arX

iv:0

810.

2095

v1 [

hep-

th]

12

Oct

200

8

DSF–19–2008

September 2008

Twisted Noncommutative Field Theory with

the Wick-Voros and Moyal Products

Salvatore Galluccio, Fedele Lizzi and Patrizia Vitale

Dipartimento di Scienze Fisiche, Universita di Napoli Federico IIand INFN, Sezione di Napoli

Monte S. Angelo, Via Cintia, 80126 Napoli, Italy

salvatore.galluccio@na.infn.it, fedele.lizzi@na.infn.it,

patrizia.vitale@na.infn.it

Abstract

We present a comparison of the noncommutative field theories builtusing two different star products: Moyal and Wick-Voros (or nor-mally ordered). For the latter we discuss both the classical and thequantum field theory in the quartic potential case, and calculate theGreen’s functions up to one loop, for the two and four points cases.We compare the two theories in the context of the noncommutativegeometry determined by a Drinfeld twist, and the comparison is madeat the level of Green’s functions and S-matrix. We find that while theGreen’s functions are different for the two theories, the S-matrix isthe same in both cases, and is different from the commutative case.

1 Introduction

It is likely that at short distances spacetime has to be described by differentgeometrical structures, and that the very concept of point and localizabilitymay no longer be adequate. This is one of the main motivations for theintroduction of noncommutative geometry [1, 2, 3]. The simplest kind ofnoncommutative geometry is the so called canonical one [4, 5]. What isusually done for the construction of a field theory on a noncommutative spaceis to deform the product among functions (and hence among fields) withthe introduction of a noncommutative ⋆ product, so that for the coordinatefunctions one has

[xi, xj ]⋆ ≡ xi ⋆ xj − xj ⋆ xi = iθij . (1.1)

In the simplest case θij is constant, i.e. it does not depend on the x’s. Thechoice of the ⋆ product compatible with (1.1) is not unique, in the followingwe will introduce two different products, the Moyal [6, 7] and Wick-Voros [8,9, 10, 11] and compare their “physical predictions”.

There are several reasons to consider field theories on a noncommutativespace equipped with the standard canonical noncommutativity, ranging fromintrinsic motivations to the localizability of events [4, 5] to string theory [12]to constructive field theories [13]. Field theories on noncommutative spaceshave interesting renormalization properties [14, 15]. For a review see [16, 17],their references and their citations. What we will compare are field theories inwhich the product among fields is substituted by the two different ⋆ products.This leads to an action in which arbitrary degree derivatives of the fields arepresent, as a series in θ. Written in terms of derivatives the two actions withthe Moyal and Wick-Voros products are different. There is however a mapwhich renders equivalent the algebras generated by the two products. Fieldtheory with the Wick-Voros product has been discussed in [18] as regularizingmodel, their conclusion (that ultraviolet divergences persist) is in agreementwith ours.

This paper originates from the consideration that one can reason in twoways: one point of view is to say that what counts is the noncommutativestructure of spacetime, and the ⋆ product is just a way to express this intimatestructure, and therefore one chooses the most convenient product. As long asone is describing the same field theory, the results should be the same, a factnoted already in [19]. Another view is to claim total lack of interest in the

1

noncommutativity of spacetime. What counts is the fact that one has a fieldtheory on ordinary spacetime, whose action contains an infinity of derivativesof arbitrary order. With this second point of view one would not in principleexpect the same physical results from the two theories. In this paper wecalculate the Green’s functions of the Wick-Voros field theory, and foundthem to be different from the Moyal case. This leads to a contradiction. Wewill see that the contradiction is only apparent. Green’s functions are notobservable quantities, what is observable is the S-matrix.

Discussions of the properties of the S-matrix often go together with theissue of Poincare invariance. Relation (1.1) is not Poincare invariant, and thiscasts doubts on its being fundamental. It is however possible to build a theorywhich is invariant under a deformation of the Poincare Lie algebra, so that thetheory becomes a twisted theory. This theory has a symmetry described bya noncommutative, non cocommutative Hopf algebra. In particular the kindof noncommutativity described by the two ⋆ products is the one generatedby a twist [20, 21, 22]. Then the theory has a twisted Poincare symmetry [23,24, 25].

The presence of a twist forces to reconsider all of the steps in a field theory,which has to be built in a coherent “twisted” way. We will see that there isequivalence between the two theories at the very end, where by “very end”we mean the calculation of the S-matrix. Prior to this, vertex, propagatorsand Green’s functions are in fact different. Moreover the equivalence is onlyobtained if a consistent procedure of twisting all products is applied. In thisway the Poincare symmetry, which appears to be broken in (1.1), is preserved,albeit in a deformed way, as a noncommutative, non cocommutative Hopfalgebra.

There is some ambiguity in the issue of twisting, and some results havebeen somewhat controversial [26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. In anoptimal world one should let experiments resolve these ambiguities. Unfor-tunately the noncommutative structure of spacetime is not yet mature for aconfrontation with experiments at such a level. What we do in this paperis to use the field theories built with the Wick-Voros and Moyal productsto check each other. We will see that using a consistent twisting procedurewe obtain that at the level of S-matrix the two theories are equivalent. Thisgives us the indication on the procedure to follow for noncommutative theo-ries coming from a twist.

2

In this paper we will consider exclusively spatial noncommutativity, i.e.time is a commuting variable. The matrix θ therefore is of the form

θij = θεij (1.2)

with ε the antisymmetric tensor of order two.

The paper is organized as follows. In Section 2 we introduce the twoproducts. In section 3 we discuss the classical free field theory for the Wick-Voros product. In section 4 we calculate the Green’s functions for the twotheories for the two and four point case to one loop, and compare the twocases. In Sect. 5 we describe the two products as twisted noncommutativegeometries. In section 6 we describe the relevant twisted products which wethen use in Sect. 7 to calculate the S-matrix. A final short section containsthe conclusions.

2 The Wick-Voros and Moyal Products

In this section we describe in a comparative way the two ⋆ products we areusing in this paper. The most well known product is the Moyal product [6, 7]

f(~x) ⋆M g(~x) = f(~x)ei

2θij←−∂i−→∂jg(~x) (2.1)

where the operator←−∂i (resp.

−→∂j ) acts on the left (resp. the right). This

product comes from a Weyl map which associates to a function on the planean operator according to:

ΩM(f) =1

2π

∫

d2ηf(η1, η2)eiθijX

iηj (2.2)

where f is the symplectic Fourier transform of the function f :

f =1

θπ

∫

d2xf(x1, x2)e−iθijxiηj (2.3)

θij is the inverse of θij, and the X are operators which satisfy the commuta-tion relation

[X i, Xj] = iθij (2.4)

3

It is useful to think of the operators X in two dimensions in an abstract way,without reference to spacetime and define them as

X1 =a + a†√

2

X2 =a− a†

i√2

(2.5)

and a is an operator which we define on a certain basis by its components as

anm =√θnδm,n+1 (2.6)

with m,n ≥ 0. Of course we are using the analogy of the commutationrelations (2.4) with the usual quantum mechanical commutation relations,and using the |n〉’s as a convenient basis.

The Moyal product is then defined as

f ⋆M g = Ω−1M

(

ΩM(f)ΩM(g))

(2.7)

From this expression is not difficult (see for example [36]) to obtain integralexpressions for the product, a few of which are collected in [37, Appendix].The standard expression is then an asymptotic expansion of the integralexpressions [38].

One important property of the Moyal product is that∫

d2xf ⋆M g =

∫

d2xfg (2.8)

and obviouslyx1 ⋆M x2 − x2 ⋆M x1 = iθ (2.9)

We now proceed to the definition of the Wick-Voros product. For thefollowing it is useful to consider the space as a complex plane defining:

z± =x1 ± ix2

√2

(2.10)

where of course z∗+ = z−. With this substitution we define the Wick-Vorosproduct as

f ⋆V g =∑

n

(

θn

n!

)

∂n+f∂

n−g = feθ

←−∂+−→∂−g (2.11)

4

where

∂± =∂

∂z±=

1√2

(

∂

∂x1∓ i

∂

∂x2

)

(2.12)

Notice that the Moyal product (2.7) may be rewritten in these coordinatesas

f ⋆M g = feθ2(←−∂+−→∂−−

←−∂−−→∂+)g (2.13)

It results

z+ ⋆V z− = z+z− + θ

z− ⋆V z+ = z+z− (2.14)

and therefore[z+, z−]⋆V = θ (2.15)

Going back to the x’s, it is possible to see that this relation gives rise againto the standard commutator among the x’s:

x1 ⋆V x2 − x2 ⋆V x1 = iθ (2.16)

With the z± coordinates the Laplacian and the d’Alembertian are respec-tively ∇2 = 2∂+∂− and = ∂2

0 − ∇2. The integral on the plane is still atrace, but the strong condition of (2.8) is not valid anymore:

∫

d2zf ⋆V g =

∫

d2zg ⋆V f 6=∫

d2zfg (2.17)

where by d2z we mean the usual measure on the plane dz+dz−. We will alsouse the notation

k± =k1 ± ik2√

2(2.18)

for a generic vector ~k.

The Wick-Voros and Moyal products can be cast in the same generalframework in that they are both coming from a generalised “Weyl map”.More precisely, as we saw in (2.2) the Moyal product comes from a map whichassociates operators to functions, with symmetric ordering. The Wick-Vorosproduct comes from a similar map, a weighted Weyl map as follows:

ΩV (f) =1

2π

∫

d2ηf(η, η)eθηa†

e−θηa (2.19)

5

An equivalent way to associate the operators ΩV (f) to a function f =∑

mn fmnzm+ zn− analytic on the plane is:

ΩV (f) =∑

mn

fmna†man (2.20)

where a has been defined in (2.6). Thus effectively the map (2.20) corre-sponds to the normal (or Wick) ordering (and is sometimes called normalordered product). In this sense the two maps correspond to two differentquantization procedures (see for example [39, 40]).

3 Classical Free Field Theory

Although the main interest of this paper is in the interacting quantum fieldtheory, we start the discussion from the classical free case. In this sectionwe discuss the kind of field theory one obtains from a deformation of thefree Klein-Gordon action based on the Wick-Voros product. Such analysisis unnecessary in the Moyal case, because in that case the action, beingquadratic in the fields, is the same as in the commutative case.

We consider a field theory described by an action which is a Wick-Vorosdeformation of a scalar field theory action, obtained inserting the star prod-uct. Consider a classical free theory and its action, Lagrangian density andLagrangian defined as:

S0 =

∫

dtL0 =

∫

dtd2z L0 =

∫

dtd2z1

2

(

∂µϕ ⋆V ∂µϕ−m2ϕ ⋆V ϕ)

(3.1)

With the help of (2.11) it may be rewritten as

S0 =

∫

dtd2z1

2

(

∂µϕeθ←−∂+−→∂−∂µϕ−m2ϕeθ

←−∂+−→∂−ϕ

)

=

∫

dtd2z1

2ϕ[

e−θ2∇2

(−∂2µ −m2)

]

ϕ (3.2)

This is a theory which contains an infinite number of the derivatives of thefields, and in principle even the Cauchy problem would not be well defined.This appears to be the biggest difference with respect to the noncommutativefield theory defined via the Moyal product. In the latter case the action being

6

the same as in the commutative case, the solution of the free theory is stillgiven by plane waves, and upon quantization the propagator is the same asin the commutative case. This time instead the action is different, the theoryis non local as it contains derivatives of arbitrary order.

The two products are equivalent in a precise technical sense: there is aninvertible map [41, 42]

T (f) =∑

n

θntn(f) (3.3)

with the tn differential operators, such that

T (f ⋆M g) = T (f) ⋆V T (g) (3.4)

whereT = e

θ4∇2

(3.5)

Therefore the two products define the same deformed algebra. This is cer-tainly true if we consider functions as formal power series in the generators.The issue can be more complicated in the realm of C∗-algebras. Startingfrom the same set of functions the completion in the supremum norm of thetwo products could in principle be different.

The fact that the algebra is the same does not mean that the two de-formations of an action are the same. Therefore let us map the free actionS0 (3.1) written with the Wick-Voros product to the corresponding actionwith the Moyal product, using (3.5), to find which Moyal theory correspondsto it.

The action (3.1)is mapped into:

S ′0 =

∫

dtd2z T−1(L0)

=

∫

dtd2z1

2

((

e−θ4∇2

∂µϕ)

⋆M

(

e−θ4∇2

∂µϕ)

−m2(

e−θ4∇2

ϕ)

⋆M

(

e−θ4∇2

ϕ))

=

∫

dtd2z1

2

((

e−θ4∇2

∂µϕ)(

e−θ4∇2

∂µϕ)

−m2(

e−θ4∇2

ϕ)(

e−θ4∇2

ϕ))

=

∫

dtd2z1

2

(

∂µϕe− θ

2∇2

∂µϕ−m2ϕe−θ2∇2

ϕ)

(3.6)

7

which is not the free action with the Moyal product. In fact in the latter casethe noncommutative product could be eliminated from the integral leavingjust the free commutative action. Therefore, the two actions being different,they could in principle give different equations of motion.

Since we are dealing with a theory involving an infinite number of deriva-tives we can ask whether we would need an infinity of boundary conditionsto solve the classical theory. This is not so, as the higher derivatives ap-pear as analytic functions of the Laplacian, and in this case the boundaryproblem is the same as in the standard case. Note that with our choice ofθµν our equation is second order in the time derivatives, so that the initialvalue problem requires knowledge of the field and its derivative as initialcondition. But also if we had deformed the time derivatives, the initial datafor the Cauchy problem would have been the same if the higher derivativehad been an analytic function of the d’Alembertian. For more details andreferences see the recent paper [43].

Let us derive the classical equations of motion starting from the variationof the action. Since the standard techniques have been developed for a theorywith a finite number of derivatives, we will proceed from first principles andstart from the infinitesimal variation of the field:

ϕ→ ϕ+ δϕ. (3.7)

It is not difficult to see that the corresponding infinitesimal variation of theaction under such a transformation is given by

δS0 =

∫

dtd2z ((∂0ϕ) ⋆V (∂0δϕ)− (∂+ϕ) ⋆V (∂−δϕ)− (∂+δϕ) ⋆V (∂−ϕ)

−m2ϕ ⋆V δϕ)

, (3.8)

where we have used the trace property of the integrals with Wick-Vorosproducts. By integrating by parts and using once again the trace propertywe obtain, up to boundary terms:

δS0 = −∫

dtd2z(δϕ) ⋆V(

∂0∂0ϕ− 2∂+∂−ϕ+m2ϕ)

, (3.9)

namely

δS0 = −∫

dtd2z∞∑

n=0

θn

n!∂n+(δϕ)∂

n−

(

∂0∂0ϕ− 2∂+∂−ϕ+m2ϕ)

. (3.10)

8

By integrating once again by parts we obtain:

δS0 = −∫

dtd2z

∞∑

n=0

(−θ)nn!

(δϕ)∂n+∂

n−

(

∂0∂0ϕ− 2∂+∂−ϕ+m2ϕ)

, (3.11)

that is

δS0 = −∫

dtd2z(δϕ)e−θ∂+∂−(

∂0∂0 − 2∂+∂− +m2)

ϕ. (3.12)

Since the variation of the action δS must be vanishing for any variation ofthe field δϕ, we obtain that the equation of motion is given by

e−θ∂+∂−(

∂0∂0 − 2∂+∂− +m2)

ϕ = 0. (3.13)

Equivalently, it can be written as:

e−θ2∇2 (

+m2)

ϕ = 0 (3.14)

As we can see, the equation of motion (3.14) differs from the classical Klein-Gordon equation

(

+m2)

ϕ = 0, (3.15)

only by the presence of the exponential of the Laplacian, an invertible op-erator. It is immediate to see that all solutions of the commutative theoryare still solutions of the noncommutative one. It is in principle possible thatthere can be solutions of the noncommutative equation (3.14) which are notsolutions of the commutative one. This is not the case, due to the invertibilityof the operator e−

θ2∇2

.

Notice that the on shell condition is not altered by the presence of thedeformation factor. In other words the dispersion relation is the same in thetwo cases. In fact in Fourier transform

ϕ(x) =

∫

d3k

(2π)3eik·xϕ(k) (3.16)

and then the equation (3.14) becomes:

e−θ2∇2 (

+m2)

∫

d3k

(2π)3eik·xϕ(k) =

∫

d3k

(2π)3e

θ2k2(

−k2 +m2)

eik·xϕ(k) = 0.

(3.17)

9

The relatione

θ2k2(

k2 −m2)

ϕ(k) = 0, (3.18)

gives the same on shell relation of the classical case since the exponentialnever vanishes.

Classically therefore the two theories have the same solutions of the equa-tions of the motion, despite the fact that the action, the Lagrangian and theequations of motion are different.

4 Green’s Functions

Let us consider a field theory described by the action:

S = S0 +g

4!

∫

dtd2z ϕ ⋆ ϕ ⋆ ϕ ⋆ ϕ (4.1)

where ⋆ is either ⋆M or ⋆V . In the following we will use a generic ⋆ forall relations and formulas valid for both products. We now calculate theFeynman rules for these field theories.

Because of property (2.8) the free theory is unchanged for the Moyal case.Therefore the Moyal propagator is the same as in the undeformed case. Inthe Wick-Voros case [18] there are differences.

To this purpose let us rewrite the action S0 in Eq. (3.2) in the form

S0 =

∫

dtd2z dt′d2z′ ϕ(t, z)K(t, t′, z, z′)ϕ(t′, z′) (4.2)

withK(t, t′, z, z′) = e−

θ2∇2

(−∂2µ −m2)δ(t− t′)δ2(z − z′). (4.3)

The quantum propagator ∆⋆V (x, y) is the two-point Green’s function ofthe free theory, that is, the inverse of the operator K,

∆⋆V (xa, xb) =

∫

d3p

(2π)3eip·(xa−xb)

e−θ2|~p|2

p2 −m2. (4.4)

10

We can read off the propagator in momentum space, and compare it withthe one in the Moyal (and undeformed) case

G(2)0M

(p) =1

p2 −m2

G(2)0V(p) =

e−θ2|~p|2

p2 −m2(4.5)

Since the poles in the propagator in momentum space are the same as inthe commutative theory, despite the change in the propagator, the free fieldtheory with the Wick-Voros product is the same as in the commutative (andMoyal noncommutative) case. This is the quantum counterpart of the pre-vious result that the solutions of the classical equations of the motion arethe same. Nevertheless the two propagators are not identical, and we willhave to take this into account in the following. Note however that for infi-nite momentum there is an essential singularity, or a zero, of the propagator,according to the sign of θ. The meaning of the essential singularity is notclear, but the oddity is that the sign of θ has no physical meaning since itcan be changed by an exchange of the sign of one of the two coordinates, ina theory which appears to be parity invariant. We will see later that, withthe proper twisting of the theory, also this paradox is solved.

We now proceed to the calculation of the interaction vertex in the Wick-Voros case, comparing it with the theory obtained with the Moyal product.In this latter case the difference with respect to the commutative case residesin the fact that the vertex acquires a phase [44]. In order to see the correctionslet us write down the Moyal product as a convolution twist in momentumspace∗:

(f ⋆M g)(x) =

∫

d3k

(2π)3d3k′

(2π)3f(k)g(k′)ei(k+k′)·xe−

i

2θ~k∧~k′ (4.6)

where f and g are the Fourier transforms of f and g and

~k ∧ ~k′ = εijkik′j (4.7)

We see that in momentum space the Moyal product is the standard convolu-tion of Fourier transforms, twisted by a phase. For the Wick-Voros product,

∗Some of the formulas of this section are specific to our 2+1 case, but the results aremore general.

11

defining k± = (k1 ± ik−)/√2 in a way analogous to (2.10) we have

(f ⋆V g)(z+, z−, t) =

∫

d3k

(2π)3d3k′

(2π)3f(k)g(k′)ei(k+k′)·xe−θk−k′+ (4.8)

Explicitly the exponent of the twist in the convolution can be expressed as

k−k′+ =

1

2

(

~k · ~k′ + i~k ∧ ~k′)

(4.9)

with the same imaginary part as in the Moyal case (4.6) plus a real part.

For a ϕ4 theory in ordinary space the four points vertex in momentumspace is the coupling constant multiplying the δ of momentum conservation:

V = −i g4!(2π)3δ3

(

4∑

a=1

ka

)

(4.10)

In the Moyal case we have that the vertex acquires a phase factor due to thetwist in the product (4.6):

V⋆M = V∏

a<b

e−i

2θijkaikbj (4.11)

The presence of the phase in the vertex makes it non invariant for a genericexchange of the momenta. This is a consequence of noncommutativity andof the fact that the integral of Moyal product of more than two functionsis not invariant for an exchange of the functions. Invariance for a cyclicrotation of the factors still survives. This gives rise to a difference betweenplanar and nonplanar graphs, and ultimately to the well known phenomenonof infrared-ultraviolet mixing [45].

In the Wick-Voros case the correction, due to (4.9), is not just a phase,but it contains a real exponent:

V⋆V = V∏

a<b

e−θka−kb+ = V∏

a<b

e−θ2( ~ka· ~kb+i ~ka∧ ~kb) (4.12)

The exponent can have both signs, and in some case it could diverge exponen-tially for large external momenta. The divergence is however compensatedby the fact that, to the four points function, there must be added the contri-bution of the four propagators, each of which comes with an exponentially

12

convergent part. These convergent parts compensate the possibly divergentcontributions of the vertex for positive θ.

We can write the vertices with an unified notation as

V⋆ = V∏

a<b

eka•kb (4.13)

where

ka • kb =

− i2θijkaikbj Moyal

−θka−kb+ Wick-Voros(4.14)

To calculate the four points Green’s function in theWick-Voros case at thetree level we must attach to the vertex four propagators (4.5), each carryingan exponential. The four points Green’s function therefore is

G(4)0V

= −ig(2π)3 e−θ(

P4a=1

ka−ka++P

a<b ka−kb+)∏4

a=1 (k2a −m2)

δ(3)

(

4∑

a=1

ka

)

(4.15)

With some simple algebraic passages we can express the exponent as

∑4a=1 ka−ka+ +

∑

a<b ka−kb+ (4.16)

= 14

(

|~k1|2 + |~k2|2 + |~k3|2 + |~k4|2 + 2i∑

a<b~ka ∧ ~kb + |~k1 + ~k2 + ~k3 + ~k4|2

)

The δ of conservation of momentum effectively kills the last term, so thatthe four point function, at tree level, is

G(4)0V

= −ig(2π)3 e− θ

4

P

4a=1|~ka|2

∏

a<b e− i

2θijkaikbj

∏4a=1 (k

2a −m2)

δ(3)

(

4∑

a=1

ka

)

(4.17)

Noticing that in the Moyal case, because of antisymmetry, it results p•p = 0,we can express in the unified notation the Green’s functions as:

G(4)0 = −ig(2π)3 e

P

a≤b ka•kb

∏4a=1 (k

2a −m2)

δ

(

4∑

a=1

ka

)

(4.18)

The presence of a real exponent for the Wick-Voros case could signify thatthe ultraviolet behaviour of the theory could be different from the Moyal(and the commutative) case. Hence we calculate the one loop correction

13

to the propagator, and verify the ultraviolet behaviour of the theory underrenormalization. The presence of the phase in the four point function in thecomplete vertex (4.17) makes it non invariant for a generic permutation ofthe external momenta, and this in turn implies that the planar and nonplanarcases are to be treated differently, this is what happens in the Moyal case aswell. Consider first the planar case in figure 1 (a). The amplitude is obtained

p

q

−p

qqqqp −p

(a) (b)

Figure 1: The planar (a) and nonplanar (b) one loop correction to the prop-agator

using three propagators (4.5), two with momentum p, one with momentumq, and the vertex (4.12) with assignments k1 = −k4 = p and k2 = −k3 = q,and of course the integration in q and the proper symmetry factor:

G(2)P = −ig

3

∫

d3q

(2π)3e−θ(2p−p++q−q+)e−θ(p−q+−p−q+−p−p+−q−q+−q−p++q−p+)

(p2 −m2)2(q2 −m2)

= −ig3

∫

d3q

(2π)3e−θp−p+

(p2 −m2)2(q2 −m2)(4.19)

where the first exponential is due to the propagators, and the second to thevertex. In this case all the contribution due to q cancel, so that there is nochange in the convergence of the integral.

We now proceed to the discussion of the nonplanar case, in Fig. 1 (b).The structure is the same as in the planar case, but this time the assignmentsare instead k1 = −k3 = p and k2 = −k4 = q, and we have:

G(2)NP = −ig

6

∫

d3q

(2π)3e−θ(2p−p++q−q+)e−θ(p−q+−p−p+−p−q+−q−p+−q−q++p−q+)

(p2 −m2)2(q2 −m2)

= −ig6

∫

d3q

(2π)3e−θ(p−p++i~p∧~q)

(p2 −m2)2(q2 −m2)(4.20)

14

This time the q contribution does not cancel completely, and there remainsthe factor

p−q+ − q−p+ = i~p ∧ ~q (4.21)

so that the phase factor of the Moyal case is reproduced. We can express inthe unified notation:

G(2)P = −ig

3

∫

d3q

(2π)3ep•p

(p2 −m2)2(q2 −m2)

G(2)NP = −ig

6

∫

d3q

(2π)3ep•p+p•q−q•p

(p2 −m2)2(q2 −m2)(4.22)

The ultraviolet divergence of the diagram is unchanged, with respect tothe commutative case, for the planar diagram. In the nonplanar case thereis the presence of the oscillating factor i~p ∧ ~q in the exponential. This factorsoftens the ultraviolet divergence, since it dampens the functions for highq, but is responsible for infrared divergences. We can conclude at this levelthat, while the Green’s functions are different, between the Moyal and Wick-Voros case, their ultraviolet behaviour is the same as far as the momentumin the loop is concerned†. This indicates that the noncommutative geometry,at the ultraviolet level, is basically described by the uncertainty principle,consequence of the commutator (1.1), which is unchanged between Wick-Voros and Moyal cases. Nevertheless the two Green’s functions are not thesame because of the p • p term which vanishes in the Moyal case, but not inthe Wick-Voros one.

We now proceed to the one-loop Green’s functions corresponding to theplanar case of Fig. 2 In the NC case the Green’s function correspondent to

k1

k2 k3

k4

q

Figure 2: The planar one loop four points diagram.

†The convergence properties can however be changed by going to a different noncom-mutative space, such as a torus [18]

15

it can easily be calculated by properly joining two vertices. It turns out thatwe have for the two cases

G(4)P =

(−ig)28

(2π)3∫

d3q

(2π)3e

P

a≤b ka•kbδ(∑4

a=1 ka)

(q2 −m2)((k1 + k2 − q)2 −m2)∏4

a=1 (k2a −m2)

(4.23)The exponent in the numerator can be rewritten as in (4.17) and we seethat the internal momentum q appears only in the denominator, so that alsoin this case the planar diagram has the same renormalization property ofthe undeformed theory. In the Moyal case the real part exponent of thenumerator is not present. In the Wick-Voros case instead there is the samecorrection encountered at tree level.

The three nonplanar cases are shown in figure 3. The calculation of their

k1 k1 k1

k2 k2 k2k3 k3

k4 k4

q q k3

k4

(1) (2) (3)

Figure 3: The non planar one loop four points diagrams.

contribution is straightforward and gives, in momentum space,

G(4)P =

(−ig)28

∫

d3qe

P

a≤b ka•kbδ(∑4

a=1 ka)

(q2 −m2)((k1 + k2 − q)2 −m2)∏4

a=1 (k2a −m2)

(4.24)

G(4)NPa

= ((−ig)2

8

∫

d3qe

P

a≤b ka•kb+Eaδ(∑4

a=1 ka)

(q2 −m2)((k1 + k2 − q)2 −m2)∏4

a=1 (k2a −m2)

(4.25)

with

E1 = q • k1 − k1 • q = i~q ∧ ~k1E2 = k2 • q − q • k2 + k3 • q − q • k3 = i(~k2 ∧ ~q + ~k3 ∧ ~q)

E3 = k1 • q − q • k1 + k2 • q − q • k2 = i(~k1 ∧ ~q + ~k2 ∧ ~q) (4.26)

16

Contrary to our expectations we find that the Green’s functions are differ-ent. The Green’s functions are not however directly measurable quantities,the S-matrix is. We will calculate it in the twist-deformed framework inSect. 7.

5 The Wick-Voros and Moyal Products as

Twisted Noncommutative Geometries

The main physical motivation to study field theory equipped with a ⋆ prod-uct is the belief that, at very short distances, the geometry of spacetime isdeformed, with the deformation dictated by a small parameter, θ in our case.In the presence of noncommutativity the concept of point is not well defined,and in fact the proper mathematical formalism should use the theory of C∗-algebras and the language of spectral triples (see for example [1, 2, 3]). Astar product deforms the commutative algebra of functions on a space into anoncommutative algebra. The proper formal definition of the mathematicalobjects involved in the definition is beyond the scope of this article. For usit suffices to know that the plane equipped Moyal product can be made intoa spectral triple [37, 46].

As we have discussed in section 3 the two products come from a differentquantization of the same Poisson structure, which classifies ⋆ products upto equivalences [47]. They can also be seen as gauge equivalent for the(infinite rank) group of gauge transformations given by field redefinition ofthe kind (3.3). Note however that the action is not invariant under the actionof this gauge group.

With the introduction of a different, but equivalent, product one canheuristically reason as follows. The presence of the noncommutativity de-scribed by (1.1) gives the noncommutative structure of space, regardless ofthe realization of the product one uses. In fact one could avoid the use of a ⋆product altogether, by considering the fields to be infinite matrices functionof the operators X defined in (2.5) and solving, for example with a path inte-gration, this matrix model. We tested this conjecture for a bosonic quantumfield theory with a ϕ4 interaction and found that the two deformations of theaction give different Green’s functions. Interestingly however the ultravioletstructure of the two theories remains the same. We are nevertheless in front

17

of a puzzle.

The element that we need consider to solve this puzzle is symmetries.The commutation relation (1.1) breaks Poincare symmetry, and this is not adesirable feature for a fundamental theory. The symmetry can be reinstatedhowever at a deformed level, considering the fact that both products can beseen as coming from a Drinfeld twist [20, 21]. The noncommutative geometrydescribed by either ⋆-product is therefore a twisted noncommutativity.

Given the Poincare Lie algebra Ξ and its universal enveloping algebra UΞ,the twist F which we will consider is an element of UΞ⊗UΞ. For the Moyaland Wick-Voros case it is respectively

FM = exp[−iθij

2∂i ⊗ ∂j ] (5.1)

FV = exp[−θ∂+ ⊗ ∂−] (5.2)

where partial derivatives stand for translation generators and have to beappropriately realized when acting on a given space. Following [48, 49, 50, 51]we will consider the following point of view: the noncommutative geometry

is a consequence of a twist of all products of the theory. Then every bilinearmap µ defined as

µ : X ⊗ Y → Z (5.3)

(where X, Y, Z are vector spaces) is consistently deformed by composing itwith the appropriate realization of the twist F . In this way we obtain adeformed version µ⋆ of the initial bilinear map µ:

µ⋆ := µ F−1 , (5.4)

The ⋆-product on the space of functions is recovered setting X = Y = Z =Fun(M). That is, if we indicate with m0 the usual pointwise product betweenfunctions ‡:

m0 : Fun(M)⊗ Fun(M) −→ Fun((M)

m0(f ⊗ g) = f · g (5.5)

‡At this level we need not specify which kind of algebra of functions we are considering.The algebra of formal series of the generators is adequate, but more restricted algebrassuch as Schwarzian functions can also be considered.

18

the noncommutative product can be seen as the composition of m0 with thetwist:

F : Fun(M)⊗ Fun(M) −→ Fun(M)⊗ Fun(M) (5.6)

so that

f ⋆M g = m0 F−1⋆M(f ⊗ g)

f ⋆V g = m0 F−1⋆V(f ⊗ g) (5.7)

Associativity of the product is ensured by normalization and cocycle con-ditions (see [48, 50] for a short introduction; see also the book [52]). We alsointroduce the universal R-matrix which represents the permutation group innoncommutative space

R := F21F−1 (5.8)

withF21(a⊗ b) = τ F τ(a⊗ b) (5.9)

and τ the usual exchange operator

τ(a⊗ b) = b⊗ a. (5.10)

For the cases at hand with the two twists given by (5.1), (5.2): it results:

R⋆V = R⋆M = F−2⋆M(5.11)

that is, the exchange operator, and therefore the statistics, are the same inthe two cases.

The presence of the twist deforms the structure of the universal envelop-ing algebra of the Poincare Lie-algebra, rendering it a noncocommutativeHopf algebra. The analysis of [23, 24, 25], made for the Moyal case, canbe repeated in the Wick-Voros case with similar conclusions. Therefore therepresentations of the undeformed Poincare algebra can still be used. We willsee later in the paper the important role that the twisted Poincare symmetrywill play in the equivalence between Moyal and Wick-Voros theories.

6 Twist-deformed Products

We have now the necessary ingredients to calculate a physical process, like theS-matrix for the elastic scattering of two particles. We recall than one of the

19

crucial ingredients in the importance of the S-matrix in physics is the issue ofLorentz and Poincare invariance. If we naively insert the Green’s functionsof Sect. 4 into the calculation of the S-matrix we would find a dependenceof it from the external momenta, something like a momentum dependenceof the coupling constant. What is more relevant for our purposes, we findthat the result would be different for the Moyal and the Wick-Voros case, incontradiction with the heuristic reasoning we made in the introduction. Wewould also find a breaking of Poincare invariance§.

The reason for the breaking of Poincare invariance is that the commuta-tor (1.1) apparently breaks this invariance. However the invariance can bereinstated if one considers it to be a quantum symmetry, i.e. the Poincarealgebra is not a cocommutative Hope algebra, but it has a nontrivial coprod-uct [23, 24].

Our purpose is to show, with an explicit calculation of scattering ampli-tudes, that the naive procedure which leads to a difference among the twocases can be corrected by a coherent twisting procedure. We will see that, ifthe twisted symmetry is properly implemented, the final, “physical” result,will be the same in the Wick-Voros and Moyal cases, despite the presence ofdifferent propagators and vertices. We will consider the elastic scattering oftwo particles, as described in Fig. 4.

k1 k3

k2 k4

Figure 4: The two particles elastic scattering

The first consequence of noncommutativity is the fact that, since the ver-tex is non invariant for noncyclic exchange of the particles, we have to twist-

§We are considering θij to be constant. Another solution which preserves Poincare in-

variance is to have it a tensor [4, 53] or to have it transform together with the product [54].The residual rotational invariance is an artifact of the two-dimensionality of the model.In higher dimensions also this invariance is broken.

20

symmetrize the incoming and outgoing states using the R-matrix. Severalaspects of this twist symmetrization and the consequences for spin and statis-tics have been discussed in [27, 55, 56]. In the commutative case the order ofthe propagators into the vertex is irrelevant, and therefore this discussion isredundant. Here there are several twists at work and we have to be carefulin considering all of them.

Since we have to properly define multiparticle states as twisted tensorproducts, and accordingly modify the definition of their scalar product, forthe remaining part of the section we will only deal with free fields solutionof the Klein Gordon equation, and free states. In the next section these willserve to define the asymptotic states.

Let us consider the two-particle state:

|ka, kb〉 = |ka〉 ⊗ |kb〉 (6.1)

Although for the comparison we are going to make later we will not actuallyuse the fact that the state has to be symmetrised, we will discuss the sym-metrisation of the states for completeness. Consider the exchange operator

τ |ka〉 ⊗ |kb〉 = |kb〉 ⊗ |ka〉 (6.2)

The symmetrized state, eigenvector of the exchange operator τ with eigen-value +1, is:

|ka, kb〉simm =|ka〉 ⊗ |kb〉+ |kb〉 ⊗ |ka〉

2(6.3)

inserting the two expressions (6.1) or (6.3) does not make a difference inthe calculation (of the connected diagrams) because of the invariance forexchange on the incoming momenta. The symmetries for identical particleschange for the noncommutative case [27, 55, 56], we have to take into accountthe fact that the tensor product is twisted, and moreover that the exchangeis twisted as well. Therefore we define

|ka, kb〉⋆ = F−1 |ka, kb〉 (6.4)

where by F we indicate the twist that this time acts in momentum space:

F−1⋆M|ka〉 ⊗ |kb〉 = e−

i

2θijkai⊗kbj |ka〉 ⊗ |kb〉

F−1⋆V|ka〉 ⊗ |kb〉 = eθka−⊗kb+ |ka〉 ⊗ |kb〉 (6.5)

21

This is not the only change we have to make to the state (6.3): the statehas to be eigenvalue of the twist-exchange, given by the R-matrix acting inmomentum space. The properly symmetrized state is therefore

|ka, kb〉simm⋆=

1

2

(

F−1 |ka〉 ⊗ |kb〉+ F−1R−1 |ka〉 ⊗ |kb〉)

=1

2

(

F−1 |ka〉 ⊗ |kb〉+ F−1FF−121 |ka〉 ⊗ |kb〉)

(6.6)

We define as usual the momentum eigenstates as created by the creationoperators ak, a

†k:

|k〉 = a†k |0〉 (6.7)

where ak, a†k are obtained in terms of the free field

ϕ(x) =

∫

d2k√

(2π)22ωk

(ake−ik·x + a†ke

ik·x) (6.8)

to be

ak =i

√

(2π)22ωk

∫

d2xeik·x↔

∂0 ϕin(x)

a†k = − i√

(2π)22ωk

∫

d2xe−ik·x↔

∂0 ϕin(x) (6.9)

The operators ak, a†k may be regarded, for fixed k, as functionals of the fields,

therefore their ⋆ product may be obtained as in [51]

a(k) ⋆M a(k′) = e−i

2θijkik

′j a(k)a(k′)

a(k) ⋆M a†(k′) = ei

2θijkik

′j a(k)a†(k′)

a†(k) ⋆M a(k′) = e−i

2θijkik′j a†(k)a(k′) (6.10)

a(k) ⋆V a(k′) = e−θk−k′+ a(k)a†(k′)

a(k) ⋆V a†(k′) = eθk−k′+ a(k)a†(k′)

a†(k) ⋆V a(k′) = e−θk−k′+ a†(k)a(k′). (6.11)

Therefore we may reexpress Eqs. (6.5) and (6.6) as

|ka, kb〉⋆ = a†ka ⋆ a†kb|0〉 (6.12)

22

and

|ka, kb〉simm⋆=

a†ka ⋆ a†kb+ a†kb ⋆ a

†ka

2|0〉 (6.13)

The next step is the twist of the inner product. We consider it as a mapfrom two copies of the Fock space of states into complex numbers. In thecommutative case, for the momentum one particle states we have:

〈· |·〉 : |k〉 ⊗ |k′〉 −→ 〈k |k′〉 = 〈0| aka†k′ |0〉 = δ(k − k′) (6.14)

We twist this product in the usual way composing it with the twist operator:⟨

·⋆∣

∣ ·⟩

: |k〉 ⊗ |k′〉 −→ 〈· |·〉 F−1 : |k〉 ⊗ |k′〉 = F−1(k, k′) 〈k |k′〉

= 〈0| ak ⋆ a†k′ |0〉 (6.15)

with F−1(k, k′) given by the exponential factor in Eqs. (6.5) for the Moyaland Wick-Voros case respectively.

We are not yet finished twisting! Let us consider the inner product in thecommutative case:

〈k1, k2 |k3, k4〉 = δ(k1 − k3)δ(k2 − k4) (6.16)

In the noncommutative case we have to twist the two-particle state accord-ing to (6.5), and then we have to twist the inner product according to thetwo-particle generalization of (6.15). In order to realise such a generalizationwe must consider the action of the twist on two-particle states. This is done,in canonical form, via the coproduct of the Hope algebra. Given a represen-tation of an element of the Hope algebra on a space, the representation ofthe element on the product of states is given (in the undeformed case) by

∆0(u)(f ⊗ g) = (1⊗ u+ u⊗ 1)(f ⊗ g) (6.17)

The coproduct is responsible for example of the Leibnitz rule. For the twistedHope algebra the coproduct is deformed according to the fact that it is theR-matrix which realizes the permutations:

∆⋆(u)(f ⊗ g) = (1⊗ u+R−1(u⊗ 1))(f ⊗ g) (6.18)

However the twists we are considering are built out of translations, whosecoproduct is undeformed:

∆0(∂i) = ∆⋆M (∂i) = ∆⋆V (∂i) = 1⊗ ∂i + ∂i ⊗ 1 (6.19)

23

Since we are acting on two-particle states we need to define also

∆0(∂i ⊗ ∂j) = ∆⋆(∂i ⊗ ∂j) = 1⊗ 1⊗ ∂i ⊗ ∂j + ∂i ⊗ ∂j ⊗ 1⊗ 1 (6.20)

Therefore the twisted inner product among two-particle states

⟨

k1k2⋆∣

∣ k3k4

⟩

= 〈· |·〉 ∆⋆(F−1)( |k1k2〉 ⊗ |k3k4〉) (6.21)

may be easily computed to be

⟨

k1, k2⋆M∣

∣ k3, k4

⟩

= ei

2θij(k1i+k2i)(k3j+k4j ) 〈k1, k2 |k3, k4〉

⟨

k1, k2⋆V∣

∣ k3, k4

⟩

= eθ(k1−+k2−)(k3++k4−) 〈k1, k2 |k3, k4〉 (6.22)

We can now calculate the twisted inner product of twisted states. Combining(6.22) with (6.5) we obtain the simple expression:

⋆M

⟨

k1, k2⋆M∣

∣ k3, k4

⟩

⋆M

= ei

2θij

P

a<b kaikbj 〈k1, k2 |k3, k4〉

⋆V

⟨

k1, k2⋆V∣

∣ k3, k4

⟩

⋆V

= eθP

a<b ka−kb+ 〈k1, k2 |k3, k4〉 (6.23)

that is

⋆

⟨

k1, k2⋆∣

∣ k3, k4

⟩

⋆

= e−P

a<b ka•kb 〈k1, k2 |k3, k4〉 (6.24)

with ka • kb defined in (4.14).

Recalling the results (6.10) and (6.11), we can cast the previous expressionin the form:

⋆

⟨

k1, k2⋆∣

∣ k3, k4

⟩

⋆

= 〈0| ak1 ⋆ ak2 ⋆ a†k3 ⋆ a†k4|0〉 (6.25)

This is in some sense also a consistency check. We could have started with thecommutative expression 〈k1, k2 |k3, k4〉 = 〈0| ak1ak2a†k3a

†k4|0〉 and twisted the

product among the creation and annihilation operators, obtaining the aboveresult. We decided to follow a longer procedure to highlight the appearanceof the various twists.

24

7 The twisted S-matrix

Let |f〉 , |i〉 denote a collection of free asymptotic states at t = ±∞ respec-tively. We also assume that we can define in some way the one particleincoming and outgoing states. This is a nontrivial assumption¶, in a theoryin which localization is impossible the concept of localization may not be welldefined. Nevertheless is it reasonable to expect that also in this theory, forsmall θ for large distances and times it will be possible to talk on incomingand outgoing states, expandable in the eigenvalues of momentum |k〉.

As in standard books in quantum field theory we define the S matrix asthe matrix which describes the scattering of the initial |i〉 states into the final|f〉 states

Sfi =in⋆

⟨

f⋆∣

∣ i

⟩

⋆out

=out⋆

⟨

f⋆∣

∣ S⋆∣

∣ i

⟩

⋆out

=in⋆

⟨

f⋆∣

∣ S⋆∣

∣ i

⟩

⋆in

(7.1)

where the twisted inner product of twisted states (6.23) is understood. Theone-particle asymptotic state is defined as in (6.7) to be

|k〉in = N⋆(k)a†k |0〉in = −N⋆(k)

i√

(2π)22ωk

∫

d2xe−ik·x↔

∂0 ϕin(x) |0〉in(7.2)

with N⋆(k) a normalization factor to be determined for the Moyal and Wick-Voros cases separately. Analogously for the out states. Moreover we assume,as in the commutative case, that the matrix elements of the interacting fieldϕ(x) approaches those of the free asymptotic field as time goes to ∓∞

limx0→±∞

〈f |ϕ(x)|i〉 = Z1/2〈f |ϕout

in(x)|i〉 (7.3)

with Z a renormalization factor. To be definite let us consider an elasticprocess of two particles in two particles. According to the previous sectionwe have then

Sfi⋆(k1, .., k4) =in⋆

⟨

k1k2⋆

| k3k4⟩

in⋆

= eP

a<b ka•kbin 〈k1k2 |k3k4〉out (7.4)

which can be expressed in terms of Green’s functions, following the sameprocedure as in the commutative case (see for example [57]). On repeatedly

¶We thank Harald Grosse for bringing this fact to our attention.

25

using (7.2) and (7.3) we arrive at

Sfi =in⋆

⟨

k1k2⋆

| k3k4⟩

out⋆

= disconnected graphs

+ N⋆(k1)N⋆(k2)N⋆(k3)N⋆(k4)(iZ−1/2)2e−

P

a<b ka•kb (7.5)

×∫

Πa d2xa

√

(2π)22ωka

e−ikaxa

(∂2µ +m2)aG(x1, x2, x3, x4)

where G(x1, x2, x3, x4) is the four-point Green’s function.

In order to fix the normalization of the asymptotic states let us computethe scattering amplitude for one particle going into one particle, at zerothorder. Up to the undeformed normalization factors N(pa), this has to give adelta function

N(k)N(p)δ2(k − p) = N∗⋆ (k)N⋆(p)in⋆

⟨

k⋆

| p⟩

out⋆

= N∗⋆ (k)N⋆(p)e−k•p

in 〈k |p〉out= N∗⋆ (k)N⋆(p)e

−k•pδ2(k − p) (7.6)

from which

N⋆M (p) = N(p)

N⋆V (p) = e−θ4|~p|2N(p). (7.7)

Let us now compute the scattering amplitude for the process above (two-particles in two particles) at one loop. We have two kinds of contributionto (7.5), one coming from the planar terms (4.24), which in spatial coordi-nates read

GP(x1, x2, x3, x4) =

∫

Πad2ka

√

(2π)22ωka

eikaxa

G(4)P (k1, k2, k3, k4) (7.8)

the other coming from non planar terms (4.25)

GNP(x1, x2, x3, x4) =

∫

Πad2ka

√

(2π)22ωka

eikaxa

G(4)NP(k1, k2, k3, k4) (7.9)

Let us do the computation for the planar case first. Substituting in (7.5)we find the same result in Moyal andWick-Voros case; moreover they coincide

26

with the undeformed result:

Sfi⋆P (k1, .., k4) =(−ig)2

8(2π)3N(k1)N(k2)N(k3)N(k4)Πae

θ4|~ka|2

e−P

a<b ka•kb

∫

Πad2xa

√

(2π)22ωka

e−ikaxa

∫

Πad2pa

√

(2π)22ωpa

eipaxa

(−p2a +m2)

∫

d3q

(2π)3e

P

a≤b pa•pbδ(∑4

a=1 pa)

(q2 −m2)((p1 + k2 − q)2 −m2)∏4

a=1 (p2a −m2)

(7.10)

The integration over the xa variables yields factors of (2π)2δ(2)(ka − pa);therefore the propagators of the external legs cancel as in the standard case,as well as the factor

Πaeθ4|~ka|2e−

P

a<b ka•kb × eP

a≤b pa•pbδ(2)(ka − pa) −→ 1 (7.11)

so that we are left with the usual commutative expression

Sfi⋆P (k1, .., k4) = Sfi(k1, .., k4) (7.12)

In the NP case instead we find

Sfi⋆NP (k1, .., k4) =(−ig)2

8(2π)3N(k1)N(k2)N(k3)N(k4)Πae

θ4|~ka|2

e−P

a<b ka•kb

∫

Πad2xa

√

(2π)22ωka

e−ikaxa

∫

Πad2pa

√

(2π)22ωpa

eipaxa

(−p2a +m2)

∫

d3q

(2π)3e

P

a≤b pa•pb+Eaδ(∑4

a=1 pa)

(q2 −m2)((p1 + k2 − q)2 −m2)∏4

a=1 (p2a −m2)

(7.13)

After integrating over xa the propagators of the external legs cancel and thesimplification (7.11) continues to hold, but we are left with the exponentialof Ea which does not simplify. Its explicit expression is given in (4.26), aswe can see it is an imaginary phase, and it has the same expression in the

Moyal and Wick-Voros case. It depends on the q variable, therefore it getsintegrated and modifies the IR and UV behaviour of the loop: this is thecorrection responsible for the UV/IR mixing [45]. Therefore we can concludethat

Sfi⋆MNP (k1, .., k4) = Sfi⋆V NP (k1, .., k4) 6= Sfi(k1, .., k4) (7.14)

27

8 Conclusions

In Giuseppe Tomasi di Lampedusa’s novel Il Gattopardo (translated The

Leopard) [58] the Prince of Salina says: “Change everything so that nothingchanges”. This sums up the situation that we faced in our analysis of the fieldtheory in the presence of the Wick-Voros and Moyal products. We startedwith different actions, coming from different Lagrangians densities alreadyat the level of the free theory. The free propagator for the Wick-Voros isdifferent from the Moyal case, but the classical theory has no new solutions,and at the quantum level the poles of the propagators are the same. Thenwe found a different vertex for the quartic theory, which led to a differentfour points function. But the differences are reabsorbed in the S-matrix,provided one recognizes the properly normalized asymptotic states. It is notanymore enough to think of a flux of particles to be identified by ordinaryplane waves described by the usual exponential wave with the customarydispersion relation. In a field theory with a different propagator such asthe one considered here the asymptotic states change. The noncommutativecases are however different from the commutative one (something has tochange!), but they describe the same “physics” among themselves.

The two noncommutative products (Moyal and Wick-Voros) are differentrealizations of the same algebra, and as such describe the same noncommu-tative geometry, and it would have been curious to find different physicalconsequences. In fact one could have studied the noncommutative geometryexclusively at the operatorial level, without the need for a deformed product.But at the end of the day, to confront with a physical theory, one has to mapthe states into physically observable states, that an experimenter (at least anideal one) can prepare. The correspondence between states and real objectsis not immediate in noncommutative geometry, and has to be handled withextreme care. This is the morale of this tale. In noncommutative geome-try, the different structure of space time forces to change the correspondencebetween mathematical objects and physical observables. This should leadto the formulation of a coherent theory of observables and measurements onnoncommutative spaces.

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Acknowledgments

We thank A.P. Balachandran, Seckin Kurkcuoglu, Renato Musto, DenjoeO’Connor and Pietro Santorelli for discussions and correspondence.

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