Generalized and Modified Quantum Mechanics  

In General
* Motivation: Comes from many different directions, such as desire to explain the collapse of the wave function interpreted as a physical phenomenon (non-linear quantum mechanics), incorporating irreversibility or Lorentz invariance (relativistic quantum mechanics) or diffeomorphism invariance, accounting for phenomena (such as interference in time...), etc.

Modified Formalisms > s.a. foundations; path integrals; relativistic quantum mechanics [including curved spaces].
@ General references: Bastin ed-70; Jordan in(72); Mielnik CMP(74); Wignall FP(88); Hoekzema 93; Budnik ht/94; Beretta MPLA(05) [constraints from consistency with thermodynamics].
@ Different Hilbert spaces: Namiki & Pascazio PLA(93), PRP(93) [many]; Myrheim qp/99 [real]; Buniy et al PLB(05)ht [discrete, from quantum spacetime]; Znojil a0910-in; > s.a. Banach Space [use of KS2].
@ Alternatives to : Anastopoulos FP(01)qp/00 [probabilities and phases].
@ Modified postulates: Sudbery SHPMP(02)qp/00, qp/00 [re quantum jumps]; Svozil qp/00 [re probabilities].
@ Non-Hamiltonian: Bolivar PRA(98); Tarasov qp/01, PLA(01)qp/03; Gitman & Kupriyanov EPJC(07)ht/06; Vol PRA(06)qp/05; > s.a. Open Systems.
@ Generalized measures: Sorkin MPLA(94)gq; Chryssomalakos & Durdevich MPLA(04)qp/03.
@ Two-state vector formalism: (Time-symmetric measurements) Aharonov et al PR(64); Vaidman a0706.
@ Non-reversible, semigroup: Petrosky & Prigogine PhyA(88), PLA(93) [for density matrices]; Castagnino & Laura PRA(97)qp/96; Bohm et al IJTP(03)ht/99, ht/99, PLA(00)ht/99, ht/99-in [Gamow vectors].
@ Pre-quantum mechanics: Adler & Millard NPB(96)ht/95; Durdevich qp/99 [C*-algebra]; Soucek qp/01; Khrennikov qp/03 [no "no-go"].
@ Event-enhanced: Blanchard & Jadczyk PLA(95), AdP(95)ht/94, qp/95, RPMP(95)qp [and measurement], FP(96)qp [relativistic].
@ Other modifications: Zambrini PRA(87) [euclidean]; Fivel PRA(94) [tests]; Petrov qp/97; Baugh et al FP(03)ht [ultraquantum theory]; Mostafazadeh PLA(04)qp/03 [time-dependent ]; Kitada qp/04 [??]; Khrennikov FPL(05) [from statistical field theory]; Aharonov et al a0712 [multiple-time states].

Different Underlying Mathematics > s.a. analysis [fractional]; differential geometry; non-standard analysis; Topos.
@ Discrete time: Bender et al PRD(05), PRD(86), PRD(87); Khorrami AP(95), AP(96); Date CQG(03)gq/02.
@ Discrete spacetime: Gudder FP(88); Lorente in(97)qp/04; Piazza ht/05-in [localized subsystems in Hilbert space]; Koehler qp/06; Odake & Sasaki a0902 [correspondence with regular Schrödinger equation, Crum's theorem]; > s.a. quantum mechanics in phase space.
@ Quaternionic: Adler 95; Adler JMP(96)ht [projective group representations]; Brumby & Joshi CSF(96)qp; Horwitz FP(96)qp; Maia ht/99 [spin]; De Leo & Ducati JMP(01)mp/00; Maia & Bezerra IJTP(01)ht [geometric phase]; De Leo & Ducati JMP(06)mp [diffusion by potential step], JMP(07)-a0706 [wave packet behavior]; de Melo & Pimentel a0809-in [variational formulation]; McKague a0911 [non-local boxes].
@ Octonionic: De Leo & Abdel-Khalek PTP(96)ht, IJTP(98)ht/99; Dzhunushaliev FPL(06)ht/05, ht/06, qp/07, AHEP(07)-a0706 [non-associative].
@ Non-Hermitian, PT-symmetric: Bender et al JMP(99), PRL(02), AJP(03)nov-ht + comment van Hameren, qp/05; Mostafazadeh qp/04; Kleefeld ht/04; Bender et al JPA(06)ht/05, comment Mostafazadeh ht/06 [vs Hermitian]; issue JPA(06)#32; Martin qp/07 [just quantum mechanics in non-orthogonal basis]; Bender RPP(07)ht [rev]; Bender & Mannheim a0804, a0902; issue JPA(08)#24; Das & Greenwood PLB(09)-a0905 [positive inner product]; Kleefeld a0906 [inner product and C operator]; Graefe et al a0910 [classical limit and modified canonical structure]; Bagchi & Fring PLA(09) [and deformed commutation relations, minimal length]; Jones-Smith & Mathur a0908 [relativistic]; > s.a. optics; quantum effects and systems; modified approaches to quantum field theory.
@ Over a Galois field: Lev ht/02, TMP(04)ht/02, ht/02, ht/02, FFTA(06)ht/03, IJMPB(06)ht; Vourdas JPA(05), AAM(06)qp, JPA(07).
@ Other proposals: Adler & Horwitz JMP(96)ht; Dragovich ITSF(98)mp/04, Djordjevic & Nesic ht/04-in [Adelic]; Fivel qp/03 [metaplectic]; Oeckl ATMP(08)ht/05 [general boundary]; Hübschmann et al CMP(09) [on a stratified space].

Other Modifications > s.a. canonical, deformation, geometric quantization; Non-Linear Quantum Mechanics; quantum collapse.
* Causal quantum mechanics: Ordinary quantum theory modified by two hypotheses, state vector reduction is a well-defined process, and strict local causality applies; The first holds in some versions of Copenhagen quantum mechanics and need not necessarily imply testable deviations from ordinary quantum mechanics; The second implies that measurement events which are spacelike separated have no non-local correlations.
* Supersymmetric quantum mechanics: Used as a powerful tool for generating new potentials with known spectra starting from a known solvable one; The Hilbert space decomposes into a direct sum of an even and an odd part, = 1 2, and the Hamiltonian is of the form H = Q2, with Q = matrix{0, q; q* q}; > s.a. classical systems; supersymmetry in field theory; quantum oscillator.
@ Causal quantum mechanics: Kent PRA(05) [collapse locality loophole].
@ Stochastic extension: & Hughston; Adler & Horwitz JMP(00); Adler & Bassi JPA(07)-a0708 [non-white noise and collapse]; > s.a. Open System; Trajectory.
@ Supersymmetric: Gendenshtein & Krive SPU(85); Boya et al PRD(87); Rota & Stein PNAS(90); Goldstein et al AJP(94)jul [examples]; Cooper et al PRP(95); Fernández IJMPA(97)qp/96 [exactly solvable]; Junker ht/96 [path-integral aspects]; Debergh JPA(97) [in curved space]; Fröhlich et al CMP(98) [and differential geometry]; Capdequi-Peyranere MPLA(99)qp/00 [duality]; Aoyama et al NPB(01)qp [n-fold]; Cooper et al 01; Daoud & Kibler mp/01-in, mp/01-in, PLA(04) [fractional supersymmetry]; issue JPA(04)#43; Spector JPA(04)qp/03 [partial supersymmetry]; Rau JPA(04)qp [extension, examples]; Parthasarathi et al JPA(04) [complex phase-space formulation]; Khare mp/04-ln [intro]; Kibler & Daoud qp/04-in [N = 2 fractional of order k]; Hong et al PRD(05)ht [particle on a sphere]; Kuznetsova et al JHEP(06)ht/05 [N-extended, irreps]; Rawat & Negi IJTP(09)ht/07 [quaternionic formulation]; Andrianov et al NPB(07), Sokolov NPB(07) [non-linear supersymmetry]; Lundholm JMP(08)-a0710 [geometry]; Dzhunushaliev JMP(08)-a0712 [octonionic extension and hidden variables]; Kuznetsova RPMP(08) [irreps]; Bagarello PLA(08)-a0904 [extended, and coherent states]; Acosta-Humanez PhD(09)-a0906 [Galois theory approach]; Fernández a0909-ln; > s.a. coherent states; relativistic quantum mechanics.
@ Relational formulation: Rovelli ht/94, IJTP(96)qp; Francis gq/05; Marlow qp/06; Giddings PRD(08).
@ Generalized framework: Fischbach et al PRL(91) [different ]; Aerts & Gabora Kyb(05)qp/04, Kyb(05)qp/04 [state-context-property theory].
> Motivated by quantum gravity: see modified uncertainty relations; non-commutative physics.

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