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]; > s.a. QCD effects; QED phenomenology.

@ __Phenomenology__: Moffat IJMPD(03) [and cosmic
rays]; Moffat 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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2017