Formulations of Quantum Theory| 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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