Vacuum energy from noncommutative models

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Vacuum

energy

from

noncommutative

models

S. Mignemi

a

,

b

,

∗

,

A. Samsarov

c

a_{DipartimentodiMatematicaeInformatica,UniversitàdiCagliari,vialeMerello92,09123Cagliari,Italy} b_{INFN,SezionediCagliari,CittadellaUniversitaria,09042Monserrato,Italy}

c_{RudjerBoškovi´cInstitute,Bijeniˇckacesta54,10002Zagreb,Croatia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received16December2017

Receivedinrevisedform8February2018 Accepted9February2018

Availableonline13February2018 Editor:M.Cvetiˇc

The vacuumenergyiscomputedforascalarfieldinanoncommutativebackgroundinseveralmodels of noncommutativegeometry.One mayexpectthat the noncommutativityintroducesanaturalcutoff ontheultravioletdivergencesoffieldtheory.Ourcalculationsshow howeverthatthisdependsonthe particular modelconsidered: insomecasesthe divergencesare suppressedandthe vacuumenergyis onlylogarithmicallydivergent,inothercasestheyarestrongerthaninthecommutativetheory.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Noncommutative models [1,2] may improve the behavior of field theoriesin theultraviolet regionby smoothing orremoving someofthesingularitiesofcommutativequantumfieldtheory.In fact,theyimplya lowerboundonthe lengthscales,givenbythe inverseofthe noncommutativityparameter

κ

, orequivalently an upperboundontheenergyscales,givenby

κ

.Thescale

κ

is usu-allyassumedto ofthe orderof thePlanck mass MP

∼

1019 GeV (butlowervaluesarenotexcluded),andmayactasanatural cut-off on the divergences ofquantum field theory,in contrast with thecommutativecasewherethecutoffmustbeimposedbyhand. This possibility may be tested in the calculation of the vac-uum energy of quantum fields. Thiscomputation has interesting implicationsoncosmology,sincethevacuumenergyisoften iden-tifiedwiththecosmologicalconstant[3]. Althoughthisargument isalmost certainlywrong, sinceit isnotbased onawell-defined theoryandpredicts avaluethatcanbe120ordersofmagnitudes greaterthan theobservedone, itcan stillbe interesting tocheck ifthenoncommutativityparametercanactasanaturalcutoffand improvethe ultraviolet behavior ofthe theory.Of course, if

κ

is ofPlanck scale, this doesnot change much the predictions from a phenomenological point of view,since in this context alsothe standardUVcutoffisusuallyassumedtohavethesamescale.

In this paper, we calculate the vacuum energy of a massless scalar field in noncommutative background, using the heat ker-nelmethod.Thismethodallowstoevaluatetheone-loopeffective

*

Correspondingauthor.

E-mailaddresses:smignemi@unica.it(S. Mignemi),andjelo.samsarov@irb.hr (A. Samsarov).

action by calculating the integral of an operator, related to the solution of the heat equation on a Euclideanmanifold. We shall followtheapproachof[4],wheremodelspresentingabreakingof Lorentzinvariancearestudied.Areviewoftheheatkernel formal-ism canbe found forexamplein[5]. Acalculation similarto the presentone,butdifferinginseveralrespectsandbasedona per-turbative expansion inthe noncommutativityparameter, hasalso beenperformedin[6].

Weinvestigateaclassofnoncommutativemodelscharacterized byadeformationoftheHeisenbergalgebra,whichinturnimplies a deformation of the Poincaré symmetry and hence of the field equations.AlsothemeasureoftheHilbertspacemustbeadapted to thenontrivialrepresentationofthedeformedHeisenberg alge-bra,andthesetwoeffectscombinetomodifythevalueoftheheat kernelintegralincomparisonwiththecommutativeone.

Weshowthat,contrarytonaiveexpectations, noncommutativ-itydoesnotcompletelyregularizethetheory,andonlyinsomeof themodelsexaminedtheUVbehaviorisimprovedwithrespectto thecommutativetheory,whileinothermodelsitcanbeworsened. Thebestimprovementoccursintheanti-Snydermodel,wherethe traceoftheheatkernelisfinite,andthedivergenceofthevacuum energyisonlylogarithmic.

2. Heatkernel

Letusconsiderafieldtheoryobeyingtheequation1

D

φ

=

F

(∂

0

, ∂

i

)φ

=

0

,

(1)

1 _{We adopt the following conventions: metric η}

μν =diag(−1,1,1,1); μ=

0,1,2,3;i=1,2,3;v2_=vμ_v μ.

https://doi.org/10.1016/j.physletb.2018.02.016

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

where

D

isadifferentialoperatorthatdeformstheusualLaplacian

∂

μ

∂

μ byaparameter

κ

,insuchawaytopreservetheinvariance

underspatialrotations.

ItisknownthatforaquantumbosonicfieldinEuclideanspace, withpartition function definedas Z

= (

det

D)

−1/2, the one-loop effectiveactionW

=

1₂ln det

D

canbewrittenintermsoftheheat kernel, W

= −

1 2 ∞



1/2 ds s K

(

s

),

(2)

wheres isarealparameterand K

(

s

)

=



dx

<

x

|

e−sD

|

x

>

isthe traceoftheheatkernel.Thecutoff1

/



2_,with

_

_

_1,atthelower limitisintroducedbecauseinstandardfieldtheorytheintegral(2) isusuallydivergentfors

→

0 (UVdivergence).We recallthatthe heatkernel K

(

s

,

x

,

x

)

= <

x

|

e−sD

|

x

>

isdefinedasasolutionof theheatequation

(∂

s

+

D

)

K

(

s

,

x

,

x

)

=

0

,

K

(

0

,

x

,

x

)

= δ(

x

,

x

).

(3) The calculations are most easily performed in momentum space, where the solution of the heat equation is trivial. It fol-lowsthat[4] K

(

s

)

=

V

(

2

π

)

d ∞



−∞ ddp e−sF(p0,pi)

_,

₍₄₎

where V is the volume of spacetime. In the special case of the undeformedLaplaceoperator,K

(

s

)

=

V

/(

4

π

s

)

d/2_.

The effective action follows fromeq. (2). The vacuumenergy density

λ

isdefinedas

λ

= −

W

V

,

(5)

and

λ

isoftenidentifiedwiththecosmologicalconstant.Ofcourse, in standard field theory the value of

λ

dependson the cutoff



introduced to regularize the UV divergences of the effective ac-tion.Inparticular,infourdimensions

λ

=



4

_/

₆₄

_π

2_.Onemayhope that innoncommutative models thesedivergences mightbe reg-ularizedbythenoncommutativityscale

κ

,sothat thecalculation givesafiniteresultwithoutneedofintroducinganartificialcutoff. Wewantto studyifthishappensinsome well-knowncases.For easeofcalculation,weconsidermasslessfields,thatmayhowever leadtoIRdivergences.Weshallalwaysunderstandthattheseare regularizedwhenoneconsidersmassivefields.

3. Noncommutativemodels

Noncommutative theories are based on the hypothesis that spacetimehasagranularstructure,implementedthroughthe non-commutativity of spacetime coordinates, with a scale of length

κ

−1_{,thatisusually(butnotnecessarily)identifiedwiththePlanck} length. Because of the presence of thisfundamental scale, most noncommutativegeometriesareassociatedtoadeformationofthe actionofLorentztransformationsonphasespace,andhenceofthe Poincaréalgebra.

Itmust be notedthat their propertiesare not completely de-terminedbythenoncommutativityofspacetimecoordinates,since thesamenoncommutativecoordinatescanbeassociatedwith dif-ferentcoproducts. Different realizations of a given noncommuta-tivegeometryare oftencalledbasesandusuallylead todifferent physicalpredictions. The different bases can be characterized by specifyingtheirdeformedHeisenbergalgebra,generatedbythe po-sition operators xμ and the momentum operators pμ. From the

knowledge of this algebra, one can obtain the coproduct of mo-mentaandtheotherrelevantquantities,usingthemethods devel-opedinref. [7].

In orderto calculate the heat kernelone mustfirst of all es-tablishthefieldequations.Thedeformedinvarianceofthetheory ispreservedifthe(momentumspace) deformedLaplaceequation isidentifiedwiththeCasimiroperatorC ofthedeformedPoincaré algebra.However, thischoice isnot unique,becauseanyfunction ofC couldbeadopted.

Moreover, a nontrivial measure must be fixed on the Hilbert space,againinvariant underdeformedLorentztransformations.To singleoutthismeasureuniquely,wealsorequirethattheposition operatorsaresymmetricintherepresentationchosen.

In this paper, we consider some specific models that lead to simplecalculationsoftheheatkernel: thefirstoneisthe Snyder model [8]. Its main peculiarity is that it preserves the standard action of the Lorentz group on phase space, andit can be seen asdualtodeSitterspacetime.ItsdeformedHeisenbergalgebra,in theoriginalSnyderbasis,isgivenby

[

xμ

,

xν

] =

i Jμν

κ

2

,

[

pμ

,

pν

] =

0

,

[

xμ

,

pν

] =

i



η

μν

+

pμpν

κ

2



,

(6) where Jμν

=

xμ pν

−

xν pμ are the generators ofthe Lorentz al-gebra.Since theLorentztransformations arenotdeformedinthis case,theCasimiroperatorissimplygivenby2

C

=

p2

≡ −

p2₀

+

p2_i

,

(7)

withm2

_{= −}

_p2

_<

_κ

2_{.Thisimpliesanupperboundfortheallowed} particlemassesinthismodel.

There is also the possibility of choosing the opposite sign in frontof

κ

2_{in(6) (anti-Snydergeometry)[8,9].Inthiscasethereis} noupperboundontheparticlemasses.However, alltherelations we shall discuss holdtrue, by simply changing the sign infront of

κ

2_.

Theother examplesbelong tothe

κ

-Poincaréclass: oneis the so-calledMagueijo–Smolinmodel[10].Thenontrivialcommutators ofitsHeisenbergalgebraintheGranikbasis[11] are

[

xi

,

x0

] =

i xi

κ

,

[

x0

,

pi

] =

i pi

κ

,

[

xi

,

pj

] =

i

δ

i j

,

[

x0

,

p0

] = −

i



1

−

p0

κ



.

(8)

ThePoincaréalgebraisnowdeformedanditsCasimiroperatoris

C

=

−

p 2 0

+

p2i



1

−

p0 κ



2

.

(9)

Inthiscase,thebound p0

<

κ

musthold.

The last one is the Majid–Ruegg (MR) model [12], describing the

κ

-Poincaré model [2] defined inthe bicrossproductbasis. Its Heisenbergalgebrareads

[

xi

,

x0

] =

i xi

κ

,

[

x0

,

pi

] =

i pi

κ

,

[

xi

,

pj

] =

i

δ

i j

,

[

x0

,

p0

] = −

i

.

(10) Also inthis casethe Poincaré algebra is deformed, with Casimir operator C

= −



2

κ

sinhp0 2

κ



2

+

ep0κ _p2 i

.

(11)

2 _{Asnotedabove,thischoiceisnotunique.SometimesthechoiceC=p}2_/(1− p2

/κ2

TheEuclideanversion ofthisclass ofnoncommutativemodels is usually obtainedby an analytic continuation ofthe Lorentzian theory, with p0

→

ip0,

κ

→

i

κ

, see discussions in [13] for the

κ

-Poincaré model. This assumption appears natural considering that

κ

plays the role ofan energy scale. Requiring

κ

→

i

κ

also appearsnecessaryinordertogetaphysicallysensible interpreta-tionof theEuclideantheory inthe

κ

-Poincaré case, whileinthe Snydercasethesituationislessclear.

Fortheclassofmodelsconsideredhere, thecalculationofthe heat kernel differs in two ways fromthe standard case: first, as discussed above, the Laplace operator is chosen proportional to theCasimiroperator, inordertobeinvariantunderthedeformed Lorentz transformations. Moreover, the measure of momentum spaceis not trivial, sincethe phase spaceoperators satisfy a de-formed Heisenberg algebra,andhence a nontrivialrealizationon theHilbertspacemustbefound.

4. Snydermodel

We define the Euclidean Snyder model by the analytic con-tinuation p0

→

ip0,

κ

→

i

κ

, asforthe

κ

-Poincaré models. Keep-ing instead

κ

→

κ

would simplyinterchange the rolesofSnyder andanti-SnyderEuclideanspaces.Ourchoicemaintainsthebound

m2

_<

_κ

2 _{alsointheEuclideancase.}

Inbothinstances,theEuclideanCasimiroperatorisgivenby

C

=

p2_E

=

p2₀

+

p2_i

.

(12)

We start by considering anti-Snyder space, because it gives rises to more interesting results. Euclidean anti-Snyder space in thebasis (6) canbe realized by asuitable choice of operators in aquantumrepresentation.Twomainchoicescan befoundinthe literature, in terms ofa standard Hilbertspace of functionsof a canonical momentum variable Pμ, with Euclidean signature:the firstonereads[8] pμ

=

Pμ

,

xμ

=

i

∂

∂

Pμ

+

i

κ

2PμPν

∂

∂

Pν

,

(13)

with

−∞

<

Pμ

<

∞

. We require that in this representation the positionoperators xμ be symmetric,i.e. that

< ψ

|

xμ

| φ > =

< φ

|

xμ

| ψ >

.Thisoccursifoneintroducesanontrivialmeasurein the P -space[14],

d

μ

=

d4P

(

1

+

P2

_/

_κ

2

₎

d+21

,

(14)

withd thedimensionofthespace.

Wecan nowproceedto computethe traceoftheheat kernel. In this representation,the Laplacian (12) is simply given by P2. Hence,ford

=

4,eq.(4) gives

K

(

s

)

=

V 16

π

4 ∞



−∞ d4_P

(

1

+

P2

_/

_κ

2

₎

5/2 e −s P2

.

(15)

Definingpolarcoordinates,withradial coordinate

ρ

=

√

P2_{, after} integratingontheangularvariables,(15) becomes

K

(

s

)

=

V 8

π

2 ∞



0

ρ

3_d

_ρ

(

1

+

ρ

2

_/

_κ

2

₎

5/2 e− sρ2

.

(16)

Performingtheintegral,oneobtains

K

(

s

)

=

κ

4_V 24

π

2



2

(

1

+

κ

2s

)

−

κ

√

π

s

(

3

+

2

κ

2s

)

eκ2serfc

(

κ

√

s

)

,

(17)

where erfc

(

x

)

is the complementary error function, erfc

(

x

)

=

2 √ π



_∞ x e− t2 dt.

Inthelimits

→

0, K

(

s

)

takesthefinitevalue ₁₂κ4V

π2,incontrast withthecommutativetheory,whereitdivergesass−2_.Alsointhe limit s

→ ∞

ittakesafinitevalue.The evaluationofthevacuum energy(5) gives thenfors

→

0 alogarithmic divergence,andthe vacuumenergydensityreads

λ

=

κ

4 12

π

2 ln



m

,

(18)

wherem isthe IRscale.The divergenceismuch milderthan the commutative result,

λ

∼



4_{,although, if one assumes}

_

_∼

_κ

_{, the} numericalvalueof

λ

isnotmuchdifferentinthetwocases.

Thesameresultscanbeobtainedusinganotherrepresentation of the relations(6), againdefinedin terms ofa standard Hilbert spaceoffunctionsofacanonicalmomentumvariablePμ [9],

pμ

=

Pμ

1

−

P2

_/

_κ

2

,

xμ

=

i



1

−

P2

_/

_κ

2

∂

∂

Pμ

,

(19) withP2

_<

_κ

2_{.Inthiscase,themeasureforwhichtheoperatorsxμ} aresymmetricisgivenby[9]

d

μ

=

d

4_P

1

−

P2

_/

_κ

2

,

(20)

independently from the dimension of the space, while p2

₌

P2 1−P2_/κ2.Hence, K

(

s

)

=

V 16

π

4



P2_<κ2 d4P

1

−

P2

_/

_κ

2 e − s P 2 1−P 2/κ2

_.

₍₂₁₎

In polar coordinates

ρ

=

√

P2_{,this becomes after integration on} theangularcoordinates

K

(

s

)

=

V 8

π

2 κ2



0

ρ

3_d

_ρ

1

−

ρ

2

_/

_κ

2 e − sρ2 1−ρ2/κ2

_.

₍₂₂₎ Byachangeofvariables

ρ

→

ρ

1−ρ2_/κ2,onefinallyrecovers(16). TheSnydermodelisobtainedbyreplacing

κ

2 _with

₋

_κ

2_in(13) and(14),butnow thecalculation ismore involved,since the in-tegral for K

(

s

)

does not converge on the boundary. In fact, the representation(13) becomesnow

pμ

=

Pμ

,

xμ

=

i

∂

∂

Pμ

−

i

κ

2PμPν

∂

∂

Pν

,

(23)

with P2

<

κ

2_,andmeasure

d

μ

=

d

4_P

(

1

−

P2

_/

_κ

2

₎

d+21

.

(24)

Theintegral(16) becomes

K

(

s

)

=

V 8

π

2 κ



0

ρ

3_d

_ρ

(

1

−

ρ

2

_/

_κ

2

₎

5/2 e −sρ2

,

(25)

that divergesat

ρ

=

κ

.One mustthereforeintroduce an UV cut-off already at this stage, for example taking as upper limit of integration



<

κ

. This gives atleading orderthe constant value

K

(

s

)

∼

κ₍5(2−2κ2/3)

κ2_−2₎3/2 . Taking the samecutoff



forthe integration overs,itfollowsthat

λ

∼ −

κ

5

₍

_

2

₋

₂

_κ

2

_/

₃

₎

(

κ

2

₋

_

2

₎

3/2 ln



m

.

(26)

Therefore,inthiscasethevacuumenergydivergesas

(

κ

−



)

−3/2, with

(

κ

−



)

1.

5. MSmodel

Thismodelbelongsto the

κ

-Poincaré classandconsiderations analogoustothose of[13] suggest that the Euclideantheory can bedefinedthroughtheprescription p0

→

ip0,

κ

→

i

κ

.Thisleads totheEuclideanLaplacian

C

=

p 2 0

+

p2i



1

−

p0 κ



2

.

(27)

The action of the 4-dimensionalrotations on phase space is de-formed andonly the action ofthe spatial rotations is preserved. However,thecalculationcanbeperformedinawaysimilartothe oneoftheprevioussection.

First ofall, we notice that the MS model can be represented onastandardHilbertspaceoffunctionsofacanonicalmomentum variablePμ as pμ

=

Pμ 1

+

P0

/

κ

,

xμ

=

i

(

1

+

P0

/

κ

)

∂

∂

Pμ

,

(28)

where

−∞

<

Pi

<

∞

, 0

<

P0

<

∞

. In this representation, the measureforwhichtheoperatorsxμ aresymmetricisgivenby

d

μ

=

d

4_P 1

+

P0

/

κ

.

(29)

Theheatkernelintegralbecomestherefore

K

=

V 16

π

4



d4P 1

+

P0

/

κ

e−s P2

,

(30)

andcanbeseparatedinto

K

=

V 16

π

4 ∞



0 d P0 1

+

P0

/

κ

e−s P20 ∞



−∞ d3Pi e−s P 2 i

_.

₍₃₁₎ Thisgives K

= −

κ

V 32

π

3/2 e−κ2s s3/2



i

π

erf

(

i

κ

√

s

)

+

Ei

(

κ

2s

)

,

(32)

whereerf

(

x

)

is theerrorfunctionandEi

(

x

)

theexponential inte-gral.

Fors

→

0, K

∼ −

₃₂κV π3/2s−

3/2



_ln

₍

_κ

2_s

₎

₊

_γ

₊

_O

₍

_s

₎



_,where

_γ

_is the Euler–Mascheroni constant. For s

→ ∞

, K vanishes. As one couldhaveguessedfromthestructureoftheintegral,inthiscase onlythe integration on p0 gives rise to a milderUV divergence, whilethespatialpartpresentstheusualdivergences−3/2_,witha further logarithmic factor. The calculation of the vacuumenergy densitygivesatleadingorder

λ

∼

1

24

π

3/2

κ

3_ln



κ

.

(33)

TheUV divergenceismilderthan inthecommutative case.With thenaturalidentification



=

κ

,thelogarithmictermvanishes,and theleadingdivergenceisgivenbythenexttermintheexpansion, withthestandard

κ

4 _behavior.

6. MRmodel

Let us consider now the MR model. As discussed before, the Euclideantheoryisobtainedforp0

→

ip0,

κ

→

i

κ

.TheEuclidean Laplacianisthen C

=



2

κ

sinhp0 2

κ



2

+

ep0κ _p2 i

.

(34)

ArepresentationoftheHeisenbergalgebraisgivenby

pμ

=

Pμ

,

x0

=

i

∂

∂

P0

−

i

κ

Pi

∂

∂

Pi

,

xi

=

i

∂

∂

Pi

.

(35)

TheseoperatorsareHermitianforthemeasure[15]

d

μ

=

e3P0κ _d4_P

_.

₍₃₆₎

Theheatkernelintegralbecomesthen

K

(

s

)

=

V 16

π

4 ∞



−∞ d4P e3P0κ _e−s[4κ2sinh2 P02κ+eP0/κ Pi2]

_,

₍₃₇₎ or K

(

s

)

=

V 16

π

4 ∞



−∞ d P0e 3P0 κ _e−4κ2s sinh2 P02κ ∞



−∞ d3Pie−s e P0/_{κ P}2 i

_,

(38) and,afterintegrationoverthespatialcoordinates,

K

(

s

)

=

V 16

π

3/2 1 s3/2 ∞



−∞ d P0e 3P0 2κ _e−4κ2s sinh2 P02κ

_.

₍₃₉₎ Thelastintegrationgives

K

(

s

)

=

V

32

π

2

_κ

2_s3

(

1

+

2

κ

2_s

_).

₍₄₀₎

Inthiscase,thedivergencefors

→

0 isworsethaninthe commu-tativecase,whiletheexpressionconvergesfors

→ ∞

.

Computingthevacuumenergydensityweobtain

λ

=



6 192

π

2

_κ

2



1

+

3

κ

2



2

.

(41)

Again,ifone identifies



with

κ

,

λ

∝

κ

4_{,likeinthestandard} the-ory.

7. Conclusions

Using the heat kernel method, we have shown that in some cases noncommutativity can regularize the behavior of the vac-uumenergyofascalarfieldtheory.Thisishowevernotauniversal property:itholdsfortheanti-Snydermodel,butnotnecessarilyin differentinstances, likethe MRorthe MSmodel. Itis important to remarkthat our resultsare independent ofthe representation chosen ina Hilbertspace. Thishasbeenshownexplicitly forthe Snyder model, but can be checked also in the other cases. It is howevercrucialtochoosethecorrectmeasureintheHilbertspace. The results obtained hereare in agreement withexplicit cal-culations of quantum field theory in Snyder space, which show an improvement of the divergences withrespect to the commu-tative case[16]. Analogous conclusions concerning the energy of thevacuuminnoncommutativetheorieshavebeenobtainedusing averydifferentapproachrelatedtotheWheeler–deWittequation, in ref. [17].After completion of thispaper, we become aware of furtherworkstreatingrelatedproblems[18,19].

Acknowledgements

A.S. would like to thank the University of Cagliari and INFN, Sezione di Cagliari, for their kind hospitality, and acknowledges support by the Croatian Science Foundation under the project (IP-2014-09-9582)anda partialsupportbythe H2020CSA Twin-ningprojectNo.692194,RBI-T-WINNING.

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