1 Introduction

Quantum Fields on Noncommutative Spacetimes: Theory and Phenomenology

\ArticleName

Quantum Fields on Noncommutative Spacetimes:
Theory and PhenomenologyThis paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html

\Author

Aiyalam P. BALACHANDRAN , Alberto IBORT , Giuseppe MARMO
and Mario MARTONE

A.P. Balachandran, A. Ibort, G. Marmo and M. Martone

Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA \EmailDbal@physics.syr.edu, mcmarton@syr.edu

Dipartimento di Scienze Fisiche, University of Napoli and INFN,
Via Cinthia I-80126 Napoli, Italy \EmailDmarmo@na.infn.it

\ArticleDates

Received March 24, 2010, in final form June 08, 2010; Published online June 21, 2010

\Abstract

In the present work we review the twisted field construction of quantum field theory on noncommutative spacetimes based on twisted Poincaré invariance. We present the latest development in the field, in particular the notion of equivalence of such quantum field theories on a noncommutative spacetime, in this regard we work out explicitly the inequivalence between twisted quantum field theories on Moyal and Wick–Voros planes; the duality between deformations of the multiplication map on the algebra of functions on spacetime and coproduct deformations of the Poincaré–Hopf algebra acting on ; the appearance of a nonassociative product on when gauge fields are also included in the picture. The last part of the manuscript is dedicated to the phenomenology of noncommutative quantum field theories in the particular approach adopted in this review. CPT violating processes, modification of two-point temperature correlation function in CMB spectrum analysis and Pauli-forbidden transition in are all effects which show up in such a noncommutative setting. We review how they appear and in particular the constraint we can infer from comparison between theoretical computations and experimental bounds on such effects. The best bound we can get, coming from Borexino experiment, is  TeV for the energy scale of noncommutativity, which corresponds to a length scale  m. This bound comes from a different model of spacetime deformation more adapted to applications in atomic physics. It is thus model dependent even though similar bounds are expected for the Moyal spacetime as well as argued elsewhere.

\Keywords

noncommutative spacetime; quantum field theory; twisted field construction; Poincaré–Hopf algebra

\Classification

81R50; 81R60

1 Introduction

A great deal of effort has been put in trying to achieve a formulation of a theory of quantum gravity.

It has long been speculated that spacetime structure changes radically at Planck scales. In 1994, in a fundamental paper, Doplicher, Fredenhaghen and Roberts argued that the commutator among the four spacetime coordinates of a physical theory in which Einstein’s theory of relativity and principles of Quantum Mechanics both coexist, should not vanish111The idea of extra-dimensions has been widely used recently. Even in those cases the description of spacetime in terms of a four dimensional manifold must be achieved in some sort of effective limit. It has been shown that the four dimensional limit of certain string theories [3] does, in fact, involve a noncommutative spacetime. [2]. This argument suggested that noncommutativity of spacetime at Planck scales, a feature seen before as just a way to regularize quantum field theory, is possibly one of the main features of quantum gravity. We will now sketch the argument, for further details we refer to the original papers [2, 4].

From quantum physics we know that to probe a spacetime region with radius of the order of the Planck length

 LP=√Gℏc3≃1.6×10−33 cm,

we need a particle of mass such that its Compton wavelength is smaller than the length scale of the spacetime region we wish to probe, namely of the Planck length:

 λC=ℏM∗c≤LP⇒M∗≥ℏLPc≃1019 Gev.

Einstein’s theory of relativity tells us that the Schwarzschild radius associated to such a mass distribution is

 RS=2GM∗c2≥2LP.

It is thus greater than the region we would like to explore. Thus probing spacetime at the Planck scale generates a paradox: in the process an event horizon (or a trapped surface) is created which now prevents us to access altogether the spacetime region we were initially interested in.

In order to avoid the collapse of the probed region, we must assume that it is not possible to simultaneously measure all four spacetime coordinates. This requirement can be incorporated in the noncommutativity of coordinates. A simple choice for this noncommutativity is

 [ˆxμ,ˆxν]=iθμν. (1.1)

Here are constants and are the coordinate functions on an -dimensional spacetime on which the Poincaré group acts in a standard manner for :

 ˆxμ(x)=xμ.

In the present paper, the relations (1.1) will be assumed. We will describe how to formulate the minimal requirements for a quantum field theory (QFT) on such a spacetime and two particular instances of quantum field theories will be constructed over such noncommutative spacetimes, the Moyal and the Wick–Voros quantum fields. We should point it out here that different proposals have been discussed in the literature (see for instance [13, 14] and references therein), however the construction provided here is constructive and their consistency and properties can be tested directly. The paper will be organised as follows. In the next section we will introduce the Drinfel’d twist and the deformation of the Poincaré Hopf algebra . Both concepts will play a crucial role in the formulation of QFT on noncommutative spacetime, which will be then explained in Section 3. Relations (1.1) do not uniquely specify the spacetime algebra. For example, both Moyal and Wick–Voros planes are compatible with (1.1). In Sections 4 and 5 we will see how this freedom gets reflected on the QFT side. In Section 6 we will present some further mathematical developments while the last section will be devoted to describe phenomenological consequences of noncommutative spacetimes. They influence the cosmic microwave background, the mass difference and the Pauli principle among others. From available data we will present the experimental bounds on the scale of spacetime noncommutativity. The best bound we find for its energy scale is  TeV. It comes from Borexino and Superkamiokande experiments. We will then conclude with some final remarks.

2 The Drinfel’d twist and deformed coproduct

At first we notice that the relations (1.1) can be implemented by deforming the product of the standard commutative algebra of functions on our -dimensional space-time ), where , are smooth functions on and is the point-wise multiplication map:

 m0(f⊗g)(x)=f(x)g(x)=g(x)f(x)=m0(g⊗f)(x).

There is a general procedure to deform such a product in a controlled way using the so-called twist deformation [5].

Let us denote as such a deformation of which leads to (1.1). Here is a noncommutative product and the twist map. The two satisfy:

 mθ(f⊗g)(x)=m0∘Fθ(f⊗g)(x):=(f⋆g)(x), (2.1) Fθ: F(M)⊗F(M)→F(M)⊗F(M)andFθ→\mathds1asθ→0.

An explicit form of is

 Fθ=expi2θ[∂x⊗∂y−∂y⊗∂x]. (2.2)

In particular the unit is preserved by the deformation. We notice that

• is one-to-one and invertible;

• acts on the tensor product in a non-factorizable manner, i.e. the action on intertwines the two factors.

The above choice of leads to the Moyal plane :

 mMθ(f⊗g)(x,y)≡f(x,y)⋅g(x,y)+i2θ[(∂xf)(∂yg)−(∂yf)(∂xg)] mMθ(f⊗g)(x,y)≡+∑n=2[i2θ(∂x⊗∂y−∂y⊗∂x)]nn!(f⊗g).

But it is not unique. Another choice, leading to the Voros plane is

 FVθ=exp12θ[∂x⊗∂x+∂y⊗∂y]FMθ=FMθexp12θ[∂x⊗∂x+∂y⊗∂y]. (2.3)

The noncommutative relations (1.1) bring with them another problem: at first sight it seems that the noncommutativity of spacetime coordinates violates Poincaré invariance: the l.h.s. of (1.1) transforms in a non-trivial way under the standard action of the Poincaré group whereas the r.h.s. does not. But there is still a way to act properly with the Poincaré group, or rather, with its group algebra , on the deformed algebras if its action is changed to the so-called twisted action [6, 7, 8]. It goes as follows. The l.h.s. of (1.1) is an element of the tensor product space followed by . The way in which acts on the tensor product space requires a homomorphism . It is not fixed by the way acts on , but is a further ingredient of the theory we need to specify. This map is called the coproduct and will be denoted by :

 Δ: CP→CP⊗CP.

Once we provide with the coproduct , that is we also specify how acts on a tensor product space, we get a new mathematical structure called a Hopf algebra . There are further formal compatibility requirements between the multiplication map of and the map , but we will not discuss them here. For further details on Hopf algebras, we refer to [9, 10, 11, 12].

Usually we assume for the coproduct the simple separable map :

 Δ0(g)=g⊗g∀g∈P. (2.4)

It extends to by linearity. But (2.4) is not the only possible choice. The idea proposed in [6, 7, 8] is that we can assume a different coproduct on , that is “twisted” or deformed with respect to , to modify the action of the Poincaré group on tensor product spaces in such a way that it does preserve relations (1.1). We must realize here that the change of is not a mere mathematical construction, as it affects the way composite systems transform under spacetime symmetries. This observation will have deep consequences in the physical interpretation of the theory as it will be shown later. This modification changes the standard Hopf algebra structure associated with the Poincaré group (the Poincaré–Hopf algebra ) to a twisted Poincaré–Hopf algebra ().

The deformed algebra is not unique. For the Moyal and Wick–Voros cases they are different, although isomorphic222The two deformations are in fact equivalent in Hopf algebra deformation theory. That is they belong to the same equivalence class in the non-Abelian cohomology that classifies Hopf algebra twist-deformations. See for details [11].. The isomorphism map is

 γ=e−14θ(∂2x+∂2y),γ: AMθ→AVθandFVθ=γ⊗γFMθΔ(γ−1).

We denote them by when we want to emphasise that we are working with Moyal and Wick–Voros spacetimes.

Still they lead to different QFT’s since the isomorphism map is not unitary. We discuss this point in detail in Sections 4 and 5.

The explicit form of the deformed coproduct of is obtained from requiring that the action of is an automorphism of the new algebra of functions on spacetime. That is, the action of has to be compatible with the new noncommutative multiplication rule (2.1). In particular, for ,

 g▹mθ(f⊗h)(x)=mθ(g▹(f⊗h))(x).

It is easy to see that the standard coproduct choice (2.4) is not compatible [6, 7, 8] with the action of on the deformed algebra . In the cases under consideration, where are twist deformations of , there is a simple rule to get deformations of compatible with . They are given by the formula:

 Δθ=(Fθ)−1Δ0Fθ, (2.5)

where is an element in and it is determined by the map introduced before, being the realisation of on .

For , the corresponding give us the Hopf algebras .

Without going deeper into the deformation theory of Hopf algebras, we just note that the deformations we are considering here are very specific ones since we keep the multiplication rules unchanged and deform only the co-structures of the underlying Hopf algebra. Thus for , we only change to leaving the group multiplication the same. For a deeper discussion on deformations of algebras and Hopf algebras, we refer again to the literature [9, 10, 11, 12].

Lastly we have to introduce the concept of twisted statistics. It is a strict consequence of the twisted action of the Poincaré group on the tensor product space (2.5). In quantum mechanics two kinds of particles, with different statistics, are admitted: fermions, which are described by fully antisymmetrised states, and bosons, which are instead completely symmetric. Let be a single particle Hilbert space. Then given a two-particle quantum state, with , we can get its symmetrised and anti-symmetrised parts as:

 α⊗S,Aβ=\mathds1±τ02α⊗β,

where the map is called the flip operator and it simply switches the elements on which it acts,

 τ0(α⊗β)=β⊗α.

From the foundations of quantum field theories it can be proved that the statistics of particles have to be superselected, that is Poincaré transformations cannot take bosons (fermions) into fermions (bosons). In other words, a symmetric (antisymmetric) state must still be symmetric (antisymmetric) after the action of any element of the Poincaré group. This requirement implies that the flip operator has to commute with the coproduct of any element of . As can be trivially checked, the action of commutes with the coproduct of , but not with . If we do not modify the flip operator, we end up with a theory in which, for example, a rotation can transform a fermion into a boson.

If the deformation of the coproduct is of the kind we have been considering so far, that is a twist deformation as in (2.5), again it is easy to find a deformation of the flip operator which commutes with :

 τ0→τθ=(Fθ)−1τ0Fθ (2.6)

and, moreover . This equation contains the -matrix of a quasi-triangular Hopf algebra. By definition, is given by

 τθ=R∘τ0. (2.7)

Hence:

 R:=F−1θ∘Fθ21,Fθ21≡τ0Fθτ−10. (2.8)

In quantum physics on a noncommutative spacetime, we then consider symmetrisation (antisymmetrisation) with respect to rather then :

 α⊗Sθ,Aθβ=\mathds1±τθ2α⊗β. (2.9)

3 Examples: Moyal and Wick–Voros planes

The preceding section explains the general features of the formalism of deformations of algebras induced by the relations (1.1) and how to deform the Hopf algebra structure of the Poincaré group to keep the theory still invariant under its action, as in (2.5). We now proceed with the study of two explicit examples: Moyal and Wick–Voros planes.

As previously indicated, we denote by and the twists and multiplication maps for the deformed algebras respectively. Then as in (2.1)

 mM,Vθ(f⊗g)≡m0∘FM,Vθ(f⊗g). (3.1)

In the following, for the sake of simplicity, we will work in two dimensions. The generalization to arbitrary dimensions will be discussed in Section 7.

In two dimensions, we can always write as

 θμν=θϵμν,ϵ01=−ϵ10=1, (3.2)

where is a constant. Then as in (2.2) and (2.3) the two twists assume the form

 FMθ=expi2θ[∂x⊗∂y−∂y⊗∂x], FVθ=exp12θ[∂x⊗∂x+∂y⊗∂y]FMθ=FMθexp12θ[∂x⊗∂x+∂y⊗∂y]. (3.3)

Hereafter we will call and the Moyal and Wick–Voros twists, and and the Moyal and Wick–Voros algebras respectively. Both deformations, and , realize the commutation relations (1.1). This fact already shows, as pointed out above, how noncommutativity of spacetime does not fix uniquely the deformation of the algebra. There are many more noncommutative algebras of functions on spacetime that realize (1.1) with different twisted products. In order to address the study of how this freedom reflects on the quantum field theory side it is enough to work with two of them. Thus hereafter we will only work with .

Given the above expressions for the twists, explicit expressions for the noncommutative product of the functions in the two cases follow immediately from (3.1):

 (f⋆Mg)(x)=mMθ(f⊗g)(x)≡f(x)ei2θαβ←−∂α⊗−→∂βg(x), (3.4)

If we let the -product to act on the coordinate functions, we get in both cases the noncommutative relations (1.1).

Once the two twists are given, following (2.5), we can immediately write down the deformations of the two coproducts as well:

 ΔM,Vθ(g)=(FM,Vθ)−1Δ0(g)FM,Vθ=(FM,Vθ)−1(g⊗g)FM,Vθ. (3.5)

Here are:

 FMθ=exp(−i2θ[Px⊗Py−Py⊗Px]), (3.6) FVθ=exp(−12θ[Px⊗Px+Py⊗Py])FMθ=FMθexp(−12θ[Px⊗Px+Py⊗Py]),

where are translation generators. Their realisation on is:

 Pμ▹f(x)≡(Pμf)(x)=−i(∂μf)(x).

We end this section with a discussion on how the gets twisted in the two cases. As is skew-symmetric and is the composition of and a symmetric part, we get:

 FVθ21=exp(−12θ[Px⊗Px+Py⊗Py])(FMθ)−1.

In the -matrix (2.8), any symmetric part of the twist cancels. Thus in both Moyal and Wick–Voros cases, the statistics of particles is twisted in the same way:

 RM,V=(FMθ)−2=exp(−iθ[Px⊗Py−Py⊗Px]).

We must point out here that the results discussed in this paper differ from those in [13] because of the differences in the construction of quantum field theories on here and in [14]. Thus while our approach is based on Hilbert spaces, operators and explicitly enforces unitarity, the other approach uses functional integrals. There are issues to be clarified in the case of functional integrals in the context of the Moyal plane as they do not fulfill reflection positivity. For a detailed discussion of this issue, see [15].

4 Quantum field theories on Moyal and Voros planes

It is time now to discuss how to quantize the two theories introduced in the previous section. Our approach to the Moyal plane is discussed in [8, 16] and to the Wick–Voros plane can be found in [17, 18]. For another approach to the latter, see [13].

The quantization procedure consists in finding a set of creation operators, and by adjointness the annihilation counterpart, which create multiparticle states providing a unitary representation of the twisted Poincaré–Hopf symmetry. Such a set of creation and annihilation operators must also implement the appropriate twisted statistics (2.7)–(2.9). Out of them we can construct the twisted quantum fields (QFs). It has been in fact proven elsewhere [8, 16] that the Hamiltonian constructed out of such twisted fields is Hopf–Poincaré invariant.

Let us first consider the Moyal case. As our previous work [8, 16] shows,

 aMp=cpexp(−i2pμθμνPν), (4.1) aM†p=c†pexp(i2pμθμνPν), (4.2)

where , are the untwisted annihilation and creation operators (we can assume all such operators to refer to in-, out- or free-operators as the occasion demands), provide the operators we were looking for. Let’s see that.

Since we are considering only deformations in which the coproduct is changed, the way in which the Poincaré group acts on a single particle state is the usual one. Therefore the creation operator (4.2) on the vacuum will act like the untwisted operator . It is then plausible that the generators of Poincaré transformations on the Hilbert space under consideration have to be the untwisted ones. We will now confirm this: if is the unitary representation of the Poincaré group, we will show that the multiparticle states created by acting with (4.2) on the vacuum transform with the Moyal coproduct (3.5).

If we consider the action of a general group element of the Poincaré group on a two-particle state , we expect

 ΔMθ((a,Λ))▹|p,q⟩θ=(FMθ)−1((a,Λ)⊗(a,Λ))FMθ▹|p,q⟩θ ΔMθ((a,Λ))▹|p,q⟩θ=|Λp,Λq⟩θe−i2(p∧q−Λp∧Λq)e−i(p+q)⋅a, (4.3)

where , and we have used the properties of a momentum eigenstate, and .

For the generators of the unitary representation of the Poincaré group on the Hilbert space under consideration, we have

 U(a,Λ)c†pU−1(a,Λ)=eip⋅ac†Λp.

Defining [23]333In the case under study, because of the twisted statistics, the creation operators, and likewise their adjoints, do not commute. The order in which they act on a state becomes then an issue. The choice made here is motivated by asking for consistency [8]. The scalar product we consider for the definition of the adjoint is the one associated with the untwisted creation and annihilation operators.

 |p,q⟩Mθ=aM†qaM†p|0⟩ (4.4)

we can now explicitly compute how acts on the two-particle state considered in (4.4):

 U(a,Λ)|p,q⟩θ=U(a,Λ)c†qU−1(a,Λ)U(a,Λ)ei2q∧Pc†p|0⟩=c†Λqc†Λp|0⟩e−i(p+q)⋅aei2q∧p U(a,Λ)|p,q⟩θ=aM†Λqei2q∧Pc†Λp|0⟩ei2q∧pe−i(p+q)⋅a=|Λp,Λq⟩e−i2(p∧q−Λp∧Λq)e−i(p+q)⋅a. (4.5)

In the above computation we have used and a relation which will be used repeatedly in what follows: . Thus (4.5) coincides with (4.3). It is a remarkable fact that the appropriate deformation of the coproduct naturally appears as the Poincaré group generator acts on the two particle states obtained by the creation operator (4.2). This result generalises to -particles states.

The two-particle state in (4.3) also fulfills the twisted statistics. From (2.7), (2.9) we expect that (throughout this paper we will only consider the bosonic case):

 |p,q⟩Mθ=\mathds1+τMθ2|p,q⟩=12(|p,q⟩+e−i→q∧→p|q,p⟩)

and

 |q,p⟩Mθ=\mathds1+τMθ2|q,p⟩=12(|q,p⟩+e−i→p∧→q|p,q⟩)=e−i→q∧→p|p,q⟩Mθ.

On the other hand using the definition (4.4) of two-particle states in terms of the creation operators ’s:

 |q,p⟩Mθ=aM†paM†q|0⟩=c†pexp(i2pμθμνPν)c†q|0⟩ |q,p⟩Mθ=e−i2→q∧→pa†qexp(−i2qμθμνPν)c†p|0⟩=e−i→q∧→p|p,q⟩θ,

where we have used the relation introduced above and the fact that and commute. So these creation operators do implement the statistics we want.

The operators defined in (4.1), (4.2) are called in the literature [19, 20, 21] as dressed operators. They are obtained from the ones by dressing them using the exponential term. Exploiting a peculiar property of the Moyal plane, which will be explained in the next section, we can obtain the Hermitian quantum field on the Moyal plane:

 (4.6)

where denotes as usual, through a similar dressing of the standard scalar field:

 φMθ=φ0e12←∂μθμνPν. (4.7)

This formula is first deduced for in-, out- or free-fields. For example,

 φM,inθ=φin0e12←∂μθμνPν.

But since the Heisenberg field becomes the ‘in’ field as ,

 φ0(x)→φin0asx0→−∞,

and is time-independent, we (at least heuristically) infer (4.7) for the fully interacting Heisenberg field.

Products of the field (4.6) have a further remarkable property which we have called self-reproducing property:

 (φMθ⋆MφMθ)(x)=[(φ0⋅φ0)(x)]e12←∂μθμνPν,

where the represents the standard point-wise product. This property generalises to products of fields

 φMθ⋆MφMθ⋆M⋯⋆MφMθN-factors=φN0e12←∂μθμνPν, (4.8)

where again indicates the -th power with respect the commutative product . This self-reproducing property plays a significant role in general theory. It is the basis for the proof of the absence of UV-IR mixing in Moyal field theories (with no gauge fields) [22, 23].

Now consider the Wick–Voros case. The twisted creation operators, which correctly create states from the vacuum transforming under the twisted coproduct, are [17]

 aV†p=c†pei2(pμθμνPν−iθpνPν), (4.9)

where uses the Euclidean scalar product. Its adjoint is

 aVp=e−i2(pμθμνPν+iθpνPν)cp. (4.10)

We prove elsewhere [24] that (4.9) and (4.10) are also dictated by the covariance of quantum fields.

It can be shown that, like in Moyal case, the states obtained by the action of reproduce the appropriate twisted statistics too.

Although we have obtained, like in Moyal, (4.9) and (4.10) by dressing the operators, the quantum field on Wick–Voros plane:

 φVθ(x)=∫dμ(p)[aVpe−p(x)+aV†pep(x)] (4.11)

cannot be obtained from an overall dressing like in (4.7). This property fails due to the fact that is not possible to factorise the same overall exponential out of both (4.9) and (4.10), since it is not possible in (4.9), (4.10) to move the exponential from right (left) to left (right), that is:

 aV†p≠exp(−i2(pμθμνPν+iθpνPν))c†p, aVp≠cpexp(−i2(pμθμνPν+iθpνPν)). (4.12)

This is because and do not commute with the exponentials in (4.9) and (4.10), in fact moving () to the right (left) will bring a factor :

 aV†p=exp(i2(pμθμνPν−iθpνPν)−θ2pνpν)c†p, aVp=cpexp(−i2(pμθμνPν+iθpνPν)−θ2pνpν).

A consequence is that we have to twist the creation-annihilation parts ( in-, out- or free-) fields separately:

 φ(+)V,Iθ=∫dμ(p)aV,I†ep=φ(+)I0e12(←∂μθμνPν−iθ←∂μPμ), φ(−)V,Iθ=∫dμ(p)aV,Ie−p=e12(→∂μθμνPν+iθ→∂μPμ)φ(−)I0, ep(x)=eip⋅x,dμ(p):=d3p2√→p2+m2,m=mass of the f\/ield φI0,

where now we have added the superscript I to , , and .

Therefore to obtain the field (4.11) we have to twist creation and annihilation parts separately

 φV,Iθ=φ(+)V,Iθ+φ(−)V,Iθ. (4.13)

A further point relates to the self-reproducting property of these Wick–Voros fields. The quantum fields are self-reproductive, but in different ways. Thus

 φ(+)V,Iθ⋆Vφ(+)V,Iθ⋆V⋯⋆Vφ(+)V,IθM-factors=(φ(+)I0)Me12(←∂μ)θμνPν−iθ←∂μPμ, φ(−)V,Iθ⋆Vφ(−)V,Iθ⋆V⋯⋆Vφ(−)V,IθM′-factors=e12(→∂μ)θμνPν+iθ→∂μPμ(φ(−)I0)M′,

where, as in (4.8), is the -th power of with respect to the commutative product . Given (4.13), it follows that the full field, , does not have the self-reproducing property.

5 Voros versus Moyal: a comparison

It is now time to compare the two theories we have previously introduced. We will follow the treatment in [18] and prove them to be inequivalent. First we want to approach this question of equivalence from a heuristic point of view. We will conclude this section proving that no similarity transformation can relate the two theories, showing their inequivalence with respect to isospectral transformations.

The first big difference among QFT’s on Moyal and Wick–Voros planes has already appeared above. Although we have not stressed it, (4.1) and (4.2) could have been changed to

 aMp=exp(−i2pμθμνPν)cp, (5.1) aM†p=exp(i2pμθμνPν)c†p. (5.2)

But in fact (4.1) equals (5.1) and (4.2) equals (5.2) because [16, 25]. In (4.12) we have instead seen that this is not the case for the Wick–Voros plane. This observation is important since it ensures that the Moyal field (4.6), which we can obtain from the field by twisting it, is Hermitian. Had we constructed the Voros field in a similar way, we would have ended up with a non-Hermitian operator.

Equation (4.12) is also the reason for the absence of the self-reproducing property on the Wick–Voros plane that by itself causes differences at the level of physics. (For example the arguments for the absence of UV/IR mixing on the Moyal plane would fail here.)

But that is not all. The states in the Wick–Voros case are not normalised in the same way as in the Moyal case. For instance

 ⟨k1,k2|p1,p2⟩≡⟨0|aV,Ik1aV,Ik2aV,I†p2aV,I†p1|0⟩=eθk1⋅k24√(→k21+m2)(→k22+m2) (5.3) I=in, out,|0⟩‘in=|0⟩out,m=mass of the f\/ield φI0.

For scattering theory, normalisation is important. It depends on the scalar product we are using. If we normalise the states as in the Moyal case, since the normalisation constant in (5.3) is momentum dependent, the normalised states no longer transform with the Wick–Voros coproduct. The factor (5.3) has been computed using the standard scalar product in the Fock space. We can try changing it [17] so that the states become correctly normalised. But then the representation ceases to be unitary. We can try to seek for another one, modifying using a non trivial operator . We could not find any such which would preserve the way the unitary representation has to act on single particle states. It then appears that the Wick–Voros plane is not suitable for quantum physics.

We now show that there is no similarity transformation taking , into , . One way to quickly see this is to examine the operators without the Moyal part of the twist. So we consider , and

 aV,I′p=e12θpνPνcIp, (5.4) aV,I′†p=cI†pe12θpνPν. (5.5)

Now

 [cIp,cI†k]=2|p0|δ3(p−k)\mathds1, p0=√→p2+m2,m=mass of the f\/ield φI0.

Had there existed an invertible operator such that

 WcIpW−1=aV,I′p,WcI†pW−1=aV,I′†p,

then we would have

 [aV,I′p,aV,I′†k]=2|p0|δ3(p−k)\mathds1. (5.6)

But a direct calculation of the l.h.s. using (5.4), (5.5) shows that (5.6) is not true.

However there exists an invertible operator which transforms to and is given by:

 S=eθ4(PμPμ+K),K=−∫dμ(k)kμkμcI†kcIk (5.7)

and a simple computation now shows that

 SaM,I†pS−1=aV,I†p.

where, as usual, I on , denotes in-, out- or free- while in and , we use the Euclidean scalar product.

But

 SaM,IpS−1=e−i2(pμθμνPν−iθpνpν)cIp=~aV,Ip≠aV,Ip.

Let us pursue the properties of this operator further.

The operator leaves the vacuum invariant and shows that certain correlators in the Moyal and Wick–Voros cases are equal. From the explicit expression (5.7) it follows also that the map induced by the operator is isospectral, but not unitary in the Fock space scalar product. It is possible to define a new scalar product which makes unitary [17]. But the previously defined would not be unitary in this scalar product as we discussed above.

Now consider the twisted fields

 φM,Iθ=∫dμ(p)[aM,Ipe−p+aM,I†pep], ~φV,Iθ=∫dμ(p)[~aV,Ipe−p+aV,I†pep],

where denotes as before. Then of course,

 (5.8)

Also

 S|0⟩=S−1|0⟩=0. (5.9)

From (5.8) and (5.9) we obtain trivially the equality of the -points correlation functions:

 ⟨φM,Iθ(x1)φM,Iθ(x2)⋯φM,Iθ(xN)⟩0=⟨~φV,Iθ(x1)~φV,Iθ(x2)⋯~φV,Iθ(xN)⟩0.

Consider simple interaction densities such as

 HMI=φM,Iθ⋆MφM,Iθ⋆M⋯⋆MφM,IθN-factorsandHVI=~φV,Iθ⋆V~φV,Iθ⋆V⋯⋆V~φV,IθN-factors

in either field. Since only acts on the operator parts of the fields, the similarity transformation in (5.8) will not map to :

 S▹HMI≠HVI.

Hence

 ⟨φM,Iθ(x1)⋯φM,Iθ(xj)HMI(xj+1)φM,Iθ(xj+2)⋯φM,Iθ(xN)⟩0 ≠⟨~