Deformed Quantum Uncertainty Relations| Deformed Quantum Uncertainty Relations |
In General
> s.a. modified coherent states [thermal, minimal-length].
@ General references: Sarris & Proto PhyA(07) [from metric phase space];
Massar & Spindel PRL(08)-a0710 [for discrete Fourier transform];
Eune & Kim MPLA(14)-a1309 [from constraints];
Iorio & Pais a1901 [from Dirac fields in graphene].
@ Related topics: Ozawa PRA(03)qp/02 [measurement disturbance],
PLA(03)qp/02 [limitations];
Slawny JMP(07) [position and length operators];
Pedram PRD(12)-a1112 [approach to quantum mechanics];
Kalogeropoulos AJSS-a1303 [and solvable Lie algebras];
Moradpour et al a2012 [and Tsallis entropy].
Gravity-Motivated Generalized Forms > s.a. phenomenology of quantum
uncertainties; spacetime geometry in quantum gravity [metric fluctuations].
* Idea: In gravity, two scales
that can be used to deform the standard uncertainty relations are the Planck
length lP and the cosmological horizon
length lh; With quantum-gravity
effects, one tends to get larger uncertainties than in standard quantum mechanics,
related by Δx Δp ≥ (\(\hbar\)/2) (1 + ...), while
fixed, discrete spacetime tends to give smaller ones, related by Δx
Δp ≥ (\(\hbar\)/2) (1 − ...).
* Generalized Uncertainty Principle (GUP):
A UV modification, motivated by the quantum-gravity idea of a smallest possible
\(\Delta x \approx l_{\rm P}^~\); One proposed form is
Δxi Δpj \(\ge (\hbar\)/2) (1 + \(\alpha^2\, l_{\rm P}^{\,2}\) Δpi2 / \(\hbar\)2) δij .
* Extended Uncertainty Principle (EUP): An IR modification taking into account curved spacetime effects, motivated by physics in anti-de Sitter space,
Δxi Δpj ≥ (\(\hbar\)/2) (1 + β2 Δxi2 / lh2) δij ;
One can also have both terms present, in an extended generalized
uncertainty principle (EGUP).
* Károlyházy uncertainty relation:
If a device is used to measure a length l, there will be a minimum uncertainty
in the measurement given by \((\delta l)^3 \sim \ell_{\rm P}^2\,l\).
@ General references: Kempf et al PRD(95)ht/94;
Kempf & Mangano PRD(97)ht/96 [regularization];
Quesne & Tkachuk Sigma(07)qp/06 [2-parameter version];
Brau & Buisseret PRD(06)ht [and gravitational quantum well];
Balasubramanian et al AP(15)-a1404 [and operator self-adjointness];
Tawfik & Diab IJMPD(14)-a1410,
RPP(15)-a1509 [rev];
Bruneton & Larena GRG(17)-a1602 [Hilbert space representations].
@ Derivations:
Bojowald & Kempf PRD(12)-a1112 [for systems with a discrete coordinate];
Casadio a1310-proc [from a "horizon wave function"];
Faizal et al PLB(17)-a1701 [from effective field theory];
Kuzmichev & Kuzmichev a1911 [with Newtonian gravity].
@ With maximal momentum:
Nozari & Etemadi PRD(12)-a1205;
Pedram PLB(12)-1110,
PLB(12)-a1210 [higher-order];
Etemadi & Nozari a1412;
Lake Gal-a1712 [dark-energy modified];
Lake et al CQG(19)-a1812 [from geometric superpositions];
Petruzziello a2010 [and no minimal length].
@ Phenomenology:
Bosso PhD(17)-a1709;
Casadio & Scardigli PLB-a2004
[and Poisson brackets, equivalence principle];
> s.a. specific applications.
@ In DSR:
Cortés & Gamboa PRD(05)ht/04;
Chung & Hassanabadi PLB(18)-a1807 [and consequences].
@ And extra dimensions: Mu et al ChPL(11)-a0909;
Köppel et al a1703-proc.
@ And Lorentz invariance:
Kim FdP(98)qp/97-proc;
Sasakura PTP(99)ht,
JHEP(00)ht;
Molotkov qp/02 [for photons];
Kim & Noz AIP(07)qp/06;
Tkachuk a1310;
Lambiase & Scardigli PRD(18)-a1709 [SME parameter \(\beta\)];
Todorinov et al AP(19)-a1810 [relativistic generalization].
@ In curved spaces: Golovnev & Prokhorov JPA(04)qp/03 [in curved spacetime];
Bambi & Urban CQG(08)-a0709 [particle in de Sitter space];
Park PLB(08)-a0709;
Cooperstock & Dupre AP(13)-a0904 [in terms of spacetime energy-momentum];
Mignemi MPLA(10)-a0909,
Ghosh & Mignemi IJTP(11)-a0911 [(anti-)de Sitter space];
Perivolaropoulos PRD(17)-a1704 [with maximum observable length];
Dąbrowski & Wagner a2006 [EUP, arbitrary spatial curvature];
Petruzziello & Wagner a2101.
@ Other backgrounds:
Marchiolli & Ruzzi a1106 [discrete phase space];
Iorio et al a1910 [in 3D gravity, and BTZ black hole].
@ Spacetime uncertainty relation: Burderi & Di Salvo PRD(16)-a1207-MG13
[Δr Δt > G\(\hbar\)/c4];
Burderi et al PRD(16)-a1603 [quantum clock];
Bolotin et al a1604 [quantum clock, and the principle of maximum force];
Singh a1910 [Károlyházy relation].
@ Related topics: Lindner et al PLA(96) [particle number-phase];
Hogan ap/07 [holographic uncertainty principle];
Das & Pramanik PRD(12)-a1205 [and path integrals];
Bosso PRD(18)-a1804 [Hamiltonian and Lagrangian classical and quantum theories];
Lake et al a1912 [for angular momentum and spin];
> s.a. Lifshitz Theories; Schwinger's
Quantum Action Principle; wigner functions.
From Deformed Algebras
> s.a. Commutation Relations; modified lorentz symmetry;
modified quantum theory; poincaré group.
* Idea: One can obtain modified
uncertainty relations from a deformation of the Poincaré and/or Heisenberg
Δxi Δpj \(\ge {1\over2}|\langle[x_i, p_j]\rangle|\) .
* Examples: From the symmetry algebra of AdS one obtains the EUP above; In string theory one gets the modified commutation relations
[xi, pj] = i\(\hbar\)[(1 + β p2) δij + β' pi pj] .
@ General references: Maggiore PLB(93)ht;
Abdelkhalek et al PRD(16)-a1607;
Faizal PLB(16)-a1605 [from supersymmetry breaking].
@ In deformation quantization:
Zhang PLA(99)ht/03;
Przanowski & Turrubiates JPA(02)m.QA;
Gerstenhaber JMP(07).
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