Modified Uncertainty Relations

In General > s.a. modified coherent states [thermal].
@ References: Sarris & Proto PhyA(07) [from metric phase space]; Massar & Spindel PRL(08)-a0710 [for discrete Fourier transform].

Gravity-Motivated Generalized Forms > s.a. locality; quantum-gravity phenomenology.
* Generalized Uncertainty Principle (GUP): Motivated by the quantum-gravity idea that there is a smallest possible x l_P,

x_i p_j (/2) (1 + ² l_P² p_i²/²) _ij .

* Extended Uncertainty Principle (EUP): Motivated by physics in anti-de Sitter space (one can also have both terms present, EGUP),

x_i p_j (/2) (1 + ² x_i²/l_h²) _ij .

* Remark: Taking into account quantum-gravity effects, one tends to get larger uncertainties than in standard quantum mechanics, related by x p (/2) (1+...), while fixed, discrete spacetime tends to give smaller ones, related by x p (/2) (1–...).
@ And Lorentz-invariance: Kim qp/97-in; Sasakura PTP(99)ht, JHEP(00)ht; Molotkov qp/02 [for photons]; Kim & Noz AIP(07)qp/06.
@ In curved spaces: Golovnev & Prokhorov JPA(04); Bambi & Urban CQG(08)-a0709 [particle in de Sitter space]; Park PLB(08)-a0709; Cooperstock & Dupre a0904 [in terms of spacetime energy-momentum]; Mignemi a0909 [in (anti-)de Sitter space].
@ And general relativity, quantum gravity: Doplicher et al PLB(94); Amelino-Camelia MPLA(97)gq, et al PAN(98)ht/97 [-deformed covariant phase space]; Adler & Santiago MPLA(99)gq, ht/99; Yoneya PTP(00)ht, IJMPA(01)ht/00 [string theory]; Camacho GRG(02)gq [from non-conformal metric fluctuations]; Shalyt-Margolin & Tregubovich gq/02 [and mixed states]; Dragovich ht/04-in [p-adic, adelic]; Bambi CQG(08)-a0804 [departures linear in l_P]; > s.a. black-hole entropy, information and thermodynamics.
@ With characteristic length: Kempf et al PRD(95)ht/94; Kempf & Mangano PRD(97)ht/96 [regularization]; Brau JPA(99) [harmonic oscillator, H atom]; Cortés & Gamboa PRD(05)ht/04 [in DSR]; Brau & Buisseret PRD(06)ht [and gravitational quantum well]; Mu et al a0909 [and extra dimensions]; > s.a. quantum oscillator.
@ Black holes: Brout et al PRD(99)ht/98; Scardigli PLB(99)ht [micro-black holes]; Xiang & Shen MPLA(04) [thermodynamics]; Maziashvili PLB(06) [remnants]; > s.a. specific types.
@ Inflation: Hassan & Sloth NPB(03)ht/02; Nozari & Akhshabi IJMPD-a0910.
@ Other cosmological models: Rama PLB(01); Nozari & Fazlpour GRG(06)gq [early-universe thermodynamics]; Battisti & Montani PLB(09)-a0808 [mixmaster universe]; Shalyt-Margolin a0807 [for cosmological-constant–spacetime-volume]; Vakili IJMPD(09)-a0811 [effects on classical and quantum dS/AdS]; Zhu et al PLB(09)-a0811 [thermodynamics].
@ Related topics: Lindner et al PLA(96) [particle number-phase]; Hogan ap/07 [holographic uncertainty principle].

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 algebra, using a prescription that can be derived from the Schwarz inequality

x_i p_j (1/2) | [x_i , p_j] | .

* Examples: From the symmetry algebra of AdS one obtaines the EUP above; In string theory one gets the modified commutation relations

[x_i, p_j] = i [(1 + p²) _ij + ' p_i p_j] .

@ General references: Maggiore PLB(93)ht.
@ String theory: Konishi et al PLB(90); Capozziello et al IJTP(00)gq/99; Benczik et al PRD(02)ht; Hossenfelder et al PLB(03)ht.
@ Non-commutative geometry: Carlen & Vilela Mendes PLA(01); Brandenberger & Ho PRD(02)ht; Bolonek & Kosinski PLB(02)ht, APPB(03)ht/02 [non-commutative quantum mechanics].
@ In deformation quantization: Zhang PLA(99)ht/03; Przanowski & Turrubiates JPA(02)m.QA; Gerstenhaber JMP(07).

Other Minimal-Length Phenomenology > s.a. fine-structure constant; modified coherent states; modified QED; types of quantum field theory.
@ General references: Ozawa PRA(03)qp/02 [measurement disturbance], PLA(03)qp/02 [limitations]; Slawny JMP(07) [position and length operators].
@ H atom: Akhoury & Yao PLB(03)hp; Benczik et al PRA(05); Stetsko & Tkachuk PRA(06).
@ Other consequences: Nozari & Azizi GRG(06)qp/05 [free particle + box]; Nozari & Mehdipour GRG(05)qp [wave packet dispersion], CSF(07)ht/06 [ideal gas thermodynamics]; Quesne & Tkachuk SIGMA(07)qp/06; Bang & Berger PRD(06) [minimum uncertainty wave packets]; Das & Vagenas CJP(09)-a0901-in [Lamb shift, reflection and transmission coefficients, ...]; > s.a. discrete spacetime.

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