Research ArticleLorentz Violation of the Photon Sector in Field Theory Models

Lingli Zhou1 and Bo-Qiang Ma12

1 School of Physics and State Key Laboratory of Nuclear Physics and Technology Peking University Beijing 100871 China2 Collaborative Innovation Center of Quantum Matter Beijing China and Center for High Energy PhysicsPeking University Beijing 100871 China

Correspondence should be addressed to Bo-Qiang Ma mabqpkueducn

Received 6 June 2014 Accepted 4 August 2014 Published 21 August 2014

Academic Editor Anastasios Petkou

Copyright copy 2014 L Zhou and B-Q Ma This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

We compare the Lorentz violation terms of the pure photon sector between two field theory models namely the minimal standardmodel extension (SME) and the standardmodel supplement (SMS) From the requirement of the identity of the intersection for thetwo models we find that the free photon sector of the SMS can be a subset of the photon sector of the minimal SME We not onlyobtain some relations between the SME parameters but also get some constraints on the SMS parameters from the SME parametersTheCPT-odd coefficients (119896

119860119865)120572 of the SME are predicted to be zeroThere are 15 degrees of freedom in the Lorentz violationmatrix

Δ120572120573 of free photons of the SMS related with the same number of degrees of freedom in the tensor coefficients (119896119865)120572120573120583] which are

independent from each other in the minimal SME but are interrelated in the intersection of the SMS and the minimal SME Withthe related degrees of freedom we obtain the conservative constraints (2120590) on the elements of the photon Lorentz violation matrixThe detailed structure of the photon Lorentz violation matrix suggests some applications to the Lorentz violation experiments forphotons

1 Introduction

Lorentz symmetry is one of the basic principles of modernphysics and it stands as one of the basic foundationsof the standard model of particle physics The minimalstandard model has achieved a great success in predictionsand explanations of various experiments Nevertheless somefundamental questions remain to be answered One of themost essential questions is whether the Lorentz invarianceholds exactly or to what extent it holds Through theoreticalresearches and recent available experiments on the Lorentzinvariance violation (LIV or LV) we can obtain a deeperinsight into the nature of Lorentz symmetry and clarify thesefundamental questions

The possible Lorentz symmetry violation (LV) effectshave been investigated for decades from various theoriesmotivated by the unknown underlying theory of quantumgravity together with various phenomenological applications[1 2] The existence of an ldquoaeligtherrdquo or ldquovacuumrdquo can bringthe breaking down of Lorentz invariance [3 4] From basic

consideration there are investigations on the concepts ofspace-time such as whether the space-time is discrete orcontinues [5ndash8] or whether a fundamental length scaleshould be introduced to replace the Newtonian constant 119866[9] The Lorentz violation can happen in many alternativetheories for example the doubly special relativity (DSR) [10ndash12] torsion in general relativity [13ndash15] noncovariant fieldtheories [16ndash19] and large extra dimensions [20 21] Amongthese theoretical investigations of Lorentz violation it is apowerful framework to discuss various LV effects based ontraditional techniques of effective field theory in particlephysics It starts from the Lagrangian of the standard modeland then includes all possible terms containing the Lorentzviolation effects The magnitudes of these LV terms can beconstrained by various experiments The standard modelextension (SME) [22] is an example within such field theoryframeworks inwhich the LV terms aremeasuredwith severaltensor fields as coupling constants and modern experimentshave built severe constraints on the relevant Lorentz violationparameters [23]

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 374572 8 pageshttpdxdoiorg1011552014374572

2 Advances in High Energy Physics

The standard model extension is an effective frameworkfor phenomenological analysis We still need a fundamentaltheory to derive the Lorentz violation terms from basicprinciples An attempt for such a purpose has been offered in[24 25] in which a more basic principle denoted as physicalinvariance or physics independence is proposed to extendthe basic principle of relativity Instead of the requirementthat the equations describing the laws of physics have thesame form in all admissible frames of reference it requiresthat the equations describing the laws of physics have thesame form in all admissible mathematical manifolds Thisprinciple leads to the following replacement of the ordinarypartial 120597

120572and the covariant derivative119863

120572

120597120572997888rarr 119872

120572120573120597120573 119863

120572997888rarr 119872

120572120573119863120573 (1)

where 119872120572120573 is a local matrix We separate it to two matriceslike 119872120572120573 = 119892120572120573 + Δ120572120573 where 119892120572120573 is the metric tensorof space-time and Δ120572120573 is a new matrix which is particle-type dependent generally Since 119892120572120573 is Lorentz invariant Δ120572120573contains all the Lorentz violating degrees of freedom from119872120572120573ThenΔ120572120573 brings new terms violating Lorentz invariancein the standard model and is called Lorentz invarianceviolation matrix This new framework can be referred to asthe Standard Model Supplemented with Lorentz ViolationMatrix or Standard Model Supplement (SMS) [24 25] forshort and it has been applied to discuss the Lorentz violationeffects for protons [24] photons [26] and neutrinos [27 28]

Before accepting the SMS as a fundamental theory onecan take the SMS as an effective framework for phenomeno-logical applications by confronting various experiments todetermine andor constrain the Lorentz violation matrix Δ120572120573for various particles From a more general sense the SMSshould be a subset of a general version of the SME Howeverin the case of theminimal version of the SME the relationshipbetween the SMS and the SME is unclear yet The purpose ofthis paper is to compare the Lorentz violation effects of thephoton sector between the two models As have been wellknown light has always played a significant role in the devel-opments of physics and the Lorentz violation in the photonsector is also under active investigations both theoreticallyand experimentally [2] The Lorentz violation parameters ofphotons in the SME have been well constrained by variousexperiments as summarized in [23] By confronting thecollected data in [23] we can obtain bounds and detailedstructure of the Lorentz violation parameters in the modelSMS [24ndash26]

This paper is organized as follows Section 2 providesthe relation between the photon Lorentz violation matrix inthe SMS and various tensor fields in the minimal SME inthe case that the two models give the same results Theserelations define the boundary of the intersection of these twotheories too In Section 3 we discuss the general structureof the photon Lorentz violation matrix in our model andits implications for the potential property of the space-timestructure for photons At the same time we obtain theconstraints on the elements of the photon Lorentz violationmatrix Then conclusion is given in the last section

2 Relations between Two Models

In the standard model extension (SME) the terms thatviolate Lorentz invariance are added by hands from someconsiderations such as gauge invariance Hermitean andpower-counting renormalizability We consider just the min-imal SME [22] here The Lorentz violation terms in theminimal SME contain many tensor fields as coupling con-stants with their magnitudes to be determinedconstrainedby experiments Though these Lorentz violation terms areallowed by some general considerations the reason for theirexistence still needs to be provided from theoretical aspectsIn the SMS [24 25] the Lorentz violation terms arise fromthe replacement of (1) which is considered as a necessaryrequirement from the basic principle of physical invarianceAs both of the two models are built within the frameworkof effective field theory in particle physics they can beconsidered as two special cases of a general standard modelextension within the effective field theory It is thereforenecessary to study the relationship between the two models

TheLagrangians of the pure photon sector in the SMS andthe minimal SME are

LSMS = minus1

4119865120572120573119865120572120573

+LGV (2)

where

LGV = minus1

2Δ120572120573Δ120583](119892120572120583120597120573119860120588120597]119860120588 minus 120597

120573119860120583120597]119860120572)

minus 119865120583]Δ120583120572120597120572119860

]

(3)

LSME = minus1

4119865120572120573119865120572120573

+LCPT-evenphoton +L

CPT-oddphoton (4)

where

LCPT-evenphoton = minus

1

4(119896119865)120572120573120583]119865120572120573119865120583]

LCPT-oddphoton =

1

2(119896119860119865

)120572120598120572120573120583]119860

120573119865120583]

(5)

We denote the matrix Δ120572120573 above as Lorentz invarianceviolation matrix whose dimension is massless When weignore the field redefinition there are 16 independent dimen-sionless degrees of freedom inΔ120572120573 generally [25] As couplingconstants the vacuum expectation value of Δ120572120573 is CPT-evenand the vacuum expectation value of its derivative 120597

120583Δ120572120573 is

CPT-odd Coefficients (119896119865)120572120573120583] and (119896

119860119865)120572 are CPT-even and

CPT-odd respectivelyThe CPT-even terms in SMEmight beunderstood as originated from somegeneral relativity consid-eration as proposed in [13 14] (119896

119865)120572120573120583] is antisymmetric for

the first pair indices 120572 and 120573 antisymmetric for the secondpair 120583 and ] and symmetric for the interchange of the twopairs of indices Hence there are 21 degrees of freedom in(119896119865)120572120573120583] With the redefinition of the gauge field there are

2 degrees of freedom to be reduced in (119896119865)120572120573120583] So there are

19 independent degrees of freedom under the redefinitionof the fields and 21 degrees of freedom in general withoutconsidering this redefinition Another 4 degrees of freedom

Advances in High Energy Physics 3

are in (119896119860119865

)120572 After all consideration there are 19 + 4 =

23 independent degrees of freedom for (119896119860119865

)120572 and (119896

119865)120572120573120583]

to consider the field redefinition and 21 + 4 = 25 oneswithout considering this redefinition in general in the purephoton sector of the minimal SME [23] Given the situationthat the Lorentz violation matrix Δ

120572120573 here is coupled withother types of fermions and bosons there are no universalredefinitions for all the fields of different particles yet So wediscuss mainly the general form of Δ120572120573 with all the 16 degreesof freedom fit data from various experiments and obtain themagnitudes or constraints by the experiments avoiding anya priori assumption on Δ120572120573

We can make a direct correspondence between the twoLagrangians in (2) and (4) (cf the table in [24]) whenconsidering the vacuum expectation values of both Δ120572120573

(CPT-even) and its derivative 120597120583Δ120572120573 (CPT-odd) as coupling

constants and Lorentz violation parameters In the case thatjust the Lorentz violation matrix Δ

120572120573 is adopted as theviolation parameters there are terms left in (2) which cannotbe covered by the Lagrangian in (5) Comparing directly (3)with (5) we cannot find a direct term-to-term equivalencebetween the Lagrangians of the free photon sector in the SMSand the minimal SME

Here we treatΔ120572120573 as its vacuum expectation value that isas coupling constants in the field theory frameworkThen anyderivatives 120597

120583Δ120572120573 vanish in the following partial integrations

during the derivationsIn the standard model supplement the motion equation

for free photons is

Π120574120588

SMS119860120588 = 0 (6)

and the Lagrangian in (2) reads also

LSMS = minus1

2119860120574Π120574120588

SMS119860120588 (7)

where

Π120574120588

SMS = minus1198921205741205881205972+ 120597120574120597120588

+ Δ120574120572120597120588120597120572+ Δ120588120572120597120574120597120572+ Δ120574120573Δ120588]120597120573120597]

minus 119892120574120588(2Δ120583120572120597120583120597120572+ 119892120572120583Δ120572120573Δ120583]120597120573120597])

(8)

From (2) to (7) partial integrations are used We use theFourier transformation 119860

120588(119909) = 119860

120588(119901) exp(minus119894119901 sdot 119909) to get

Π120574120588

SMS (119901) = 1198921205741205881199012minus 119901120574119901120588

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

+ 119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

(9)

For the free photon in the minimal SME the Lagrangian issimilar

LSME = minus1

2119860120574Π120574120588

SME119860120588 (10)

where

Π120574120588

SME = minus1198921205741205881205972+ 120597120574120597120588+ 2(119896119865)120574120572120573120588

120597120572120597120573

+ 2(119896119860119865

)120572120598120574120572120573120588

120597120573

(11)

and the representation in momentum space is

Π120574120588

SME (119901) = 1198921205741205881199012minus 119901120574119901120588minus 2(119896119865)120574120572120573120588

119901120572119901120573

minus 2119894(119896119860119865

)120572120598120574120572120573120588

119901120573

(12)

We see that Π120574120588

SMS(119901) and Π120574120588

SME(119901) are the inverse of thephoton propagator in the momentum space The propagatordetermines the propagating properties of photons

When the two Lagrangians equations (2) and (4) areequivalent to each other for free photons that is we considerthe common part (intersection) between the two modelssome enlightenments are expected to come We can getΠ120574120588

SMS = Π120574120588

SME Then the matrix equation is satisfied

119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

= minus2(119896119865)120574120572120573120588

119901120572119901120573minus 2119894(119896

119860119865)120572120598120574120572120573120588

119901120573

(13)

Making derivative with respect to momentum 119901120572 for twotimes we obtain

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)120574(120572120573)120588

minus 4119894(119896119860119865

)120583120598120583120574120588(120572119897119904

120573

)

(14)

We accept conventions of general relativity for the notationof indices here and in the following derivations The coef-ficient 119897119904

120573is introduced here and its dimension is [length]

or [mass]minus1 119897119904120573represents the characteristic length of the

physical process Based on the symmetricantisymmetricproperties of indices 120574 and 120588 we get two matrix equationsfurther

minus119894(119896119860119865

)120583120598120583120574120588(120572119897

119904

120573)= (119896119865)[120574|(120572120573)|120588]

= 0 (15)

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)(120574|(120572120573)|120588)

= minus4(119896119865)120574(120572120573)120588

(16)

The general formula (16) here demonstrates the relations ofthe Lorentz violationmatrix Δ

120572120573and the coefficient (119896

119865)120574120572120573120588

for the intersection of the SMS and the minimal SME

4 Advances in High Energy Physics

When we take 119896119860119865

and 119896119865of (15) and (16) into the

Lagrangians of the minimal SME of (5) we find that theLagrangian of the minimal SME of (4) can be converted tothat of the SMS of (2)This tells us that the free photon sectorof the SMS can be considered as a subset of the minimalSME provided that the coefficients (119896

119860119865)120583 and (119896

119865)120574120572120573120588

areconstrained by (15) and (16)

The identity equation (15) tells us the relations between(119896119865)120574120572120573120588

and (119896119860119865

)120583 The constraints mean that the violation

coefficient (119896119860119865

)120583 (CPT-odd) vanishes in the photon sector of

the minimal SME that is

(119896119860119865

)120583= 0 (17)

There is a maximal sensitivity 10minus42 sim 10minus43 GeV for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11in Table 2

These four parameters are defined in terms of coefficient(119896119860119865

)120572 in Table 3 In (16) tensor (119896

119865)120574120572120573120588

is antisymmetricfor indices 120574 120572 and antisymmetric for indices 120573 120588 At thesame time (119896

119865)120574120572120573120588

= (119896119865)120573120588120574120572

So there are 21 degrees offreedom in (119896

119865)120574120572120573120588

generally and 21 minus 6 = 15 independentelements in tensor (119896

119865)120574(120572120573)120588

We have already known thatthere are 16 degrees of freedom for Δ120572120573 without the fieldredefinitions being consideredTherefore any one ofΔ120572120573 and(119896119865)120574120572120573120588

cannot completely determine the other one In (16)there are 15 degrees of freedomof the Lorentz violationmatrixΔ120572120573

related with 15 ones of the tensor (119896119865)120574120572120573120588

A definiteΔ120572120573can determine at most 15 degrees of freedom of (119896

119865)120574120572120573120588

andvice versa

We consider an example that Δ120572120573

is a symmetric matrixThen we can get the explicit form for it Multiplying 119892120574120588 onboth sides of (16) we obtain

3119892120583]Δ120583120572Δ ]120573 + 6Δ

(120572120573)= minus2(119896

119865)120572120573 (18)

where we define (119896119865)120572120573

= 119892120574120588(119896119865)120574120572120573120588

The tensor (119896119865)120574120572120573120588

has most of the properties of the Riemann curvature tensorSo (119896119865)120572120573

is a ldquoRicci tensorrdquo and satisfies that (119896119865)120572120573

= (119896119865)120573120572

Aswe know that (119896119865)120572120573

≪ 1 andΔ120572120573

≪ 1 the solution ofΔ120572120573

in terms of (119896119865)120572120573 to the second order is

Δ120572120573

= minus1

3(119896119865)120572120573

minus1

18119892120583](119896119865)120583120572(119896119865)]120573 (19)

With the assumption of Δ120572120573

being a symmetric matrix thereare 10 independent elements in it Then the Lorentz invari-ance violation matrix for the free photon can be obtainedfrom the tensor (119896

119865)120574120572120573120588

completely and can be consideredas somewhat a kind of Ricci tensor In the following partthe general case of Δ

120572120573with all the 16 degrees of freedom is

considered

3 Lorentz Violation Matrix of Photons

The Lorenz violation parameters of the minimal SME havebeen constrained from various recent experiments The datacan also provide bounds on themagnitudes of the elements ofthe Lorentz violation matrix of photons in the SMS through(15) and (16) above The Lorentz violation parameters ofthe SME commonly used in experiments are four matrices(120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 in [23]

(120581119890+)119895119896

= minus(119896119865)01198950119896

+1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

(120581119890minus)119895119896

= minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896

(120581119900+)119895119896

= minus1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(120581119900minus)119895119896

=1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(20)

The same indices mean summation More parameters relatedwith the Lorentz violation matrix are listed in Table 3

In terms of (119896119865)120572120573120583] the four matrices can be rewritten as

follows

(120581119890+)119895119896

= (

minus (119896119865)0101

+ (119896119865)2323

minus (119896119865)0102

+ (119896119865)2331

minus (119896119865)0103

+ (119896119865)2312

rdquo minus (119896119865)0202

+ (119896119865)3131

minus (119896119865)0203

+ (119896119865)3112

rdquo rdquo minus (119896119865)0303

+ (119896119865)1212

) (21)

which is symmetric for the indices 119895 and 119896

(120581119890minus)119895119896

= (

minus (119896119865)0101

minus (119896119865)2323

+ 120572 minus (119896119865)0102

minus (119896119865)2331

minus(119896119865)0103

minus (119896119865)2312

rdquo minus(119896119865)0202

minus (119896119865)3131

+ 120572 minus(119896119865)0203

minus (119896119865)3112

rdquo rdquo minus(119896119865)0303

minus (119896119865)1212

+ 120572

) (22)

Advances in High Energy Physics 5

which is symmetric for the indices 119895 and 119896 and 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 (119896119865)0131

minus (119896119865)0223

(119896119865)0112

minus (119896119865)0323

rdquo 0 (119896119865)0212

minus (119896119865)0331

rdquo rdquo 0

)

(23)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

2(119896119865)0123

(119896119865)0131

+ (119896119865)0223

(119896119865)0112

+ (119896119865)0323

rdquo 2(119896119865)0231

(119896119865)0212

+ (119896119865)0331

rdquo rdquo 2(119896119865)0312

)

(24)

which is symmetric for the indices 119895 and 119896 The symbol rdquoabove means that the matrix element is not written explicitlyfor brevity and it can be obtained from the property ofsymmetryantisymmetry of the corresponding matrix

With (16) we can replace the tensor (119896119865)120572120573120583] with the

Lorentz violation matrix Δ120572120573 So the four matrices above arerewritten like

(120581119890+)119895119896

= (

1

2Δ

1

2(Δ10Δ20

+ Δ23Δ13)

1

2(Δ10Δ30

+ Δ32Δ12)

rdquo 1

2Δ

1

2(Δ20Δ30 + Δ31Δ21)

rdquo rdquo 1

2Δ

)

(25)

which is symmetric for the indices 119895 and 119896 with Δ = Δ120572120573119892120572120573

(120581119890minus)119895119896

= (

1

2Δ00 +

1

6Δ11 minus

5

6Δ22 minus

5

6Δ33

1

2(Δ10Δ20 minus Δ23Δ13)

1

2(Δ10Δ30 minus Δ32Δ12)

rdquo 1

2Δ00 +

1

6Δ22 minus

5

6Δ11 minus

5

6Δ33

1

2(Δ20Δ30 minus Δ31Δ21)

rdquo rdquo 1

2Δ00 +

1

6Δ33 minus

5

6Δ11 minus

5

6Δ22

) (26)

which is symmetric for the indices 119895 and 119896 with 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 minus1

2(Δ02Δ32 + Δ01Δ31)

1

2(Δ03Δ23 + Δ01Δ21)

rdquo 0 minus1

2(Δ03Δ13

+ Δ02Δ12)

rdquo rdquo 0

)

(27)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

1

2(Δ02Δ32 minus Δ01Δ31)

1

2(Δ01Δ21 minus Δ03Δ23)

rdquo 1

2(Δ03Δ13 minus Δ02Δ12)

rdquo rdquo

)

(28)

which is symmetric for the indices 119895 and 119896 The metrictensor 119892

120572120573= diag(minus1 1 1 1) is used here The symbol

ldquordquo in matrix (120581119900minus)119895119896 denotes that the involved elements

(119896119865)0123 (119896

119865)0231 and (119896

119865)0312 cannot be written in terms

of the Lorentz violation matrix Δ120572120573 through (16) We haveknown from the preceding section that only 15 degrees offreedom inmatrixΔ120572120573 are interrelated with the same number

of independent degrees of freedom in tensor (119896119865)120574120572120573120588

and viceversa Besides the representation of (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896

and (120581119900minus)119895119896 in terms of Δ120572120573 the relations between other

Lorentz violation parameters commonly used in the minimalSMEand the Lorentz violationmatrix here are summarized inTable 3There are 15 independent expressions in terms of Δ120572120573

appearing in the four matrices (120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and

(120581119900minus)119895119896 These 15 independent expressions help to determine

15 degrees of freedom of the Lorentz violation matrix Δ120572120573With the recent maximal sensitivities attained from cur-

rent experiments for Lorentz violation parameters (120581119890+)119895119896

(120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 of the free photon sector in the

minimal SME (see Table 2) we get the maximal sensitivitiesor the conservative bounds from experiments for Lorentzinvariance matrix Δ120572120573photon

(

3Δ33 + 10minus17 10minus5 10minus5 10minus6

10minus9 Δ33 + 10minus17 10minus9 10minus9

10minus9 10minus9 Δ33 + 10minus17 10minus9

10minus9 10minus8 10minus8 Δ33

) (29)

in the Sun-centered inertial reference frame [23 29] Thepublication [23] claimed a 2120590 limit on Lorentz violationcoefficients (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 (120581119900minus)119895119896 and so forth in

Table 2 The 15 independent degrees of freedom of Δ120572120573 aredetermined and there is still one freedom Δ33 remainingunclear The maximal sensitivity for the elements of Δ120572120573 islisted in Table 1

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

2 Advances in High Energy Physics

The standard model extension is an effective frameworkfor phenomenological analysis We still need a fundamentaltheory to derive the Lorentz violation terms from basicprinciples An attempt for such a purpose has been offered in[24 25] in which a more basic principle denoted as physicalinvariance or physics independence is proposed to extendthe basic principle of relativity Instead of the requirementthat the equations describing the laws of physics have thesame form in all admissible frames of reference it requiresthat the equations describing the laws of physics have thesame form in all admissible mathematical manifolds Thisprinciple leads to the following replacement of the ordinarypartial 120597

120572and the covariant derivative119863

120572

120597120572997888rarr 119872

120572120573120597120573 119863

120572997888rarr 119872

120572120573119863120573 (1)

where 119872120572120573 is a local matrix We separate it to two matriceslike 119872120572120573 = 119892120572120573 + Δ120572120573 where 119892120572120573 is the metric tensorof space-time and Δ120572120573 is a new matrix which is particle-type dependent generally Since 119892120572120573 is Lorentz invariant Δ120572120573contains all the Lorentz violating degrees of freedom from119872120572120573ThenΔ120572120573 brings new terms violating Lorentz invariancein the standard model and is called Lorentz invarianceviolation matrix This new framework can be referred to asthe Standard Model Supplemented with Lorentz ViolationMatrix or Standard Model Supplement (SMS) [24 25] forshort and it has been applied to discuss the Lorentz violationeffects for protons [24] photons [26] and neutrinos [27 28]

Before accepting the SMS as a fundamental theory onecan take the SMS as an effective framework for phenomeno-logical applications by confronting various experiments todetermine andor constrain the Lorentz violation matrix Δ120572120573for various particles From a more general sense the SMSshould be a subset of a general version of the SME Howeverin the case of theminimal version of the SME the relationshipbetween the SMS and the SME is unclear yet The purpose ofthis paper is to compare the Lorentz violation effects of thephoton sector between the two models As have been wellknown light has always played a significant role in the devel-opments of physics and the Lorentz violation in the photonsector is also under active investigations both theoreticallyand experimentally [2] The Lorentz violation parameters ofphotons in the SME have been well constrained by variousexperiments as summarized in [23] By confronting thecollected data in [23] we can obtain bounds and detailedstructure of the Lorentz violation parameters in the modelSMS [24ndash26]

This paper is organized as follows Section 2 providesthe relation between the photon Lorentz violation matrix inthe SMS and various tensor fields in the minimal SME inthe case that the two models give the same results Theserelations define the boundary of the intersection of these twotheories too In Section 3 we discuss the general structureof the photon Lorentz violation matrix in our model andits implications for the potential property of the space-timestructure for photons At the same time we obtain theconstraints on the elements of the photon Lorentz violationmatrix Then conclusion is given in the last section

2 Relations between Two Models

In the standard model extension (SME) the terms thatviolate Lorentz invariance are added by hands from someconsiderations such as gauge invariance Hermitean andpower-counting renormalizability We consider just the min-imal SME [22] here The Lorentz violation terms in theminimal SME contain many tensor fields as coupling con-stants with their magnitudes to be determinedconstrainedby experiments Though these Lorentz violation terms areallowed by some general considerations the reason for theirexistence still needs to be provided from theoretical aspectsIn the SMS [24 25] the Lorentz violation terms arise fromthe replacement of (1) which is considered as a necessaryrequirement from the basic principle of physical invarianceAs both of the two models are built within the frameworkof effective field theory in particle physics they can beconsidered as two special cases of a general standard modelextension within the effective field theory It is thereforenecessary to study the relationship between the two models

TheLagrangians of the pure photon sector in the SMS andthe minimal SME are

LSMS = minus1

4119865120572120573119865120572120573

+LGV (2)

where

LGV = minus1

2Δ120572120573Δ120583](119892120572120583120597120573119860120588120597]119860120588 minus 120597

120573119860120583120597]119860120572)

minus 119865120583]Δ120583120572120597120572119860

]

(3)

LSME = minus1

4119865120572120573119865120572120573

+LCPT-evenphoton +L

CPT-oddphoton (4)

where

LCPT-evenphoton = minus

1

4(119896119865)120572120573120583]119865120572120573119865120583]

LCPT-oddphoton =

1

2(119896119860119865

)120572120598120572120573120583]119860

120573119865120583]

(5)

We denote the matrix Δ120572120573 above as Lorentz invarianceviolation matrix whose dimension is massless When weignore the field redefinition there are 16 independent dimen-sionless degrees of freedom inΔ120572120573 generally [25] As couplingconstants the vacuum expectation value of Δ120572120573 is CPT-evenand the vacuum expectation value of its derivative 120597

120583Δ120572120573 is

CPT-odd Coefficients (119896119865)120572120573120583] and (119896

119860119865)120572 are CPT-even and

CPT-odd respectivelyThe CPT-even terms in SMEmight beunderstood as originated from somegeneral relativity consid-eration as proposed in [13 14] (119896

119865)120572120573120583] is antisymmetric for

the first pair indices 120572 and 120573 antisymmetric for the secondpair 120583 and ] and symmetric for the interchange of the twopairs of indices Hence there are 21 degrees of freedom in(119896119865)120572120573120583] With the redefinition of the gauge field there are

2 degrees of freedom to be reduced in (119896119865)120572120573120583] So there are

19 independent degrees of freedom under the redefinitionof the fields and 21 degrees of freedom in general withoutconsidering this redefinition Another 4 degrees of freedom

Advances in High Energy Physics 3

are in (119896119860119865

)120572 After all consideration there are 19 + 4 =

23 independent degrees of freedom for (119896119860119865

)120572 and (119896

119865)120572120573120583]

to consider the field redefinition and 21 + 4 = 25 oneswithout considering this redefinition in general in the purephoton sector of the minimal SME [23] Given the situationthat the Lorentz violation matrix Δ

120572120573 here is coupled withother types of fermions and bosons there are no universalredefinitions for all the fields of different particles yet So wediscuss mainly the general form of Δ120572120573 with all the 16 degreesof freedom fit data from various experiments and obtain themagnitudes or constraints by the experiments avoiding anya priori assumption on Δ120572120573

We can make a direct correspondence between the twoLagrangians in (2) and (4) (cf the table in [24]) whenconsidering the vacuum expectation values of both Δ120572120573

(CPT-even) and its derivative 120597120583Δ120572120573 (CPT-odd) as coupling

constants and Lorentz violation parameters In the case thatjust the Lorentz violation matrix Δ

120572120573 is adopted as theviolation parameters there are terms left in (2) which cannotbe covered by the Lagrangian in (5) Comparing directly (3)with (5) we cannot find a direct term-to-term equivalencebetween the Lagrangians of the free photon sector in the SMSand the minimal SME

Here we treatΔ120572120573 as its vacuum expectation value that isas coupling constants in the field theory frameworkThen anyderivatives 120597

120583Δ120572120573 vanish in the following partial integrations

during the derivationsIn the standard model supplement the motion equation

for free photons is

Π120574120588

SMS119860120588 = 0 (6)

and the Lagrangian in (2) reads also

LSMS = minus1

2119860120574Π120574120588

SMS119860120588 (7)

where

Π120574120588

SMS = minus1198921205741205881205972+ 120597120574120597120588

+ Δ120574120572120597120588120597120572+ Δ120588120572120597120574120597120572+ Δ120574120573Δ120588]120597120573120597]

minus 119892120574120588(2Δ120583120572120597120583120597120572+ 119892120572120583Δ120572120573Δ120583]120597120573120597])

(8)

From (2) to (7) partial integrations are used We use theFourier transformation 119860

120588(119909) = 119860

120588(119901) exp(minus119894119901 sdot 119909) to get

Π120574120588

SMS (119901) = 1198921205741205881199012minus 119901120574119901120588

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

+ 119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

(9)

For the free photon in the minimal SME the Lagrangian issimilar

LSME = minus1

2119860120574Π120574120588

SME119860120588 (10)

where

Π120574120588

SME = minus1198921205741205881205972+ 120597120574120597120588+ 2(119896119865)120574120572120573120588

120597120572120597120573

+ 2(119896119860119865

)120572120598120574120572120573120588

120597120573

(11)

and the representation in momentum space is

Π120574120588

SME (119901) = 1198921205741205881199012minus 119901120574119901120588minus 2(119896119865)120574120572120573120588

119901120572119901120573

minus 2119894(119896119860119865

)120572120598120574120572120573120588

119901120573

(12)

We see that Π120574120588

SMS(119901) and Π120574120588

SME(119901) are the inverse of thephoton propagator in the momentum space The propagatordetermines the propagating properties of photons

When the two Lagrangians equations (2) and (4) areequivalent to each other for free photons that is we considerthe common part (intersection) between the two modelssome enlightenments are expected to come We can getΠ120574120588

SMS = Π120574120588

SME Then the matrix equation is satisfied

119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

= minus2(119896119865)120574120572120573120588

119901120572119901120573minus 2119894(119896

119860119865)120572120598120574120572120573120588

119901120573

(13)

Making derivative with respect to momentum 119901120572 for twotimes we obtain

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)120574(120572120573)120588

minus 4119894(119896119860119865

)120583120598120583120574120588(120572119897119904

120573

)

(14)

We accept conventions of general relativity for the notationof indices here and in the following derivations The coef-ficient 119897119904

120573is introduced here and its dimension is [length]

or [mass]minus1 119897119904120573represents the characteristic length of the

physical process Based on the symmetricantisymmetricproperties of indices 120574 and 120588 we get two matrix equationsfurther

minus119894(119896119860119865

)120583120598120583120574120588(120572119897

119904

120573)= (119896119865)[120574|(120572120573)|120588]

= 0 (15)

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)(120574|(120572120573)|120588)

= minus4(119896119865)120574(120572120573)120588

(16)

The general formula (16) here demonstrates the relations ofthe Lorentz violationmatrix Δ

120572120573and the coefficient (119896

119865)120574120572120573120588

for the intersection of the SMS and the minimal SME

4 Advances in High Energy Physics

When we take 119896119860119865

and 119896119865of (15) and (16) into the

Lagrangians of the minimal SME of (5) we find that theLagrangian of the minimal SME of (4) can be converted tothat of the SMS of (2)This tells us that the free photon sectorof the SMS can be considered as a subset of the minimalSME provided that the coefficients (119896

119860119865)120583 and (119896

119865)120574120572120573120588

areconstrained by (15) and (16)

The identity equation (15) tells us the relations between(119896119865)120574120572120573120588

and (119896119860119865

)120583 The constraints mean that the violation

coefficient (119896119860119865

)120583 (CPT-odd) vanishes in the photon sector of

the minimal SME that is

(119896119860119865

)120583= 0 (17)

There is a maximal sensitivity 10minus42 sim 10minus43 GeV for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11in Table 2

These four parameters are defined in terms of coefficient(119896119860119865

)120572 in Table 3 In (16) tensor (119896

119865)120574120572120573120588

is antisymmetricfor indices 120574 120572 and antisymmetric for indices 120573 120588 At thesame time (119896

119865)120574120572120573120588

= (119896119865)120573120588120574120572

So there are 21 degrees offreedom in (119896

119865)120574120572120573120588

generally and 21 minus 6 = 15 independentelements in tensor (119896

119865)120574(120572120573)120588

We have already known thatthere are 16 degrees of freedom for Δ120572120573 without the fieldredefinitions being consideredTherefore any one ofΔ120572120573 and(119896119865)120574120572120573120588

cannot completely determine the other one In (16)there are 15 degrees of freedomof the Lorentz violationmatrixΔ120572120573

related with 15 ones of the tensor (119896119865)120574120572120573120588

A definiteΔ120572120573can determine at most 15 degrees of freedom of (119896

119865)120574120572120573120588

andvice versa

We consider an example that Δ120572120573

is a symmetric matrixThen we can get the explicit form for it Multiplying 119892120574120588 onboth sides of (16) we obtain

3119892120583]Δ120583120572Δ ]120573 + 6Δ

(120572120573)= minus2(119896

119865)120572120573 (18)

where we define (119896119865)120572120573

= 119892120574120588(119896119865)120574120572120573120588

The tensor (119896119865)120574120572120573120588

has most of the properties of the Riemann curvature tensorSo (119896119865)120572120573

is a ldquoRicci tensorrdquo and satisfies that (119896119865)120572120573

= (119896119865)120573120572

Aswe know that (119896119865)120572120573

≪ 1 andΔ120572120573

≪ 1 the solution ofΔ120572120573

in terms of (119896119865)120572120573 to the second order is

Δ120572120573

= minus1

3(119896119865)120572120573

minus1

18119892120583](119896119865)120583120572(119896119865)]120573 (19)

With the assumption of Δ120572120573

being a symmetric matrix thereare 10 independent elements in it Then the Lorentz invari-ance violation matrix for the free photon can be obtainedfrom the tensor (119896

119865)120574120572120573120588

completely and can be consideredas somewhat a kind of Ricci tensor In the following partthe general case of Δ

120572120573with all the 16 degrees of freedom is

considered

3 Lorentz Violation Matrix of Photons

The Lorenz violation parameters of the minimal SME havebeen constrained from various recent experiments The datacan also provide bounds on themagnitudes of the elements ofthe Lorentz violation matrix of photons in the SMS through(15) and (16) above The Lorentz violation parameters ofthe SME commonly used in experiments are four matrices(120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 in [23]

(120581119890+)119895119896

= minus(119896119865)01198950119896

+1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

(120581119890minus)119895119896

= minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896

(120581119900+)119895119896

= minus1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(120581119900minus)119895119896

=1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(20)

The same indices mean summation More parameters relatedwith the Lorentz violation matrix are listed in Table 3

In terms of (119896119865)120572120573120583] the four matrices can be rewritten as

follows

(120581119890+)119895119896

= (

minus (119896119865)0101

+ (119896119865)2323

minus (119896119865)0102

+ (119896119865)2331

minus (119896119865)0103

+ (119896119865)2312

rdquo minus (119896119865)0202

+ (119896119865)3131

minus (119896119865)0203

+ (119896119865)3112

rdquo rdquo minus (119896119865)0303

+ (119896119865)1212

) (21)

which is symmetric for the indices 119895 and 119896

(120581119890minus)119895119896

= (

minus (119896119865)0101

minus (119896119865)2323

+ 120572 minus (119896119865)0102

minus (119896119865)2331

minus(119896119865)0103

minus (119896119865)2312

rdquo minus(119896119865)0202

minus (119896119865)3131

+ 120572 minus(119896119865)0203

minus (119896119865)3112

rdquo rdquo minus(119896119865)0303

minus (119896119865)1212

+ 120572

) (22)

Advances in High Energy Physics 5

which is symmetric for the indices 119895 and 119896 and 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 (119896119865)0131

minus (119896119865)0223

(119896119865)0112

minus (119896119865)0323

rdquo 0 (119896119865)0212

minus (119896119865)0331

rdquo rdquo 0

)

(23)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

2(119896119865)0123

(119896119865)0131

+ (119896119865)0223

(119896119865)0112

+ (119896119865)0323

rdquo 2(119896119865)0231

(119896119865)0212

+ (119896119865)0331

rdquo rdquo 2(119896119865)0312

)

(24)

which is symmetric for the indices 119895 and 119896 The symbol rdquoabove means that the matrix element is not written explicitlyfor brevity and it can be obtained from the property ofsymmetryantisymmetry of the corresponding matrix

With (16) we can replace the tensor (119896119865)120572120573120583] with the

Lorentz violation matrix Δ120572120573 So the four matrices above arerewritten like

(120581119890+)119895119896

= (

1

2Δ

1

2(Δ10Δ20

+ Δ23Δ13)

1

2(Δ10Δ30

+ Δ32Δ12)

rdquo 1

2Δ

1

2(Δ20Δ30 + Δ31Δ21)

rdquo rdquo 1

2Δ

)

(25)

which is symmetric for the indices 119895 and 119896 with Δ = Δ120572120573119892120572120573

(120581119890minus)119895119896

= (

1

2Δ00 +

1

6Δ11 minus

5

6Δ22 minus

5

6Δ33

1

2(Δ10Δ20 minus Δ23Δ13)

1

2(Δ10Δ30 minus Δ32Δ12)

rdquo 1

2Δ00 +

1

6Δ22 minus

5

6Δ11 minus

5

6Δ33

1

2(Δ20Δ30 minus Δ31Δ21)

rdquo rdquo 1

2Δ00 +

1

6Δ33 minus

5

6Δ11 minus

5

6Δ22

) (26)

which is symmetric for the indices 119895 and 119896 with 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 minus1

2(Δ02Δ32 + Δ01Δ31)

1

2(Δ03Δ23 + Δ01Δ21)

rdquo 0 minus1

2(Δ03Δ13

+ Δ02Δ12)

rdquo rdquo 0

)

(27)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

1

2(Δ02Δ32 minus Δ01Δ31)

1

2(Δ01Δ21 minus Δ03Δ23)

rdquo 1

2(Δ03Δ13 minus Δ02Δ12)

rdquo rdquo

)

(28)

which is symmetric for the indices 119895 and 119896 The metrictensor 119892

120572120573= diag(minus1 1 1 1) is used here The symbol

ldquordquo in matrix (120581119900minus)119895119896 denotes that the involved elements

(119896119865)0123 (119896

119865)0231 and (119896

119865)0312 cannot be written in terms

of the Lorentz violation matrix Δ120572120573 through (16) We haveknown from the preceding section that only 15 degrees offreedom inmatrixΔ120572120573 are interrelated with the same number

of independent degrees of freedom in tensor (119896119865)120574120572120573120588

and viceversa Besides the representation of (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896

and (120581119900minus)119895119896 in terms of Δ120572120573 the relations between other

Lorentz violation parameters commonly used in the minimalSMEand the Lorentz violationmatrix here are summarized inTable 3There are 15 independent expressions in terms of Δ120572120573

appearing in the four matrices (120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and

(120581119900minus)119895119896 These 15 independent expressions help to determine

15 degrees of freedom of the Lorentz violation matrix Δ120572120573With the recent maximal sensitivities attained from cur-

rent experiments for Lorentz violation parameters (120581119890+)119895119896

(120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 of the free photon sector in the

minimal SME (see Table 2) we get the maximal sensitivitiesor the conservative bounds from experiments for Lorentzinvariance matrix Δ120572120573photon

(

3Δ33 + 10minus17 10minus5 10minus5 10minus6

10minus9 Δ33 + 10minus17 10minus9 10minus9

10minus9 10minus9 Δ33 + 10minus17 10minus9

10minus9 10minus8 10minus8 Δ33

) (29)

in the Sun-centered inertial reference frame [23 29] Thepublication [23] claimed a 2120590 limit on Lorentz violationcoefficients (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 (120581119900minus)119895119896 and so forth in

Table 2 The 15 independent degrees of freedom of Δ120572120573 aredetermined and there is still one freedom Δ33 remainingunclear The maximal sensitivity for the elements of Δ120572120573 islisted in Table 1

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

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Advances in High Energy Physics 3

are in (119896119860119865

)120572 After all consideration there are 19 + 4 =

23 independent degrees of freedom for (119896119860119865

)120572 and (119896

119865)120572120573120583]

to consider the field redefinition and 21 + 4 = 25 oneswithout considering this redefinition in general in the purephoton sector of the minimal SME [23] Given the situationthat the Lorentz violation matrix Δ

120572120573 here is coupled withother types of fermions and bosons there are no universalredefinitions for all the fields of different particles yet So wediscuss mainly the general form of Δ120572120573 with all the 16 degreesof freedom fit data from various experiments and obtain themagnitudes or constraints by the experiments avoiding anya priori assumption on Δ120572120573

We can make a direct correspondence between the twoLagrangians in (2) and (4) (cf the table in [24]) whenconsidering the vacuum expectation values of both Δ120572120573

(CPT-even) and its derivative 120597120583Δ120572120573 (CPT-odd) as coupling

constants and Lorentz violation parameters In the case thatjust the Lorentz violation matrix Δ

120572120573 is adopted as theviolation parameters there are terms left in (2) which cannotbe covered by the Lagrangian in (5) Comparing directly (3)with (5) we cannot find a direct term-to-term equivalencebetween the Lagrangians of the free photon sector in the SMSand the minimal SME

Here we treatΔ120572120573 as its vacuum expectation value that isas coupling constants in the field theory frameworkThen anyderivatives 120597

120583Δ120572120573 vanish in the following partial integrations

during the derivationsIn the standard model supplement the motion equation

for free photons is

Π120574120588

SMS119860120588 = 0 (6)

and the Lagrangian in (2) reads also

LSMS = minus1

2119860120574Π120574120588

SMS119860120588 (7)

where

Π120574120588

SMS = minus1198921205741205881205972+ 120597120574120597120588

+ Δ120574120572120597120588120597120572+ Δ120588120572120597120574120597120572+ Δ120574120573Δ120588]120597120573120597]

minus 119892120574120588(2Δ120583120572120597120583120597120572+ 119892120572120583Δ120572120573Δ120583]120597120573120597])

(8)

From (2) to (7) partial integrations are used We use theFourier transformation 119860

120588(119909) = 119860

120588(119901) exp(minus119894119901 sdot 119909) to get

Π120574120588

SMS (119901) = 1198921205741205881199012minus 119901120574119901120588

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

+ 119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

(9)

For the free photon in the minimal SME the Lagrangian issimilar

LSME = minus1

2119860120574Π120574120588

SME119860120588 (10)

where

Π120574120588

SME = minus1198921205741205881205972+ 120597120574120597120588+ 2(119896119865)120574120572120573120588

120597120572120597120573

+ 2(119896119860119865

)120572120598120574120572120573120588

120597120573

(11)

and the representation in momentum space is

Π120574120588

SME (119901) = 1198921205741205881199012minus 119901120574119901120588minus 2(119896119865)120574120572120573120588

119901120572119901120573

minus 2119894(119896119860119865

)120572120598120574120572120573120588

119901120573

(12)

We see that Π120574120588

SMS(119901) and Π120574120588

SME(119901) are the inverse of thephoton propagator in the momentum space The propagatordetermines the propagating properties of photons

When the two Lagrangians equations (2) and (4) areequivalent to each other for free photons that is we considerthe common part (intersection) between the two modelssome enlightenments are expected to come We can getΠ120574120588

SMS = Π120574120588

SME Then the matrix equation is satisfied

119892120574120588(2Δ120583120572119901120583119901120572+ 119892120572120583Δ120572120573Δ120583]119901120573119901])

minus Δ120574120572119901120588119901120572minus Δ120588120572119901120574119901120572minus Δ120574120573Δ120588]119901120573119901]

= minus2(119896119865)120574120572120573120588

119901120572119901120573minus 2119894(119896

119860119865)120572120598120574120572120573120588

119901120573

(13)

Making derivative with respect to momentum 119901120572 for twotimes we obtain

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)120574(120572120573)120588

minus 4119894(119896119860119865

)120583120598120583120574120588(120572119897119904

120573

)

(14)

We accept conventions of general relativity for the notationof indices here and in the following derivations The coef-ficient 119897119904

120573is introduced here and its dimension is [length]

or [mass]minus1 119897119904120573represents the characteristic length of the

physical process Based on the symmetricantisymmetricproperties of indices 120574 and 120588 we get two matrix equationsfurther

minus119894(119896119860119865

)120583120598120583120574120588(120572119897

119904

120573)= (119896119865)[120574|(120572120573)|120588]

= 0 (15)

2119892120574120588(Δ120572120573

+ Δ120573120572

+ 119892120583]Δ120583120572Δ ]120573)

minus Δ120574120572119892120588120573

minus Δ120574120573119892120588120572

minus Δ120588120572119892120574120573

minus Δ120588120573119892120574120572

minus Δ120574120572Δ120588120573

minus Δ120574120573Δ120588120572

= minus4(119896119865)(120574|(120572120573)|120588)

= minus4(119896119865)120574(120572120573)120588

(16)

The general formula (16) here demonstrates the relations ofthe Lorentz violationmatrix Δ

120572120573and the coefficient (119896

119865)120574120572120573120588

for the intersection of the SMS and the minimal SME

4 Advances in High Energy Physics

When we take 119896119860119865

and 119896119865of (15) and (16) into the

Lagrangians of the minimal SME of (5) we find that theLagrangian of the minimal SME of (4) can be converted tothat of the SMS of (2)This tells us that the free photon sectorof the SMS can be considered as a subset of the minimalSME provided that the coefficients (119896

119860119865)120583 and (119896

119865)120574120572120573120588

areconstrained by (15) and (16)

The identity equation (15) tells us the relations between(119896119865)120574120572120573120588

and (119896119860119865

)120583 The constraints mean that the violation

coefficient (119896119860119865

)120583 (CPT-odd) vanishes in the photon sector of

the minimal SME that is

(119896119860119865

)120583= 0 (17)

There is a maximal sensitivity 10minus42 sim 10minus43 GeV for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11in Table 2

These four parameters are defined in terms of coefficient(119896119860119865

)120572 in Table 3 In (16) tensor (119896

119865)120574120572120573120588

is antisymmetricfor indices 120574 120572 and antisymmetric for indices 120573 120588 At thesame time (119896

119865)120574120572120573120588

= (119896119865)120573120588120574120572

So there are 21 degrees offreedom in (119896

119865)120574120572120573120588

generally and 21 minus 6 = 15 independentelements in tensor (119896

119865)120574(120572120573)120588

We have already known thatthere are 16 degrees of freedom for Δ120572120573 without the fieldredefinitions being consideredTherefore any one ofΔ120572120573 and(119896119865)120574120572120573120588

cannot completely determine the other one In (16)there are 15 degrees of freedomof the Lorentz violationmatrixΔ120572120573

related with 15 ones of the tensor (119896119865)120574120572120573120588

A definiteΔ120572120573can determine at most 15 degrees of freedom of (119896

119865)120574120572120573120588

andvice versa

We consider an example that Δ120572120573

is a symmetric matrixThen we can get the explicit form for it Multiplying 119892120574120588 onboth sides of (16) we obtain

3119892120583]Δ120583120572Δ ]120573 + 6Δ

(120572120573)= minus2(119896

119865)120572120573 (18)

where we define (119896119865)120572120573

= 119892120574120588(119896119865)120574120572120573120588

The tensor (119896119865)120574120572120573120588

has most of the properties of the Riemann curvature tensorSo (119896119865)120572120573

is a ldquoRicci tensorrdquo and satisfies that (119896119865)120572120573

= (119896119865)120573120572

Aswe know that (119896119865)120572120573

≪ 1 andΔ120572120573

≪ 1 the solution ofΔ120572120573

in terms of (119896119865)120572120573 to the second order is

Δ120572120573

= minus1

3(119896119865)120572120573

minus1

18119892120583](119896119865)120583120572(119896119865)]120573 (19)

With the assumption of Δ120572120573

being a symmetric matrix thereare 10 independent elements in it Then the Lorentz invari-ance violation matrix for the free photon can be obtainedfrom the tensor (119896

119865)120574120572120573120588

completely and can be consideredas somewhat a kind of Ricci tensor In the following partthe general case of Δ

120572120573with all the 16 degrees of freedom is

considered

3 Lorentz Violation Matrix of Photons

The Lorenz violation parameters of the minimal SME havebeen constrained from various recent experiments The datacan also provide bounds on themagnitudes of the elements ofthe Lorentz violation matrix of photons in the SMS through(15) and (16) above The Lorentz violation parameters ofthe SME commonly used in experiments are four matrices(120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 in [23]

(120581119890+)119895119896

= minus(119896119865)01198950119896

+1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

(120581119890minus)119895119896

= minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896

(120581119900+)119895119896

= minus1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(120581119900minus)119895119896

=1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(20)

The same indices mean summation More parameters relatedwith the Lorentz violation matrix are listed in Table 3

In terms of (119896119865)120572120573120583] the four matrices can be rewritten as

follows

(120581119890+)119895119896

= (

minus (119896119865)0101

+ (119896119865)2323

minus (119896119865)0102

+ (119896119865)2331

minus (119896119865)0103

+ (119896119865)2312

rdquo minus (119896119865)0202

+ (119896119865)3131

minus (119896119865)0203

+ (119896119865)3112

rdquo rdquo minus (119896119865)0303

+ (119896119865)1212

) (21)

which is symmetric for the indices 119895 and 119896

(120581119890minus)119895119896

= (

minus (119896119865)0101

minus (119896119865)2323

+ 120572 minus (119896119865)0102

minus (119896119865)2331

minus(119896119865)0103

minus (119896119865)2312

rdquo minus(119896119865)0202

minus (119896119865)3131

+ 120572 minus(119896119865)0203

minus (119896119865)3112

rdquo rdquo minus(119896119865)0303

minus (119896119865)1212

+ 120572

) (22)

Advances in High Energy Physics 5

which is symmetric for the indices 119895 and 119896 and 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 (119896119865)0131

minus (119896119865)0223

(119896119865)0112

minus (119896119865)0323

rdquo 0 (119896119865)0212

minus (119896119865)0331

rdquo rdquo 0

)

(23)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

2(119896119865)0123

(119896119865)0131

+ (119896119865)0223

(119896119865)0112

+ (119896119865)0323

rdquo 2(119896119865)0231

(119896119865)0212

+ (119896119865)0331

rdquo rdquo 2(119896119865)0312

)

(24)

which is symmetric for the indices 119895 and 119896 The symbol rdquoabove means that the matrix element is not written explicitlyfor brevity and it can be obtained from the property ofsymmetryantisymmetry of the corresponding matrix

With (16) we can replace the tensor (119896119865)120572120573120583] with the

Lorentz violation matrix Δ120572120573 So the four matrices above arerewritten like

(120581119890+)119895119896

= (

1

2Δ

1

2(Δ10Δ20

+ Δ23Δ13)

1

2(Δ10Δ30

+ Δ32Δ12)

rdquo 1

2Δ

1

2(Δ20Δ30 + Δ31Δ21)

rdquo rdquo 1

2Δ

)

(25)

which is symmetric for the indices 119895 and 119896 with Δ = Δ120572120573119892120572120573

(120581119890minus)119895119896

= (

1

2Δ00 +

1

6Δ11 minus

5

6Δ22 minus

5

6Δ33

1

2(Δ10Δ20 minus Δ23Δ13)

1

2(Δ10Δ30 minus Δ32Δ12)

rdquo 1

2Δ00 +

1

6Δ22 minus

5

6Δ11 minus

5

6Δ33

1

2(Δ20Δ30 minus Δ31Δ21)

rdquo rdquo 1

2Δ00 +

1

6Δ33 minus

5

6Δ11 minus

5

6Δ22

) (26)

which is symmetric for the indices 119895 and 119896 with 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 minus1

2(Δ02Δ32 + Δ01Δ31)

1

2(Δ03Δ23 + Δ01Δ21)

rdquo 0 minus1

2(Δ03Δ13

+ Δ02Δ12)

rdquo rdquo 0

)

(27)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

1

2(Δ02Δ32 minus Δ01Δ31)

1

2(Δ01Δ21 minus Δ03Δ23)

rdquo 1

2(Δ03Δ13 minus Δ02Δ12)

rdquo rdquo

)

(28)

which is symmetric for the indices 119895 and 119896 The metrictensor 119892

120572120573= diag(minus1 1 1 1) is used here The symbol

ldquordquo in matrix (120581119900minus)119895119896 denotes that the involved elements

(119896119865)0123 (119896

119865)0231 and (119896

119865)0312 cannot be written in terms

of the Lorentz violation matrix Δ120572120573 through (16) We haveknown from the preceding section that only 15 degrees offreedom inmatrixΔ120572120573 are interrelated with the same number

of independent degrees of freedom in tensor (119896119865)120574120572120573120588

and viceversa Besides the representation of (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896

and (120581119900minus)119895119896 in terms of Δ120572120573 the relations between other

Lorentz violation parameters commonly used in the minimalSMEand the Lorentz violationmatrix here are summarized inTable 3There are 15 independent expressions in terms of Δ120572120573

appearing in the four matrices (120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and

(120581119900minus)119895119896 These 15 independent expressions help to determine

15 degrees of freedom of the Lorentz violation matrix Δ120572120573With the recent maximal sensitivities attained from cur-

rent experiments for Lorentz violation parameters (120581119890+)119895119896

(120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 of the free photon sector in the

minimal SME (see Table 2) we get the maximal sensitivitiesor the conservative bounds from experiments for Lorentzinvariance matrix Δ120572120573photon

(

3Δ33 + 10minus17 10minus5 10minus5 10minus6

10minus9 Δ33 + 10minus17 10minus9 10minus9

10minus9 10minus9 Δ33 + 10minus17 10minus9

10minus9 10minus8 10minus8 Δ33

) (29)

in the Sun-centered inertial reference frame [23 29] Thepublication [23] claimed a 2120590 limit on Lorentz violationcoefficients (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 (120581119900minus)119895119896 and so forth in

Table 2 The 15 independent degrees of freedom of Δ120572120573 aredetermined and there is still one freedom Δ33 remainingunclear The maximal sensitivity for the elements of Δ120572120573 islisted in Table 1

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

When we take 119896119860119865

and 119896119865of (15) and (16) into the

Lagrangians of the minimal SME of (5) we find that theLagrangian of the minimal SME of (4) can be converted tothat of the SMS of (2)This tells us that the free photon sectorof the SMS can be considered as a subset of the minimalSME provided that the coefficients (119896

119860119865)120583 and (119896

119865)120574120572120573120588

areconstrained by (15) and (16)

The identity equation (15) tells us the relations between(119896119865)120574120572120573120588

and (119896119860119865

)120583 The constraints mean that the violation

coefficient (119896119860119865

)120583 (CPT-odd) vanishes in the photon sector of

the minimal SME that is

(119896119860119865

)120583= 0 (17)

There is a maximal sensitivity 10minus42 sim 10minus43 GeV for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11in Table 2

These four parameters are defined in terms of coefficient(119896119860119865

)120572 in Table 3 In (16) tensor (119896

119865)120574120572120573120588

is antisymmetricfor indices 120574 120572 and antisymmetric for indices 120573 120588 At thesame time (119896

119865)120574120572120573120588

= (119896119865)120573120588120574120572

So there are 21 degrees offreedom in (119896

119865)120574120572120573120588

generally and 21 minus 6 = 15 independentelements in tensor (119896

119865)120574(120572120573)120588

We have already known thatthere are 16 degrees of freedom for Δ120572120573 without the fieldredefinitions being consideredTherefore any one ofΔ120572120573 and(119896119865)120574120572120573120588

cannot completely determine the other one In (16)there are 15 degrees of freedomof the Lorentz violationmatrixΔ120572120573

related with 15 ones of the tensor (119896119865)120574120572120573120588

A definiteΔ120572120573can determine at most 15 degrees of freedom of (119896

119865)120574120572120573120588

andvice versa

We consider an example that Δ120572120573

is a symmetric matrixThen we can get the explicit form for it Multiplying 119892120574120588 onboth sides of (16) we obtain

3119892120583]Δ120583120572Δ ]120573 + 6Δ

(120572120573)= minus2(119896

119865)120572120573 (18)

where we define (119896119865)120572120573

= 119892120574120588(119896119865)120574120572120573120588

The tensor (119896119865)120574120572120573120588

has most of the properties of the Riemann curvature tensorSo (119896119865)120572120573

is a ldquoRicci tensorrdquo and satisfies that (119896119865)120572120573

= (119896119865)120573120572

Aswe know that (119896119865)120572120573

≪ 1 andΔ120572120573

≪ 1 the solution ofΔ120572120573

in terms of (119896119865)120572120573 to the second order is

Δ120572120573

= minus1

3(119896119865)120572120573

minus1

18119892120583](119896119865)120583120572(119896119865)]120573 (19)

With the assumption of Δ120572120573

being a symmetric matrix thereare 10 independent elements in it Then the Lorentz invari-ance violation matrix for the free photon can be obtainedfrom the tensor (119896

119865)120574120572120573120588

completely and can be consideredas somewhat a kind of Ricci tensor In the following partthe general case of Δ

120572120573with all the 16 degrees of freedom is

considered

3 Lorentz Violation Matrix of Photons

The Lorenz violation parameters of the minimal SME havebeen constrained from various recent experiments The datacan also provide bounds on themagnitudes of the elements ofthe Lorentz violation matrix of photons in the SMS through(15) and (16) above The Lorentz violation parameters ofthe SME commonly used in experiments are four matrices(120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 in [23]

(120581119890+)119895119896

= minus(119896119865)01198950119896

+1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

(120581119890minus)119895119896

= minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896

(120581119900+)119895119896

= minus1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(120581119900minus)119895119896

=1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902

(20)

The same indices mean summation More parameters relatedwith the Lorentz violation matrix are listed in Table 3

In terms of (119896119865)120572120573120583] the four matrices can be rewritten as

follows

(120581119890+)119895119896

= (

minus (119896119865)0101

+ (119896119865)2323

minus (119896119865)0102

+ (119896119865)2331

minus (119896119865)0103

+ (119896119865)2312

rdquo minus (119896119865)0202

+ (119896119865)3131

minus (119896119865)0203

+ (119896119865)3112

rdquo rdquo minus (119896119865)0303

+ (119896119865)1212

) (21)

which is symmetric for the indices 119895 and 119896

(120581119890minus)119895119896

= (

minus (119896119865)0101

minus (119896119865)2323

+ 120572 minus (119896119865)0102

minus (119896119865)2331

minus(119896119865)0103

minus (119896119865)2312

rdquo minus(119896119865)0202

minus (119896119865)3131

+ 120572 minus(119896119865)0203

minus (119896119865)3112

rdquo rdquo minus(119896119865)0303

minus (119896119865)1212

+ 120572

) (22)

Advances in High Energy Physics 5

which is symmetric for the indices 119895 and 119896 and 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 (119896119865)0131

minus (119896119865)0223

(119896119865)0112

minus (119896119865)0323

rdquo 0 (119896119865)0212

minus (119896119865)0331

rdquo rdquo 0

)

(23)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

2(119896119865)0123

(119896119865)0131

+ (119896119865)0223

(119896119865)0112

+ (119896119865)0323

rdquo 2(119896119865)0231

(119896119865)0212

+ (119896119865)0331

rdquo rdquo 2(119896119865)0312

)

(24)

which is symmetric for the indices 119895 and 119896 The symbol rdquoabove means that the matrix element is not written explicitlyfor brevity and it can be obtained from the property ofsymmetryantisymmetry of the corresponding matrix

With (16) we can replace the tensor (119896119865)120572120573120583] with the

Lorentz violation matrix Δ120572120573 So the four matrices above arerewritten like

(120581119890+)119895119896

= (

1

2Δ

1

2(Δ10Δ20

+ Δ23Δ13)

1

2(Δ10Δ30

+ Δ32Δ12)

rdquo 1

2Δ

1

2(Δ20Δ30 + Δ31Δ21)

rdquo rdquo 1

2Δ

)

(25)

which is symmetric for the indices 119895 and 119896 with Δ = Δ120572120573119892120572120573

(120581119890minus)119895119896

= (

1

2Δ00 +

1

6Δ11 minus

5

6Δ22 minus

5

6Δ33

1

2(Δ10Δ20 minus Δ23Δ13)

1

2(Δ10Δ30 minus Δ32Δ12)

rdquo 1

2Δ00 +

1

6Δ22 minus

5

6Δ11 minus

5

6Δ33

1

2(Δ20Δ30 minus Δ31Δ21)

rdquo rdquo 1

2Δ00 +

1

6Δ33 minus

5

6Δ11 minus

5

6Δ22

) (26)

which is symmetric for the indices 119895 and 119896 with 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 minus1

2(Δ02Δ32 + Δ01Δ31)

1

2(Δ03Δ23 + Δ01Δ21)

rdquo 0 minus1

2(Δ03Δ13

+ Δ02Δ12)

rdquo rdquo 0

)

(27)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

1

2(Δ02Δ32 minus Δ01Δ31)

1

2(Δ01Δ21 minus Δ03Δ23)

rdquo 1

2(Δ03Δ13 minus Δ02Δ12)

rdquo rdquo

)

(28)

which is symmetric for the indices 119895 and 119896 The metrictensor 119892

120572120573= diag(minus1 1 1 1) is used here The symbol

ldquordquo in matrix (120581119900minus)119895119896 denotes that the involved elements

(119896119865)0123 (119896

119865)0231 and (119896

119865)0312 cannot be written in terms

of the Lorentz violation matrix Δ120572120573 through (16) We haveknown from the preceding section that only 15 degrees offreedom inmatrixΔ120572120573 are interrelated with the same number

of independent degrees of freedom in tensor (119896119865)120574120572120573120588

and viceversa Besides the representation of (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896

and (120581119900minus)119895119896 in terms of Δ120572120573 the relations between other

Lorentz violation parameters commonly used in the minimalSMEand the Lorentz violationmatrix here are summarized inTable 3There are 15 independent expressions in terms of Δ120572120573

appearing in the four matrices (120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and

(120581119900minus)119895119896 These 15 independent expressions help to determine

15 degrees of freedom of the Lorentz violation matrix Δ120572120573With the recent maximal sensitivities attained from cur-

rent experiments for Lorentz violation parameters (120581119890+)119895119896

(120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 of the free photon sector in the

minimal SME (see Table 2) we get the maximal sensitivitiesor the conservative bounds from experiments for Lorentzinvariance matrix Δ120572120573photon

(

3Δ33 + 10minus17 10minus5 10minus5 10minus6

10minus9 Δ33 + 10minus17 10minus9 10minus9

10minus9 10minus9 Δ33 + 10minus17 10minus9

10minus9 10minus8 10minus8 Δ33

) (29)

in the Sun-centered inertial reference frame [23 29] Thepublication [23] claimed a 2120590 limit on Lorentz violationcoefficients (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 (120581119900minus)119895119896 and so forth in

Table 2 The 15 independent degrees of freedom of Δ120572120573 aredetermined and there is still one freedom Δ33 remainingunclear The maximal sensitivity for the elements of Δ120572120573 islisted in Table 1

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

which is symmetric for the indices 119895 and 119896 and 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 (119896119865)0131

minus (119896119865)0223

(119896119865)0112

minus (119896119865)0323

rdquo 0 (119896119865)0212

minus (119896119865)0331

rdquo rdquo 0

)

(23)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

2(119896119865)0123

(119896119865)0131

+ (119896119865)0223

(119896119865)0112

+ (119896119865)0323

rdquo 2(119896119865)0231

(119896119865)0212

+ (119896119865)0331

rdquo rdquo 2(119896119865)0312

)

(24)

which is symmetric for the indices 119895 and 119896 The symbol rdquoabove means that the matrix element is not written explicitlyfor brevity and it can be obtained from the property ofsymmetryantisymmetry of the corresponding matrix

With (16) we can replace the tensor (119896119865)120572120573120583] with the

Lorentz violation matrix Δ120572120573 So the four matrices above arerewritten like

(120581119890+)119895119896

= (

1

2Δ

1

2(Δ10Δ20

+ Δ23Δ13)

1

2(Δ10Δ30

+ Δ32Δ12)

rdquo 1

2Δ

1

2(Δ20Δ30 + Δ31Δ21)

rdquo rdquo 1

2Δ

)

(25)

which is symmetric for the indices 119895 and 119896 with Δ = Δ120572120573119892120572120573

(120581119890minus)119895119896

= (

1

2Δ00 +

1

6Δ11 minus

5

6Δ22 minus

5

6Δ33

1

2(Δ10Δ20 minus Δ23Δ13)

1

2(Δ10Δ30 minus Δ32Δ12)

rdquo 1

2Δ00 +

1

6Δ22 minus

5

6Δ11 minus

5

6Δ33

1

2(Δ20Δ30 minus Δ31Δ21)

rdquo rdquo 1

2Δ00 +

1

6Δ33 minus

5

6Δ11 minus

5

6Δ22

) (26)

which is symmetric for the indices 119895 and 119896 with 120572 equiv

(23)(119896119865)01198970119897

(120581119900+)119895119896

= (

0 minus1

2(Δ02Δ32 + Δ01Δ31)

1

2(Δ03Δ23 + Δ01Δ21)

rdquo 0 minus1

2(Δ03Δ13

+ Δ02Δ12)

rdquo rdquo 0

)

(27)

which is antisymmetric for the indices 119895 and 119896 and

(120581119900minus)119895119896

= (

1

2(Δ02Δ32 minus Δ01Δ31)

1

2(Δ01Δ21 minus Δ03Δ23)

rdquo 1

2(Δ03Δ13 minus Δ02Δ12)

rdquo rdquo

)

(28)

which is symmetric for the indices 119895 and 119896 The metrictensor 119892

120572120573= diag(minus1 1 1 1) is used here The symbol

ldquordquo in matrix (120581119900minus)119895119896 denotes that the involved elements

(119896119865)0123 (119896

119865)0231 and (119896

119865)0312 cannot be written in terms

of the Lorentz violation matrix Δ120572120573 through (16) We haveknown from the preceding section that only 15 degrees offreedom inmatrixΔ120572120573 are interrelated with the same number

of independent degrees of freedom in tensor (119896119865)120574120572120573120588

and viceversa Besides the representation of (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896

and (120581119900minus)119895119896 in terms of Δ120572120573 the relations between other

Lorentz violation parameters commonly used in the minimalSMEand the Lorentz violationmatrix here are summarized inTable 3There are 15 independent expressions in terms of Δ120572120573

appearing in the four matrices (120581119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 and

(120581119900minus)119895119896 These 15 independent expressions help to determine

15 degrees of freedom of the Lorentz violation matrix Δ120572120573With the recent maximal sensitivities attained from cur-

rent experiments for Lorentz violation parameters (120581119890+)119895119896

(120581119890minus)119895119896 (120581119900+)119895119896 and (120581

119900minus)119895119896 of the free photon sector in the

minimal SME (see Table 2) we get the maximal sensitivitiesor the conservative bounds from experiments for Lorentzinvariance matrix Δ120572120573photon

(

3Δ33 + 10minus17 10minus5 10minus5 10minus6

10minus9 Δ33 + 10minus17 10minus9 10minus9

10minus9 10minus9 Δ33 + 10minus17 10minus9

10minus9 10minus8 10minus8 Δ33

) (29)

in the Sun-centered inertial reference frame [23 29] Thepublication [23] claimed a 2120590 limit on Lorentz violationcoefficients (120581

119890+)119895119896 (120581119890minus)119895119896 (120581119900+)119895119896 (120581119900minus)119895119896 and so forth in

Table 2 The 15 independent degrees of freedom of Δ120572120573 aredetermined and there is still one freedom Δ33 remainingunclear The maximal sensitivity for the elements of Δ120572120573 islisted in Table 1

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

Table 1 Maximal sensitivities (2120590) for the Lorentz violation matrixof photons

Coefficient SensitivityΔ00minus 3Δ33

10minus17

Δ11 minus Δ33 10minus17

Δ22 minus Δ33 10minus17

Δ01 10minus5

Δ02

10minus5

Δ03

10minus6

Δ10

10minus9

Δ20 10minus9

Δ30 10minus9

Δ12 10minus9

Δ13

10minus9

Δ21 10minus9

Δ23 10minus9

Δ31

10minus8

Δ32 10minus8

Table 2 Maximal sensitivities (2120590) for the photon sector (from[23]) The superscripts 119883119884 and 119885 there are converted to 1 2 and3 here respectively for consistence with the notation of the Lorentzviolation matrix

Coefficient Sensitivity(120581119890+)12

10minus32

(120581119890+)13

10minus32

(120581119890+)23

10minus32

(120581119890+)11minus (120581119890+)22

10minus32

(120581119890+)33

10minus32

(120581119900minus)12

10minus32

(120581119900minus)13

10minus32

(120581119900minus)23

10minus32

(120581119900minus)11minus (120581119900minus)22

10minus32

(120581119900minus)33

10minus32

(120581119890minus)12

10minus17

(120581119890minus)13

10minus17

(120581119890minus)23

10minus17

(120581119890minus)11minus (120581119890minus)22

10minus17

(120581119890minus)33

10minus16

(120581119900+)12

10minus13

(120581119900+)13

10minus14

(120581119900+)23

10minus14

120581tr 10minus14

119896(3)

(119881)0010minus43 GeV

119896(3)

(119881)1010minus42 GeV

Re 119896(3)(119881)11

10minus42 GeVIm 119896(3)

(119881)1110minus42 GeV

Equation (29) demonstrates that the Lorentz violationmatrixΔ120572120573 does not need to be a symmetricmatrix in generalThe nonsymmetric structure of the photon Lorentz violationmatrix suggests preferred directions and potential anisotropy

of space-time [26 30] for propagating of the free photon evenin the case of no gravitationMore experiments will givemoredetails for Δ120572120573

There are different representations for the Lorentz vio-lation matrix Δ

120572120573 in different coordinate systems Theserepresentations are related with each other by a coordinatetransformation matrix 119879120572120573 in group SO(13) When the rel-ative velocity between these two coordinate systems is muchsmaller than the light speed the element of119879120572120573 is either order119874(1) or close to zero An element of the Lorentz violationmatrix in a coordinate system is the linear combinations ofthe elements of Δ120572120573 in the other coordinate system Thenthe magnitudes of the Lorentz violation matrix in these twocoordinates are not different too much from each other Sowe expect that the upper bound on the violation parametersappearing in [26] is compatible with the maximal sensitivityshown in (29) The limit of order 10minus14 in [26] for the photonLorentz violationmatrix is indeed compatible with the bound10minus8 hereFrom (29) we find that the trace Δ equiv tr(Δ120572120573) = 119892

120572120573Δ120572120573 ≃

10minus17 A competitive upper bound 16 times 10minus14 on the photonLorentz matrix Δ120572120573 was obtained in [26] In that article [26]we made an assumption about the form of the matrix Δ120572120573

theoretically for the analysis on the data there There is no apriori assumptions here about the general structure of Δ120572120573We see that the maximal attained sensitivity 10minus17 for thetrace of the Lorentz violation matrix is stronger than theupper limit 10minus14 gotten in [26] Compared with the stringentbound on the trace Δ the maximal attained sensitivities putlooser limits on the nondiagonal elements of Δ120572120573 shown in(29)

Through this work we have seen that the two theories ofthe SMS and the minimal SME can give same results for freephotons Equation (16) shows the correlations between theLorentz invariance violationmatrixΔ120572120573 of ourmodel and thecoupling tensor (119896

119865)120572120573120583] appearing in the photon sector of the

minimal SME The relations of the violation parameters Δ120572120573with the parameters (119896

119865)120572120573120583] uncover the detailed structure

of the Lorentz violation matrix of free photons in (29) Up tonow there have been no compelling experimental evidencesfor the existence of Lorentz violation for photons All that wehave gotten so far are the theoretical analysis and themaximalsensitivities attained from the recent experiments

4 Conclusion

Two Lorentz violation models the minimal standard modelextension (SME) and the standard model supplement (SMS)are compared here for the photon sector For all the termsin the Lagrangians of the pure photon sector there is nodirect one-to-one correspondence between the twomodels ingeneral However some interesting results can be obtained bythe requirement that the two models are identical with eachother in the intersection We find that the free photon sectorof the SMS can be a subset of the minimal SME providedwith some connections in the SME parameters (i) Weconsider the photon sector of the two models and two main

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

Table 3 Definitions for the photon sector in the minimal SME together with relations with the Lorentz violation matrix of the SMS (the twoleft columns are from [23])

Symbol Definition Relation(120581119890+)119895119896

minus(119896119865)01198950119896

+1

4120598119895119901119902120598119896119903119904(119896

119865)119901119902119903119904 (25)

(120581119890minus)119895119896

minus(119896119865)01198950119896

minus1

4120598119895119901119902

120598119896119903119904

(119896119865)119901119902119903119904

+2

3(119896119865)01198970119897

120575119895119896 (26)

(120581119900+)119895119896

minus1

2120598119895119901119902(119896

119865)0119896119901119902

+1

2120598119896119901119902(119896

119865)0119895119901119902 (27)

(120581119900minus)119895119896 1

2120598119895119901119902

(119896119865)0119896119901119902

+1

2120598119896119901119902

(119896119865)0119895119901119902 (28)

120581tr minus2

3(119896119865)01198970119897

minus2

3Δ11 +

1

3Δ

1198961

(119896119865)0213

1198962 (119896119865)0123

1198963 (119896119865)0202

minus (119896119865)1313

minus1

2Δ

1198964 (119896119865)0303

minus (119896119865)1212

minus1

2Δ

1198965

(119896119865)0102

+ (119896119865)1323

minus1

2(Δ10Δ20+ Δ23Δ13)

1198966

(119896119865)0103

minus (119896119865)1223

minus1

2(Δ10Δ30+ Δ32Δ12)

1198967 (119896119865)0203

+ (119896119865)1213

minus1

2(Δ20Δ30 + Δ31Δ21)

1198968 (119896119865)0112

+ (119896119865)0323

minus1

2(Δ03Δ23 minus Δ01Δ21)

1198969

(119896119865)0113

minus (119896119865)0223 1

2(Δ01Δ31minus Δ02Δ32)

11989610

(119896119865)0212

minus (119896119865)0313 1

2(Δ03Δ13minus Δ02Δ12)

119896(3)

(119881)00minusradic4120587(119896

119860119865)0 0

119896(3)

(119881)10minusradic

4120587

3(119896119860119865)3 0

Re 119896(3)(119881)11

radic2120587

3(119896119860119865)1 0

Im 119896(3)

(119881)11minusradic

2120587

3(119896119860119865)2 0

equations are obtained between Δ120572120573 (119896119865)120572120573120583] and (119896

119860119865)120572

through the propagator of photons in the momentum space(ii) These equations suggest that the CPT-odd coefficients(119896119860119865

)120572 are zero Such a suggestion is supported by available

experimental bounds for example there is the maximalsensitivity 10

minus42 sim 10minus43 GeV from experiments for thecoefficients 119896(3)

(119881)00 119896(3)(119881)10

Re 119896(3)(119881)11

and Im 119896(3)

(119881)11 (iii) There

are 15 degrees of freedom in the Lorentz violation matrix Δ120572120573

and the samenumber of degrees of freedom in tensor (119896119865)120572120573120583]

to be interrelated We got the conservative bound equation(29) on the detailed structure of the photon Lorentz violationmatrix in our model The bounds on Δ120572120573 are gotten from thelimits on the Lorentz parameters of the minimal SME Thedetailed structure of the photon Lorentz violation matrix canplay an important role for applications to Lorentz violationexperiments

For Δ120572120573 of free photons due to the factor that a universalconstant can be absorbed into the gauge field 119860120583 there are15 independent degrees of freedom in Δ120572120573 of free photons todescribe Lorentz violation In the paper we do not use these15 independent degrees of freedom to derive the magnitudes

of all the 16 elements ofΔ120572120573 but use the relations of it with theparameters in the minimal SME to get the constraints on Δ120572120573

of free photons from the constraints of various experimentson the minimal SME

The strong constraints on the matrix elements of Δ120572120573

mean that Lorentz violation is small for photons if it existsThematrixΔ120572120573 is not symmetric generallyThenonsymmetryproperty of Δ120572120573 implies that the space-time for free photonscan be not isotropic very well even in the case of nogravitation To date there has been theoretical analysis onLorentz violation and there is no strong experimental evi-dence supporting Lorentz violation for photons Generallywe should study the minimal standard model extensionand the standard model supplement separately and thendetermine whether these two models are equivalent to eachother by directly confronting relevant experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

Acknowledgments

The work was supported by National Natural Science Foun-dation of China (Grant no 11035003 and no 11120101004) andby the Research Fund for the Doctoral Program of HigherEducation China

References

[1] G Amelino-Camelia ldquoQuantum-spacetime phenomenologyrdquoLiving Reviews in Relativity vol 16 article 5 2013

[2] L Shao andB-QMa ldquoLorentz violation effects on astrophysicalpropagation of very high energy photonsrdquo Modern PhysicsLetters A vol 25 p 3251 2010

[3] P A M Dirac ldquoIs there an AEligtherrdquo Nature vol 168 pp 906ndash907 1951

[4] J D Bjorken ldquoA dynamical origin for the electromagnetic fieldrdquoAnnals of Physics vol 24 pp 174ndash187 1963

[5] M Planck ldquoUber irreversible Strahlungsvorgangerdquo Sitzungsb-erichte der Koniglich Preussischen Akademie der Wissenschaftenzu Berlin vol 5 article 440 1899

[6] H S Snyder ldquoQuantized space-timerdquo Physical Review vol 71pp 38ndash41 1947

[7] H S Snyder ldquoThe electromagnetic field in quantized space-timerdquo Physical Review vol 72 p 68 1947

[8] Y Xu andB-QMa ldquoParticles and fields gravitation cosmologyand nuclear physicsrdquo Modern Physics Letters A vol 26 no 28article 2101 2011

[9] L Shao and B-QMa ldquoNote on a new fundamental length scalel instead of the Newtonian constant Grdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1771ndash1774 2011

[10] G Amelino-Camelia ldquoRelativity in spacetimes with short-distance structure governed by an observer-independent(Planckian) length scalerdquo International Journal of ModernPhysics D vol 11 no 1 pp 35ndash59 2002

[11] JMagueijo andL Smolin ldquoLorentz invariancewith an invariantenergy scalerdquo Physical Review Letters vol 88 Article ID 1904032002

[12] X Zhang L Shao and B-Q Ma ldquoPhoton gas thermodynamicsin doubly special relativityrdquo Astroparticle Physics vol 34 pp840ndash845 2011

[13] W-T Ni ldquoYangrsquos gravitational field equationsrdquo Physical ReviewLetters vol 35 no 5 pp 319ndash320 1975

[14] W-T Ni ldquo Searches for the role of spin and polarization ingravityrdquo Reports on Progress in Physics vol 73 Article ID056901 2010

[15] M L Yan ldquoThe renormalizabilty of the general gravity theorywith torsion and the spontaneous breaking of lorentz grouprdquoCommunications inTheoretical Physics vol 2 no 4 article 12811983

[16] H B Nielsen and M Ninomiya ldquo120573-function in a noncovariantYang-Mills theoryrdquo Nuclear Physics B vol 141 no 1-2 pp 153ndash177 1978

[17] S Chadha and H B Nielsen ldquoLorentz invariance as a lowenergy phenomenonrdquo Nuclear Physics B vol 217 no 1 pp 125ndash144 1983

[18] H B Nielsen and I Picek ldquoThe Redei-like model and testingLorentz invariancerdquo Physics Letters B vol 114 p 141 1982

[19] H B Nielsen and I Picek ldquoLorentz non-invariancerdquo NuclearPhysics B vol 211 pp 269ndash296 1983

[20] V Ammosov and G Volkov ldquoCan neutrinos probe extradimensionsrdquo httparxivorgabshep-ph0008032

[21] H Pas S Pakvasa and T J Weiler ldquoSterile-active neutrinooscillations and shortcuts in the extra dimensionrdquo PhysicalReview D vol 72 Article ID 095017 2005

[22] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standard modelrdquo Physical Review D vol 58 Article ID116002 1998

[23] V A Kostelecky and N Russell ldquoData tables for Lorentz andCPT violationrdquo Reviews of Modern Physics vol 83 p 11 2011

[24] L Zhou and B Ma ldquoLorentz invariance violation matrix froma general principlerdquo Modern Physics Letters A Particles andFields Gravitation Cosmology Nuclear Physics vol 25 no 29pp 2489ndash2499 2010

[25] L-L Zhou andB-QMa ldquoNew theory of Lorentz violation froma general principlerdquo Chinese Physics C vol 35 no 11 pp 987ndash991 2011

[26] L Zhou and B-Q Ma ldquoA theoretical diagnosis on light speedanisotropy fromGRAAL experimentrdquoAstroparticle Physics vol36 pp 37ndash41 2012

[27] L Zhou and B-Q Ma ldquoNeutrino speed anomaly as signal ofLorentz violationrdquoAstroparticle Physics vol 44 pp 24ndash27 2013

[28] B-Q Ma ldquoThe phantom of the opera superluminal neutrinosrdquoModern Physics Letters A vol 27 Article ID 1230005 2012

[29] R Bluhm V A Kostelecky C D Lane andN Russell ldquoProbingLorentz and CPT violation with space-based experimentsrdquoPhysical Review D vol 68 Article ID 125008 2003

[30] J P Bocquet D Moricciani and V Bellini ldquoLimits on light-speed anisotropies from compton scattering of high-energyelectronsrdquo Physical Review Letters vol 104 Article ID 2416012010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of