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v3 [

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2010

CPHT-RR 123.1109LPT ORSAY 09.103

Curing the UV/IR mixing for field theorieswith translation-invariant ⋆ products

Adrian Tanasaa,b∗

and Patrizia Vitalec,d†

aCentre de Physique Theorique, CNRS UMR 7644,

Ecole Polytechnique, 91128 Palaiseau, France

bInstitutul de Fizica si Inginerie Nucleara Horia Hulubei,

P. O. Box MG-6, 077125 Magurele, Romania

cLaboratoire de Physique Theorique, Universite Paris XI

91405 Orsay Cedex, France

dDipartimento di Scienze Fisiche, Universita di Napoli Federico II

and INFN, Sezione di Napoli, Via Cintia 80126 Napoli, Italy

November 5, 2018

Abstract

The ultraviolet/infrared (UV/IR) mixing of noncommutative field theories has beenrecently shown to be a generic feature of translation- invariant associative products.In this paper we propose to take into account the quantum corrections of the modelto modify in this way the noncommutative action. This idea was already used tocure the UV/IR mixing for theories on Moyal space. We show that in the presentframework also, this proposal proves successful for curing the mixing. We achieve thistask by explicit calculations of one and higher loops Feynman amplitudes. For thesake of completeness, we compute the form of the new action in the matrix base forthe Wick-Voros product.

Keywords: noncommutative scalar field theory, translation invariance, UV/IR mixing,quantum corrections

∗e-mail: adrian.tanasa@ens-lyon.org†e-mail:vitale@na.infn.it

1

1 Introduction and motivation

Noncommutative geometry [1] is an appealing framework for the quantization of gravity. Atthe Planck scale, the quantum nature of the underlying spacetime replaces a local interactionby a specific nonlocal effective interaction in the ordinary Minkowski space [2] .

Noncommutative quantum field theories (for general reviews, see [3] and [4]) can be inter-preted as limits of matrix models or of string theory models. The first use of noncommutativegeometry in string theory was in the formulation of open string theory [5]. Noncommutativ-ity is here natural just because an open string has two ends and an interaction which involvestwo strings joining at their end points shares all the formal similarities to noncommutativematrix multiplication. In this context, one also has the Seiberg-Witten map [6], which mapsthe noncommutative vector potential to a conventional Yang-Mills vector potential, explicitlyexhibiting the equivalence between these two classes of theories.

But probably the simplest context in which noncommutativity arises is in a limit inwhich a large background antisymmetric tensor potential dominates the background metric.In this limit, the world-volume theories of Dirichlet branes become noncommutative [7],[8]. Noncommutativity was also recently proved to arise as some limit of loop quantumgravity models. There, the effective dynamics of matter fields coupled to 3−dimensionalquantum gravity is described after integration over the gravitational degrees of freedom bysome noncommutative quantum field theory [9] . In a different context, some 3−dimensionalnoncommutative space emerging in the context of 3−dimensional Euclidean quantum gravitywas also studied in [10].

In condensed matter physics, noncommutative theories can be of particular interest whendescribing effective non-local interactions, as is the case, for example, of the fractional quan-tum Hall effect. where different authors proposed that a good description of this phenomenoncan be obtained using noncommutative rank 1 Chern-Simons theory [11].

Nevertheless, when going from commutative to noncommutative theories, locality is lostand one can wonder, in this situation, if renormalizability can be restored. Indeed, whendescribing theories on the noncommutative Moyal space (the most studied noncommutativespace), a new type of non-local divergence occurs, the UV/IR mixing [12]. This new di-vergence is non-local and cannot be absorbed by counterterms at the level of the 2-pointfunction.

Despite this important difficulty, solutions exist for renormalizability to be restored. Thisis achieved for the φ4 theory by modifying the propagation part of the initial action, suchthat this new type of divergence is cured. A first type of modification adds a harmonicoscillator term in the propagator [13]. A different type of modification was proposed in [14],where the quantum correction 1/p2 was included in the bare action in momentum space.The physical interpretation of this model in the long distance regime was studied in [15].Such a modified scalar propagator appears also in recent work on non-abelian gauge theoryin the context of the Gribov-Zwanziger result [16]. Both these noncommutative models wereproved renormalizable at any order in perturbation theory, but the latter is also manifestlytranslation-invariant. Several field theoretical properties have been further investigated, forboth these noncommutative models (see [17]-[29] and references therein). Furthermore, somealgebraic geometrical properties of the parametric representation of the Grosse-Wulkenhaarmodels have been investigated in [30].

2

A part from the Moyal product, other noncommutative products have been investigatedto construct noncommutative field theories on flat space-time. One of them is the Wick-Voros(WV) product [31] which corresponds to normal ordering in deformation quantization [32],as opposed to symmetric ordering which is instead related to the Moyal product. Recently,a scalar φ4 theory was studied, with this kind of noncommutativity [33]. Computing thenon-planar tadpole Feynman amplitude, it was shown that the UV/IR mixing appears inthis framework as well, although the propagator and the vertex have different forms than inthe Moyal case. This result was extended to generic translation invariant products on flatspace-time [34], showing in a one-loop calculation that the UV/IR mixing stays unmodifiedfor the whole class of products.

In this paper we propose to modify the Euclidean action of scalar field theories with quar-tic interaction on noncommutative R

d, with generalized translation-invariant star products,along the lines of [14], in order to cure the UV/IR mixing. Namely, we compute the one-loopquantum correction for the propagator and we modify the scalar action accordingly. Wethen compute, for the modified action, the one-loop quantum corrections for the propagatorand for the vertex and we show that the mixing has disappeared when inserting the modifiednon-planar tadpoles into ”bigger” non-planar graphs.

At tree level we show that the propagator may be decomposed as a sum of Klein-Gordonpropagators, some of which with negative sign. In a commutative setting this is a signal ofillness of the theory since, when performing a Wick rotation to the Minkowski space, thesenew fields lead to negative norm states, which in turn can be rephrased into loss of unitarityof the S matrix. Whether or not the same conclusions can be drawn for our model is aninteresting open problem, mainly because the Minkowskian analogue of an Euclidean theoryis not uniquely defined in the NC setting, but also because there is no general agreementon the definition of particle states, commutation relations and the S matrix formalism itself(cfr [35, 33] and refs. therein). We will come back to this argument in the paper.

Quantum field theories (QFT) with the WV product (which, as already stated, are aparticular class of the field theories we treat here) have already been investigated in theliterature [41]. This product has been studied within the coherent states framework [42] andin relation to matrix models and Chern-Simons theory [43], [44]. Black holes have also beendefined using such a noncommutative product [45]. Finally, let us also state that a differentapproach for studying QFT with WV product has been undertaken in [46].

From a mathematical point of view, translation-invariant ⋆ products on the (hyper-)planeare all equivalent to the Moyal product in the sense of formal series [47] as they all share thesame underlying Poisson bracket. Nevertheless, they are not a priori physically equivalent,as they yield QFTs with different quantum actions. Furthermore, one cannot relate theseQFTs by simple field redefinitions, as we will show in the sequel.

The paper is structured as follows. In the next section we recall the definition and somebasic properties of translation-invariant generalizations of the Moyal product on EuclideanR

d. The results of [34] are recalled, showing the appearance of the UV/IR mixing. In section3 we write the action of the modified model we propose. We also derive the associatedFeynman rules, namely the modified propagator and modified vertex. We also discuss theissue of ghost states in a Minkowskian formulation.

We then compute the one-loop quantum corrections for the propagator and the vertex

3

of the proposed model. This allows to show that, inserting non-planar tadpoles into somehigher loop graphs leads to IR convergent Feynman amplitudes, thus curing the problemof the UV/IR mixing. In section 4 we recall the matrix basis for the WV product and wecompute the expression of the modified action in this basis. Finally, the last section presentssome conclusions and perspectives.

2 Translation-invariant products

In this section we review translation-invariant star products on the space Rd, as derived in

[34]. As we will see, examples of such products are the Moyal product and the less knownWV one (or normal ordered product). For the present purposes we present the products interms of their integral kernel in Fourier space, although other forms are available. A genericstar product on R

d may be represented as

(φ ⋆ ψ)(x) =1

(2π)d2

∫ddp ddq ddk eip·xφ(q)ψ(k)K(p, q, k), (2.1)

where K can be a distribution and φ(q) is the Fourier transform of f = φ. The product ofd-vectors is understood with the Minkowskian or Euclidean metric: p · x = pix

i. The usualpointwise product is also of this kind for K(p, q, k) = δd(k − p + q). Translation invariancerequires that the product obey

Tb(φ) ⋆ Tb(ψ) = Tb(φ ⋆ ψ), (2.2)

where Tb(f)(x) = f(x+ b) represents the translation by the vector b. At the level of Fouriertransform we have

Tbφ(q) = eibpφ(q). (2.3)

It may be seen that, for the product (2.1) to be invariant, the kernel must be of the form

K(p, q, k) = eα(p,q)δ(k − p+ q), (2.4)

where α is a generic function of p and q, further constrained by associativity and cyclicity.We therefore consider products that can be expressed as

(φ ⋆ ψ)(x) =1

(2π)d2

∫ddp ddq eip·xφ(q)ψ(p− q)eα(p,q). (2.5)

Except for the commutative case, d has to be even because of translation invariance (besidesdegenerate cases, where one of the dimensions commutes with the other ones).

Let us also emphasize here that translation invariance requires as well the commutatorof coordinates to be constant, as in the Moyal case. We explictly show this at the end ofthis section, in equation (2.31).

When α = 0, one has the usual pointwise product. One then has two important exam-ples of noncommutative associative products which are of the form above. They are bothborrowed by ordinary phase space quantization and correspond to different ordering choices.

4

One is the Moyal product, quite well studied in the literature. It corresponds to symmet-ric ordering in deformation quantization of phase space (Weyl quantization) and in Fouriertransform it acquires the form

(φ ⋆M ψ)(x) =1

(2π)d2

∫ddp ddq φ(q)ψ(p− q)eip·xe

i

2piθΣ

ijqj (2.6)

thus giving

αM(p, q) = − i

2θΣijqipj. (2.7)

We have denoted by Σ the d× d block-diagonal antisymmetric matrix

Σ =

0 1−1 0

..

and by θ some constant noncommutativity parameter.The other example is the WV product. It corresponds to normal ordering in deformation

quantization of phase space and in Fourier transform it reads

(φ ⋆WV ψ)(x) =1

(2π)d2

∫ddp ddq φ(q)ψ(p− q)eip·xe−θq

−·(p+−q+) (2.8)

with

pi± =pi1 ± ipi2√

2, i = 1, ..., d/2 (2.9)

the function α is therefore given by

αWV (p, q) = −θq− · (p+ − q+). (2.10)

Further restrictions on K come from the associativity requirement which reads∫

ddkK(p, k, q)K(k, r, s) =

∫ddkK(p, r, k)K(k, s, q) (2.11)

This is nothing but the usual cocycle condition in the Hochschild cohomology. For moredetails on cohomological aspects we refer the interested reader to [34]. In terms of α, equation(2.11) reads

α(p, q) + α(q, r) = α(p, r) + α(p− r, q − r) (2.12)

¿From this cocycle relation then follow:

α(p, p) = α(0, 0) = α(p, 0)

α(0, p) = α(0,−p)α(p, q) = −α(q, p) + α(0, q − p) (2.13)

andα(p+ q, p) + α(−p− q,−q).+ α(0, p+ q)− α(0, p)− α(0, q) = 0. (2.14)

5

The second relation of (2.13) ensures also the trace property. We have indeed∫

ddx φ ⋆ ψ =

∫ddx ddp ddq eα(p,q)eip·xφ(q)ψ(p− q)

=

∫ddq eα(0,q)φ(q)ψ(−q) =

∫ddx ψ ⋆ φ. (2.15)

For the product to be commutative α has to satisfy the condition

α(p, q) = α(p, p− q), (2.16)

which may be regarded as a coboundary condition (see again [34]). Finally, let us noticethat the Moyal and WV products are related by the following relation:

αM(p, q) = αWV (p, q)−θ

2q · (p− q). (2.17)

As already stated above, translation-invariant products have been introduced in [34] inthe context of noncommutative scalar field theories with quartic interaction, on R

d withMinkowskian metric. Here we choose to work with Euclidean metric. We keep the notationd for the number of space-time dimensions; nevertheless, when explicit divergence analysisis performed, we refer to the case

d = 4.

The action reads

S =

∫ddx

[1

2(−∂µφ ⋆ ∂µφ+m2φ ⋆ φ) +

λ

4!φ ⋆ φ ⋆ φ ⋆ φ

]. (2.18)

In momentum space the propagator is thus

G(2)0 (p) =

e−α(0,p)

p2 +m2(2.19)

whereas for the vertex we have

V⋆ = V0eα(k1+k2,k1)+α(k3+k4,k3)+α(0,k1+k2), (2.20)

and we have denoted by V0 the ordinary commutative vertex

V0 =λ

4!(2π)dδd

(4∑

a=1

ka

). (2.21)

Interestingly, a propagator of the form (2.19) with the function α specified in (2.10) wasalready found in [40] in a different approach.

To obtain the four-point correlation function at tree level we just attach to the vertexfour propagators. We thus have (up to a constant):

G(4)0 =

eα(k1+k2,k1)+α(k3+k4,k3)+α(0,k1+k2)−∑4

a=1α(0,ka)

∏4a=1(k

2a +m2)

δ

(4∑

a=1

ka

). (2.22)

6

Let us now make the following important remark. In order to obtain the usual Feynmanrules of a Moyal QFT, one can try by reabsorbing the phase e−α(0,p) of (2.19) by a properfield redefinition. Nevertheless, this would not reproduce the vertex form of Moyal QFT. Wethus conclude that, by a simple field redefinition one does not have equivalence between thegeneral class of QFTs we deal with in this paper and the Moyal one. In [33] it was shownthat an equivalence between Moyal and WV Minkowskian QFTs can be established at thelevel of the S-matrix only by implementing the appropriate twisted Poincare symmetry foreach of them.

Before going further we also give some explanations on the planarity of the Feynmangraphs used in this work. As in the Moyal case, the vertex has a symmetry under cyclicpermutation of the incoming/outgoing fields at some vertex. Furthermore, one can also usesome matrix base to re-expresses these products (for the WV case, see section 4). For allthese reasons, an appropriate way to represent Feynman graphs is through ribbon graphs.If the genus of the manifold on which the respective graph is denoted is vanishing, therespective graph is planar. Furthermore, another important notion is the one of faces brokenby external legs. In the rest of the paper, by a slight abuse of language we will call non-planargraphs also the planar graphs with more than one face broken by external legs.

Consider now the two graphs of Figures 1 and 2.

p

q

Figure 1: The planar tadpole graph.

p

q

Figure 2: The non-planar tadpole graph.

For the planar case of Figure 1 the correction is obtained using three propagators of theform (2.19), one with momentum p, one with momentum −p, one with momentum q andthe vertex (2.20) with assignments k1 = −k4 = p and k2 = −k3 = q and, of course, theintegration in q. We have (up to a constant)

G(2)1, P =

e−α(0,p)

(p2 +m2)2

∫ddq

eσ(p,q)

(q2 +m2)(2.23)

7

where we have used the fact that the three exponential factors of the vertex combine withtwo out of three exponential factors of the propagators to yield

σ(p, q) = α(p+ q, p) + α(−p− q,−q) + α(0, p+ q)− α(0, q)− α(0, p). (2.24)

Using the cocycle condition (2.14), we have then

σ(p, q) = 0 (2.25)

Notice that, with respect to the commutative case, the only correction is in the factore−α(0,p) which is the correction of the free propagator. The ultraviolet divergences here arethus identical to the commutative and to the Moyal case.

Consider now the non-planar case in Figure 2. The structure is the same as in the planarcase, but this time the assignments are

k1 = −k3 = p and k2 = −k4 = q. (2.26)

We have (up to a constant)

G(2)1, NP =

∫ddq

e−α(0,p)+α(p+q,p)−α(p+q,q)eσ(p,q)

(p2 +m2)2(q2 +m2). (2.27)

with the same notation as above. Therefore σ(p, q) = 0 and the one-loop corrections to thepropagator in the non-planar case can be rewritten as

G(2)1,NP =

e−α(0,p)

(p2 +m2)2,

∫ddq

eω(p,q)

(q2 +m2), (2.28)

where we have introduced the antisymmetric function

ω(p, q) = α(p+ q, p)− α(p+ q, q). (2.29)

For the Moyal product this term is the oscillating phase. For general translation invariantproducts this function has been computed explicitly in [34], using the properties of thefunction α. It does not depend on the specific translation invariant product but only on thecohomology class of α. Except for numerical factors, one has, for all translation-invariantproducts

ω(p, q) = ipiθΣijqj = 2αM(p, q). (2.30)

Details on the derivation may be found in [34]. The function α(0, p) which appears in (2.28)is not integrated in the loop, therefore it does not influence the convergence properties ofthe graphs.

In fact the function ω is nothing but the Poisson structure of the underlying classicalspace, the germ of deformation of the commutative product towards the star product. Itdetermines the noncommutativity of space-time coordinates. A straightforward calculationgives

xi ⋆ xj − xj ⋆ xi = − ∂2α

∂pi∂qj(0, 0) +

∂2α

∂pj∂qi(0, 0) = ωij = iθΣij . (2.31)

Using (2.30), one can prove that the Feynman integral (2.28) is UV finite but has a

C1

(θp)2+m2C2log(θ p)

2 + F (p) (2.32)

behavior in the IR regime of the external momentum p. We have denoted by C1 and C2 someconstants and by F (p) some analytic function at p = 0 (see [14] for a detailed analysis).

8

3 The proposed model and quantum corrections - cur-

ing the UV/IR mixing

In this section we write the action for the proposed model in the noncommutative settingdescribed previously. We then compute quantum corrections of the modified model (one andhigher number of loops) which show the way in which the UV/IR is manifestly cured.

As already stated in the Introduction, the modification we propose for the model of theprevious section is dictated by the quantum corrections. Thus, we add to the action thesupplementary term:

δS[φ] =a

2θ2

∫ddx (∂µ)

−1φ ⋆ (∂µ)−1φ. (3.1)

The complete action, in coordinate space, reads then

S =

∫ddx

[1

2(∂µφ ⋆ ∂µφ+

a

θ2∂−1µ φ ⋆ ∂−1

µ φ+m2φ ⋆ φ) +λ

4!φ ⋆ φ ⋆ φ ⋆ φ

], (3.2)

with

∂−1µ φ(x) =

∫dµx′ φ(x′). (3.3)

The supplementary term is better understood in momentum space. Observing that for thestar product (2.5) ∫

ddx f(x) ⋆ g(x) =

∫ddp f(p)g(−p)eα(0,p) (3.4)

we have

δS[φ] =a

2θ2

∫ddp

eα(0,p)

p2φ(p)φ(−p). (3.5)

This leads to a modification of the propagator (2.19) in the form

G(2)0 (p) =

e−α(0,p)

p2 +m2 + a(θp)2

. (3.6)

Note that, as in [14] the new parameter a is taken to be positive (such that the propagator(3.6) is positively defined. The vertex contribution remains the same as in (2.20) (since theterm (3.5) is only quadratic in the field).

A few remarks are to be done here. The first one is related to the possibility of decom-posing the propagator (3.6) as a sum of conventional Klein-Gordon propagators. We use theformula

1

A+B=

1

A− 1

AB

1

A +B, (3.7)

forA = p2 +m2, B =

a

θ2p2. (3.8)

Thus, the propagator (3.6) writes

e−α(0,p)

p2 +m2− e−α(0,p)

p2 +m2

a

θ2p2(p2 +m2) + a

9

=e−α(0,p)

p2 +m2− e−α(0,p)

p2 +m2

a

θ2(p2 +m21)(p

2 +m22), (3.9)

where −m21 and −m2

2 are the roots of the denominator of the second term in the LHSconsidered as a second order equation in p2, namely

m21 =

θ2m2 −√θ4m4 − 4θ2a

2θ2

m22 =

θ2m2 +√θ4m4 − 4θ2a

2θ2(3.10)

with 0 < a < θ2m4/4. Also

m2 > m22 >

m2

2> m2

1 > 0 (3.11)

Note that this decomposition was already made (for the Moyal case) in [24]. We now gofurther and decompose the second term on the RHS of (3.9); after some algebra this finallyleads to rewrite the propagator (3.6) as an alternate sum

G(2)0 (p) =

a/θ2

m22 −m2

1

(1

m21(p

2 +m22)

− 1

m22(p

2 +m21)

)(3.12)

In similar situations, in the commutative framework (for example QFT with higher deriva-tives) an equivalent description is introduced in terms of KG fields. The ones responsible forthe negative propagators are the ghost fields. They lead, when analitically continuing themodel to the Minkowskian setting, to states of negative norm. They are not independentfields, therefore one could resort to the original formulation in terms of one field (for scalartheories as the one considered here) but this usually causes the loss of uninitarity for the Smatrix. Some care is needed in order to extend this analysis to our model. The first maindifference between commutative and noncommutative field theory is that the Minkowskianversion of a NC field theory is not uniquely defined. Important features, such as the UV/IRmixing of the Euclidean formulation may be completely absent in some Minkowskian formu-lations [36].. The second important point is that the very concept of particle state is not welldefined in a nonlocal theory. This remark is particularly important for our model, where theasymptotic regime is not attained for small noncommutativity [15]. However, let us stick tothe usual Wick rotation. Then, it was shown in [37] that time-space noncommutativity leadsto perturbative nonunitarity, independently from the details of the model, although a morecareful analysis indicates that unitarity can be restored [38] . Later on it has been arguedwith a nonperturbative study [39] that unitarity loss is a direct consequence of the UV/IRmixing. Whether this problem persists in our model, where the modification to the kineticterm was introduced precisely to cancel the UV/IR mixing, it is a delicate issue and we don’tknow the answer. In order to see the consequences of the ghost fields appearing in (3.12) inthe appropiate Minkowskian formulation, one should carefully define the S matrix and thenstudy its properties. This is an interesting problem which deserves further investigation.

Let us also make a second remark with respect to the Euclidean theory treated here.Even though, for massive theories, the quantum correction of the propagator (2.32) hassub-leading logarithmic divergence, one does not need to take it into consideration when

10

proposing a model which may be perturbatively renormalizable. We will come back to thispoint at the end of the next section.

Since we have modified the propagator (2.19) to (3.6) but not the vertex (2.20), theone-loop corrections to the propagator represented by the graphs of Figures 1 and 2 may bederived as in (2.23), (2.27) and we find respectively:

G(2)1, P =

e−α(0,p)

(p2 +m2 + aθ2p2

)2

∫ddq

eσ(p,q)

(q2 +m2 + aθ2q2

), (3.13)

G(2)1, NP =

e−α(0,p)

(p2 +m2 + aθ2p2

)2

∫ddq

eω(p,q)+σ(p,q)

(q2 +m2) + aθ2q2

(3.14)

with σ(p, q) = 0.Let us consider now the one-loop correction to 4-point functions. For the planar case in

Fig. 3 we have

K

q

k

Figure 3: Planar one loop four point graph of incoming momentum K

G(4)1, P =

λ2

8G

(4)0

∫ddq ddk

δ(K − q − k)eσ(q,k)

(q2 +m2 + aθ2q2

)(k2 +m2 + aθ2k2

),

(3.15)

with G40 the 4-points Schwinger function at tree level, given by (2.22), while σ(q, k) = 0 (with

σ defined in (2.24)). Note that we have also denoted by K the total incoming momentum.For the non-planar case, one possible graph is shown in Fig. 4. We have then

G(4)1, NP =

λ2

8G4

0

∫ddq ddk

δ(K − q − k)eω(q,k)+σ(k,q)

(q2 +m2 + aθ2q2

)(k2 +m2 + aθ2k2

), (3.16)

with ω(q, k) defined in (2.29), (2.30) while the remaining exponential factors rearrange insuch a way to yield σ(k, q) vanishing. The other non-planar graphs are obtained similarly,with a relabeling of the external paths.

For d = 4, the last integral behaves like a logarithm in the external momenta. Indeed, thiscan be seen from the fact that, in the UV regime of the loop momentum q, the correctionsin a/(θ2q2) are neglectable. Solving the δ function, the integral (3.16) behaves like

∫d4q

eω(q,k)

q4. (3.17)

11

K

q

k

Figure 4: A non-planar one loop four points graph of total incoming momentum K

Because of the form (2.30) of the factor ω(q, k) one can easily obtain the logarithm behaviourin the external momenta. Nevertheless, these logarithms are harmless for perturbative renor-malization. We will come back on this point at the end of this section.

Let us now compute some 2−loop contributions to the 4−point Schwinger function. Thefirst of them is the one due to the graph in Fig 5. This graph is obtained by inserting the

K

k

q

p

Figure 5: A 2−loop graph obtained by inserting into the bubble graph of Fig. 3 a non-planartadpole of momentum p.

non-planar tadpole (3.14) of momentum p into the planar graph in fig 3. One has

G(4)2, P = G

(4)0

∫d4p

eω(p,q)+σ(p,q)

p2 +m2 + aθ2p2

∫ddq d4k

δ(K − q − k)eσ(q,k)

(q2 +m2 + aθ2q2

)2(k2 +m2 + aθ2k2

), (3.18)

and σ = 0. Let us emphasize that again, the cancellation of the exponentials in σ is obtainedthanks to the cocycle condition (2.14). It is this cancellation that makes that the Feynmanintegrals have the same behaviour as in the Moyal case. Let us now investigate the behaviourof the Feynman amplitude of the more general graph obtained from inserting a chain of Nnon-planar tadpoles in the planar graph of Fig. 3 (see Fig. 6).

The integral to investigate writes indeed

∫ N∏

i=1

ddpieω(pi,q)+σ(pi,q)

p2i +m2 + aθ2p2i

∫ddq ddk (

1

q2 +m2 + aθ2q2

)N+1 δ(K − q − k)eσ(q,k)

k2 +m2 + aθ2q2

(3.19)

12

k

K

q

Figure 6: A non-planar graph obtained from the insertion into the bubble graph of Fig. 3of a chain of non-planar tadpoles of momenta pi (i = 1, . . . , N).

with σ(pi, q) = σ(q, k) = 0. This is the generalization of the Feynman amplitude (3.18).Let us now have a closer look at the structure of the divergences of this general integral.For d = 4, when performing the integrations in the momenta pi (i = 1, . . . , N) and placingourselves in the IR regime of the momentum q, each of these integrals leads to a 1/θ2q2

behaviour (as proved above). The integral (3.19) thus becomes

∫d4q(

1

θ2q2)N(

1

q2 +m2 + aθ2q2

)N+1 1

(q −K)2 +m2 + aθ2(q−K)2

(3.20)

Note that if a = 0 this integral is IR divergent for N > 1 (for N = 1 the mass m preventsthe divergence to appear). Nevertheless, if a 6= 0, in the IR regime of q the dominant termis the a/θ2q2 in the propagators and the integral leads to an IR finite behaviour.

Let us now discuss the appearance of some logarithmic divergences at the level of the 2−and 4−point functions, logarithms on the external momenta of the respective graph. Thisdivergences are harmless when dealing with perturbative renormalizability. To illustrate this,let us consider the example of Fig. 7, where one has to deal with a chain of N non-planarbubble graphs inserted into some “bigger” non-planar graph. Since, as already explainedabove, the correction proposed here in the propagator is irrelevant in the UV regime (wherethis analysis is now performed), the Feynman integral gives the same behaviour as in acommutative theory, namely:

∫d4p

1

(p2 +m2)3logN

p2

m2≈ N ! (3.21)

This is a (large) finite number which appears as a difficulty in summing perturbation theory- the renormalon problem (see for example [48]). The situation is analogous for the non-planar tadpoles insertions. These logarithms should however be taken into considerationwhen defining a model which is requested to be non-perturbatively renormalizable. Let usalso remark that in [14], these logarithm divergences have not appeared because of the useof some appropriate scale decomposition (the multi-scale analysis being used there for theproof of perturbative renormalization).

13

p

pp

Figure 7: Insertion of a chain of bubble graph into some bigger graph.

4 The modified action for the WV product in the ma-

trix basis

For the WV product it is possible to rewrite the model in a suitably defined matrix basis. Thebasis is a variation of the one described in [49] for the Moyal product and it was introduced,up to our knowledge, in [43] where the WV product was used on R

2 to build a fuzzy version ofthe disc. In the following we review the derivation of the matrix basis as in [43] and we adaptit to the present notation; we then derive the model under consideration in the matrix basis.It is convenient to consider the plane as a complex space with z = (x+iy)/

√2. The quantized

versions of z and z are the usual annihilation and creation operators, a = (x + iy)/√2 and

a† = (x− iy)/√2 with a slightly unusual normalization, so that their commutation rule is

[a, a†] = θ. (4.1)

Given the function φ(z, z) consider its Taylor expansion:

φ(z, z) =∞∑

m,n=0

φTaymn z

mzn. (4.2)

To this function we associate the operator

Ωθ(φ) := φ =∞∑

m,n=0

φTaymn a

†man. (4.3)

We have thus “quantized” the plane using a normal ordering prescription. The map Ωθ isinvertible. It can be efficiently expressed defining the coherent states:

a |z〉 = z |z〉 , (4.4)

One then hasΩ−1

θ (φ) = φ(z, z) = 〈z| φ |z〉 . (4.5)

14

The maps Ω and Ω−1 yield a procedure of going back and forth from functions to operators.Moreover, the product of operators being noncommutative, a noncommutative ⋆ productbetween functions is implicitly defined as

(φ ⋆ φ′) (z, z) = Ω−1 (Ω(φ) Ω(φ′)) . (4.6)

It is possible to see that the WV product (2.8) is exactly of this form, namely

(φ ⋆WV ψ)(z, z) = 〈z| φψ |z〉 . (4.7)

There is another useful basis on which it is possible to represent the operators and hence thefunctions - the matrix basis. As we already stated above, it is similar to the one introducedfor the Moyal product in [49] and subsequently used in [13]. Consider the number operator

N = a†a, (4.8)

and its eigenvectors which we indicate by |n〉:

N |n〉 = nθ |n〉 . (4.9)

We can then express the operators within a density matrix notation:

φ =∞∑

m,n=0

φmn |m〉 〈n| . (4.10)

Applying the dequantization map (4.5) to φ, we associate to it

φ(z, z) = 〈z| φ |z〉 =∞∑

m,n=0

φmn 〈z |m〉 〈n |z〉 :=∞∑

m,n=0

φmnvmn(z, z) (4.11)

to be compared with (4.2), the same function in a different basis. Observing that

〈z |n〉 = e−zz2θ

z√n!θn

(4.12)

we have

vnm(z, z) = e−zz2θ

znzm√n!m!θn+m

. (4.13)

The elements of the density matrix basis have a very simple multiplication rule

|m〉 〈n |p〉 〈q| = δnp |m〉 〈q| ; (4.14)

this leads tovmn(z, z) ⋆WV vpq(z, z) = δnpvmq(z, z). (4.15)

Moreover, one has: ∫d2z vnm(z, z) = δnmθπ. (4.16)

15

The functions vnm(z, z) thus form an orthogonal basis in the noncommutative algebra offunctions on the plane, with the WV product, the matrix basis (in analogy with its operatorcounterpart, (4.14)). The connection between the expansions (4.3) and (4.10) is given by:

a =∞∑

n=0

√(n+ 1) θ |n〉 〈n + 1| (4.17)

a† =∞∑

n=0

√(n+ 1) θ |n+ 1〉 〈n| . (4.18)

Thus, looking at their symbols

z = 〈z| a |z〉 =∑

n

√(n+ 1)θ vnn+1(z, z)

z = 〈z| a† |z〉 =∑

n

√(n+ 1)θ vn+1n(z, z), (4.19)

we then have

φTaymn =

minm,n∑

l=0

(−1)lφm−l,n−l

l!√

(m− l)!(n− l)!θm+n. (4.20)

In the density matrix basis, using (4.13), the product (2.8) (or (4.7)) simplifies to an infiniterow by column matrix multiplication:

(φ ⋆WV ψ)mn =∞∑

k=1

φmkψkn. (4.21)

Using the expansion (4.11) and (4.16) it is easy to see that

∫d2z φ(z, z) = πθ TrΦ = πθ

∞∑

n=0

φnn, (4.22)

where we have introduced the infinite matrix Φ with entries φij.Let us generalize this basis to higher (even) dimensions. In d dimensions we need d/2

copies of the WV plane, with d/2 pairs of complex coordinates zi, zi. As for the Moyalhyperplane, coordinates describing different 2-planes, commute among themselves. We define

v~m~n(z1, ..., zd/2, z1, ..., zd/2) =⟨z1, ...zd/2

∣∣m1, ..., md/2

⟩ ⟨n1, ..., nd/2

∣∣z1, ...zd/2⟩= Π

d/2i=1v

imini

(4.23)with

vimini= 〈zi |mi〉 〈ni |zi〉 = e−

zizi2θ

zmi

i zni

i√ni!mi!θni+mi

(4.24)

The functions v~m~n form an orthogonal basis in the algebra of noncommutative functions onthe hyperplane, as it may be easily checked that

v~m~n ⋆WV v~p~q = v~m~qδ~n~p. (4.25)

16

Moreover, one has ∫ddz v~m~n = (πθ)d/2δ~m~n. (4.26)

The field φ is therefore expanded as

φ(z, z) =

∞∑

mi,ni=0

i=1,...,d

φ~m~nv~m~n(z, z). (4.27)

Moreover, one has

φ(z, z) ⋆WV ψ(z, z) =∑

φ~m~nψ~n~qv~m~q (4.28)

and also ∫ddz φ(z, z) = (πθ)d

∑φ~m~nδ~m~n. (4.29)

Having established this basis it is now possible to express the action of our model (3.2) inmatrix notation. The mass and the interaction terms are just row by column multiplication.We have

m2

2

∫ddz φ ⋆WV φ+

λ

4!

∫ddz φ ⋆WV φ ⋆WV φ ⋆WV φ

= (πθ)d/2(µ2

2

∑φ~m~nφ~n~qδ~m~q +

λ

4!

∑φ~m~nφ~n~qφ~q~pφ~p~rδ~m~r

)(4.30)

Let us consider the kinetic term. We first observe that

1

2

∫ddz ∂µφ ⋆WV ∂µφ =

∫ddz

∑

i

∂ziφ ⋆WV ∂ziφ (4.31)

and

∂ziφ =1

θ[zi, φ]⋆WV

(4.32)

∂ziφ =1

θ[zi, φ]⋆WV

. (4.33)

Eqs. (4.17), (4.18), suitably generalized to the d case, imply in turn that zi and zi areexpanded in the matrix basis as

zi =∑

~n

√(ni + 1)θ vini ni+1Πj 6=iv

jnj nj

(4.34)

zi =∑

~n

√(ni + 1)θ vini+1ni

Πj 6=ivjnj nj

. (4.35)

We then have

∂ziφ =1

θ

∑

~m~n

φ~m~n

(√mi + 1vimi+1ni

−√niv

imi ni−1

)Πj 6=iv

imi ni

∂ziφ =1

θ

∑

~m~n

φ~m~n

(√miv

imi−1ni

−√ni + 1vimi ni+1

)Πj 6=iv

imi ni

(4.36)

17

The kinetic term (4.31) becomes

(πθ)d√θ

∑

~m~n

~p~q

φ~m~nφ~p~q

∑

i

(δmiqiδnipi(−mi − ni − 1)

+δmi+1qiδni pi−1√qipi + δni−1 piδmiqi+1

√(qi + 1)(pi + 1)

)Πj 6=iδmjqjδnjpj (4.37)

Let us consider now the supplementary term we have added in (3.2)

1

θd

∫ddz

(∫ddz′ φ(z′, z′)

)⋆WV

(∫ddz′ φ(z′, z′)

). (4.38)

To compute the two indefinite integrals above we use the Taylor expansion (4.3). We thusarrive at the following expression:

1

θd

∑

~m~n

~r~s

φTay~m~nφ

Tay~r~s Πi

1

(mi + 1)(ni + 1)(ri + 1)(si + 1)

∫ddz zm+1

i zn+1i ⋆WV z

r+1i zs+1

i (4.39)

We then use the matrix basis to evaluate the star product and we obtain:

1

θd

∑

~m~n

~r~s

φTay~m~nφ

Tay~r~s

∑

~p

∫ddzΠi

zpi+mi

i zpi+ni−ri+sii e−

ziziθ

(mi + 1)(ni + 1)(ri + 1)(si + 1)

(pi + ni + 1)!√

(pi +mi + 1)!(pi + ni − ri + si + 1)!θmi+ni+ri+si+4

pi!(pi + ni − ri)!√(pi +mi)!(pi + ni − ri + si)!θ2pi+mi+ni−ri+si

. (4.40)

The integral may be easily performed and, after some algebra, we obtain for the supplemen-tary term the expression:

∑

~m~n

~r~s

φTay~m~nφ

Tay~r~s

∑

~p

Πi(pi +mi + 1)!(pi + ni + 1)!θri−mi−pi

(mi + 1)(ni + 1)(ri + 1)(si + 1)pi!(pi + ni − ri)!δmi+ri,ni+si. (4.41)

Finally, using (4.20) to rewrite the coefficients φTay, we get

∑

~m~n

~r~s ~p

Πi(pi +mi + 1)!(pi + ni + 1)!θri−mi−pi

(mi + 1)(ni + 1)(ri + 1)(si + 1)pi!(pi + ni − ri)!δmi+ri,ni+si

min~m,~n∑

~l=0

min~r,~s∑

~k=0

Πj

(−1)lj+kjθ−mj−rjφ~m−~l~n−~lφ~r−~k~s−~k

lj!kj!√

(mj − lj)!(nj − lj)!(rj − kj)!(sj − kj)!. (4.42)

where, with an abuse of notation,∑min~m,~n

~l=0stands for

∑minmi,nili=0

i=1,..,d

. Finally, summing all

the contributions that we obtain from (4.30), (4.31) and (4.42), the complete action (3.2) of

18

our model with supplementary term, is rewritten in the matrix basis as

S[φ] = (πθ)d/2(m2

2

∑φ~m~nφ~n~qδ~m~q +

λ

4!

∑φ~m~nφ~n~qφ~q~pφ~p~rδ~m~r

)

+(πθ)d/2√

θ

∑

~m~n

~p~q

φ~m~nφ~p~q

∑

i

(δmiqiδnipi(−mi − ni − 1)

+δmi+1qiδni pi−1√qipi + δni−1 piδmiqi+1

√(qi + 1)(pi + 1)

)Πj 6=iδmjqjδnjpj

+∑

~m~n

~r~s ~p

Πi(pi +mi + 1)!(pi + ni + 1)!θri−mi−pi

(mi + 1)(ni + 1)(ri + 1)(si + 1)pi!(pi + ni − ri)!δmi+ri,ni+si

min~m,~n∑

~l=0

min~r,~s∑

~k=0

Πj

(−1)lj+kjθ−mj−rjφ~m−~l~n−~lφ~r−~k~s−~k

lj !kj!√

(mj − lj)!(nj − lj)!(rj − kj)!(sj − kj)!. (4.43)

5 Conclusion and perspectives

In this paper we have proposed a solution for curing the UV/IR mixing which appearswhen implementing field theories using a translation-invariant product on R

4. This solutiongeneralizes the one proposes in [14] for curing the UV/IR mixing on Moyal space. Weexplicitly compute the 1−loop Feynman amplitudes of the proposed model, as well as higherloops amplitudes of some graphs obtained by an insertion of non-planar tadpoles. Our resultis mainly due to the cocycle condition (2.13).

An immediate perspective is to obtain a proof of the perturbative renormalization of theproposed model at any order in perturbation theory. This could be achieved by investigatingthe general form of the factor generalizing the Moyal oscillating phase of some Feynmanamplitude.

As already stated in this paper, the noncommutative products that we have worked hereare equivalent in the formal series sense of Kontsevich. We thus explicitly show that animportant field theoretical result - curing the UV/IR mixing - can be obtained applyingthe same recipe as in Moyal field theory. It is interesting to further understand the explicitrelation between this formal series equivalence of the noncommutative products and theequivalence of the Euclidean field theories thus implemented.

Nevertheless, as already argued in [33], when doing Minkowskian field theory, the situa-tion is manifestly different, because the different factors on the external propagator can leadto a different S matrix form, unless properly implementing the quantum symmetry of themodel as a twisted symmetry.

Acknowledgment: Adrian Tanasa acknowledges the CNCSIS grant “Idei” 454/2009, ID-44. Patrizia Vitale acknowledges the European Science Foundation Exchange Grant 2595under the Research Networking Program “Quantum Geometry and Quantum Gravity” . Theauthors would also like to warmly thank LPT Orsay for the hospitality.

19

References

[1] Alain Connes and Matilde Marcolli, ”Noncommutative Geometry, Quantum Fields andMotives”.

[2] S. Doplicher, K. Fredenhagen and J. E. Roberts, “The Quantum structure of space-time at the Planck scale and quantum fields,” Commun. Math. Phys. 172, 187 (1995)[arXiv:hep-th/0303037].

[3] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Phys. Rept. 378, 207(2003) [arXiv:hep-th/0109162].

[4] M. R. Douglas and N. A. Nekrasov, “Noncommutative field theory,” Rev. Mod. Phys.73, 977 (2001) [arXiv:hep-th/0106048].

[5] E. Witten, “Noncommutative Geometry And String Field Theory,” Nucl. Phys. B 268,253 (1986).

[6] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 9909,032 (1999) [arXiv:hep-th/9908142].

[7] Connes A, Douglas M. R., Schwarz A.: ”Noncommutative Geometry and Matrix Theory:Compactification on Tori”, JHEP 9802, 3-43 (1998)

[8] M. R. Douglas and C. M. Hull, “D-branes and the noncommutative torus,” JHEP 9802,008 (1998) [arXiv:hep-th/9711165].

[9] L. Freidel and E. R. Livine, “Effective 3d quantum gravity and non-commutative quan-tum field theory,” Phys. Rev. Lett. 96, 221301 (2006) [arXiv:hep-th/0512113].

[10] E. Joung, J. Mourad and K. Noui, “Three Dimensional Quantum Geometry and De-formed Poincare Symmetry,” J. Math. Phys. 50, 052503 (2009) [arXiv:0806.4121 [hep-th]].

[11] Susskind L.: The Quantum Hall Fluid and Non-Commutative Chern Simons Theory.hep-th/0101029; Polychronakos A. P.: Quantum Hall states on the cylinder as unitarymatrix Chern-Simons theory. JHEP, 06, 70-95 (2001); Hellerman S., Van RaamsdonkM.: Quantum Hall physics equals noncommutative field theory. JHEP 10, 39-51 (2001).

[12] S. Minwalla, M. Van Raamsdonk and N. Seiberg, “Noncommutative perturbative dy-namics,” JHEP 0002, 020 (2000) [arXiv:hep-th/9912072].

[13] H. Grosse and R. Wulkenhaar, “Renormalisation of phi**4 theory on noncommutativeR**2 in the matrix base,” JHEP 0312, 019 (2003) [arXiv:hep-th/0307017]. Renormal-izationof φ4-theory on noncommutative R

4 in the matrix base, Commun. Math. Phys.256, 305-374 (2005)

[14] R. Gurau, J. Magnen, V. Rivasseau and A. Tanasa, “A translation-invariant renor-malizable non-commutative scalar model,” Commun. Math. Phys. 287, 275 (2009)[arXiv:0802.0791 [math-ph]].

20

[15] R. C. Helling, J. You, “Macroscopic Screening of Coulomb Potentials From UV/IR-Mixing”, JHEP 0806, 067 (2008) [arXiv:0707.1885 [hep-th]].

[16] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel and H. Verschelde, Phys. Rev. D78, 065047 (2008) [arXiv:0806.4348 [hep-th]].

[17] V. Rivasseau and A. Tanasa, “Parametric representation of ’covariant’ noncommutativeQFT models,” arXiv:math-ph/0701034, Commun. Math. Phys. 279, 355 (2008)

[18] R. Gurau, A. P. C. Malbouisson, V. Rivasseau and A. Tanasa, “Non-CommutativeComplete Mellin Representation for Feynman Amplitudes,” Lett. Math. Phys. 81, 161(2007) arXiv:0705.3437 [math-ph].

[19] R. Gurau and A. Tanasa, “Dimensional regularization and renormalization of non-commutative QFT,” Annales Henri Poincare, 9, 655 (2008) arXiv:0706.1147 [math-ph].

[20] A. Tanasa and F. Vignes-Tourneret, “Hopf algebra of non-commutative field theory,”J. Noncomm. Geom. 2 , arXiv:0707.4143 [math-ph].

[21] A. de Goursac, A. Tanasa and J. C. Wallet, “Vacuum configurations for renormalizablenon-commutative scalar models,” Eur. Phys. J. C 53, 459 (2008) arXiv:0709.3950 [hep-th].

[22] A. Tanasa, “Feynman amplitudes in renormalizable non-commutative quantum field the-ory,” sollicited by Modern Encyclopedia Mathematical Physics arXiv:0711.3355 [math-ph].

[23] A. Tanasa, “Overview of the parametric representation of renormalizable non-commutative field theory”, J. Phys. Conf. Series 103, 012012 (2008) [arXiv:0709.2270[hep-th]].

[24] J. B. Geloun and A. Tanasa, “One-loop β functions of a translation-invariant renormal-izable noncommutative scalar model,” Lett. Math. Phys. 86, 19 (2008) [arXiv:0806.3886[math-ph]].

[25] A. Tanasa, “Parametric representation of a translation-invariant renormalizable non-commutative model,” J. Phys. A 42, 365208 (2009) [arXiv:0807.2779 [math-ph]].

[26] J. Magnen, V. Rivasseau and A. Tanasa, “Commutative limit of a renormalizable non-commutative model,” Europhys. Lett. 86, 11001 (2009) [arXiv:0807.4093 [hep-th]].

[27] A. Tanasa, Rom. J. Phys. 53, 1207 (2008) [arXiv:0808.3703 [hep-th]].

[28] T. Krajewski, V. Rivasseau, A. Tanasa and Z. Wang, “Topological Graph Polynomialsand Quantum Field Theory, Part I: Heat Kernel Theories,” arXiv:0811.0186 [math-ph].

[29] A. Tanasa and D. Kreimer, “Combinatorial Dyson-Schwinger equations in noncommu-tative field theory,” arXiv:0907.2182 [hep-th].

21

[30] P. Aluffi and M. Marcolli, “Feynman motives of banana graphs,” arXiv:0807.1690 [hep-th].

[31] A. Voros, “Wentzel-Kramers-Brillouin method in the Bargmann representation”,Phys. Rev. A40, 6814 (1989). M. Bordemann and S. Waldmann, “A Fedosov StarProduct of Wick Type for Kahler Manifolds,” Lett. Math. Phys. 41, 243 (1997).arXiv:q-alg/9605012. M. Bordemann and S. Waldmann, “Formal GNS Constructionand States in Deformation Quantization,” Comm. Math. Phys. 195, 549 (1998).arXiv:q-alg/9607019.

[32] F. Bayen, in Group Theoretical Methods in Physics, ed. E. Beiglbock , et. al. [Lect.Notes Phys. 94, 260 (1979)];

[33] S. Galluccio, F. Lizzi and P. Vitale, “Twisted Noncommutative Field Theory with theWick-Voros and Moyal Products,” Phys. Rev. D 78, 085007 (2008) [arXiv:0810.2095[hep-th]].

[34] S. Galluccio, F. Lizzi and P. Vitale, “Translation Invariance, Commutation Relationsand Ultraviolet/Infrared Mixing,” JHEP 0909, 054 (2009) [arXiv:0907.3640 [hep-th]].

[35] P. Aschieri, F. Lizzi and P. Vitale, Phys. Rev. D 77, 025037 (2008) [arXiv:0708.3002[hep-th]].

[36] D. Bahns, “The ultraviolet-finite Hamiltonian approach on the noncommuta-tive Minkowski space,” Fortsch. Phys. 52, 458 (2004) [arXiv:hep-th/0401219].D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli, “Ultraviolet finite quan-tum field theory on quantum spacetime,” Commun. Math. Phys. 237, 221 (2003)[arXiv:hep-th/0301100].

[37] J. Gomis and T. Mehen, Space-time noncommutative field theories and unitarity, Nucl.Phys. B 591, 265 (2000) [arXiv:hep-th/0005129]. L. Alvarez-Gaume, J. L. Barbonand R. Zwicky, Remarks on time-space noncommutative field theories, JHEP 0105, 057(2001) [arXiv:hep-th/0103069]. D. Bahns, S. Doplicher, K. Fredenhagen, G. Piacitelli,”On the unitarity problem in space/time noncommutative theories“, Phys. Lett. B 533,178 (2002)

[38] Y. Liao and K. Sibold, “Time-ordered perturbation theory on noncommutative space-time. II. Unitarity,” Eur. Phys. J. C 25, 479 (2002) [arXiv:hep-th/0206011].

[39] C. S. Chu, J. Lukierski and W. J. Zakrzewski, Hermitian analyticity, IR/UV mixing andunitarity of noncommutative field theories, Nucl.Phys. B 632, 219 (2002) [arXiv:hep-th/0201144].

[40] A. Smailagic and E. Spallucci, “Feynman path integral on the noncommutative plane,”J. Phys. A 36, L467 (2003) [arXiv:hep-th/0307217].

[41] M. Chaichian, A. Demichev and P. Presnajder, “Quantum field theory on noncommu-tative space-times and the persistence of ultraviolet divergences,” Nucl. Phys. B 567(2000) 360 [arXiv:hep-th/9812180].

22

[42] A. B. Hammou, M. Lagraa and M. M. Sheikh-Jabbari, “Coherent state induced star-product on R(lambda)**3 and the fuzzy sphere,” Phys. Rev. D 66, 025025 (2002)[arXiv:hep-th/0110291]. A. Pinzul and A. Stern, A new class of two-dimensional non-

commutative spaces, JHEP 0203, 039 (2002) [arXiv:hep-th/0112220];Absence of the holographic principle in noncommutative Chern-Simons theory, JHEP0111, 023 (2001) [arXiv:hep-th/0107179].

G. Alexanian, A. Pinzul and A. Stern, Generalized Coherent State Approach to Star

Products and Applications to the Fuzzy Sphere, Nucl. Phys. B 600, 531 (2001)[arXiv:hep-th/0010187]. M. Daoud, “Extended Voros product in the coherent statesframework,” Phys. Lett. A 309, 167 (2003).

[43] F. Lizzi, P. Vitale and A. Zampini, “The fuzzy disc,” JHEP 0308, 057 (2003)[arXiv:hep-th/0306247].“The beat of a fuzzy drum: fuzzy Bessel functions for the disc,” JHEP 0509, 080 (2005)[arXiv:hep-th/0506008].

[44] A.P. Balachandran, K, Gupta and S. Kurkcuoglu, Edge Currents in Noncommutative

Chern–Simons Theory from a new Matrix Model, Syracuse Preprint SU-4252-783, SahaPreprint SINP/TNP/03-19, [arXiv:hep-th/0306255].

[45] R. Banerjee, S. Gangopadhyay and S. K. Modak, “Voros product, NoncommutativeSchwarzschild Black Hole and Corrected Area Law,” arXiv:0911.2123 [hep-th].

[46] A. P. Balachandran and M. Martone, “Twisted Quantum Fields on Moyal and Wick-Voros Planes are Inequivalent,” Mod. Phys. Lett. A 24, 1721 (2009) [arXiv:0902.1247[hep-th]]; A. P. Balachandran, A. Ibort, G. Marmo and M. Martone, “Inequivalenceof QFT’s on Noncommutative Spacetimes: Moyal versus Wick-Voros,” arXiv:0910.4779[hep-th].

[47] M. Kontsevich, “Deformation quantization of Poisson manifolds, I,” Lett. Math. Phys.66 (2003) 157 [arXiv:q-alg/9709040].

[48] V. Rivasseau, ”From perturbative to constructive renormalization“, Princeton Univer-sity Press, 1991.

[49] J. M. Gracia-Bondia and J. C. Varilly, Algebras of Distributions Suitable For PhaseSpace Quantum Mechanics. 1, J. Math. Phys. 29 (1988) 869.

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