Generalized and Modified Quantum Mechanics  

In General > s.a. canonical quantization; geometric quantization; hilbert space; modified formalisms; 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]; > s.a. Relationalism.
@ Other probabilistic models, correlations: Barnum et al EPTCS(15)-a1507 [non-signaling composites of probabilistic models based on euclidean Jordan algebras]; Krumm et al NJP(17)-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]; Banks a2001; Majid a2002; > 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]; interference [higher-order interference]; matter phenomenology in quantum gravity; modified uncertainty relations [with minimal length]; non-commutative physics.

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]; Bagarello & Feinberg a2001 [bicoherent-state path integrals]; Ashida et al a2006-AiP [rev].
@ 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]; Bender et al a2103 [and renormalization]; Mannheim a2104-proc [and the ghost problem].
@ 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]; Zhang a2005 [reformulation]; > 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]; Ayad a1911-PhD; > s.a. relativistic quantum mechanics.
@ Different approaches: Acosta-Humanez PhD(09)-a0906 [Galois theory approach]; Castellani et al AHP(18)-a1706 [integral form formalism]; Troost a2004.
@ 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].

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