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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