA(14)-a1405 [based on density matrices]; Hickey & Gour JPA(18)-a1801 [measures of imaginarity]; > s.a. quantum systems [non-commutative variables]; quaternions.
> Canonical and related approaches: see Affine and Algebraic Quantum Theory; canonical quantization; hilbert space [including rigged]; operator theory.
> Other main approaches: see histories-based; modified versions [including non-Hamiltonian]; path integrals; quantum computing; quantum theory [books].

Techniques and Related Concepts > s.a. clifford algebra; geometric aspects; observers; quantum states and systems.
* Information theory: 2015, Two approaches have been pursued with the goal of understanding quantum theory in information-theoretic terms, the "device-independent" framework characterizing quantum correlations in terms of conditional probability distributions, and the characterization of quantum theory among the allowed "general probabilistic theories".
* Ambiguities: Ambiguities in quantization 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; Cisło & Łopuszański JMP(01)mp/00 [1+1 sho with different Ls]; de Souza Dutra JPA(06)-a0705 [orderings and representations]; > s.a. duality; quantum 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)mar, Hojman & Shepley JMP(91) [need]; Acatrinei JPA(04)ht/02 [examples without]; Sharan & Chingangbam qp/03 [as connection 1-form]; Deriglazov PLA(09) [singular Lagrangian]; Wharton a1301.
@ Linearity: Jordan PRA(06)qp/05; Holman qp/06 [assessment of arguments]; Jordan JPCS(09)qp/07; Ercolessi et al IJMPA(07)-a0706 [alternative linear structures on TQ]; Jordan PRA(10)-a1002 [comment on tests]; Bassi et al a1212-FQXi [is it an exact principle?]; > s.a. non-linear quantum mechanics.
@ With gauge freedom: Wawrzycki CMP(04)mp/03, mp/03-conf [covariance]; Isidro & de Gosson MPLA(07)qp/06 [Abelian gerbe over phase space]; > s.a. gauge.
@ 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]; Fernández de Córdoba et al a1304 [thermodynamical approach, gravitationally-induced irreversibility]; > s.a. stochastic process and quantization.
@ From equations of motion: Ho et al PRL(07) [and model phase transition]; Kochan IJGMP(10)-ht/07.
@ From classical ensemble: Parwani JPA(05) [using uncertainty measure]; Hegseth a0704 [using imperfect information]; > s.a. quantum foundations; Rokhsar-Kivelson Point.
@ Topological quantization: Nettel et al RPMP(09)-a0801 [based on Maupertuis' formalism for classical mechanics]; Arciniega et al JGSP(12)-a1201 [for a free massive bosonic field]; > s.a. quantum oscillators.
@ Integral quantization: Tao CTP(11)-a1010 [and Berry phase]; Bergeron & Gazeau AP(14) [two basic examples].
@ Related topics: Zabey FP(75) [reconstruction theorems]; Gudder IJTP(92) [in terms of 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 AIP-qp/05-ln [with picture calculus]; Mohrhoff IJQI(09)qp/06 [from stability of matter]; Hewitt-Horsman & Vedral NJP(07) [Deutsch-Hayden approach]; Müller & Ududec PRL(12)-a1110 [computational reversibility, self-duality, and non-locality]; Hardy PTRS(12)-a1201 [operator tensor formulation]; Vacca & Zambelli PRD(12)-a1208 [effective Hamiltonian action]; Ziaeepour a1305 [in symmetry language]; Brooker a1308 [theory of discrete extension]; Feldmann a1312 [information theory]; Baez & Pollard a1311 [statistical analogy, quantropy]; Palmer a1502, a1605 [invariant set theory]; Chiribella & Yuan I&C(16)-a1504 [information-theoretic approaches]; Karuvade et al a2101 [a changing inner product].
> Related concepts: see axioms; causality; entropy; locality; momentum; origin of quantum theory; probability in physics; topology; Unitarity.
> Techniques: see categories in physics; computational physics; graph theory in physics; green functions; Groupoid [Schwinger's picture]; hamilton-jacobi theory; logic [quantum logic]; matrix; Perturbation Methods; Propagator; Schwinger's Action Principle; symmetry in quantum physics [including reduction].

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

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