Wedge-Local Quantum Fields and Noncommutative Minkowski Space

# Wedge-Local Quantum Fields and Noncommutative Minkowski Space

Harald Grosse
Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria
E-mail:
Gandalf Lechner
International Erwin Schrödinger Institute for Mathematical Physics,
Boltzmanngasse 9, A-1090 Vienna, Austria,
E-mail:
###### Abstract:

Within the setting of a recently proposed model of quantum fields on noncommutative Minkowski space, the consequences of the consistent application of the proper, untwisted Poincaré group as the symmetry group are investigated. The emergent model contains an infinite family of fields which are labelled by different noncommutativity parameters, and related to each other by Lorentz transformations. The relative localization properties of these fields are investigated, and it is shown that to each field one can assign a wedge-shaped localization region in Minkowski space. This assignment is consistent with the principles of covariance and locality, i.e. fields localized in spacelike separated wedges commute.

Regarding the model as a non-local, but wedge-local, quantum field theory on ordinary (commutative) Minkowski spacetime, it is possible to determine two-particle S-matrix elements, which turn out to be non-trivial. Some partial negative results concerning the existence of observables with sharper localization properties are also obtained.

Space-Time Symmetries, Non-Commutative Geometry, Field Theories in Higher Dimensions, Integrable Field Theories

## 1 Introduction

In relativistic quantum field theories, Einstein causality is implemented by requiring that the observables of spacelike separated observers are represented by commuting operators. This principle of locality is usually assumed to hold for arbitrarily small spacelike distances. However, all current approaches to quantum physics trying to incorporate effects of quantum gravity, like string theory [1], quantum field theory on noncommutative spacetimes [2, 3] or loop quantum gravity [4], show some kind of non-local behaviour. In fact, in these theories locality is usually a meaningful concept only in some large scale limit.

The probably simplest examples of such theories are quantum field theories on a deformed, noncommutative Minkowski space, on which the coordinates satisfy a commutation relation of the form

 [^xμ,^xν]=iQμν. (1)

Here the noncommutativity parameter is some real, antisymmetric -matrix. Spacetime models of this form can be motivated by considering the restrictions on event measurements suggested by classical gravity and the uncertainty principle [5], or, in certain cases, as low-energy limits of string theory [6].

In this paper, we study a specific model on noncommutative Minkowski space from a new point of view, and investigate its locality properties, which turn out to be quite different from what is usually expected.

As our starting point we take in Section 2 the scalar massive free field on noncommutative Minkowski space, with a fixed noncommutativity parameter . Formulating this field as an operator on Fock space, we then generate a whole family of non-local quantum fields from it by acting on with the usual second quantized representation of the proper Poincaré group associated to the free scalar field of mass .

The emerging model has similarities with other systems studied in the literature: For fixed , the -point functions of the field coincide with the -point functions recently proposed by Fiore and Wess [7] and Chaichian et. al. [8]. However, we do not use a twisted version of the Poincaré group as these authors do. Rather, the use of a representation of the untwisted symmetry forces us to consider a large class of fields , labelled by an orbit of noncommutativity parameters, and related to each other by Lorentz transformations.

Our model has also connections to the free field model studied by Doplicher, Fredenhagen and Roberts in [5], since also there, a whole spectrum of noncommutativity parameters is considered. However, due to a different action of the Poincaré group, there exist also essential differences between the two models, which are spelled out in detail in Section 2.

In Section 3, we prove our main result stating that the fields , albeit non-local, are far from being completely delocalized with respect to each other. These relative locality properties are proven with the help of a novel construction, associating to each an infinitely extended, wedge-shaped spacetime region . We define a bijection between a set of wedges and a set of noncommutativity parameters, and consider the corresponding fields . It is then shown that this association of spacetime regions to field operators is completely consistent with the principles of covariance and locality:

Under the adjoint action of the representation , they transform according to

 U(y,Λ)ϕW(x)U(y,Λ)−1 =ϕΛW(Λx+y),(y,Λ)∈P↑+. (2)

In dimensions, we make the well-known observation that each field alone transforms covariantly only under the subgroup SO of the Lorentz group. However, the whole family respects the full Lorentz symmetry.

Furthermore, for two wedges and two spacetime points , we find (wedge-) local commutativity in the form

 [ϕW(x),ϕ~W(y)]=0if(W+x)isspaceliketo(~W+y). (3)

Therefore may be interpreted as a field configuration localized in the region instead of the point set . Such a type of localization is similar to the string-local quantum fields studied by Mund, Schroer and Yngvason [9], but different from the usual uniform nonlocality of quantum fields on noncommutative Minkowski space.

This difference in localization is due to the fact that in comparison to usual quantum field theories on Minkowski space with fixed noncommutativity , we consider here a model encompassing a whole Lorentz orbit of noncommutativites, and study the relations of the corresponding subtheories with respect to each other.

The fields introduced here can also be understood as wedge-local quantum fields on genuine, “commutative” Minkowski space , . From this point of view, our analysis fits into the ongoing research aiming at constructions of quantum field theory models with the help of wedge-localized operators [10, 11, 12, 13, 14, 15]. In this context, such fields are mostly used as auxiliary quantities, which, due to their weakened locality properties, can be constructed more easily than point-local Wightman fields, also in the presence of non-trivial interactions [12].

It is shown in Section 4 that from this perspective, the model defined by the fields has a number of similarities to completely integrable quantum field theories in two dimensions. In particular, the algebra of the creation and annihilation parts of the free field on noncommutative Minkowski space is quite similar [16] to the Zamolodchikov-Faddeev algebra [17], which underlies integrable models with factorizing S-matrices [18]. Making use of the model-independent results about wedge-local operators found by Borchers, Buchholz and Schroer [19], we calculate two-particle S-matrix elements and show that the model under consideration describes non-trivial interaction.

In any spacetime dimension , we therefore arrive at a theory of wedge-local, interacting quantum fields. This observation requires an investigation of the question if there exist also observables with sharper localization properties, like localization in bounded spacetime regions, in this setting. First results indicating that there probably is no strictly local quantum field theory corresponding to our wedge-local construction are presented in Section 4.

The article ends in Section 5 with a discussion of the results and an account of open questions.

## 2 Noncommutative Minkowski spacetime and twisted CCR algebras

The simplest example of a noncommutative spacetime is the noncommutative counterpart of Minkowski space (of dimension ), which is usually described by a -algebra of selfadjoint coordinate operators , , satisfying

 [^xμ,^xν]=iQμν, (4)

with a fixed, real, antisymmetric -matrix [3], called the noncommutativity parameter or simply the noncommutativity.

Considering the most interesting four-dimensional case, this algebra can be realized in its Schrödinger representation [5], where the are operators on (a dense domain in) ,

 (^x0ψ)(s1,s2) =κes1⋅ψ(s1,s2), (^x1ψ)(s1,s2) =−i(∂s1ψ)(s1,s2), (^x2ψ)(s1,s2) =κms2⋅ψ(s1,s2), (^x3ψ)(s1,s2) =−i(∂s2ψ)(s1,s2).

Here and are arbitrary real parameters measuring the strength of noncommutative effects, and the matrix takes its standard form

 Q=⎛⎜ ⎜ ⎜⎝0κe00−κe000000κm00−κm0⎞⎟ ⎟ ⎟⎠. (5)

If , the relations (4) are not invariant under the natural action of the full Lorentz group on the vector , and therefore Lorentz covariance is broken from the outset if is taken to be a fixed matrix. Recently, proposals were made how to formulate a “twisted” action of the Lorentz group in this setting, leading to a different concept of covariance [7, 8, 20].

However, such a deformation of the symmetry group is not necessary if one allows for a richer operator form of the commutator , encompassing a whole spectrum of numerical matrices . This approach was taken by Doplicher, Fredenhagen and Roberts and shown to lead to models with better covariance properties [5].

In the present article, we work in a somewhat similar framework, and consider a family of quantum fields which depend explicitly on the noncommutativity parameter , and are related to each other by Lorentz transformations.

To describe the construction of such fields and their relations to similar models studied in the literature, let us introduce some notation. We consider the free scalar quantum field of mass on ordinary (“commutative”) -dimensional Minkowski spacetime, . The energy of a particle with momentum is denoted , and the upper mass shell by . Generally, we shall use the letters for on-shell momenta, and the boldface letters for their respective spatial components.

Defined as an operator-valued distribution, acts on its domain in the Bosonic Fock space over the single particle space , where is the Lorentz invariant measure on .

On , we have the usual (anti-) unitary second quantized representation of the Poincaré group with spin zero and mass . The proper orthochronous transformations , and the total reflections mapping to and leaving the other coordinates unchanged, are represented as (, )

 (U(y,Λ)Ψ)n(p1,...,pn) =ei∑nl=1pl⋅y⋅Ψn(Λ−1p1,...,Λ−1pn), (6a) (U(0,j0)Ψ)n(p1,...,pn) =¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Ψn(−j0p1,...,−j0pn), (6b) (U(0,jk)Ψ)n(p1,...,pn) =Ψn(jkp1,...,jkpn),k=1,...,d−1. (6c)

In particular, the total spacetime reflection acts by complex conjugation on each -particle space.

The generators of the translations will be denoted by , and the corresponding vacuum vector by .

The free field is defined with the help of the standard representation of the CCR algebra on , i.e. we have the creation/annihilation operators , , which satisfy

 a(p)a(q) =a(q)a(p), (7) a(p)a∗(q) =a∗(q)a(p)+ω\scriptsize\boldmath{p}δ(\boldmath{p}−\boldmath{q})idH. (8)

These operators give the field

 ϕ0(x):=∫dμ(p)(eip⋅xa∗(p)+e−ip⋅xa(p)), (9)

which after smearing in becomes an unbounded operator on containing the dense subspace of finite particle number in its domain.

As is well known, has a counterpart on noncommutative Minkowski space, which can be realized on the tensor product of the representation space of the coordinate operators and Fock space as [5]

 ϕ⊗(Q,x)=∫dμ(p)(eip⋅xa∗⊗(Q,p)+e−ip⋅xa⊗(Q,p)), (10)

with the creation/annihilation operators

 a⊗(Q,p):=e−ip⋅^x⊗a(p),a∗⊗(Q,p):=eip⋅^x⊗a∗(p), (11)

taking values in the operators on . Here satisfies (4) for some arbitrary , the space of real, antisymmetric -matrices. We indicate the dependence on explicitly in our notation, since will be allowed to vary later on.

As a consequence of (4), the satisfy the commutation relations

 a⊗(Q,p)a⊗(Q,p′) =e−ipQp′a⊗(Q,p′)a⊗(Q,p),pQp′:=pμQμνp′ν, (12a) a∗⊗(Q,p)a∗⊗(Q,p′) =e−ipQp′a∗⊗(Q,p′)a∗⊗(Q,p), (12b) a⊗(Q,p)a∗⊗(Q,p′) =e+ipQp′a∗⊗(Q,p′)a⊗(Q,p)+ω\scriptsize\boldmath{p}δ(\boldmath{p}−%\boldmath$p$′)idV⊗H. (12c)

It has been realized by a number of authors (see, for example, [20, 21]) that this algebra can also be represented on instead of by using the following distributions, containing the energy-momentum operators ,

 a(Q,p):=ei2pQPa(p),a∗(Q,p):=e−i2pQPa∗(p). (13)

Using

 ei2pQPa(p′)=e−i2pQp′⋅a(p′)ei2pQP (14)

and the antisymmetry of , it is easy to show that and that the also satisfy the relations (12).

Let us denote the corresponding field operators by

 ϕ(Q,x) :=∫dμ(p)(eip⋅xa∗(Q,p)+e−ip⋅xa(Q,p)). (15)

In the context of these or similar fields, some authors propose to work with a “twisted” Poincaré algebra [7, 8, 20] to arrive at a covariant formulation despite being constant. We take here a different point of view and and use the well-known representation (6) of the untwisted Poincaré group to implement the relativistic symmetry.

Since the adjoint action of on the field induces also a transformation of (see Lemma 2.1 below), it is necessary to consider a whole family of fields labelled by noncommutativity parameters. It is thus of interest to determine the commutation relations between the commutation/annihilation operators and for , generalizing (12). For a somewhat related discussion, see [7].

By straightforward calculation, one finds the following exchange relations, valid for arbitrary on-shell momenta and matrices :

 a(Q,p)a(Q′,p′) =e−i2p(Q+Q′)p′a(Q′,p′)a(Q,p), a∗(Q,p)a∗(Q′,p′) =e−i2p(Q+Q′)p′a∗(Q′,p′)a∗(Q,p), (16) a(Q,p)a∗(Q′,p′) =ei2p(Q+Q′)p′a∗(Q′,p′)a(Q,p)+ω\scriptsize\boldmath{p}δ(% \boldmath{p}−\boldmath{p}′)ei2p(Q−Q′)P.

Starting from this “twisted” CCR algebra, we consider the quantum fields (15), depending not only on the spacetime points (in the sense of distributions), but also on the matrices .

In the remainder of this section we analyze the transformation behaviour of these fields under Poincaré transformations. Afterwards, a comparison to other models [5, 7, 8] will be presented.

For the trivial noncommutativity parameter , we see that coincides with the free field on commutative Minkowski space. Consequently, enjoys the well-known covariance and locality properties of a Wightman field.

If , however, the field is neither local nor does it transform covariantly under the full Lorentz group. To see its nonlocality explicitly, we compute the two-particle contribution of the field commutator applied to the vacuum. Using the antisymmetry of , we find

 2i∫dμ(p)∫dμ(q)ei(px+qy)sin(qμQμνpν2)a∗(p)a∗(q)Ω. (17)

This expression does not vanish for spacelike separated except for the case .

To study the transformation behaviour of the fields under Poincaré transformations, we consider the action of the Poincaré group on the algebra (2).

It is shown in the lemma below that the adjoint action of on the , where is an element of the Lorentz group , induces on the transformation

 Q⟼γΛ(Q) :={ΛQΛT;Λ∈L↑−ΛQΛT;Λ∈L↓,Q∈IR−d×d. (18)

Here, as usual, and denote the sets of orthochronous and anti-orthochronous Lorentz transformations, respectively, and corresponding notations are used for the associated subsets of the Poincaré group. Note that in view of the structure of the Lorentz group, is an -action, i.e. , .

###### Lemma 2.1

(Transformation properties of the twisted CCR algebra)
The operator-valued distributions , , , transform under the adjoint action of (6) according to, , ,

 U(y,Λ)a∗(Q,p)U(y,Λ)−1 =e±iΛp⋅ya∗(γΛ(Q),±Λp), (19) U(y,Λ)a(Q,p)U(y,Λ)−1 =e∓iΛp⋅ya(γΛ(Q),±Λp), (20)

where the first sign is valid for and the second sign holds for . Hence the fields (15) satisfy

 U(y,Λ)ϕ(Q,x)U(y,Λ)−1 =ϕ(γΛ(Q),Λx+y),(y,Λ)∈P. (21)

Proof. We begin by looking at orthochronous Poincaré transformations and find

 U(y,Λ)a∗(Q,p)U(y,Λ)−1 =eiΛp⋅yU(y,Λ)e−i2pμQμνPνU(y,Λ)−1a∗(Λp) =eiΛp⋅ye−i2(Λp)μ(ΛQΛT)μνPνa∗(Λp) =eiΛp⋅ya∗(ΛQΛT,Λp)=eiΛp⋅ya∗(γΛ(Q),Λp).

In complete analogy, one shows (20) for . For transformations involving time reflection , we take into account that (6b) is a conjugate linear operator, which leads to a change of sign in the exponent of . Since , time reflection acts on according to

 U(0,j0)a∗(Q,p)U(0,j0)−1 =e+i2(−j0p)μ(j0QjT0)μνPνa∗(−j0p)=a∗(γj0(Q),−j0p),

as claimed in (19). The argument for (20) is completely analogous, and the transformation behaviour (21) of follows directly from (19, 20).

In the following proposition, we show that despite the violation of the usual covariance and locality properties of quantum field theory, the fields satisfy the remaining Wightman axioms [22] for a scalar field. Furthermore, the Reeh-Schlieder property (cyclicity of the vacuum for the field algebra, see [22]) holds.

We consider the fields smeared with Schwartz testfunctions , i.e. the distributions

 f⟼ϕ(Q,f) =∫ddxf(x)ϕ(Q,x)=a∗(Q,f+)+a(Q,f−), (22) a#(Q,f±) :=∫dμ(p)f±(p)a#(Q,p), (23) f±(p) :=∫ddxf(x)e±ip⋅x,p=(ω% \scriptsize\boldmath{p},\boldmath{p})∈H+m. (24)
###### Proposition 2.2

(Wightman properties of the field operators )
Let and .

1. The dense subspace of vectors of finite particle number is contained in the domain of any and is stable under the action of these operators.

2. For ,

 f⟼ϕ(Q,f)Ψ (25)

is a vector-valued tempered distribution.

3. For one has

 ϕ(Q,f)∗Ψ =ϕ(Q,¯¯¯f)Ψ, (26)

and for real , is essentially selfadjoint on .

4. The Reeh-Schlieder property holds: For any non-empty open subset ,

 DQ(O) :=span{ϕ(Q,f1)⋯ϕ(Q,fn)Ω:n∈\fa N0,f1,...,fn∈S(O)} (27)

is dense in .

Proof. Since the proofs of these statements are very similar to the corresponding arguments for the well-known free field , we can be brief about them.

a) The fact that and follows directly from the definition of this operator (22).

b) Since the operators give only rise to multiplication by phases, the smeared creation/annihilation operators satisfy the familiar bounds, ,

 (28)

In the topology of , the right hand side depends continuously on . Hence the temperateness of , , follows.

c) The first statement is a straightforward consequence of the fact that is real. The essential selfadjointness for real can be proven along the same lines as [23, Prop. 5.2.3] by showing that every is an entire analytic vector for .

d) Making use of the spectral properties of the representation of the translations, one can apply the standard Reeh-Schlieder argument [22] to show that is dense in if and only if is dense. Choosing such that the supports of the Fourier transforms do not intersect the lower mass shell, one notes that contains the vectors

 ϕ(Q,f1)⋯ϕ(Q,fn)Ω=a∗(Q,f+1)⋯a∗(Q,f+n)Ω =√n!Pn(Sn(f+1⊗...⊗f+n)),

where denotes the orthogonal projection from onto its totally symmetric subspace , and is the operator multiplying with

 Sn(p1,...,pn) =∏1≤l

Varying the testfunctions within the above limitations gives rise to dense sets of one particle wavefunctions in . Since is a unitary operator on , the vectors which can be obtained in this way form a total set in . Hence the image of this set under the projection is total in , which implies the density of in view of the Fock structure of .

Before we analyze the localization properties of the fields in the following sections, we point out some relations to other models studied in the literature.

In the recent preprint [7], Fiore and Wess consider a framework for quantum field theories on noncommutative Minkowski space (with fixed noncommutativity ), in which coordinate differences are represented by commuting operators. This leads them to consider -point functions of the form

 WQn(x1,...,xn):=⟨Ω,ϕ(x1)⋆Q⋯⋆Qϕ(xn)Ω⟩, (30)

where denotes a Wightman field on commutative Minkowski space and is a Moyal-like product. For (and analogously, for tempered distributions), is defined as

 (f⋆Qg)(x1,...,xn,y1,...,ym)=n∏l=1m∏k=1exp(−i2∂∂xμlQμν∂∂yνk)f(x1,...,xk)g(y1,...,yl).

In a more ad hoc manner, the same -point functions have also been proposed in [24].

Taking in (30) for the free field , and considering a fixed , the exchange relations of the imply

 ϕ(Q,x1)⋯ϕ(Q,xn)Ω =∏1≤l

A proof of this equation can be carried out using induction in the field number . In particular, the vacuum expectation values of the field products coincide with the distributions (30).

This observation clarifies the meaning of our model as a theory containing the fields studied by Fiore and Wess [7] and Chaichian et. al. [8] as subtheories, for different values of the noncommutativity . Put differently, what we analyze in this article are not the individual subtheories given by fixed , but rather the relative properties of these fields with respect to each other.

We would also like to compare our fields to the model formulated in terms of the tensor product field operators (10) [5] in more detail.

As mentioned before, the creation and annihilation operators (11) and (13) satisfy the same algebraic relations if is fixed (12). More precisely, the -algebra generated by the smeared fields , , is isomorphic to the -algebra generated by the fields . One can easily calculate that this isomorphism is implemented by the following unitary operator.

Given and , , we define ,

 (V(n)Q,ξΨn)(p1,...,pn) :=Ψn(p1,...,pn)⋅ei(p1+...+pn)^xξ,Ψn∈Hn. (31)

The direct sum of these operators, , is a unitary mapping to , and

 VQ,ξFQV∗Q,ξ =FQ⊗,VQ,ξa#(Q,p)V∗Q,ξ=a#⊗(Q,p), (32)

where the second equation holds in the sense of distributions.

In particular, it follows from that the following -point functions coincide, ,

 ⟨(ξ⊗Ω),ϕ⊗(Q,x1)⋯ϕ⊗(Q,xn)(ξ⊗Ω)⟩ =⟨Ω,ϕ(Q,x1)⋯ϕ(Q,xn)Ω⟩. (33)

So in the context of the fields , we are working with a vector state of the form . The choice of is irrelevant as long as , since is evaluated only on the identity operator in due to momentum conservation in the second tensor factor.

The use of such a state differs drastically from the approach taken in [5], where the concepts of noncommutative geometry were applied to identify pure states on the spacetime algebra as the noncommutative analogues of points in the undeformed spacetime manifold. This analogy suggests to consider different states on each factor , in contrast to (33).

Besides this different choice of state, there are also differences in the algebraic structure between the fields considered here and the fields considered in [5], if different values of are taken into account. In [5], the action of a Lorentz transformation on the coordinate operators is simply given by . The transformed therefore have the commutator matrix , which coincides with for orthochronous .

However, the commutation relations of the transformed creation/annihilation operators are different from those found in (19). For example,

 (eip⋅Λ^x⊗a∗(p))(eip′⋅^x⊗a∗(p′)) =e−i2p(ΛQ)p′⋅(eip′⋅^x⊗a∗(p′))(eip⋅Λ^x⊗a∗(p)), (34)

instead of (2), which gives the exchange factor . Put differently, the isomorphism mentioned before does not carry over to the algebras and generated by and for different noncommutativity parameters .

## 3 Wedges and wedge-local quantum fields

We now set out to analyze the localization properties of the fields (15). As has been shown before, these are non-local fields in the sense that they violate the usual point-like localization of Wightman fields. However, we will argue that they are not completely delocalized, either: It turns out that is localized in an infinitely extended, wedge-shaped region of Minkowski space.

The wedge is localized in depends on both, the spacetime point and the matrix . To establish the wedge-locality of these fields, the essential step is to construct a map from a set of wedges (defined below) to a set of antisymmetric matrices such that is localized in the shifted wedge . This construction is carried out in the first Subsection 3.1.

Afterwards, in Subsection 3.2, the concept of covariant, wedge local quantum fields will be introduced, and the properties of the operators will be explained.

To avoid confusion, we emphasize that the wedges considered here have no relation to the so-called “lightwedge” sometimes mentioned in the literature [25].

### 3.1 Symmetries of wedges and noncommutativity parameters

To define a correspondence between noncommutativity parameters and localization regions, we first recall some well-known definitions and facts about wedges in -dimensional Minkowski space.

As our reference wedge region, we take

 W1:={x∈IRd:x1>|x0|}, (35)

often called the right wedge in the literature. With respect to the coordinate reflections , this region has the symmetry properties

 j0W1=W1,j1W1=−W1=W′1,jkW1=W1,k>1, (36)

where denotes the causal complement of .

The set of all wedges in is defined as the set of all Poincaré transforms of , i.e. . We will mostly work with a subset , consisting only of the Lorentz transforms of ,

 W0:=LW1⊂W. (37)

contains precisely all those wedges which have the origin in their edges, see also [26, 27]. Note that in view of the symmetries (36), and since commutes with , there holds

 W′=jW=−W,W∈W0. (38)

Moreover, it follows from (36) that for spacetime dimension ,

 W0=L↑+W1,(d>2). (39)

Hence is a homogeneous space for the proper orthochronous Lorentz group when equipped with the natural action ,

 ιΛ(W):=ΛW. (40)

As is well known, the corresponding stabilizer group of is , where the factor contains the Lorentz boosts in -direction and describes rotations in the edge of , i.e. in the subspace .111Whereas the invariance of under spatial rotations around the -axis is obvious from (35), the SO-invariance follows from the fact that a boost in -direction has the lightlike vectors as eigenvectors, with positive eigenvalues.

In the following, we will consider the subgroup which is generated by the proper orthochronous part of and the total spacetime reflection . (In even spacetime dimensions, coincides with the proper Lorentz group , while in odd dimensions, .) The corresponding subgroup of the Poincaré group will be denoted .

Considering as a homogeneous space for the larger group , the stabilizer of arises from by adjoining the reflection if is even and by adjoining if is odd.

In the two-dimensional case, all transformations in leave invariant, and hence

 W0={W1,−W1},(d=2). (41)

This two-element set forms an -homogeneous space, too, and we have the stabilizer groups for .

The various stabilizers are collected in Table 1.

To set up the desired correspondence between wedges and antisymmetric matrices, we construct in the following an -homogeneous space which is isomorphic to . Since we want to understand the covariance properties (21) of the fields , we want to endow with the action (18), restricted to , i.e.

 Q⟼γΛ(Q) :={ΛQΛT;Λ∈L↑+−ΛQΛT;Λ∈jL↑+,Q∈IR−d×d. (42)

We do not take the full Lorentz group as our symmetry group here because, as is shown below, a homomorphism does not exist in the most important four-dimensional case if these spaces are treated as homogeneous spaces for all of .

As a homogeneous space, must consist of a single -orbit under , i.e. there exists such that

 Q:={γΛ(Q1):Λ∈^