Generalized and Modified Quantum Mechanics  

In General > s.a. canonical quantization; geometric quantization; quantum collapse; sub-quantum theories.
* Motivation: Comes from many different directions, such as the 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; More recent motivations include quantum information and some approaches to quantum gravity; > s.a. quantum mechanics.
* Non-equilibrium quantum theory: A proposal, inspired by pilot-wave theory and developed mainly by Valentini, in which the probability density is not necessarily |ψ|2 and the Born rule arises only in equilibrium; > s.a. pilot-wave theory.
* 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.
@ General references: Hnilo a1212-conf [transient deviation from quantum mechanics].
@ Causal quantum mechanics: Kent PRA(05) [collapse locality loophole]; Kent PRS(18)-a1807 [implications].
@ And gravity: Hu JPCS(14)-a1402 [gravitational decoherence and semiclassical gravity]; Penrose FP(14) [conformal cyclic cosmology].
@ Stochastic extension: & Hughston; Adler & Horwitz JMP(00); Adler & Bassi JPA(07)-a0708 [non-white noise and collapse]; > s.a. Open System; Trajectory.
@ Relational formulation: Rovelli ht/94, IJTP(96)qp; Francis gq/05; Marlow qp/06; Giddings PRD(08); Brown BJPS(09) [determinacy problem]; van Fraassen FP(10); Dorato a1309 [philosophical implications].
@ Other probabilistic models, correlations: Barnum et al EPTCS(15)-a1507 [non-signaling composites of probabilistic models based on euclidean Jordan algebras]; Krumm et al a1608 [generalized probabilistic theories and thermodynamics].
@ Discrete quantum mechanics: Gudder & Naroditsky IJTP(81); Jagannathan et al IJTP(81); Buniy et al PLB(05)ht; Sasaki PTRS(10)-a1004; Odake & Sasaki JPA(11)-a1104; 't Hooft a1204; Louko et al PRD(14)-a1309 [singularity resolution]; Ellerman a1310 [QM/sets]; Luoma & Piilo JPB(16)-a1509 [and non-Markovianity]; Arik & Ildes PTEP(16)-a1510 [space with a finite number of points]; > s.a. formulations of quantum mechanics.
> No-go results: see Colbeck-Renner Theorem.
> Some types of modifications motivated: see non-linear quantum mechanics; relativistic quantum mechanics; Super-Quantum Theory.
> Quantum-gravity motivated: see deformation quantization [modified commutation relations]; matter phenomenology in quantum gravity; modified uncertainty relations [with minimal length]; non-commutative physics.

Modified Formalisms > s.a. foundations; path integrals; quantum theory on generalized backgrounds [including curved spaces].
* Different Hilbert spaces: Quantum theories may be formulated in real, complex or quaternionic Hilbert spaces only, but there are physical reasons for ruling out real Hilbert spaces that rely on Heisenberg's principle (results cited by Oppio & Moretti).
@ 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]; Asorey et al JPA(13)-a1301 [with rapidly changing boundary conditions].
@ Different Hilbert spaces: Namiki & Pascazio PLA(93), PRP(93) [many]; Myrheim qp/99 [real]; Znojil a0910-conf; Oppio & Moretti RVMP(17)-a1611 [real Hilbert space]; > s.a. Banach Space [use of \(\mathbb{KS}^2\)].
@ Alternatives to Hilbert spaces: Anastopoulos FP(01)qp/00 [probabilities and phases]; Niestegge FP(14)-a1402 [ordered Banach spaces, and absence of third-order interference]; Zohar JPA(17)-a1607 [stators, half states and half operators].
@ Modified postulates: Sudbery SHPMP(02)qp/00, qp/00-conf [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; Sergi JCP(07)qp/05; > s.a. Open Systems.
@ Generalized measures: Sorkin MPLA(94)gq [quantum measure]; Chryssomalakos & Durdevich MPLA(04)qp/03; > s.a. measure theory.
@ Time-symmetric theory: Aharonov et al PR(64); Vaidman a0706-en; Aharonov et al PT(10)nov; Bopp FP(17)-a1604 [and macroscopic physics]; Kastner a1607; Leifer & Pusey PRS-a1607 [impossibility of a non-retrocausal time-symmetric ontology].
@ 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-proc [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].
@ Generalized frameworks: Fischbach et al PRL(91) [different \(\hbar\)]; Aerts & Gabora Kyb(05)qp/04, Kyb(05)qp/04 [state-context-property theory]; Hardy & Wootters FP(12)-a1005 [theories more holistic than quantum theory, bilocally tomographic]; Kisil IJTP(12)-a1005 [general representations of the Heisenberg group, including classical and quantum mechanics]; Moldoveanu a1311 [hyperbolic quantum mechanics, non-viability]; Brody JPA(14) [biorthogonal quantum mechanics]; Gould & Afshordi FP(15)-a1407 ["real ensemble" non-local description]; Krause & Arenhart a1510 [based on non-reflexive logic]; Gogioso a1703 [bestiary of exotic examples]; Thas a1712 [over general division rings with involution]; Bisio & Perinotti PRS(19)-a1905 [higher-order quantum theory].
@ Evolving Hilbert spaces: Mostafazadeh PLA(04)qp/03; Artacho & O'Regan PRB(17)-a1608.
@ Other modifications: Zambrini PRA(87) [euclidean]; Adler & Horwitz JMP(96)ht; Fivel PRA(94) [tests]; Petrov qp/97; Baugh et al FP(03)ht [ultraquantum theory]; Fivel qp/03 [metaplectic]; Kitada qp/04 [??]; Khrennikov FPL(05) [from statistical field theory]; Aharonov et al PRA(09)-a0712 [multiple-time states]; Ghirardi & Romano PRL(13)-a1301, FP(13)-a1301 [extension of quantum theory, predictively inequivalent ontological model]; Sellier & Kapanova a1704 [signed quantum mechanics, and the H atom]; Strumia a1709 [with indefinite norm]; Grassi & Mariño Sigma(19)-a1806 [exactly solvable deformation].
> Related topics: see Adelic Structures; analysis [fractional]; categories in physics; clifford algebra; differential geometry; Galois Field [finite field]; Groupoids; Hypercomplex Numbers; Modal Quantum Theory; Non-Associative Algebras; non-standard analysis; Octonions; p-Adic Structures; quaternions; Topos.

Non-Hermitian, PT-Symmetric Quantum Mechanics > s.a. approaches to quantum field theory; relativistic quantum mechanics.
* Idea: A generalization in which the Hamiltonian is not invariant under Hermitian conjugation, but under PT, the combination of parity reflection and time reversal; It gives new classes of complex Hamiltonians whose spectra are still real and positive; 2014, The theory violates the non-signaling principle of relativity.
@ General references: Bender et al JMP(99), PRL(02), AJP(03)nov-ht + comment van Hameren, CP(05)qp; Mostafazadeh qp/04; Kleefeld ht/04; issue JPA(06)#32; Bender RPP(07)ht [rev]; Bender & Mannheim PRD(08)-a0804, a0902; issue JPA(08)#24; Das & Greenwood PLB(09)-a0905 [positive inner product]; Kleefeld a0906 [inner product and C operator]; Mannheim PTRS-a0912; Graefe et al JPA(10)-a0910 [classical limit and modified canonical structure]; issue IJTP(11)#4; Brody JPA(16)-a1508 [consistency]; Mannheim PRD(18)-a1708 [inner product].
@ Relationship with standard, Hermitian quantum theory: Bender et al JPA(06)ht/05, comment Mostafazadeh ht/06; Martin qp/07 [it is just quantum mechanics in a non-orthogonal basis]; Nagao & Nielsen PTP(11)-a1009 [effective Hermiticity emerges automatically]; Lee a1312; Girardelli BJP-a1502 [nothing new].
@ And quantum field theory: Bender et al PRL(14)-a1408 [inequivalent theories from one Lagrangian]; Alexandre et al PRD(18)-a1805 [spontaneous symmetry breaking].
@ Specific systems: Znojil JMP(09) [square well]; Dasarathy et al PRA(13)-a1708 [box]; > s.a. quantum oscillators; systems with special potentials.
@ Related topics: Bagchi & Fring PLA(09) [and deformed commutation relations, minimal length]; Jones-Smith & Mathur PRD(14)-a0908 [relativistic]; Schomerus PRA(11) [spontaneous PT-symmetry breaking]; Bender & Klevansky PRA(11)-a1104 [fermionic algebras]; Znojil CJP(12)-a1205 [exactly solvable model on a quantum graph]; Bender & Weir JPA(12)-a1206 [unbroken-broken PT-symmetry phase transition]; Bender et al AJP(13)mar [phase transition in a simple mechanical system]; Lee & Mead a1303-wd [critical view]; Lee et al PRL(14)-a1312, comment Znojil a1404 [non-signaling principle violation]; Mead & Garfinkle a1610 [selection rule for transitions]; > s.a. optics; quantum effects; quantum phase transitions; statistical mechanical systems.

Supersymmetric Quantum Mechanics
* Idea: Used as a powerful tool for generating new potentials with known spectra starting from a known solvable one; The Hilbert space \(\cal H\) decomposes into a direct sum of an even and an odd part, \(\cal H\) = \(\cal H\)1 ⊕ \(\cal H\)2, and the Hamiltonian is of the form H = Q2, with Q = matrix{0, q; q* q}; > s.a. supersymmetry in field theory.
@ General references: Gendenshtein & Krive SPU(85); Boya et al PRD(87); Rota & Stein PNAS(90); Cooper et al PRP(95); 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; issue JPA(04)#43; Spector JPA(04)qp/03 [partial supersymmetry]; Parthasarathi et al JPA(04) [complex phase-space formulation]; Khare AIP(04)mp [intro]; Lundholm JMP(08)-a0710 [geometry]; Kuznetsova RPMP(08) [irreducible representations]; Bagarello PLA(08)-a0904 [extended, and coherent states]; Fernández AIP(10)-a0909; Gangopadhyaya et at 11 [r CP(12)]; Fernández a1811-in [rev]; > s.a. relativistic quantum mechanics.
@ Different approaches: Acosta-Humanez PhD(09)-a0906 [Galois theory approach]; Castellani et al AHP(18)-a1706 [integral form formalism].
@ Models, applications: Goldstein et al AJP(94)jul [examples]; Fernández IJMPA(97)qp/96 [exactly solvable]; Rau JPA(04)qp [extension, examples]; Hong et al PRD(05)ht [particle on a sphere]; Bittner & Kouri a1005-conf [applications]; Smilga JHEP(13)-a1301 [obtaining SQM model]; > s.a. classical systems; coherent states; examples of entangled states; Painlevé Equations; quantum oscillator.
@ Related topics: Daoud & Kibler mp/01-ln, mp/01-conf, PLA(04) [fractional supersymmetry]; Kibler & Daoud qp/04-in [N = 2 fractional of order k]; Kuznetsova et al JHEP(06)ht/05 [N-extended, irreducible representations]; Rawat & Negi IJTP(09)ht/07 [quaternionic formulation]; Andrianov et al NPB(07), Sokolov NPB(07) [non-linear supersymmetry]; Dzhunushaliev JMP(08)-a0712 [octonionic extension and hidden variables]; Baumgartner & Wenger NPB(15)-a1412, NPB(15)-a1503 [on the lattice]; Coffey a1501 [generalized raising and lowering operators].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 26 jul 2019