Quantum field theory in generalised Snyder spaces

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Quantum

field

theory

in

generalised

Snyder

spaces

S. Meljanac

a

,

D. Meljanac

a

,

S. Mignemi

b

,

c

,

∗

,

R. Štrajn

b

,

c a_{RudjerBoškovi´cInstitute,Bijeniˇckacesta54,10002Zagreb,Croatia}

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

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received19January2017 Accepted24February2017 Availableonline7March2017 Editor:M.Cvetiˇc

WediscussthegeneralisationoftheSnydermodelthatincludesallpossibledeformationsofthe Heisen-berg algebracompatiblewithLorentzinvariance and investigateits properties.We calculate perturba-tively thelawofaddition ofmomentaandthestarproductinthegeneralcase.Wealsoundertakethe construction ofascalarfieldtheoryonthesenoncommutativespacesshowing thatthefree theoryis equivalenttothecommutativeone,likeinothermodelsofnoncommutativeQFT.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Snyder spacetime [1] was introduced in 1947 as an attempt to avoidUV divergences inQFT. Byassuming a noncommutative structure of spacetime, and hence a deformation of the Heisen-bergalgebra,itwas possibleto definea discretespacetime with-out breaking the Lorentz invariance, opening the possibility to smoothentheshort-distancebehaviorofquantumfieldtheory.

The proposal was then forgotten for many years, until more recenttimes,whennoncommutativegeometryhasbecomean im-portantfieldofresearch

[2]

.Newmodelshavebeenintroduced,in particulartheMoyalplane

[3]

and

κ

-Minkowskigeometry

[4]

,and theformalismofHopfalgebrashasbeenappliedtotheirstudy[5]. However,contrarytoSnyder’s,thenewmodelseitherbreakor de-formtheLorentzgroupactiononphasespace.

Inspiteoftherenewedinterest inspacetime noncommutativ-ityandofitspreservationofspacetimesymmetries,relativelyfew investigationshavebeendedicatedtotheoriginalproposalof Sny-der fromthe point ofview of noncommutative geometry,except foraseriesofpapers

[6–8]

,wherethemodelwasextendedto in-cludemoregeneralLorentz-invariantmodels,astheoneproposed byMaggiore

[9]

,andthestarproduct,coproductandantipodesof itsHopfalgebrawerecalculated.Themodelwasalsoinvestigated in

[10]

,whereitwasconsideredfromageometricalpointofview asacosetinmomentumspace,withresultsequivalenttothoseof refs.[6,7].Alsotheconstruction ofQFT onSnyderspacetimewas undertakeninthesepapers.

*

Correspondingauthor.

E-mailaddress:smignemi@unica.it(S. Mignemi).

However, somebasicpropertiesoftheHopfalgebraformalism forSnyderspaceshavenotyetbeeninvestigated:forexamplethe twist hasnot been explicitlycalculated. Alsothe investigation of QFT has onlybeen sketched.In [7]a scalar field theory was de-finedintermsofthestarproduct,butnoexplicitcalculationwas carried on. Moreover, a non-Hermitian representation was used, that complicates the formalism. Other approachesto Snyder QFT arebasedonafive-dimensionalformalism

[11,10]

.

Severaleffortshavealsobeendevotedtothestudyofthe clas-sicalandquantumaspectsofthemodelfromaphenomenological pointofview,withoutresortingtotheformalismof noncommuta-tive geometry, especially in the nonrelativistic 3D limit [12–14]. The most interesting results in this context are the clarification of its lattice-like properties,leading to deformed uncertainty re-lations, and the study of the corrections induced on the energy spectrumofsomesimplephysicalsystems.

Inthisletter,weextendpreviousinvestigationsonthe noncom-mutativegeometryoftheSnydermodelintwodirections:first,we furthergeneralisethemodeltoincludeinthedefining commuta-tion relations all the terms compatiblewithundeformed Lorentz invariance.Amongthesegeneralisations,somehavepeculiar prop-erties,forexampleitispossibletoconstructmodelsthatdescribe acommutativespacetime,butneverthelessdisplaynontrivial com-mutationrelationsbetweenpositionsandmomenta,leadingto de-formedadditionrulesformomentaandnonlocalbehaviorinfield theory.

Moreover,weimprovetheresultsofref.[7]onQFT,adoptinga Hermitian representation ofthenoncommutative coordinatesand showingthat withthisdefinitionthefree fieldtheory ofa scalar

http://dx.doi.org/10.1016/j.physletb.2017.02.059

0370-2693/©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

particle is equivalent to the one in noncommutative spacetime, similarlytootherwell-knownmodels

[15,16]

.

2. Snyderspaceanditsgeneralisation

We define generalised Snyder space as a deformation of or-dinary phase space, generated by noncommutative coordinates

¯

xμ and momenta pμ that span a deformed Heisenberg algebra

¯

H(¯

x

,

p

)

,

[¯

xμ

,

¯

xν

] =

iβMμν

ψ (β

p2

),

[

pμ

,

pν

] =

0

,

[

pμ

,

¯

xν

] = −

i

ϕ

μν

(β

p2

),

(1)

togetherwithLorentzgeneratorsMμν thatsatisfythestandard re-lations

[

Mμν

,

Mρσ

] =

i



η

μρMνσ

−

η

μσMνρ

+

η

νρMμσ

−

η

νσMμρ



,

[

Mμν

,

pλ

] =

i



η

μλpν

−

η

λνpμ



,

[

Mμν

,

x

¯

λ

] =

i



η

μλx

¯

ν

−

η

νλx

¯

μ



,

(2)

wherethefunctions

ψ(β

p2

₎

_and

_ϕ

μν

(β

p2

)

areconstrainedsothat

theJacobi identities hold,

β

is a constantof theorder of1

/

M2_Pl, and

η

μν

=

diag

(

−

1

,

1

,

1

,

1

)

. The commutation relations (1)–(2)

generalise those of the Snyder spaces originally investigated in

[6,7],thatarerecoveredfor

ψ

=

const.SpecialcasesaretheSnyder realisation

[1]

,andtheMaggiorerealisation

[9]

.

We recall that in its undeformed version, the Heisenberg al-gebra

H(

x

,

p

)

is generated by commutative coordinates xμ and

momentapμ,satisfying

[

xμ

,

xν

] = [

pμ

,

pν

] =

0

,

[

pμ

,

xν

] = −

i

η

μν

.

(3)

The actionof xμ and pμ on functions f

(

x

)

belonging tothe en-velopingalgebra

A

generatedbythexμ isdefinedas

xμ



f

(x)

=

xμf

(x),

pμ



f

(x)

= −

i

∂

f

(x)

∂

xμ

.

(4)

Thenoncommutativecoordinatesxμ and

¯

theLorentzgenerators

Mμν in

(1)

–(2)canbeexpressedintermsofcommutative coordi-natesxμ andmomentapμ as

[6,7]

¯

xμ

=

xμ

ϕ

1

(β

p2

)

+ β

x

·

p pμ

ϕ

2

(β

p2

)

+ β

pμ

χ

(β

p2

),

(5)

Mμν

=

xμpν

−

xνpμ

.

(6)

Notice that the function

χ

does not appear in the defining re-lations (1)–(2), but takes into account ambiguities arising from operatororderingofxμ and pμ inequation

(5)

.

Intermsoftherealisation

(5)

,thefunctions

ϕ

μν in

(1)

read

ϕ

μν

=

η

μν

ϕ

1

+ β

pμpν

ϕ

2

,

(7)

whiletheJacobiidentitiesaresatisfiedif

ψ

= −

2

ϕ

1

ϕ

1

+

ϕ

1

ϕ

2

−

2

β

p2

ϕ

1

ϕ

2

,

(8) wheretheprimedenotesaderivativewithrespectto

β

p2.In par-ticular,thefunction

ψ

doesnotdependonthefunction

χ

.

Inserting (8) into

(1)

, it is easy to check that the coordinates

¯

xμ arecommutativefor

ϕ

2

=

2ϕ  1ϕ1 ϕ1−2βp2_ϕ

1

,andcorrespondtoSnyder spacesfor

ϕ

2

=

1+2ϕ1ϕ1 ϕ1−2βp2ϕ1

.Inparticular,theSnyderrealisation[1]

isrecoveredfor

ϕ

1

=

ϕ

2

=

1,andtheMaggiorerealisation

[9]

for

ϕ

1

=



1

− β

p2_,

_ϕ

2

=

0. Another interesting exact realisation of generalisedSnyderspacesisobtainedfor

ψ

=

s

=

const,andreads

¯

xμ

=

xμ

+

β

s

4 Kμ

,

(9)

whereKμ

=

xμ p2

₋

_2x

_·

_{ppμ arethegeneratorsofspecialconformal} transformationsinmomentumspace,satisfying

[

Kμ

,

Kν

]

=

0.

Thealgebra

(1)

–(2)includesasspecialcasesbothcommutative spaces,

ψ

=

0,andSnyderspaces,

ψ

=

1.SincetheLorentz trans-formations are not deformed,its Casimir operatoris thesameas forthePoincaréalgebra,

C =

p2_.

3. Coproductandstarproduct

The generalised Snyder spaces defined above can be investi-gatedusingtheHopf-algebraformalismdevelopedinrefs.[17–19]

andshortlyreviewedin

[20]

,towhichwe referformoredetails.1 In thisway,one candeducethe propertiesofthealgebra associ-atedtoSnyderspacestartingfromitsrealisation

(5)

.

ItcanbeshownthatforageneralHopfalgebra

A

[17–19], eik·¯x



eiq·x

=

eiP(k,q)·x+iQ(k,q)

,

(10)

and

eik·¯x



1

=

eiK(k)·x+iL(k)

,

(11)

whereeqs.

(10)

and

(11)

canbeseenasthedefiningrelationsfor thefunctions

P

,

Q

,

K

and

L

.

Itiseasilyseenthat

P

μ

(λ

k

,

q

)

satisfiesthedifferentialequation [17–19]

d

P

μ

(λk,q)

dλ

=

kα

ϕ

μ

α



_P

_(λk,q)



_,

₍₁₂₎

where

λ

isarealparameterand

P

μ

(k,

0

)

=

K

μ

(k),

P

μ

(

0

,q)

=

qμ

.

(13)

Analogously,itcanbeshownthat

Q

μ

(λ

k

,

q

)

satisfiesthe

differ-entialequation d

Q

(λk,q)

dλ

=

kα

χ

α



_P

_(λk,q)



_,

₍₁₄₎ with

χ

α

_{≡ β}

_pα

_χ

_(β

_p2

₎

_and

Q

(k,

0

)

=

L

(k),

Q

(

0

,q)

=

0

.

(15)

Thegeneralisedadditionofmomentakμ andqμ isthendefined as

kμ

⊕

qμ

=

D

μ

(k,q),

with

D

μ

(k,

0

)

=

kμ

,

D

μ

(

0

,q)

=

qμ

,

(16)

wherethefunction

D

μ

(

k

,

q

)

canbeobtainedfrom

P

μ

(

k

,

q

)

as

[7,

8,22]

D

μ

(k,q)

=

P

μ

(

K

−1

(k),q),

(17)

and the function

K

−1

μ

(

k

)

is the inverse map of

K(

k

)

, i.e.

K

−1

μ

(K(

k

))

=

kμ. Remarkably, from (12) and (7) it follows that

P

μ

(

k

,

q

)

and hence

D

μ

(

k

,

q

)

do not depend on the function

χ

in(5).

From

(11)

,itisalsopossibletocalculatethestarproductoftwo planewaves,thatturnsouttobe

eik·x



eiq·x

=

eiD(k,q)·x+iG(k,q)

,

(18)

where

G

(k,q)

=

Q

(

K

−1

(k),

q)

−

Q

(

K

−1

(k),

0

).

(19)

Notethat

G

vanishesif

χ

=

0.

1 _{Amorerigoroustreatment,includingthefullphasespace,isgivenbytheHopf} algebroidformalism[21].However,weshallnotneeditinthefollowing.

Finally,thecoproductforthemomenta



pμ canbewrittenas usualintermsof

D

μ

(

k

,

q

)

as

p

μ

=

D

μ

(p

⊗

1

,

1

⊗

p). (20)

Theprevious definitionsimplythat theadditionofmomentaand thecoproductdonot dependon

χ

.Followingthe stepssketched above,thecoproductsofmomentawerefoundforspecialcasesin ref.[7]:forexample,fortheSnyderrealisation

[1]

,

p

μ

=

1 1

− β

pα

⊗

pα



pμ

⊗

1

−

β

1

+



1

+ β

p2pμpα

⊗

p α

+



1

+ β

p2

_⊗

_p μ

.

(21)

Finally,werecall thatthe antipodesforSnyder spaceare triv-ial[7],

S(pμ

)

= −

pμ

,

S(Mμν

)

= −

Mμν

.

(22)

4. Firstorderexpansion

ThestudyoftheHopfalgebraforthegeneralisedSnydermodel isdifficult, but can be tackled usinga perturbative approach, by expandingthe realisation

(5)

ofthe noncommutative coordinates inpowersof

β

.Theexpansiongives

¯

xμ

=

xμ

+ β (

s1xμp2

+

s2x

·

p pμ

+

cpμ

)

+

O

(β

2

),

(23)

withindependentparameterss1,s2,c.Thecommutationrelations donotdependontheparameterc andtofirstorderaregivenby

[¯

xμ

,

x

¯

ν

] =

iβsMμν

+

o(β2

),

[

p_μ

,

x

¯

_ν

] = −

i

η

μν

(

1

+ β

s1p2

)

+ β

s2pμpν



+

O

(β

2

),

(24)

wheres

=

s2

−

2s1.

Themodelsofrefs.[6,7] arerecoveredfors2

=

1

+

2s1. More-over,fors1

=

0,s2

=

1,eqs.

(23)

–(24)reproducetheexactSnyder realisation,whilefors1

= −

1₂,s2

=

0 theygive thefirst-order ex-pansion of the Maggiore realisation. For s2

=

2s1, spacetime is commutative to first order in

β

, while for s1

= −

s

/

4, s2

=

s

/

2,

c

=

0 onegetstheexactrealisation

(9)

.

From (12)one can calculatethe firstorder expression forthe function

P

μ

(

k

,

q

)

inthegeneralcase,whichreads

P

μ

(k,q)

=

kμ

+

qμ

+ β

 

s1q2

+



s1

+

s2 2



k

·

q

+

s1

+

s2 3 k 2

kμ

+

s2



k

·

q

+

k 2 2

qμ

+

O

(β

2

),

(25)

fromwhereitfollowsthat

K

−1 μ

(k)

=

kμ

−

β

3

(s

1

+

s2

)k

2_k μ

+

O

(β

2

).

(26)

Theseresultsallowonetowritedownthegeneralisedaddition lawofthemomentakμ andqμ tofirstorder

(k

⊕

q)μ

=

D

μ

(k,q)

=

kμ

+

qμ

+ β



s2k

·

q qμ

+

s1q2kμ

+



s1

+

s2 2

k

·

q kμ

+

s2 2k 2_q μ



+

O

(β

2

).

(27)

Itisinterestingtoremarkthatfors2

=

2s1

=

0,althoughspacetime iscommutativeuptothefirstorderin

β

,theadditionofmomenta isdeformed,

(k

⊕

q)μ

=

kμ

+

qμ

.

(28)

The Lorentz transformations of momenta are instead not de-formed,anddenotingthemby

(ξ,

p

)

,with

ξ

therapidity param-eter,thelawofadditionofmomentaimpliesthat

(ξ,

k

⊕

q)

= (ξ

1

,k)

⊕ (ξ

2

,q)

(29)

issatisfiedfor

ξ

1

= ξ

2

= ξ

.Hencetherearenobackreactionfactors inthesenseofrefs.[23,24].Thismeansthatincompositesystems the boosted momenta of the single particles are independent of the momenta ofthe other particles inthe system. This confirms theresultsobtainedfromgeneralargumentsin

[25]

and

[20]

.

Thecoproducttofirstordercanbereadfrom

(27)

andisgiven by

p

μ

= 

0pμ

+ β



s1pμ

⊗

p2

+

s2pα

⊗

pαpμ

+



s1

+

s2 2

pμpα

⊗

pα

+

s2 2 p 2

_⊗

_p μ



+

O

(β

2

).

(30)

5. FieldtheoryfortheSnyderrealisation

The scalar field theory on Snyder spacetime was investigated in[7]usingtheSnyderrealisation

[1]

,

¯

xμ

=

xμ

+ β

x

·

p pμ

.

(31)

However, thisrealisationisnot Hermitian,andhencealsothe re-sultingactionfunctionalisnotHermitian.Thiscausessome prob-lems,inparticularin thedefinitionofameasurefortheintegral, while,asweshallshow,withaHermitianrealisationthefreefield actionreducesto theusualcommutative form.Similar considera-tionshavebeenmadein

[16]

forthe

κ

-Minkowskicase.

The realisation

(31)

canbe madeHermitian by addingits ad-joint,yielding

¯

xμ

=

xμ

+

β

2



x

·

p pμ

+

pμp

·

x



=

xμ

+ β

x

·

p pμ

−

5i 2

β

p μ

_,

(32)

wherewehaveusedthecanonicalcommutationrelationsto rear-rangetheexpression.

FortheSnyderrealisation,oneobtains

[7]

P

μ

(k,

q)

=

qμ

+



sinβk2  βk2

+

k·q k2



cos



βk

2

₋

₁



_k μ cos



βk

2

₋

k·q k2



βk

2_sin



_βk

2

,

(33) and

K

μ

(k)

=

tan



βk

2



βk

2 kμ

,

(34)

fromwhichitfollowsthat

D

μ

(k,

q)

=

1 1

− β

k

·

q



1

−

β

k

·

q 1

+



1

+ β

k2

kμ

+



1

+ β

k2_q μ



,

(35)

Usingeqs.

(14)

and

(19)

,onecancomputethefunctions

Q

and

G

fortheHermitian realisation

(32)

,obtaining

Q

(k,q)

=

5i 2 ln



cos



βk

2

₋

k

·

q k2



βk

2_sin



βk

2



,

(36)

and

G

(k,q)

=

5i

2 ln [1

− β

k

·

q]. (37)

Accordingto

(18)

,thestarproductforplanewavesinthe Her-mitian realisation

(32)

istherefore

eik·x



eiq·x

=

e

iD(k,q)·x

(

1

− β

k

·

q)5/2

,

(38) with

D

μ

(

k

,

q

)

givenby

(35)

.

Theactionofanoncommutative scalarfield

¯φ(¯

x

)

onthe iden-tity(groundstate)of

A

isdefinedas

¯φ(¯

x)



1

= φ(

x),

¯φ(¯

x)2



1

= (φ  φ)(

x). (39)

Theactionfunctionalforanoninteractingmassiverealscalarfield canthenbedefinedas

S

[φ] =

1 2



d4x

¯

(∂

μ

¯φ ∂

μ

¯φ +

m2

¯φ

2

)



1

=

1 2



d4x

(∂

μ

φ  ∂

μ

φ

+

m2

φ  φ)

(40)

Towritetheactioninsimplerform,wecomputethestar prod-uctoftworealscalarfields

φ (

x

)

and

ψ(

x

)

byexpandinginFourier series,

φ (x)

=



d4k

˜φ(

k)eikx

.

(41) Then



d4x

ψ (x)  φ (x)

=



d4x



d4k d4q

˜ψ(

k) ˜

φ(q)

eik·x



eiq·x

=



d4k d4q

˜ψ(

k) ˜

φ(q)



d4x e iD(k,q)·x

(

1

− β

k

·

q)5/2

=



d4k d4q

˜ψ(

k) ˜

φ(q)

δ

(4)



_D

_(k,q)



(

1

− β

k

·

q)5/2

.

(42) Now,since

D

μ

(

k

,

q

)

vanishesonlyforqμ

= −

kμ,

δ

(4)



D

(k,

q)



=

δ

(4)

_(q

₊

_k)





det



_∂D μ(k,q) ∂qν





_q=−k

.

(43)

Ontheotherhand,

∂

D

μ

(k,

q)

∂q

ν







q=−k

=

1 1

+ β

k2



1

+ β

k2

_δ

ν μ

−

βk

μkν 1

+



1

+ β

k2



,

(44) andthen





det



∂

D

μ

(k,

q)

∂q

_ν





_q=−k

=

1

(

1

+ β

k2

₎

5/2

.

(45) Thetermin

(45)

cancelswiththeonecomingfromthestar prod-uctin

(42)

,andfinallyoneobtains



d4x

ψ (x)  φ (x)

=



d4x

ψ (x) φ (x).

(46)

Hence,theintegralofastarproductoftwofieldsintheHermitian realisationcan be reducedto theintegral ofan ordinary product of commutative functions. The same happens in the Moyal [15]

and

κ

-Minkowskicase [16]. We conjecture that this is a univer-salpropertyofnoncommutativemodelsinaHermitianrealisation, althoughwearenotabletoproveit.

Inparticular,forafreescalarfield,theactionS

[φ]

canthenbe writtenas S

[φ] =

1 2



d4x



∂

μ

φ  ∂

μ

φ

+

m2

φ  φ



=

1 2



d4x



∂

μ

φ ∂

μ

φ

+

m2

φ

2



.

(47)

6. Fieldtheoryforthelinearisedtheory

The previous calculations can be extended to the generalised Snydermodelsby aperturbativeexpansion in

β

up tofirstorder. Thisisagoodapproximationforsmallmomenta,althoughcannot betrustedintheultravioletregion.

For simplicity, we shall limit ourconsideration to the Snyder modelsofref.[7],neglectingthegeneralisationofsection

2

.Inthis cases2

=

1

+

2s1,andtherealisation

(23)

reducesto

¯

xμ

=

xμ

(

1

+ β

s1p2

)

+ β(

1

+

2s1

)

x

·

p pμ

+

O

(β

2

).

(48) TheSnyderrealisationdiscussedintheprevioussectionisobtained fors1

=

0.Ingeneral

(48)

isnotHermitian,butaHermitian reali-sationcanbeobtainedasbeforebyredefining

¯

xμ

=

xμ

(

1

+ β

s1p2

)

+ β(

1

+

2s1

)

x

·

p pμ

−

iβ



5 2

+

6s1

pμ

+

O

(β

2

).

(49)

Forthisrealisation,eq.

(25)

reducesto

P

μ

(k,q)

=

kμ

+

qμ

+ β



s1q2

+



1 2

+

2s1

k

·

q

+



1 3

+

s1

k2



kμ

+ β(

1

+

2s1

)



k2 2

+

k

·

q



qμ

+

O

(β

2

),

(50) and

K

μ

(k)

=



1

+ β



1 3

+

s1

k2



kμ

+

O

(β

2

),

(51)

fromwhichitfollowsthat

D

μ

(k,q)

=

kμ

+

qμ

+ β



s1q2

+



1 2

+

2s1

k

·

q



kμ

+ β(

1

+

2s1

)



k2 2

+

k

·

q



qμ

+

O

(β

2

).

(52)

One can now compute the linearised

Q

and

G

from (14)

and(19),obtaining

Q

(k,

q)

= −

iβ



5 2

+

6s1



k2 2

+

k

·

q

+

O

(β

2

),

(53) and

G

(k,q)

= −

iβ



5 2

+

6s1

k

·

q

+

O

(β

2

).

(54)

ThestarproductfortheHermitianrealisation

(49)

istherefore

eik·x



eiq·x

=



1

+ β



5 2

+

6s1

k

·

q



eiD(k,q)·x

+

O

(β

2

),

(55)

where

D

isgivenineq.

(52)

.

Theproductoffieldscannowbecomputedasbeforeexpanding innoncommutativeplanewaves,andgives



d4x

ψ (x)  φ (x)

=



d4k d4q

˜ψ(

k) ˜

φ(q)



1

+ β



5 2

+

6s1

k

·

q

+

O

(β

2

)



δ

(4)

(

D

(k,q)).

(56)

Now,itiseasytoseethat

D(

k

,

q

)

vanishesonlyforqμ

= −

kμ,and then

δ

(4)



D

(k,q)



=

δ

(4)

_(q

₊

_k)





det



_δD μ(k,q) ∂qν





_q=−k

.

(57)

Ontheotherhand,

∂

D

μ

(k,

q)

∂

qν





q=−k

= δ

ν μ

− β



1 2

+

s1

k2

δ

ν_μ

+



1 2

+

2s1

kμkν



+

O

(β

2

),

(58) andhence





det



∂

D

μ

(k,q)

∂

qν





q=−k

=

1

− β



5 2

+

6s1

k2

+

O

(β

2

).

(59)

Again thecorrectionscoming from

(55)

and(59)cancel andone recovers

(46)

,showingthatalsoforgeneralrealisationsatthe lin-earleveltheintegralofthestarproductinaHermitianrealisation is equivalent to the ordinary product. In particular, eq. (47) still holdsandthe freefield propagatorcoincides withthe commuta-tiveone.

Interactionterms canbe addedto theactionthrough thestar product;forexample,acubicinteractioncanbedescribedby

S(3)

=



d4x

φ  (φ  φ).

(60)

Noticethatbecauseofthenonassociativityofthestarproductthe ordering of the products is important; one may also define the interactionusingsymmetrisedformsof

(60)

.

The computation of the star products in (60) involves the evaluation of the vertex operator

D

(_μ3)

(

k1

,

k2

,

k3

)

=

D

μ

(

k3

,

D(

k2

,

D(

k1

)))

.Thiscan beeasily doneusingtheresultsoftheprevious sections.Weleave thecalculationofthesetermsandofloop cor-rectionstofutureinvestigations.

7.Conclusions

In this paper we have extended the study of the Snyder model to its most general realisations compatible with unde-formedLorentzinvariance.Someofthenewmodelshavepeculiar properties,for example they can display commutative spacetime geometry,butnontrivialHeisenberg algebra,obeyingnonstandard lawsfortheadditionofmomenta.

WehavealsoconstructedascalarQFT onSnyderspacesalong thelinesof

[7]

.Themainimprovementofourapproachhasbeen

the use of a Hermitian representation for the noncommutative spacetimecoordinates.Thishasallowedustoshowthatthe non-interactingtermsintheactioncanbereducedtotheordinary non-commutative form, as in other noncommutative models [15,16]. This might be a universal property of noncommutative models withHermitian action,and itwould beinteresting tofurther in-vestigatetheoriginofthisproperty.Theinteractingtheorycanalso be studiedinour formalism,andwe planto pursuethistopicin futurework.

Acknowledgements

The work of S.Meljanac has beensupported by Croatian Sci-ence Foundation under the project IP-2014-09-9582, as well as by the H2020 Twinning project No. 692194, “RBI-T-WINNING”. S. MignemiwishestothanktheRudjer Boškovi ´cInstitutefor hos-pitalityduringthepreparationofthiswork.

References

[1]H.S.Snyder,Phys.Rev.71(1947)38.

[2]S.Doplicher,K.Fredenhagen,J.E.Roberts,Phys.Lett.B331(1994)39. [3]V.P.Nair,A.P.Polychronakos,Phys.Lett.B505(2001)267;

L.Mezincescu,arXiv:hep-th/0007046.

[4]J.Lukierski,H.Ruegg,A.Novicki,V.N.Tolstoi,Phys.Lett.B264(1991)331; J.Lukierski,A.Novicki,H.Ruegg,Phys.Lett.B293(1992)344.

[5]S.Majid,FoundationofQuantum GroupTheory,CambridgeUniversityPress, 1995.

[6]M.V.Battisti,S.Meljanac,Phys.Rev.D79(2009)067505. [7]M.V.Battisti,S.Meljanac,Phys.Rev.D82(2010)024028.

[8]S.Meljanac,D.Meljanac,A.Samsarov,M.Stoji ´c,Mod.Phys.Lett.A25(2010) 579;

S.Meljanac,D.Meljanac,A.Samsarov,M.Stoji ´c,Phys.Rev.D83(2011)065009. [9]M.Maggiore,Phys.Lett.B319(1993)83.

[10]F.Girelli,E.L.Livine,J.HighEnergyPhys.1103(2011)132.

[11]J.C.Breckenridge,V.Elias,T.G.Steele,Class.QuantumGravity12(1995)637. [12]L.N.Chang,D.Mini ´c,N.Okamura,T.Takeuchi,Phys.Rev.D65(2002)125027;

S.Benczik,L.N.Chang,D.Mini ´c,N.Okamura,S.Rayyan,T.Takeuchi,Phys.Rev. D66(2002)026003.

[13]S.Mignemi,Phys.Rev.D84(2011)025021; S.Mignemi,R.Štrajn,Phys.Rev.D90(2014)044019;

B.Iveti ´c,S.Mignemi,A.Samsarov,Phys.Rev.A93(2016)032109. [14]LeiLu,A.Stern,Nucl.Phys.B854(2011)894;

LeiLu,A.Stern,Nucl.Phys.B860(2012)186. [15]R.J.Szabo,Phys.Rep.378(2003)207.

[16]S.Meljanac,A.Samsarov,Int.J.Mod.Phys.A26(2011)1439. [17]S.Meljanac,Z.Škoda,D.Svrtan,SIGMA8(2012)013. [18]D.Kovacevic,S.Meljanac,J.Phys.A45(2012)135208.

[19]D.Kovacevic,S.Meljanac,A.Samsarov,Z.Škoda,Int.J.Mod.Phys.A30(2015) 1550019.

[20]S.Meljanac,D.Meljanac,F.Mercati,D.Pikutic,Phys.Lett.B766(2017)181. [21]T.Juric,S.Meljanac,R.Štrajn,Int.J.Mod.Phys.A29(2014)145022. [22]T.Juric,S.Meljanac,D.Pikutic,Eur.Phys.J.C75(2015)528. [23]G.Gubitosi,F.Mercati,Class.QuantumGravity20(2013)145002.

[24]S.Majid,AlgebraicapproachtoquantumgravityII:noncommutativespacetime, in:D.Oriti(Ed.),ApproachestoQuantumGravity,CambridgeUniversityPress, 2009.