Modified Uncertainty Relations  

Generalized Uncertainty Principle 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].

Relativistic Relation, in Quantum Gravity > s.a. locality; quantum gravity phenomenology.
* Common forms: One often sees the GUP written as

x p (/2) (1 + 2 lP2 p2/2) ,

but one that treats x and p in a more symmetric way is

x p (/2) (1 + 2 lP2 p2/2 + 2 x2/lP2) ;

The precise form for a pair of operators is obtained from their commutator using the Schwarz inequality (?).
* 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 qp/06-in.
@ In curved spaces: Golovnev & Prokhorov JPA(04); Bambi & Urban CQG(08)-a0709 [particle in de Sitter]; Park a0709.
@ 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; Scardigli PLB(99)ht [micro-black holes]; 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 lP]; > 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]; > s.a. quantum oscillator.
@ Related topics: Lindner et al PLA(96) [particle number-phase]; Hogan ap/07 [holographic uncertainty principle].

From Deformed Algebras > s.a. modified lorentz symmetry; poincaré group.
* Idea: One can obtain modified uncertainty relations from a deformation of the Poincaré and/or Heisenberg algebra, for example the modified commutation relations one gets in string theory,

[xi, pj] = i [(1 + p2) ij + ' pi pj] .

@ 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).

Minimal Length and 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].
@ Cosmology: Rama PLB(01); Hassan & Sloth NPB(03)ht/02 [inflation]; Nozari & Fazlpour GRG(06)gq [early universe thermodynamics].
@ Black holes: Brout et al PRD(99)ht/98; Xiang & Shen MPLA(04) [thermodynamics]; Maziashvili PLB(06) [remnants]; > s.a. specific types.
@ 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].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 5 jul 2008