# Non-commutative waves for gravitational anyons

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## Abstract

We revisit the representation theory of the quantum double of the universal cover of the Lorentz group in 2 + 1 dimensions, motivated by its role as a deformed Poincaré symmetry and symmetry algebra in (2 + 1)-dimensional quantum gravity. We express the unitary irreducible representations in terms of covariant, infinite-component fields on curved momentum space satisfying algebraic spin and mass constraints. Adapting and applying the method of group Fourier transforms, we obtain covariant fields on (2 + 1)-dimensional Minkowski space which necessarily depend on an additional internal and circular dimension. The momentum space constraints turn into differential or exponentiated differential operators, and the group Fourier transform induces a star product on Minkowski space and the internal space which is essentially a version of Rieffel’s deformation quantisation via convolution.

## Keywords

Anyons Quantum double 3d quantum gravity Non-commutative spacetime Group Fourier transform## Mathematics Subject Classification

16T05 81R60 81T75 83C65## 1 Introduction

The possibility of anyonic statistics in two spatial dimensions lies at the root of the peculiarity and intricacy of planar phenomena in quantum physics, ranging from the quantum Hall effect to potential uses of anyons in topological quantum computing [1]. The mathematical origin of anyonic statistics is the infinite connectedness of the planar rotation group \(\mathrm {SO}(2)\) which is topologically a circle. The goal of this paper, in brief, is to explore the consequences of this fact for quantum gravity in 2 + 1 dimensions, where the infinite connectedness is doubled and appears in both momentum space and the rotation group.

Our strategy in pursuing this goal is to extend and generalise the method developed in [2, 3], which proceeds from a definition of the symmetry algebra, via a covariant formulation of its irreducible unitary representations (UIRs) in momentum space to, finally, covariant fields in spacetime obeying differential or finite-difference equations. One of the upshots of our work is a construction of non-commutative plane waves for gravitational anyons via a group Fourier transform. While this transform arose in relatively recent literature in quantum gravity [4, 5, 6, 7], we note that it is essentially an example and extension of Rieffel’s deformation quantisation of the canonical Poisson structure on the dual of a Lie algebra via convolution [8].

Anyonic behaviour occurs in both non-relativistic and relativistic physics, and for the same topological reason. The proper and orthochronous Lorentz group \(L_3^{+\uparrow }\) in 2 + 1 dimensions retracts to \(\mathrm {SO}(2)\), and is therefore infinitely connected. Its universal cover, which cannot be realised as a matrix group, governs the properties of relativistic anyons. As we shall review, the double cover of \(L_3^{+\uparrow }\) is the matrix group \(\mathrm {SU}(1,1)\) or, equivalently, \( \mathrm {SL}(2,\mathbb {R})\). In this paper, we write the universal cover as \(\widetilde{\mathrm {SU}}(1,1)\).

In the spirit of Wigner’s classification of particles [9], relativistic anyons are classified by the UIRs of the universal cover of the (proper, orthochronous) Poincaré group in 2 + 1 dimensions, which is the semi-direct product of \(\widetilde{\mathrm {SU}}(1,1)\) with the group of spacetime translations. It is natural to identify the latter with the dual of the Lie algebra of \(\mathrm {SU}(1,1)\). Then, the universal cover of the Poincaré group is \(\widetilde{\mathrm {SU}}(1,1)\ltimes \mathfrak {su}(1,1)^*\), with the first factor acting on the second by the co-adjoint action. While the UIRs of the Poincaré group in 2 + 1 dimensions are labelled by a real mass parameter and an integer spin parameter, the corresponding mass and spin parameters for the universal cover both take arbitrary real values [10]. In other words, going to the universal cover puts mass and spin on a more equal footing.

The canonical construction of the UIRs of the Poincaré group in 2 + 1 dimensions leads to states being realised as functions on momentum space, obeying constraints [11]. Writing these constraints in a Lorentz-covariant way, and Fourier transforming leads to standard wave equations of relativistic physics like the Klein-Gordon or Dirac equation. Relativistic wave equations for anyons have also been constructed, but for spins which are not half-integers they require infinite-component wave functions, and the derivation of the equations is not straightforward [12, 13, 14, 15].

Our treatment will naturally lead us to an equation derived via a different route by Plyushchay in [14, 15]. It makes use of the discrete series UIR of \(\widetilde{\mathrm {SU}}(1,1)\), but, as pointed out by Plyushchay, it is essentially a dimensional reduction of an equation already studied by Majorana [16, 17].

The focus of this paper is a deformation of the universal cover of the Poincaré group to a quantum group which arises in (2 + 1)-dimensional quantum gravity. In 2 + 1 dimensions, there are no propagating gravitational degrees of freedom, and the phase space of gravity interacting with a finite number of particles and in a universe where spatial slices are either compact or accompanied by suitable boundary conditions at spatial infinity is finite dimensional. In those cases where the resulting phase space could be quantised, the Hilbert space of the quantum theory can naturally be constructed out of unitary representations of the quantum double of \(L_3^{+\uparrow }\) or one of its covers [18, 19, 20]. In a sense that can be made precise, this double is a deformation of the Poincaré group in 2 + 1 dimensions [21, 22], with the linear momenta in \(\mathfrak {su}(1,1)^*\) (the generators of translations) being replaced by functions on \(L_3^{+\uparrow }\) or one of its covers.

In the quantum double of \(L_3^{+\uparrow }\), Lorentz transformations and translation are implemented via Hopf algebras which are in duality, namely the group algebra of \(L_3^{+\uparrow }\) and the dual algebra of functions on \(L_3^{+\uparrow }\). The quantum double is a ribbon Hopf algebra whose *R*-matrix can be given explicitly. The unitary irreducible representations describing massive particles are labelled by an integer spin and a mass parameter taking values on a circle.

*s*is quantised in units which depends on its mass

*m*according to

*G*is Newton’s gravitational constant, see [22, 23].

Covering the Lorentz transformations without changing the momentum algebra would destroy the duality between the two. It is therefore more natural to also consider a universal covering of the momentum algebra, i.e. to identify the momentum algebra with the function algebra on \(\widetilde{\mathrm {SU}}(1,1)\). The resulting quantum double of the universal cover of the Lorentz group, called Lorentz double in [22], is a ribbon Hopf algebra and has UIRs describing massive particles which are labelled by a spin parameter *s* and a mass parameter \(\mu \) for which (1.1) makes sense.

In this paper, we consider the Lorentz double and derive a new formulation of its UIRs in terms of infinite-component functions on momentum space \(\widetilde{\mathrm {SU}}(1,1)\) obeying Lorentz-covariant constraints. We extend and use the notion of group Fourier transforms [4, 5, 6, 7] to derive covariant wave equations on Minkowski space equipped with a \(\star \)-product.

Our method extends earlier work in [3] where analogous relativistic wave equations for massive particles were obtained from the UIRs of the quantum double of the Lorentz group. The transition to the universal cover poses two separate challenges. Our wave functions in momentum space now live on \(\widetilde{\mathrm {SU}}(1,1)\), and they take values in infinite-dimensional UIRs of \(\widetilde{\mathrm {SU}}(1,1)\) called the discrete series. The change in momentum space leads to a particularly natural version of the group Fourier transform, essentially because group-valued momenta can be parametrised bijectively via the exponential map and one additional integer label. In order to include the integer in our Fourier transform, we are forced to introduce a dual circle on the spacetime side. The emergence of a compact additional dimension is a remarkable and intriguing aspect of our construction.

Fourier transforming the algebraic spin and mass constraints from momentum space to Minkowski space produces equations on non-commutative Minkowski space which involve either differential operators or the exponential of a first-order differential operator. We end the paper with a short discussion of these non-commutative wave equations for gravitational anyons, leaving a detailed study for future work. Non-commutative waves for anyons have previously been discussed in the literature [24]. However, the discussion there is in the context of non-relativistic limits rather than the inclusion of gravity, and the mathematics is rather different.

The paper is organised as follows. In Sect. 2, we introduce our notation and review the definition and parametrisation of the universal cover \(\widetilde{\mathrm {SU}}(1,1)\) as well as the discrete series UIRs. We also revisit the UIRs of the Poincaré group and briefly summarise the covariantisation of the UIRs and their Fourier transform, following [3]. In Sect. 3, we generalise the covariantisation procedure to the universal cover of the Poincaré group, thus obtaining wave equations for infinite-component anyonic wave functions. Our version of these equations is essentially that considered by Plyushchay [14, 15] but our derivation of them appears to be new. Section 4 extends the analysis to the Lorentz double. We derive a Lorentz-covariant form of the UIRs, and point out that one of the defining constraints, called the spin constraint, can be expressed succinctly in terms of the ribbon element of the Lorentz double. The group Fourier transform on \(\widetilde{\mathrm {SU}}(1,1)\) requires a parametrisation of this group via the exponential map, and we discuss this in some detail. We use the Fourier transform to derive non-commutative wave equations, and then use the short final Sect. 5 to discuss our results and to point out avenues for further research.

## 2 Poincaré symmetry and massive particles in 2 + 1 dimensions

We review the symmetry group of (2 + 1)-dimensional Minkowski space: its Lie algebra, its various covering groups and their representation theory. Unfortunately, there is no single book or paper which covers these topics in conventions which are convenient for our purposes. We therefore adopt a mixture of the conventions in the papers [22] and [3] and the book [25], which are key references for us.

### 2.1 Minkowski space and the double cover of the Poincaré group

In quantum mechanics, classical symmetries described by a Lie group *G* are implemented by projective representations, which, in the case of the Lorentz group, may equivalently be described by unitary representations of the universal covering group. In the case of relativistic symmetries in \(3+1\) dimensions, the universal cover of the Lorentz group is its double cover and is isomorphic to \(\mathrm {SL(2,\mathbb {C})}\). In 2 + 1 dimensions, the double cover of \(L_3^{+\uparrow }\) is isomorphic to \(\mathrm {SL}(2,\mathbb {R})\) or, equivalently, \(\mathrm {SU}(1,1)\), but this is not the universal cover. Therefore, a choice has to be made as to which covering group one should implement in the quantum theory. The paper [3], to which we will refer frequently for details, works with the double cover in the realisation \(\mathrm {SL}(2,\mathbb {R})\).

*a*,

*b*which satisfy \(|a|^2 -|b|^2 =1\) via

*D*is the open unit disk. Note that an \(\mathrm {SU}(1,1)\) element parametrised via (2.8) can be written as

*h*and therefore determined by the trace (which, for determinant one, fixes the eigenvalues up to ordering) plus additional data. Elements \(u\in \mathrm {SU}(1,1)\) with absolute value of the trace less than 2 are called elliptic elements. They are conjugate to rotations of the form

*G*via \(\lambda =8\pi G\).

*p*in 2 + 1 dimensions now naturally live in \((\mathfrak {su}(1,1)^*)^*\simeq \mathfrak {su}(1,1)\). For consistency with (2.17), we set

*p*of momentum space may be expressed as

*p*and

*q*in \(\mathfrak {su}(1,1)\) requires the trace. We define the inner product as

*p*, i.e.

*s*and

*c*satisfy a generalised version of the Pythagorean formula:

### 2.2 The universal cover the Lorentz group and the discrete series representation

^{1}:

The unitary irreducible representations (UIRs) of \(\widetilde{\mathrm {SU}}(1,1)\) have been classified into a number of infinite families and are given in detail in [25] and also [10]. The particular class that is used to model anyonic particles in [13, 14, 15] is called the discrete series. We briefly review this here, using [25] as a main reference but adapting notation used in [10]. In particular, we label the UIRs in the discrete series by \(l \in \mathbb {R}^+\) and a sign.

*f*and

*g*are anti-holomorphic and therefore given as power series of the form \(f=\sum _{n=0}\alpha _n\bar{z}^n\) and \(g=\sum _{n=0}\beta _n\bar{z}^n\).

We can parametrise the set of all timelike lines in 2 + 1 dimensional Minkowski space in terms of a timelike vector \({\varvec{q}}\), normalised so that \({\varvec{q}}^2=1\) and giving the direction of the line, and a vector \({\varvec{k}}\) which lies on the line and which can be chosen to satisfy \({\varvec{k}}\cdot {\varvec{q}}=0\) without loss of generality. Geometrically, \({\varvec{q}}\) lies on the two-sheeted hyperboloid, and \({\varvec{k}}\) lies in the tangent space at \({\varvec{q}}\).

*z*in the unit disk via

*w*via

*z*.

### 2.3 Massive \(\tilde{P}_3\) representations

There are various ways of getting from the UIRs of the Poincaré group to the covariant wave equations of relativistic physics. In [2, 3], a procedure was developed which is also effective when the Poincaré group is deformed to the quantum double of the Lorentz group or one of its covers. We briefly review the method here in a convenient form for extension to the anyonic case. However, relative to [3] we change the sign convention for mass and spin to agree with the one used in [22].

*m*, and in the \(p_0<0\) half space for negative

*m*. Thus, we have two equivalent characterisations of the adjoint orbit corresponding to massive particles:

*U*(1) are one-dimensional and labelled by \(s\in \frac{1}{2}\mathbb {Z}\) in our conventions.

*s*can be given in two equivalent ways. Either one considers functions on \(\mathrm {SU}(1,1)\) which satisfy an equivariance condition or sections of associated vector bundles over the homogeneous space \(\mathrm {SU}(1,1)/N^T \simeq O^T_m\). We focus on the former method here but refer the reader to [22] for a discussion of their equivalence in the context of 3d gravity and to [26] for a general reference. Adopting the conventions of [22], we define the carrier space as

*a*is interpreted as an element of \(\mathfrak {su}(1,1)^*\), and \(\langle \cdot , \cdot \rangle \) is the pairing between elements of \(\mathfrak {su}(1,1)^*\) and \(\mathfrak {su}(1,1)\) introduced and discussed in Sect. 2.1. We have attached the superscript ‘eq’ to distinguish this equivariant formulation from the later covariant version.

### 2.4 Covariant field representations

*V*is a carrier space for a (usually finite dimensional) representation of the Lorentz group. In general, such fields do not form irreducible representations of the Poincaré group and, as a result, additional constraints need to be imposed to achieve this. For fields defined on momentum space, these constraints are algebraic, but after Fourier transform they yield the familiar wave equations for a field of definite spin.

*n*of the stabiliser is precisely cancelled by the action of \(\rho ^{|s|}(n)\) on the state \(\left| \right. |s|,-s\left. \right\rangle \). This construction works for massive particles since \(\rho ^{|s|}(s^0)\) has imaginary eigenvalues. However, this is not the case for the momentum representatives on massless and tachyonic orbits and hence the above procedure is limited to particles with timelike momentum.

In [3], it is also shown that the above covariant fields produce UIRs of \(\tilde{P}_3\) for the familiar cases of spin \(s=0,\frac{1}{2}, 1\) and that the mass constraint for spin zero and the spin constraints for \(s=\frac{1}{2}\) and \(s=1\) produce the momentum space versions of the Klein-Gordon equation, Dirac equation and of field equations which square to the Proca equation.^{2}

We refer the reader to [3] for details of the Fourier transform of the spin constraints to relativistic field equations in spacetime. We now turn to the anyonic case, where we will discuss both the covariant formulation of the UIRs and the Fourier transform.

## 3 Anyonic wave equations

Anyons are quantum particles with fractional spin which occur in systems confined to two spatial dimensions. In the relativistic case, the theoretical possibility of anyonic particles is a consequence of the infinite connectedness of the Lorentz group \(L_3^{+\uparrow }\).

### Definition 3.1

*Anyonic covariant field*) The anyonic covariant field \(\tilde{\phi }_{\pm }\) associated with an equivariant field \(\psi \in V^A_{ms}\) is the map

Thus, we should use \(\tilde{\phi }_+\) to describe positive spin particles and \(\tilde{\phi }_-\) for negative spin.

### Lemma 3.1

The anyonic covariant fields (3.6) are well defined.

### Proof

### Lemma 3.2

### Proof

Note that, unlike in the finite-dimensional case, we do not also need to impose the constraint that \(p^0\) and *m* have the same sign. This follows from our conventions (2.56) and from the fact that \(id_{l+}(s^0)\) has only positive eigenvalues and \(i d_{l-}(s^0)\) only negative eigenvalues. This property was one of the motivations for Majorana to construct his infinite-component fields in [16].

We now show that the representations \(V^{A}_{ms}\) and \(W^{A}_{ms}\) are isomorphic. This implies that the covariant fields subject to the mass and spin constraints form UIRs of the universal cover \(P^\infty _3\) of the Poincaré group.

### Theorem 3.1

(Irreducibility of the carrier space \(W^{A}_{ms}\)) The covariant representation \(\pi ^{\tiny {\text { co}}}_{ms}\) of \(P^\infty _3\) on \(W^{A}_{ms}\) defined in (3.9) is unitarily equivalent to the equivariant representation \(\pi ^{\tiny {\text { eq}}}_{ms}\) on \(V^{A}_{ms}\) defined in (3.4). In particular, it is therefore irreducible.

### Proof

*p*, satisfies the spin constraint and has support entirely on the orbit \(O^T_m\), so that the mass constraint is also satisfied. Thus, the maps \(L_\pm \) are well defined.

*p*, since

It is worth stressing that, for \(s \in \frac{1}{2} \mathbb {N}\), our anyonic equation (3.27) does not reduce to Eq. (2.65) for an equivariant field with finitely many components. These equations are not equivalent as they are characterised by different irreducible representations.

## 4 Gravitising anyons

### 4.1 The Lorentz double and its representations

We now extend and apply our method for deriving wave equations from Lorentz-covariant UIRs of the Poincaré group to a deformation of the Poincaré symmetry to the quantum double of the universal cover of the Lorentz group, or Lorentz double for short. As reviewed in our Introduction, this is motivated by results from the study of 3d gravity and a general interest in understanding possible quantum deformations of standard wave equations. Referring to [27] and [28] for reviews, we sum up evidence for the emergence of quantum doubles in the quantisation of 3d gravity.

*Deformation of Poincaré symmetry:* As explained in [21] and [22] for the, respectively, Euclidean and Lorentzian case, the quantum double of the rotation and Lorentz group is a deformation of the group algebra of, respectively, the Euclidean and Poincaré group.

*Gravitational scattering:* The *R*-matrix of the Lorentz double can be used to derive a universal scattering cross section for massive particles with spin by treating gravitational scattering in 2 + 1 dimensions as a non-abelian Aharonov–Bohm scattering process [22]. This universal scattering cross section agrees with previously computed special cases, like the quantum scattering of a light spin 1/2 particle on the conical spacetime generated by a heavy massive particle, in suitable limits—see [29, 30].

*Combinatorial quantisation:* The quantum double of the Lorentz group arises naturally in the combinatorial quantisation of the Chern–Simons formulation of 3d gravity with vanishing cosmological constant. The classical limit of the quantum *R*-matrix is a classical *r*-matrix which is compatible with the non-degenerate bilinear symmetric and invariant pairing used in the Chern–Simons action [21, 22], and the Hilbert space of the quantised theory can be constructed from unitary representations of the Lorentz double [18, 19].

*Independent derivations:* Quantum doubles also emerges in approaches to 3d quantum gravity which do not rely on the combinatorial quantisation programme. In [20], the quantum double is shown to play the role of quantum symmetry in 3d loop quantum gravity. In [4], it appears in a path integral approach to 3d quantum gravity.

In analogy with our treatment of the Poincaré group in 2 + 1 dimensions, we consider the double cover \(\mathrm {SU}(1,1)\) and the universal cover \(\widetilde{\mathrm {SU}}(1,1)\) of the identity component of the Lorentz group. Our goal is to obtain a deformation of the wave equation by covariantising and then Fourier transforming, in a suitable sense, the UIRs of the quantum double of \(\widetilde{\mathrm {SU}}(1,1)\). This extends the results obtained in [3] for the double cover \({\mathrm {SU}}(1,1)\). As we shall see, the universal cover is technically more involved but also conceptually more interesting.

The quantum double of a Lie group can be defined in several ways. We follow [31, 32] with the conventions used in [2, 3]. In this approach, we view the quantum double \(\mathcal {D}(G)\) as the Hopf algebra which, as a vector space, is the space of continuous complex-valued functions \(C(G\times G)\). In order to exhibit the full Hopf algebra structure, we need to adjoin singular \(\delta \)-distributions.

*S*, \(*\)-structure and ribbon element

*c*is then as follows:

*g*for the left Haar measure on the group.

The representation theory of the double is given in [32]. In the case of \(\mathcal {D}(\mathrm {\mathrm {SU(1,1)})}\), the UIRs are classified by conjugacy classes in \(\mathrm {SU(1,1)}\) and UIRs of associated stabiliser subgroups. This should be viewed as a deformation of the discussion of \(\tilde{P}_3\), where we had adjoint orbits in the linear momentum space. In the gravitational case, the group itself is interpreted as momentum space and orbits are conjugacy classes. Here, we encounter the idea of curved momentum space discussed in the outline.

*f*generalising the function \(\exp (i\langle a, \cdot \rangle )\).

In [3], local covariant fields are introduced for \(\mathcal {D}(\mathrm {SU(1,1)})\) and deformed momentum space (spin) constraints are derived. After Fourier transform, these constraints are interpreted as deformed relativistic wave equations. In [3], this is explicitly done for particles of spin \(s=0, \frac{1}{2}\) and 1. We will not review these results here, but derive the analogues for the anyonic case, where we need to consider the quantum double \(\mathcal {D}(\widetilde{\mathrm {SU}}(1,1))\). The UIRs are discussed in [22]. We only recall the UIRs describing massive particles at this point, though we will need all conjugacy classes when we consider the Fourier transform in the next section.

*s*.

*m*and spin

*s*is \(V^A_{m s}\) as defined in (3.3). Elements

*F*of the double \(\mathcal {D}(\widetilde{\mathrm {SU}}(1,1))\) act on \(\psi \in V^A_{ms}\) according to

### Definition 4.1

*Deformed anyonic covariant field*) The deformed anyonic covariant field \(\tilde{\phi }_{\pm }\) associated with an equivariant field \(\psi \in V^A_{ms}\) is the map

*v*is chosen so that \(u=v(\lambda m,0)v^{-1}\) and \(s=l\) for \(\tilde{\phi }_+\) and \(s=-l\) for \(\tilde{\phi }_-\).

### Lemma 4.1

The covariant fields \(\tilde{\phi }_\pm \) are well defined.

### Proof

*v*, i.e. that

### Lemma 4.2

### Proof

### Theorem 4.1

(Irreducibility of the carrier space \(W^{GA}_{ms}\)) The covariant representation \(\varPi ^{\tiny {\text { co}}}_{ms}\) of \(\mathcal {D}(\widetilde{\mathrm {SU}}(1,1))\) on \(W^{GA}_{ms}\) defined in (4.19) is unitarily equivalent to the equivariant representation \(\varPi ^{\tiny {\text { eq}}}_{ms}\) on \(V^{A}_{ms}\) defined in (4.15). In particular, it is therefore irreducible.

### Proof

*u*. Thus, \(\psi \in V^A_{ms}\).

### 4.2 Group Fourier transforms

We now turn to the promised Fourier transform of the covariant UIRs of the Lorentz double. Our discussion here will be less complete and rigorous than our treatment so far. In particular, we do not survey different approaches to Fourier transforms and differential calculus in the context of Hopf algebras, but note that some relevant references are collected in [3]. Instead, we only show how ideas first proposed by Rieffel in [8] and recently pursued in the quantum gravity community under the heading of group Fourier transforms can be used to translate the algebraic mass and spin constraints in the definition (4.26) into differential and difference equations.

*G*with Lie algebra \(\mathfrak {g}\). Concentrating for simplicity on complex-valued (rather than Hilbert-space-valued) functions, the standard Fourier transform (3.23) is a map

This is precisely the situation considered by Rieffel in [8], where he observed that, if the exponential map can be used to identify the Lie group with the Lie algebra, one can transfer the convolution product of functions on *G* to functions on \(\mathfrak {g}\) and then, by Fourier transform, to functions on \(\mathfrak {g}^*\). This induces a non-commutative \(\star \)-product on functions on \(\mathfrak {g}^*\) which is a strict deformation quantisation of the canonical Poisson structure on \(\mathfrak {g}^*\). This works globally for nilpotent groups, but, as explained in [8], still makes sense, in an appropriate way, more generally. For details, we refer the reader to Rieffel’s excellent exposition in the paper [8] which also contains comments on the relation to other quantisation methods, such as Kirillov’s co-adjoint orbit method.

Ideas very similar to Rieffel’s have, more recently and apparently independently, been considered by a number of authors in the context of quantum gravity [4, 5, 6, 7]. This work has resulted in a general framework called group Fourier transforms. In developing our Fourier transform for gravitised anyons, we essentially need to adapt and extend the ideas of Rieffel and the concept of a group Fourier transforms to \(G=\widetilde{\mathrm {SU}}(1,1)\). We have found it convenient to use the terminology and notation used in the discussion of group Fourier transforms, particularly in [6, 7], which we review briefly.

*d*is the dimension of

*G*and \(\delta _e\) is the Dirac \(\delta \)-distribution at the group identity element

*e*with respect to the left Haar measure d

*g*.

*G*:

*G*and a function \(\eta :G \rightarrow \mathbb {C}\) so that, up to a set of measure zero, the plane waves take the form

### 4.3 Non-commutative waves for \(\widetilde{\mathrm {SU}}(1,1)\) and anyonic wave equations

Our proposal for a Fourier transform on \(\widetilde{\mathrm {SU}}(1,1)\) is based on the parametrisation of group elements summarised in the following proposition.

### Proposition 4.1

*n*introduced in the Proposition is the same integer which appears in the decomposition (4.9) of the rotation angle \(\mu \). This follows since, for \((\omega ,\gamma )\in E(\mu ) \),

### Proof

*u*or the element \(-u\) is in the image of the exponential map in \(\mathrm {SU}(1,1)\), i.e. there is a \(p \in \mathfrak {su}(1,1)\) and a choice of sign so that

*n*for the positive sign in (4.44) and odd

*n*for the negative sign.

We first consider the case where either *p* or \(p'\) vanishes. If one, say *p*, did then (4.47) would imply \( \exp (p')= \pm \text {id}\), but under the restriction on \(p'\), this is only possible if the upper sign holds and \(p'=0\), so that \(n=n'\) follows.

Both parabolic and hyperbolic elements have the property that, if such an element is in the image of the exponential map, its negative is not. For such elements, we must therefore have the upper sign in (4.47). Moreover, one checks from the expressions (2.25) that parabolic and hyperbolic elements which are in the image of the exponential map have a unique logarithm, so that we conclude \(p=p'\) and hence \(n=n'\).

*p*and \(p'\) must be multiples of each other. By the assumption that both lie in the forward light cone, we can deduce (recalling the sign conventions (2.56)) that

The decomposition (4.41) can be visualised and illustrated by thinking of \(\widetilde{\mathrm {SU}}(1,1)\) as an infinite cylinder, with \(\omega \) plotted along the vertical axis and \(\gamma \) parametrising the horizontal slices. In Fig. 1, we show a vertical cross section of this cylinder and display the conjugacy classes and the exponential curves.

*p*is a fixed element of \(\mathfrak {su}(1,1)\), and \(n\in \mathbb {Z}\). In other words, these are images of the exponential map with chosen initial tangent vector

*p*translated by \(\varOmega ^n\). We stress that the cross section we are showing suppresses the three-dimensional nature of these curves. To illustrate this, we show three-dimensional plots of some exponential curves starting at the identity in Fig. 2. Note that the spacelike and lightlike curves approach the boundary of the cylinder, but that the timelike curve winds round the axis of the cylinder, carrying out a complete rotation when \(\omega \) increases by \(2\pi \)

*c*and

*s*for readability. Thus

*n*. It is clear that a suitable non-commutative wave cannot depend only on a dual variable \(x\in \mathfrak {su}(1,1)^*\). It also requires an argument which is dual to the integer

*n*. The most natural candidate is an angular coordinate, parametrising a circle \(S^1\). The necessity of a fourth and circular dimension to describe the spacetime dual to \(\widetilde{\mathrm {SU}}(1,1)\) is a surprise. We will introduce it and explore its consequences at this point, postponing a discussion to our final section.

### Definition 4.2

*u*via the decomposition (4.41), and \(\varphi \in [0,2\pi )\) is an angular coordinate on the circle \(S^1\).

*p*and

*n*(see also B in [7]),

*n*in the spirit of particle physics as a label for different kinds of particles in the theory. Timelike momenta

*p*with \(n=-1\) may then be viewed as describing antiparticles. Lightlike and spacelike momenta for \(n=0\) have the usual interpretation as momenta of massless or (hypothetical) tachyonic particles. The other values of

*n*describe additional types of massive, massless and tachyonic particles. Their existence is required by the fusion rules obeyed by the plane waves, which follow from the star product

Thus, we think of the plane waves for \(\widetilde{\mathrm {SU}}(1,1)\) as describing kinematic states of particles in a theory with an invariant mass scale \(m_p\) and with infinitely many different types of particles which combine according to the fusion rules encoded in the star product.

Since the field \(\phi \) takes values in the Hilbert space \(\mathcal {H}_{l\pm }\), the extension of the \(\star \)-product (4.62) to products of such fields requires a careful tensor product decomposition. We will not pursue this here, but note that the decomposition of tensor products in the equivariant formulation of the UIRs for quantum doubles of compact Lie groups was studied in detail in [33]. A full study of the \(\star \)-product for covariant fields \(\phi \) will require an extension to non-compact groups and an adaptation of the results of that paper to our covariant formulation.

*n*in the decomposition (4.41).

Similar exponential operators have been considered in a more general context in [34], where it was stressed that they are essentially finite-difference operators. The appearance of difference-differential equations was first mentioned in relation to (2 + 1) gravity in [35].

## 5 Summary and outlook

This paper was motivated by the observation that, in the context of 2 + 1 dimensional quantum gravity, the spin quantisation (1.1) forces one to consider the universal cover of the Lorentz group and that, in order to preserve the duality between momentum space and Lorentz transformations in the quantum double, it is natural to take the universal cover in momentum space, too.

We showed how the representation theory of the quantum double of the universal cover \(\widetilde{\mathrm {SU}}(1,1)\) can be cast in a Lorentz-covariant form, and can be Fourier transformed. In this process, the universal covering of the Lorentz group necessitates the use of infinite-component fields, but the universal covering of momentum space has more interesting and far-reaching consequences.

The first of these, exhibited both in the decomposition (4.9) and in the parametrisation (4.41), is the extension of the range of the allowed mass. The fractional part \(\mu _0\) of \(\mu =8\pi Gm\) is the conventional mass of a particle, which manifests itself in classical (2 + 1)-dimensional gravity as a conical deficit angle in the spacetime surrounding the particle. The integer label *n* in the decomposition \(\mu = \mu _0 +2\pi n\) appears to be a purely quantum observable with no classical analogue. It manifests itself, for example, in the Aharonov–Bohm scattering cross section of two massive particles, as discussed in [22]. It is a rather striking illustration of the concept of ‘quantum modular observables’ as introduced in [36], extensively discussed in the textbook [37] and recently applied to the notion of spacetime in [38].

We have chosen to interpret *n* as a label of different types of particles or matter in (2 + 1)-dimensional quantum gravity. These particles can be converted into each other during interactions, according to fusion rules determined by the group product in \(\widetilde{\mathrm {SU}}(1,1)\) and the decomposition of factors and products according (4.41).

The second and surprising consequence of the universal covering of momentum space is the appearance of an additional and compact dimension on the dual side, in spacetime. This is needed to define the group Fourier transform, and allows for a simple expression of the constraint determining the particle type *n* as a differential condition.

Our results raise a number of questions and suggest avenues for future research. As discussed at the end of the previous section, the exponentiated differential operators which generically appear as group Fourier transforms of the spin constraint should be studied using rigorous analysis. One expects these to be natural operators, possibly best defined as difference operators, not least because they are, by Lemma 4.2, essentially Fourier transforms of the ribbon element of the quantum double.

It seems clear that our Hilbert-space-valued fields \(\phi \) on Minkowski space equipped with a \(\star \)-product fit rather naturally into the framework of braided quantum field theories, defined in [39] and studied, in a Euclidean setting, in [4, 40]. Our paper is only concerned with a single particle, and we only looked at simple examples of fusion rules for two spinless particles. However, braided quantum field theory naturally provides the language for discussing the gravitational interactions of several gravitational anyons in a spacetime setting. This provides an alternative viewpoint to existing momentum space discussions, with the non-commutative \(\star \)-product and the universal *R*-matrix of the quantum double encoding the quantum-gravitational interactions.

It is worth stressing that the \(\star \)-product on Minkowski space considered here preserves Lorentz covariance and that any braided quantum field theory constructed from it would similarly be Lorentz covariant. This follows essentially from the Lorentz invariance of the mass scale (4.61) and the associated Planck length scale [28]. It reflects the important fact that the Lorentz double deforms Poincaré symmetry by introducing a mass scale while preserving Lorentz symmetry, which is a challenge for any theory of quantum gravity.

Finally, it would be interesting to repeat the analysis of this paper with the inclusion of a cosmological constant. This leads to a *q*-deformation of the quantum double of \(\widetilde{\mathrm {SU}}(1,1)\) [28], and there should similarly be a *q*-deformation of the spacetime picture of the representations. Some remarks on how this might work are made in [2], but none of the details have been worked out. In the Lorentzian context, a positive cosmological constant will lead to real deformation parameter *q* while a negative cosmological constant will lead to *q* lying on the unit circle [28]. It would clearly be interesting to understand how this change in *q* captures the radically different physics of the two regimes.

## Footnotes

- 1.
The group product is essentially the one given in [25] except that we have parametrised the \(S^1\) in \(\mathrm {SU}(1,1)\) with \([0,4\pi )\) rather than \([0,2\pi )\). This ensures that a final projection to the Lorentz group results in a complete spatial rotation having an angle of \(2\pi \).

- 2.

## Notes

### Acknowledgements

SI acknowledges support through an EPSRC doctoral training grant. Several of the results in this paper were reported by BJS at the Workshop on Quantum Groups in Quantum Gravity, University of Waterloo 2016. BJS thanks the organisers for the invitation and acknowledges discussions with participants at the workshop. We thank Peter Horvathy for drawing our attention to Plyushchay’s work on anyonic wave equations and to their joint work on non-commutative waves.

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