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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