Modifications and Violations of Lorentz Invariance |

**In General **> s.a. discrete spacetime;
reference frames [preferred]; tests of lorentz invariance
and phenomenology.

* __Motivations for violation__:
Violations of Lorentz symmetry were suggested long ago by discrete models for quantum gravity;
Lorentz invariance is spontaneously broken in QED because of infrared effects; More generally,
a cutoff scale associated with a violation could regularize quantum field theory and black-hole
entropy calculations; Lorentz violations could explain the existence of ultra high-energy
cosmic rays beyond the GZK cutoff, and the observations of multi-TeV gamma rays from Mkn 501;
Motivations from quantum theory include Bell's inequalities for hidden variable theories.

* __Mechanisms__: Lorentz violations occur
in theories in which either not all inertial frames are equivalent (spacetime discrete structure
and fluctuations at Planck scales with an effectively preferred frame), or the kinematical
Lorentz group transformations from one frame to another are non-linearly deformed ("doubly
special relativity''), as happens naturally in some versions of quantum gravity (but not in
lqg!), string theory, and with quantum field theory radiative corrections in non-trivial
backgrounds; They can also arise in some semiclassical gravity models in which Minkowski space
is unstable and there is a phase transition to de Sitter space; Another way to integrate
deviations from Lorentz invariance into quantum field theory is achieved is an extension
of the standard model in which photons in quantum superpositions can travel at different
speeds without singling out preferred rest frames.

@ __Reviews__: González-Mestres phy/00-conf [high-energy astrophysics];
Gambini & Pullin gq/01-proc [canonical quantum gravity];
Kostelecký hp/01-conf [and CPT];
Amelino-Camelia AIP(01)gq [with *l* and *v* scales],
gq/02
[lqg and non-commutative geometry],
gq/03-MGX [approaches];
Colladay AIP(03)hp;
Lehnert PRD(03)gq [framework];
Bluhm pw(04)mar;
Kostelecký SA(04)aug;
Jacobson et al SPP(05)gq/04,
LNP(05)hp/04,
AP(06)ap/05;
Lehnert gq/06-conf,
a0711v1/JPA;
Iorio JPCS(07)phy/06;
Amelino-Camelia et al IJMPA(08)-proc [at Planck scale];
Liberati & Maccione ARNPS(09)-a0906 [and constraints];
Lehnert AIP(11)-a1102.

@ __Scales__: Amelino-Camelia PLB(01)ht/00 [minimum *L*];
Collins et al PRL(04)gq [size of effects];
Belenchia et al JHEP(16)-a1601 [low-energy percolation of Lorentz violation, toy model].

@ __Related topics__: Greenberg PRL(02)hp [and CPT];
Barbero & Villaseñor PRD(03)gq [effective actions and Euclidean general relativity];
Hinterleitner PRD(05)gq/04 [canonical methods];
Amelino-Camelia ap/04-proc [test theories];
Libanov & Rubakov JHEP(05)ht [infrared modification, ghosts-tachyons];
Jacobson & Wall FP(10)-a0804-GRF [emergence required by second law of thermodynamics];
Potting AIP(06)-a0902 [and masslessness of gravitons];
Visser PRD(09) [as quantum field theory regulator];
Hagar SHPMP(09) [conceptual];
Ganguly et al a1010 [Lorentz-preserving fields];
Chaichian et al PLB(11) [does not imply CPT violation, and viceversa];
Hossenfelder FP(12)-a1207 [quantum superpositions].

@ __Types__: Zhou & Ma ChPC(11)-a1109 [Lorentz-invariance violation matrix formalism];
Baccetti et al JHEP(12)-a1112 [relativity principle modifications and "minimalist" Lorentz violations];
Carmona et al a1702 [without energy-dependent photon time delays].

> __Related topics__: see finsler structures;
lorentz-violating models and theories; renormalization.

**Spontaneous Breaking** > s.a. Goldstone
Boson; symmetry breaking.

@ __General references__: Yokoi PLB(01)ht/00 [2+1];
Colladay hp/01-proc;
Jenkins PRD(04)ht/03;
Graesser et al PLB(05)ht [in gravitational couplings];
Dvali et al PRD(07)ht [without spacetime geometry effects];
Bluhm a0704-MGXI,
et al PRD(08)-a0712,
Bluhm PoS-a0801 [Goldstone modes];
Armendáriz-Picón et al JHEP(10)-a1004 [effective-theory approach];
Faizal JPA(11)
[by ghost condensation, in perturbative quantum gravity];
Gomes & Gomes PRD(12) [theories with anisotropic scaling];
Balachandran & Vaidya a1302 [in gauge theories];
Balachandran et al EPJC(15)-a1406 [infrared QED];
Balachandran MPLA(16)-a1509 [in QCD];
Bonder & Corral a1808 [local translations];
> s.a. QCD effects; QED phenomenology.

@ __Phenomenology__: Moffat IJMPD(03) [and cosmic rays];
Moffat Ent(15)-a1407 [in cosmology, and gravitational entropy].

**Modifications of the Lorentz Group** > s.a. branes;
finsler geometry; Fock-Lorentz Symmetry;
non-commutative geometry; poincaré group.

* __Theories with a preferred vector field__:
A preferred *W*^{ a} can be used to write
dispersion relations or field theory Lagrangians of the form

*p*^{a }*p*_{a}
= *ξ*_{γ}(*m*_{P})^{–1} (*W*^{ a} *p*_{a})^{3} , \(\cal L\) = *ξ*_{φ}(*m*_{P})^{–1} *φ* (*W*^{ a} ∂_{a})^{3} *φ* ;

An example of such a theory is Einstein-Aether Gravity.

@ __Maximal acceleration__: Nesterenko et al PRD(99) [and regularization];
Lambiase IJTP(01);
Papini PLA(02) [stellar stability];
Feoli IJMPD(03)gq/02;
Rama MPLA(03)ht/02 [*κ*-deformed Poincaré];
Toller ht/03 [geometries];
> s.a. kerr spacetime.

@ __Deformations and quantum version__: Finkelstein JMP(96), LMP(00)ht [spin-statistics];
Azcárraga et al ZPC(97)qa [h-deformation];
Buffenoir & Roche CMP(99)qa/97;
Lagraa JGP(00);
Lukierski & Nowicki IJMPA(03)ht/02,
APPB(02)ht,
ht/02-proc;
Amelino-Camelia et al CQG(04)
[*κ*-Poincaré algebras, and cosmological constant];
Daszkiewicz et al MPLA(08) [and 4-momentum addition];
> s.a. minkowski space; modified QED [at cosmological
scales]; Kasprzak JNCG(10)-a1009 [from Rieffel deformation of SL(2, \(\mathbb C\))].

@ __Related topics__: Bruno et al PLB(01)ht [boosts that saturate];
Lindesay mp/03 [extension];
> s.a. sound.

**Modified Action of the Lorentz Group** > s.a. poincaré group;
doubly special relativity.

@ __References__: Manida gq/99 [precursor];
Yokoi PLB(01)ht/00 [non-linear];
Toller MPLA(03)ht [transformation of momentum-energy];
Broekaert FP(05)gq/03-conf [gravitationally modified];
Tsabary & Censor mp/04.

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