On the representation theory of the Bondi-Metzner-Sachs group and its variants in three space-time dimensions

TL;DR

This paper develops an extended representation theory for the BMS group and its variants in three-dimensional space-time, crucial for understanding asymptotic symmetries in General Relativity.

Contribution

It introduces an extended Wigner-Mackey theory to classify irreducible unitary representations of BMS-like groups in 3D, including non-locally compact cases.

Findings

Irreducible representations are induced from compact, cyclic little groups of even order.

The extended theory applies to groups in any space-time dimension d ≥ 3.

The classification aids in understanding asymptotic symmetries in gravitational theories.

Abstract

The original Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space-times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B-invariant-systems in terms of strongly continuous irreducible unitary repesentations (IRS) of B. Here we introduce the analogue B(2,1) of the BMS group B in 3 space-time dimensions. B(2,1) itself admits thirty-four analogues both real in all signatures and in complex space-times. In order to find the IRS of both B(2,1) and its analogues we need to extend Wigner-Mackey's theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2,1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space-time dimensions, d is greater or equal to 3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2,1): The main results of the representation theory are: The IRS are induced from little groups which are compact. The finite little groups are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2,1) is not locally compact in the employed Hilbert topology.

Equations

$$\mathcal{B}=\mathcal{S}(A,K){⋊}_{T}G,$$

$$\mathcal{C}=\mathcal{S}{⋊}_{T}G,$$

$$x.y=x^0{y}^0-x^1{y}^1-x^2{y}^2.$$

$$N^{+}=\left\{x\in R^{2,1}|x.x=0,x^0>0\right\}.$$

$$f(tx)=tf(x)$$

$$(T(\Lambda )f)(x)=f(x\Lambda ).$$

$$B^{2,1}(N^{+})=F_1(N^{+}){⋊}_{T}SO(2,1).$$

$$(f_1,{\Lambda }_1)(f_2,{\Lambda }_2)=(f_1+T({\Lambda }_1)f_2,{\Lambda }_1{\Lambda }_2).$$

$$R_{+}^{\ast }\times N^{+}\to N^{+};(t,x)\mapsto tx,$$

$$F_1(N^{+})\leftrightarrow F(P(N^{+}))$$

$$B^{2,1}(N^{+})=F(P(N^{+})){⋊}_{T}SO(2,1).$$

$$(x^1{)}^2+(x^2{)}^2=1.$$

$${\mathrm{M}}_s(2,R)\times G\to {\mathrm{M}}_s(2,R)\mathrm{with}(m,g)\mapsto g^{\top }mg,$$

$$\mu=\left[\begin{array}[]{cc}x^{\rm o}-x^{1}&x^{2}\\ x^{2}&x^{o}+x^{1}\end{array}\right]$$

$$b(x)=\left[\begin{array}[]{cc}x^{\rm o}-x^{1}&x^{2}\\ x^{2}&x^{o}+x^{1}\end{array}\right],$$

$$\det (g^{\top }mg)=\det (g^{\top })\det (m)\det (g)=\det (m).$$

$$\det (b(x))=x\cdot x.$$

$$\gamma :G\to SO(2,1),$$

$$G=SO(2,1{)}_c.$$

$$\widetilde{N^{+}}=\left\{\mu \in {\mathrm{M}}_s(2,R),\mid x^0>0,\det \mu =0\right\}.$$

$$\mu \in \widetilde{N^{+}}\iff \mu ={\mathbf{\sigma }}^{\top }\mathbf{\sigma }$$

$$R_{+}^{\ast }\times \widetilde{N^{+}}\to \widetilde{N^{+}};(t,\mu )\mapsto t\mu ,$$

$$b:N^{+}\rightarrow\widetilde{N^{+}}\$$

$$P(\widetilde{N^{+}})=P_1(R)=S^1/Z_2\simeq S^1,$$

$$B^{2,1}(N^{+}{)}_c=F(P(\widetilde{N^{+}}){⋊}_{T}G,$$

$$B^{2,1}(N^{+}{)}_c=F(P_1(R)){⋊}_{T}G.$$

$$S^1=\{\mathbf{m}\in I||\mathbf{m}|=1\}.$$

$$\beta (\mathbf{m})=\beta (-\mathbf{m}).$$

$$B_e^{2,1}(N^{+}{)}_c=F_e(S^1){⋊}_{T}G.$$

$$B^{2,1}(N^{+}{)}_c=F(P_1(R)){⋊}_{T}G,$$

Full text

1 Introduction

The BMS group $B$ is the common asymptotic group of all curved real Lorentzian radiating space$-$times which are asymptotically flat in future null directions Bondi ; Sachs1 , and is the best candidate for the universal symmetry group of G.R..

The BMS group $B$ of general relativity was first discovered Bondi ; Sachs1 not as a transformation group of (exact) global diffeomorphisms of a fixed manifold, but as a pseudo$-$group of local diffeomorphisms (“asymptotic isometries”) of the asymptotic region of Lorentzian radiating space$-$times which are asymptotically flat in lightlike future directions.

However, Penrose Pen1 ; Pen2 ; Pen3 showed that, by “going to infinity”, $B$ could be interpreted as an (exact) global transformation group $B\times\Im^{+}\longrightarrow\Im^{+}$ of the “future null boundary” $\Im^{+}$, or, of the so called “future null infinity”, of the space$-$times concerned. Furthermore, he gave Pen1 ; Pen2 a geometric structure to $\Im^{+}$, the “strong conformal geometry” for which the transformation group is precisely the group of automorphisms Pen3 .

In 1939 Wigner laid the foundations of special relativistic quantum mechanics Wigner and relativistic quantum field theory by constructing the Hilbert space IRS of the (universal cover) of the Poincare group $P$.

The universal property of $B$ for G.R. makes it reasonable to attempt to lay a similarly firm foundation for quantum gravity by following through the analogue of Wigner’s programme with $B$ replacing $P$. Some years ago McCarthy constructed explicitly mac3 ; mac8 ; mac4 ; mac5 ; mac6 ; mac7 ; Crampin2 ; Mac the IRS of $B$ for exactly this purpose. This work was based on G.W.Mackey’s pioneering work on group representations Mackey0 ; Mackey00 ; Mackey ; Mackey1 ; Simms ; Isham ; in particular it was based on an extension mac6 of G.W.Mackey’s work to the relevant infinite dimensional case.

It is difficult to overemphasize the importance of Piard’s results Piard1 ; Piard2 who soon afterwards proved that *all *the IRS of $B$, when this is equipped with the Hilbert topology, are derivable by the inducing construction. This proves the exhaustivity of McCarthy’s list of representations and renders his results even more important.

However, in quantum gravity, complexified or euclidean versions of G.R. are frequently considered and the question arises: Are there similar symmetry groups for these versions of the theory? McCarthy constructed mac1 , in abstract form, all possible analogues of $B$, both real and in any signature, or complex, with all possible notions of asymptotic flatness “near infinity”. There are, in fact, forty$-$one such groups. One of them is $B(2,2)$, the asymptotic symmetry group of all curved real ultrayperbolic space$-$times which are asymptotically flat in null directions.

These abstract constructions were given in a quantum setting; the paper mac1 was concerned with finding the IRS of these groups in Hilbert spaces (especially for the complexification $\mathcal{C}\mathcal{B}$ of $B$ itself). It has been argued mac1 ; Mel1 that these Hilbert space representations are related to elementary particles and quantum gravity (via gravitational instantons). The IRS of $B(2,2)$ were constructed in macMel ; Mel5 . HB, a subgroup of $B(2,2)$ which arises naturally in the construction of the analogues of $B$ by McCarthy mac1 , and which, surprisingly, remained unnoticed by him, was introduced in Melas1 . The representation theory of HB was commenced in Mel6 ; Mel7 .

Here, we initiate this programme for 3$-$dim G.R.. We define the analogue $B(2,1)$ of the Bondi$-$Metzner$-$Sachs (BMS) group in 3 space$-$time dimensions. We construct the IRS of of $B(2,1)$ in the Hilbert topology. McCarthy and Crampin argued convincingly in Crampin2 that in the case of the original BMS group $B$, in 4 space$-$time dimensions, IRS in the Hilbert topology describe *bounded *sources Crampin2 , i.e., $B-$elemenatry particles. Moreover they pointed out that passing from the coarser Hilbert topology to the finer nuclear topology the effect is to obtain more IRS which now also accommodate scattering states of $B-$elemenatry particles.

Their arguments apply also in the case of $B(2,1)$, in 3 space$-$time dimensions, so we expect IRS of $B(2,1)$ in the Hilbert topology to describe bounded sources, i.e., $B(2,1)-$elemenatry particles, whereas IRS in the nuclear topology, to describe also scattering states of $B(2,1)-$elemenatry particles. Besides the Hilbert and the nuclear topologies there is a wide range of “reasonable” topologies by which we can endow $B(2,1)$ with. These topologies are elaborated in Section 6. $B(2,1)$ admits generalizations both real in all signatures and in complex space$-$times with all possible notions of asymptotic flatness “near infinity”. There are, in fact, thirty$-$five such groups including $B(2,1)$ Mel8 .

These thirty$-$five groups have the general structure Mel8

[TABLE]

where $A$ is a locally compact space, $K$ is the field of real or complex numbers, and $G$ is the group of global, conformal, orientation$-$preserving transformations of the space $A$ onto itself. $\mathcal{S}(A,K)$ is the space of supertranslations, i.e., of generalized position$-$dependent translations. The supertranslations $\mathcal{S}(A,K)$ are $K-$valued functions defined on $A$, and they constitute a normal abelian subgroup of $\mathcal{B}.$ Moreover, $\mathcal{S}(A,K)$ has a vector space structure, over the field $K$, under point$-$wise addition and scalar multiplication.

In the case of $B(2,1)$ $A$ is the circle $S^{1}$, $K$ is the field of real numbers, $G$ is the group $SL(2,R)$ of all real $2\times 2$ matrices with determinant one, and $\mathcal{S}(A,K)$ is the space of real$-$valued functions defined on $P_{1}(R)\simeq S^{1}$. $P_{1}(R)$ is the one$-$dimensional real projective space (the circle quotient the antipodal map). Alternatively, as it is explained in Section 3, in a different realization of $B(2,1)$, $S(A,K)$ is the space of “even” real$-$valued defined on $S^{1}$.

Ashtekhar $\&$ Hansen, in a unified treatment of asymptotic flatness in both null and spacelike directions for real lorentzian 4$-$dim space$-$times ass , derived a BMS$-$like group, the *Spi *group, which acts on the spacelike analogue of “null infinity” $\Im^{+}$, the “spacelike infinity”. “Spacelike infinity” is a 4$-$dim manifold, as opposed to “null infinity” $\Im^{+}$ which is a 3$-$dim manifold. One of the thirty$-$five generalizations of $B(2,1)$, is the group $B_{S}(2,1)=\mathcal{S}(S^{1}\times I,R)\rtimes_{T}SL(2,R)$, which is the analogue of the *Spi *group in 3$-$dim space$-$times. $I$ is an open interval of $R,$ and $\mathcal{S}(S^{1}\times I,R)$ is the space of real$-$valued functions defined on the product $S^{1}\times I$ which is homeomorphic to a cylinder in $R^{3}$. In the case of 3$-$dim space$-$times “spacelike infinity” is a 3$-$dim manifold, whereas “null infinity” is a 2$-$dim manifold.

By following Penrose Pen1 ; Pen2 ; Pen3 we can interpret $B(2,1)$ as a transformation group $B(2,1)\times\Im^{+}$ of the “future null infinity” of the Lorentzian 3$-$dim space$-$times involved. Furthermore, we can give a geometric structure to $\Im^{+}$, the “strong conformal geometry” for which the transformation group is precisely the group of automorphisms. The assignment of the “strong conformal geometry” to the 2$-$dim $\Im^{+}$, in the case of 3$-$dim space$-$times, follows Penrose’s original beautiful construction Pen1 ; Pen2 ; Pen3 of the “strong conformal geometry” in the case of 4$-$dim space$-$times.

However, the definition of the “strong conformal geometry”, in the case of 3$-$dim space$-$times, does not come as an immediate corollary of Penrose’s original construction of the “strong conformal geometry”, in the case of 4$-$dim space$-$times, simply by following through Penrose’s construction and by reducing on the way the number of spatial dimensions by one. There are some subtleties involved when we pass from 4 space$-$time dimensions to 3 space$-$time dimensions. In particular some care is needed, in 3 space$-$time dimensions, in the appropriate definition of the notion of the “null angle”, introduced by Penrose Pen2 ; Pen3 , in order to establish the “strong conformal geometry”. Here we do not elaborate further on this issue but we rather restrict our attention to the representation theory of $B(2,1)$ and its subtle underpinnings.

As it is explained in Section 5 in order to find the IRS of any BMS group $B$, as well as the IRS of any of its analogues, in any number $d\geq 3$ of space$-$time dimensions, we cannot apply Wigner$-$Mackey’s theory as it is. We need to extend it. The required extension, is reduced to the solution of three problems, described in detail in Subsection 5.1. These three problems are named the First, the Second, and the Third problem, and they are correspondingly, the “little groups” problem, the irreducibility problem, and the exhaustivity problem.

In Subsection 5.4 we find a solution to the First problem. The solution applies to any topology $B$ and its analogues come equipped with, in any number $d\geq 3$ of space$-$time dimensions. The solution to the Second problem, given in Subsection 5.5, applies to the Hilbert and to the nuclear topologies. Finally, the solution to the Third problem, given in Subsection 5.5, applies to the Hilbert topology only. Therefore solution to all three problems posed by the required extension of Wigner$-$Mackey’s theory are given in the Hilbert topology, and consequently, a complete list of IRS of $B(2,1)$ can be constructed in this topology.

The extension of Wigner$-$Mackey’s theory we introduce in Section 5 in order to find the IRS of $B(2,1)$ is not applicable only to $B(2,1)$ but it is also relevant and applicable to the thirty$-$four generalizations Mel8 of $B(2,1)$, both real in all signatures and in complex space$-$times. In fact the relevance and applicability of the extended theory permeates also the representation theory of the original BMS group $B$, as well as the representation theory of its forty$-$one generalizations mac1 , both real in all signatures and in complex space$-$times.

In fact by using the solutions to the three problems in the Hilbert topology, and thus the extended Wigner$-$Mackey’s theory given in Section 5, we can construct, in the Hilbert topology, all the IRS of any BMS group $B$, as well as the IRS of any of its (supersymmetric shaw ) analogues, in any number $d\geq 3$ of space$-$time dimensions.

In the other topologies for $B$, described in Section 6, the solution to the “little groups” problem given in Subsection 5.4, allows us to pursue Wigner-Mackey’s inducing construction and obtain a list of representations for $B.$ We do not know if the representations so obtained are irreducible, and moreover, we do not know if the inducing construction gives all the representations of $B.$

In order to find if the representations are irreducible we need to solve in each case (apart from the Hilbert topology case and the nuclear topology case) the irreducibility problem which is stated in Subsection 5.1 and solved in Subsection 5.5 for the Hilbert and nuclear topologies. Furthermore, in order to find if we obtain all the representations of $B$ by using Wigner$-$Mackey’s inducing construction we need to solve in each case (apart from the Hilbert topology case) the exhaustivity problem which is stated in Subsection 5.1 and solved in Subsection 5.5 for the Hilbert topology.

To the best of our knowledge, the group theoretical approach to quantum gravity advocated here, is the only approach, in which a complete list of IRS of the appropriate symmetry group, i.e., of $B$ and of any of its analogues, in any number $d\geq 3$ of space$-$time dimensions, is constructed in a given (i.e. in the Hilbert) topology.

The other approaches we are aware of Isham ; IshKak0 ; IshKak ; Bar3 ; Bar4 are afflicted by the inapplicability of the Wigner$-$Mackey’s inducing construction which manifests itself in the appearance of the “little groups” problem, the irreducibility problem, and the exhaustivity problem, which persistently occur, in this or the other form, in any group theoretical approach to quantum gravity.

The main results of the representation theory are: The IRS of $B(2,1)$ are induced from IRS of *compact *“little groups”. The “little groups” are of two types:

1. Infinite connected Lie groups. 2. Non$-$connected finite discrete groups. The non$-$connected finite discrete “little groups” are cyclic groups of even order, which are symmetry groups of regular polygons in ordinary euclidean 2$-$space. The inducing construction is exhaustive notwithstanding the fact that $B(2,1)$ is *not *locally compact in the employed Hilbert topology.

A terminological remark is here in order. Whenever, hereafter, mention is made of a representation of a group, it will always be taken to mean *strongly continuous *representation. If $\mathcal{G}$ is a topological group we say that a representation $\mathcal{U}$ of $\mathcal{G}$ on a Hilbert space $\mathcal{H}$ is strongly continuous if $g_{n}\rightarrow g$, $n=1,2,3,...$, in $\mathcal{G}$, implies $\|\mathcal{U}(g_{n})\psi\rightarrow\mathcal{U}(g)\psi\|$ for all $\psi\in\mathcal{H}.$

This paper is organized as follows. In Section 2 relation to other approaches is discussed. In Section 3 $B(2,1)$ is introduced. In Section 4 the bare essentials of Wigner$-$Mackey’s theory are given. In Section 5 an extension of Wigner$-$Mackey’s theory, necessary to construct the IRS of $B(2,1),$ and in fact the IRS of any BMS group $B$, as well as the IRS of any of its (supersymmetric) analogues, in any number $d\geq 3$ of space$-$time dimensions, is developed. In Section 6 a heuristic proof of the compactness of the “little groups” is given, and it is pointed out that besides the Hilbert topology, considered in this paper, there are other “reasonable” choices for the topology of the supertranslation space. In Section 7 the necessary data in order to construct the operators of the IRS of $B(2,1)$ are given.

2 Relation to other approaches

The BMS group $B$ can be derived, in general Hol ; Wald ; Tan1 ; Tan2 ; Tan3 , in any number $d\geq 3$ of space$-$time dimensions, by using the four methods 111There is also Bramson’s derivation Bram who defines the BMS group $B$ to be the group of mappings of “cone space”, introduced by him, onto itself which acts linearly on the space of position vectors and which preserves the norm of each position vector. Bramson’s definition of $B$ is illuminating and interesting but is not relevant to this study and so we do not include it in the methods of derivation of $B.$ specified below. The order of exposition reflects the chronological order in which they appeared Bondi ; Pen1 ; mac8 ; Ger .

As a pseudo$-$group of local diffeomorphisms which preserve the asymptotic form of the Bondi$-$Van Der Berg$-$Metzner class of metrics Bondi of Lorentzian radiating space$-$times which are asymptotically flat in lightlike future directions. 2. 2.

As an exact global transformation group $B\times\Im^{+}\longrightarrow\Im^{+}$ of the “future null boundary” $\Im^{+}$ of the Bondi$-$Van Der Berg$-$Metzner space$-$times. This group is defined to be the group of automorphisms of the “strong conformal geometry”, a geometric structure given by Penrose Pen1 , to the “future null boundary” $\Im^{+}$ of the space$-$times concerned. 3. 3.

As the group which results from the Poincare group when the translations, which can be put into bijective correspondence with a finite parameter family of homogeneous functions of degree one, are replaced by an appropriate infinite parameter family of homogeneous functions of degree one mac8 . 4. 4.

As a group of diffeomorphisms of the “future null boundary” $\Im^{+}$ of the Bondi$-$Van Der Berg$-$Metzner space$-$times which leaves invariant the “universal structure” Ger . The “universal structure” is equivalent to the “strong conformal geometry” introduced by Penrose. The key point in this method is that the invariance of the “universal structure” under the action of $B$ on $\Im^{+}$ is expressed not in terms of the elements of the group $B$ itself, but instead, *in terms of the linearization of the group $B$ close to its identity element, i.e., in terms of the generators of the Lie algebra of the group $B$. *

The three first methods of derivation give the group $B$, whereas the fourth method gives the Lie algebra of $B$. A word of caution is here in order. Both $B$ and its (supersymmetric) analogues in any number $d\geq 3$ of space$-$time dimensions are infinite dimensional Lie groups. For infinite dimensional Lie groups, the exponential map, when it exists, from a neighbourhood of 0 in the Lie algebra of $B$ to a neighbourhood of $e$ in $B$, is *not *surjective. Thus exponentiation of the Lie algebra of $B$ does not give, in general, the connected component of the identity of $B$. Therefore any conclusions drawn for $B$ by studying its Lie algebra should be drawn with this word of caution in mind.

One can also derive the variants of $B$, as well as their supersymmetric counterparts, in any number $d\geq 3$ of space$-$time dimensions, with the use of the four methods stated above, by appropriately modifying in each case, the class of metrics in the first method, the notion of “strong conformal geometry” in the second method, the appropriate infinite parameter family of homogeneous functions of degree one in the third method, and finally, the notion of “universal structure” in the fourth method.

Each method has its own merits and drawbacks. For example, it appears Wald , that in odd space$-$time dimensions, the second and the fourth method of derivation are not applicable in the case of *radiative *space$-$times. This restriction is not applicable to the case of three space$-$time dimensions we are studying in this paper, because in three space$-$time dimensions there are no propagating degrees of freedom, i.e., there is no gravitational radiation.

Here we introduce the analogue $B(2,1)$ of the BMS group $B$ in 3 space$-$time dimensions by using for the first time the algebraic way of derivation. To the best of our knowledge, the asymptotic symmetry group in 3 space$-$time dimensions was firstly considered in Bij , a work which is entirely classical in aims and scope. In particular no IRS were considered in this study.

In this investigation the authors studied gravitational waves with a space-translation Killing field. In this case, they noticed, the 4$-$dimensional Einstein vacuum equations are equivalent to the 3$-$dimensional Einstein equations with certain matter sources. The key point in their study which determined the structure of the asymptotic symmetry group they derived, is the following. The vacuum gravitational wave solutions they considered in 4 space$-$time dimensions are not asymptotically flat either at spatial or null infinity. This led to weaker fall off conditions for the matter sources in the equivalent system of Einstein equations with matter sources.

The weaker fall off conditions led in turn to the enlargement of the asymptotic symmetry group they derived. It led in particular to the enlargement of $SL(2,R)$ to the full group $Diff(S^{1})$ of the orientation preserving diffeomorphisms of the circle. $SL(2,R)$ acts on $P_{1}(R)\simeq S^{1}$, the one-dimensional real projective space (the circle quotient the antipodal map), this is going to be of interest in Subsection 3.3. As a result of this, the group obtained by the authors, by using the fourth method of derivation, is not the semidirect product of supertranslations, i.e., of the additive group of functions on the circle, with $SL(2,R)$, which is the group $B(2,1)$ considered in this paper, but it is instead the semidirect product of the supertranslations with $Diff(S^{1})$.

The analogue of the BMS group $B$ in 3 space$-$time dimensions with the first method of derivation was given in Bar2 and Bar3 . In Bar1 the authors advocated the view put forward e.g. in Str ; Str1 ; Graham ; Aha ; Boer ; Sol , namely that physics should be constrained by infinitesimal symmetry transformations that are not necessarily globally well defined.

With this motivation the authors initiated a research program in Bar1 whose objective is to materialize this thesis in Einstein’s G.R. in asymptotically flat space$-$times. Firstly they considered the 4$-$dim case Bar1 ; Bar2 , and then they examined the 3$-$dim case Bar2 ; Bar3 ; Bar4 .

In particular, in three space$-$time dimensions, the authors studied Bar3 ; Bar4 , by using Wigner$-$Mackey’s theory, the projective representations of BMS3, the semidirect product of $Diff(S^{1})$ with the supertranlations, the additive group of functions on the circle. Thus both factors of BMS3 are infinite dimensional. There is no overlap between this paper and Bar3 ; Bar4 because the group $B(2,1)$ we are considering in this paper is different from the group BMS3 whose representation theory is undertaken in Bar3 ; Bar4 .

The representation theory of BMS3 is also afflicted by the inapplicability of the Wigner$-$Mackey’s inducing construction which persistently occurs, as we pointed out in Section 1, in this or the other form, in any group theoretical approach to quantum gravity.

In the case of BMS3 the inapplicability of the Wigner$-$Mackey’s inducing construction manifests itself in the appearance of the exhaustivity problem and in the construction of quasi$-$invariant$-$measures in the appropriate orbits Bar4 . No solution is given by the authors to these problems and are left for future consideration Bar4 . In the case of $B(2,1)$ solution to the exhaustivity problem in the Hilbert topology is given in Subsection 5.5 and quasi$-$invariant$-$measures are constructed in Subsection 7.5.

A last remark is here in order regarding $B(2,1)$ and BMS3. $B(2,1)$ is a proper subgroup of BMS3. However, this does not render the study of the representations of $B(2,1)$ superfluous. In general, the following holds: Let $K$ be a group and $H$ be a subgroup of $K$. Let $T(k)$ be an irreducible representation of the group $K$. Let $T_{H}(k)$ be the restriction of $T(k)$ on the subgroup $H$. In general, the representation $T_{H}(k)$ is not irreducible. Moreover, there are irreducibles of $H$ which cannot be extended to the whole group $K$ .

As we pointed out, nuclear topology IRS of $B$ appear to allow scattering states, i.e., unbounded sources, possibly with infinite energy, and also distributional metric solutions to Einstein’s equations mac6 . It is interesting to note that a completely different group theoretic approach to quantum gravity has been developed by Isham Isham ; IshKak0 ; IshKak . In this approach IRS of the infinite dimensional canonical group $\mathcal{C}$ for quantum gravity are studied. This fascinating work also arrives, independently and from another point of view, at distributional metrics. In Isham’s work, however, these metrics occupy the centre of the stage.

$\mathcal{C}$ has the semidirect product structure

[TABLE]

where $\mathcal{S}$ is an Abelian subgroup of $\mathcal{C},$ and $G$ is a *not *locally compact group. The linear representation of $G$ on $\mathcal{S}$ which specifies this semidirect product is not relevant here. Isham does not succeed in obtaining a complete list of IRS of the canonical group. His efforts in doing so are hindered by two problems:

The exhaustivity problem: The canonical group is not locally compact, Mackey’s theorems (see e.g. Mackey1 ) ensuring the exhaustivity of the IRS constructed by inducing cannot be invoked, and, as a result, there is no clear idea, of how many IRS of the canonical group can be obtained by Wigner$-$Mackey’s inducing construction. 2. 2.

The problem, which proves to be the central mathematical problem in Isham’s approach, of finding G$$-quasi$-$invariant measures on the various G$$-orbits in $\mathcal{S}^{\prime}$, $\mathcal{S}^{\prime}$ is the topological dual of $\mathcal{S}$ . The actual construction of quasi$-$invariant measures in Isham’s approach cannot be attained by following the usual method of projecting down the Haar measure from $G$ to the G$$-orbit $\mathcal{O}\simeq G/L_{f}$, $L_{f}$ is the “little group” of a representative point $f$ of $\mathcal{O},$ since $G$ is not locally compact and hence has no Haar measure.

The exhaustivity problem in our approach is described in detail in Subsection 5.1 and its solution, in the Hilbert topology, is given in Subsection 5.5. One key difference between Isham’s approach and our approach is that in our approach the group $G$ is locally compact. As a result, the method of projecting down the Haar measure from $G$ to the G$$-orbit $\mathcal{O}\simeq G/L_{f}$ can be invoked, and G$$-quasi$-$invariant measures on the orbits $G/L_{f}$ can be constructed. For the case of $B(2,1)$ this construction is carried out in Subsection 7.5.

Both problems in Isham’s approach have to do with the applicability of Wigner$-$Mackey’s induced representation theory. In our approach in order to apply Wigner$-$Mackey’s theory we need to extend it. As we stated in the previous Section the required extension is given in Subsection 5.1 and it is reduced to the solution of three problems, and the solution to the three problems is examined in Subsection 5.5.

3 The group $B(2,1)$

We turn now to the study of $B(2,1),$ the analogue of $B$ in three space$-$time dimensions.

3.1 The group $B^{2,1}(N^{+})$

Recall that the $2+1$ Minkowski space is the vector space $R^{3}$ of row vectors with 3 real components, with the inner product defined as follows.  Let $x,y\in R^{3}$ have components $x^{\mu}$ and $y^{\mu}$ respectively, where $\mu=0,1,2$.  Define the inner product $x.y$ between $x$ and $y$ by

[TABLE]

Then the $2+1$ Minkowski space, sometimes written $R^{2,1}$, is just $R^{3}$ with this inner product. The “2,1” refers to the one plus and two minus signs in the inner product. Let $SO(2,1)$ be the (connected component of the identity element of the) group of linear transformations preserving the inner product.  Matrices $\Lambda\in SO(2,1)$ are taken as acting by matrix multiplication from the right, $x\longmapsto x\Lambda,$ on row vectors $x\in R^{2,1}$.

The future null cone $N^{+}\subset R^{2,1}$ is just the set of nonzero vectors with zero length and $x^{0}>0$:

[TABLE]

Let $R^{*}_{+}$ denote the multiplicative group of all *positive *real numbers.  Obviously, if $x\in N^{+}$, then $tx\in N^{+}$ for any $t\in R^{*}_{+}$.  Let $F_{1}(N^{+})$ denote the vector space (under pointwise addition) of all functions $f:N^{+}\rightarrow R$ satisfying the homogeneity condition

[TABLE]

for all $x\in N^{+}$ and $t\in R^{*}_{+}$.

Define a representation $T$ of $SO(2,1)$ on $F_{1}(N^{+})$ by setting, for each $x\in N^{+}$ and $\Lambda\in SO(2,1)$,

[TABLE]

Now let $B^{2,1}(N^{+})$ be the semidirect product

[TABLE]

That is to say, $B^{2,1}(N^{+})$ is, as a set, just the product $F_{1}(N^{+})\times SO(2,1)$, and the group multiplication law for pairs is

[TABLE]

Let $R^{*}_{+}$ be the multiplicative group of positive real numbers. The orbits of the dilatation action

[TABLE]

$t\in R^{*}_{+},\ x\in N^{+},$ are just the open half lines in $N^{+}$ from the origin of $R^{2,1}$. The projective null cone $P(N^{+})$ is precisely the space of orbits of this action.

We can replace the set of all homogeneous functions, $F_{1}(N^{+})$, on $N^{+}$, by the set $F(P(N^{+}))$ of all “free” functions on $P(N^{+})$, since the homogeneity constraint fixes the behaviour of functions $f\in F_{1}(N^{+})$ on these half lines. This gives a bijection

[TABLE]

by means of which we can find an expression for the representation $T$ in the space $F(P(N^{+}))$, giving us a new realization

[TABLE]

Each half$-$line $tx,$ $t\in R^{*}_{+},\ x\in N^{+},$ intersects the 2$-$plane $x^{0}=1$ exactly once; $x^{0}$ is the zero component of $x$. Since $x=(x^{1},x^{2},x^{0})\in N^{+}$, $x$ is a null vector, therefore, $(x^{1})^{2}+(x^{2})^{2}=(x^{0})^{2}.$ We conclude that the homeomorphic type of the space of orbits $P(N^{+})$ is a circle, in the model for $P(N^{+})$ constructed here, given by

[TABLE]

Rather than work with (11), it is more convenient to pass to a “spinor” version of $B^{2,1}(N^{+})$ which double covers this group.

3.2 The spinor version of $B^{2,1}(N^{+})$

$SL(2,R)$ is sometimes denoted by $G$ below. Let ${\rm M}_{s}(2,R)$ be the set of all $2\times 2$ symmetric real matrices. We define a right action of $G$ on ${\rm M}_{s}(2,R)$,

[TABLE]

where the superscript ${\top}$ means transpose.

Clearly any element $\mu\in{\rm M}_{s}(2,R)$ can be parameterized as follows:

[TABLE]

where $x^{\rm o},x^{1},x^{2}\in R$. We now consider the map $b:R^{2,1}\rightarrow{\rm M}_{s}(2,R)$ defined by

[TABLE]

where the $x^{\mu}$ are the components of $x\in R^{2,1}$.

This map is a linear bijection, so the right action of $G$ on ${\rm M}_{s}(2,R)$ induces a linear right action of $G$ on $R^{2,1}$. The right action of $G$ preserves determinants:

[TABLE]

Moreover,

[TABLE]

Therefore the right action (13) of $G$ on ${\rm M}_{s}(2,R)$, corresponds, via the linear bijection $R^{2,1}\leftrightarrow{\rm M}_{s}(2,R)$, to linear maps preserving lengths $x\cdot x.$ In fact, all linear maps $\Lambda\in SO(2,1)$ are obtained in this way, and the action (13) gives a homomorphism

[TABLE]

which is onto, and has kernel $Z_{2}=\{Id,-Id\}$ in $G$, $Id$ denoting the identity element of $G$. Thus $\gamma$ identifies $G$ as the double cover of $SO(2,1),$

[TABLE]

The null cone $N^{+}$ becomes, under the identification $R^{2,1}\leftrightarrow{\rm M}_{s}(2,R),$ $\widetilde{N^{+}}=b(N^{+})$ with

[TABLE]

Thus $\mu\in\widetilde{N^{+}}$ if and only if $\mu$ has rank exactly $1$ and $x^{0}>0$. Let ${I}$ be the set of all non$-$zero real two$-$component row vectors $\bm{\sigma}$; ${I}=R^{2}-\{0\}$. From the rank condition and the requirement $x^{0}>0$, it follows that

[TABLE]

where $\bm{\sigma}\in{I}$.

Equation (20) gives a parametrization of the null cone $\widetilde{N^{+}}$ by means of spinors. However, this parametrisation, though (by construction) onto, is not one$-$one. It is onto because every $\mu\in\widetilde{N^{+}}$ can be parameterized in this form but it is not one$-$one because both $\bm{\sigma}$ and $-\bm{\sigma}$ give rise to the same $\mu\in\widetilde{N^{+}}.$

There is a natural left action of $R^{*}_{+}$ on $\widetilde{N^{+}}$

[TABLE]

$t\in R^{*}_{+},\ \mu\in\widetilde{N^{+}}.$ The aforementioned linear bijection

[TABLE]

is preserved by the left $R^{*}_{+}$ action on $N^{+}$ and on $\widetilde{N^{+}}.$

Therefore the orbits of the $R^{*}_{+}$ action on $N^{+}$ are in bijective correspondence with the orbits of the $R^{*}_{+}$ action on $\widetilde{N^{+}}.$ Consequently, with this identification, we can use as a model for the orbits of the $R^{*}_{+}$ action on $N^{+}$, the orbits of the $R^{*}_{+}$ action on $\widetilde{N^{+}}$. Let $\mathcal{O}=t\mu,$ $t\in R^{*}_{+},\ \mu\in\widetilde{N^{+}},$ be an orbit of the $R^{*}_{+}$ action on $\widetilde{N^{+}}.$ Let $\mathcal{O}_{r}$ be any representative of this orbit. $\mathcal{O}_{r}$ can be parameterized in terms of spinors, and, as (20) indicates, if $\bm{\sigma}$ corresponds to $\mathcal{O}_{r}$ then $-\bm{\sigma}$ also corresponds to $\mathcal{O}_{r}.$

We conclude that the projective space $P(\widetilde{N^{+}})$ of the orbits of the $R^{*}_{+}$ action on $\widetilde{N^{+}}$ is

[TABLE]

The fact that both $\bm{\sigma}$ and $-\bm{\sigma}$ correspond to the *same *orbit $\mathcal{O}_{r}$ provides the starting point for a more rigorous proof Melas of the homeomorphic type of $P(\widetilde{N^{+}}).$

Consequently the spinor version $B^{2,1}(N^{+})_{c}$ of $B^{2,1}(N^{+})$ which double covers $B^{2,1}(N^{+})$ is

[TABLE]

where $G=SL(2,R).$ Interestingly enough $F(P(\widetilde{N^{+}})$ can be realized in two different ways which give rise to two different isomorphic realizations of $B^{2,1}(N^{+})_{c}.$

$\bf 1^{st}\ realization.$    $F(P(\widetilde{N^{+}})$ is the space of “free” real$-$valued functions $F(P_{1}(R))$ defined on $P_{1}(R)$. Thus in this case $B^{2,1}(N^{+})_{c}$ is realized as

[TABLE]

$\bf 2^{nd}\ realization.$    Let $S^{1}$ be the curve $S^{1}\subset I$ defined by

[TABLE]

$F(P(\widetilde{N^{+}})$ is the space of “even” real$-$valued $\beta:S^{1}\rightarrow R.$ That is, these functions satisfy the even$-$ness condition

[TABLE]

We denote this space by $F_{e}(S^{1}).$ Thus in this case $B^{2,1}(N^{+})_{c}$ is realized as

[TABLE]

We can use either of these realizations to develop the representation theory of $B^{2,1}(N^{+})$. The representation theory which has as point of departure the first realization cannot be mapped onto the representation theory which has as point of departure the second realization. Each representation theory presents its own difficulties, problems and subtleties. However, as expected, the results of the representation theory which stems from the first realization can be naturally mapped onto the results of the representation theory which stems from the second realization. Here we restrict attention to the representation theory which originates from the first realization.

3.3 The group $B(2,1)$

In the $1^{\rm{st}}$ realization $F(P(\widetilde{N^{+}})$ is realized as the space $F(P_{1}(R))$ of “free” real$-$valued functions defined on $P_{1}(R)$. As a result we have the following Theorem:

Theorem 3.1

The group $B^{2,1}(N^{+})_{c}$ can be realized as

[TABLE]

with semidirect product specified by

[TABLE]

$G$ is the group SL(2,$R$), $g\in G$, $\alpha\in F(P_{1}(R))$. Moreover, if

[TABLE]

then the components $\sigma_{1},\sigma_{2}$ of $\bm{\sigma}\in R^{2}-\{0\}$ transform linearly under $g$, so that the ratio $\sigma={\sigma_{1}}/{\sigma_{2}},\ \sigma_{2}\neq 0$,   transforms fraction linearly under $g$. Writing $\sigma g$  for the transformed ratio,

[TABLE]

The ratio $\sigma={\sigma_{1}}/{\sigma_{2}},\ \sigma_{2}\neq 0$, is a local inhomogeneous coordinate of $P_{1}(R)$. The factor $\kappa_{g}(\sigma)$ on the right hand side of (28) turns out Melas to be

[TABLE]

In analogy to $B$, it is natural to choose a measure $\lambda$ on $P_{1}(R)$ which is invariant under the maximal compact subgroup $SO(2)$ of $G$; so we choose $\lambda$ to be the standard normalized Lesbegue measure ${\rm d}\lambda=\frac{{\rm d}\theta}{2\pi}$. For physical applications, it is necessary to give $F(P_{1}(R))$ additional structure. For reasons discussed in detail in McCarthy mac1 , the supertranslation space $F(P_{1}(R))$ is restricted to be the separable Hilbert space $L^{2}(P_{1}(R),\lambda,R)$ of real$-$valued functions defined on $P_{1}(R)\simeq S^{1}$; functions square integrable with respect to $\lambda$. $B^{2,1}(N^{+})_{c}$ can now be realized in terms of $L^{2}(P_{1}(R),\lambda,R)$ Melas as follows:

Theorem 3.2

The group $B^{2,1}(N^{+})_{c}$ can be realised as

[TABLE]

with semidirect product specified, as before, by

[TABLE]

In the last relation $\alpha\in L^{2}(P_{1}(R),\lambda,R),$ and $\kappa_{g}(\sigma)$ is given, as before, by (32). We denote our final realization of our group by $B(2,1)$ to distinguish it from the previous realizations $B^{2,1}(N^{+})$ and $B^{2,1}(N^{+})_{c}$.

Reisz$-$Fr$\acute{e}$chet theorem for Hilbert spaces Rudin states that the topological dual of a Hilbert space can be identified with the Hilbert space itself, so that we have ${L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)\simeq L^{2}(P_{1}(R),\lambda,R).$ In particular, given a continuous linear functional $f\in{L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$, we can write, for $\alpha\in L^{2}(P_{1}(R),\lambda,R)$

[TABLE]

where the function $f\in L^{2}(P_{1}(R),\lambda,R)$ on the right is uniquely determined by (and denoted by the same symbol as) the linear functional $f\in{L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$ on the left.

The representation theory of $B(2,1)$ is governed by the dual action $T^{\prime}$ of $G$ on the topological dual ${L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$ of $L^{2}(P_{1}(R),\lambda,R).$ The dual action $T^{\prime}$ is defined by:

[TABLE]

A short calculation gives

[TABLE]

Now, this action $T^{\prime}$ of $G$ on ${L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$, given explicitly above is, like the action $T$ of $G$ on $L^{2}(P_{1}(R),\lambda,R)$, continuous. The “little group” $L_{f}$ of any $f\in{L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$ is the stabilizer

[TABLE]

4 Wigner$-$Mackey’s theory

Our construction of the IRs of $B(2,1)$ is based on G.W.Mackey’s pioneering work on group representations (see, for example, Mackey0 ; Mackey00 ; Mackey ; Mackey1 ); in particular, is based on an extension to the relevant infinite dimensional case mac6 ; Mel5 of his semidirect product theory. One of the sources of Mackey’s work was Wigner’s classic paper Wigner in which he broke new grounds in mathematics by giving the first explicit treatment of of infinite dimensional representations of a Lie group.

We will give the bare essentials of Wigner$-$Mackey’s theory in order to construct explicitly the operators of the induced representations of $B(2,1).$ We will not follow the functional analytic treatment of induced representations but instead we shall adopt the more geometrical fibre bundle dominated scheme. Standard references are (Wigner ,Mackey ,Mackey1 ,Simms , Isham and references therein).

Let $A$ and $\mathcal{G}$ be topological groups, and let $T$ be a given homomorphism from $\mathcal{G}$ into the group of automorphisms ${\rm Aut}(A)$ of $A.$ Suppose $A$ is abelian and $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ is the semidirect product of $A$ and $\mathcal{G},$ specified by the continuous action $T:\mathcal{G}\longrightarrow{\rm Aut}(A).$ In the product topology of $A\times\mathcal{G},$ $\mathcal{B}$ then becomes a topological group. It is assumed that it becomes a separable locally compact topological group.

The irreducible unitary continuous representations of $A$ (characters) are continuous homomorphisms

[TABLE]

of $A$ into the multiplicative group of complex numbers of unit modulus. A composition law

[TABLE]

on the characters and a multiplication by scalars

[TABLE]

turns the characters into a vector space over the reals. In particular the composition law (39) turns the characters into an abelian group, the dual group $\hat{A}$ of $A$.

Writing the character $\phi$ as

[TABLE]

one easily finds Melas that the real$-$valued function $f$ is a continuous linear functional on $A$, that is, $f$ belongs to the topological dual $A^{\prime}$ of $A$. The map

[TABLE]

provides an isomorphism between $\hat{A}$ and $A^{\prime}$, as real vector spaces, and in particular as abelian groups. We shall identify, if required, the sets $\hat{A}$ and $A^{\prime}$.

The action $T$ of $\mathcal{G}$ on $A$ induces a dual action on $\hat{A}.$ Indeed, any bijective map $b:A\longrightarrow A$, induces a map $\hat{A}\longrightarrow\hat{A}$, $\phi\longmapsto b\phi$, defined by

[TABLE]

It is to be noted that the bijective correspondence

[TABLE]

is preserved pointwise by the $\mathcal{G}$ action if and only if the action of $\mathcal{G}$ on $A^{\prime}$ is defined by

[TABLE]

In this way an action $\alpha\longmapsto h\alpha h^{-1}$ of $\mathcal{B}$ on $A$ induces an action $\phi\longmapsto h\phi$ of $\mathcal{B}$ on $\hat{A}$,

[TABLE]

For a given character $\phi\in\hat{A},$ the largest subgroup $L_{\phi}$ of $\mathcal{G}$ which leaves $\phi$ fixed is called the ”little group” of $\phi$, and the set of characters which can be reached from $\phi$ by the $\mathcal{B}$ action is called the orbit of $\phi$, denoted $\mathcal{B}\phi.$ The largest subgroup of $\mathcal{B}$ which leaves $\phi$ fixed is the semidirect product $\mathcal{B}_{\phi}=A\rtimes_{T}L_{\phi}\;$. $\mathcal{B}_{\phi}$ is a closed subgroup of $\mathcal{B}.$

Let U be a continuous irreducible unitary representation of the ”little group” $L_{\phi}$ on a Hilbert space $D.$ Then

[TABLE]

where $a\in A$ and $l\in L_{\phi},$ is a continuous unitary irreducible representation of the group $\mathcal{B}_{\phi}=A\rtimes_{T}L_{\phi}$ in $D.$

Let $\mathcal{B}_{\phi}\rightarrow\mathcal{B}\rightarrow\mathcal{B}/\mathcal{B}_{\phi}$ be the principal fibre bundle with fibre $\mathcal{B}_{\phi},$ base space the coset space $\mathcal{B}/\mathcal{B}_{\phi}$ and the smooth projection map $\pi:\mathcal{B}\rightarrow\mathcal{B}/\mathcal{B}_{\phi}$ is the usual function $\pi(h):=h\mathcal{B}_{\phi}$ which associates with each $h\in\mathcal{B}$ the coset $h\mathcal{B}_{\phi}$ to which it belongs. $\;\mathcal{B}_{\phi}$ acts upon $D$ (on the left) via

[TABLE]

There is a right action of $\mathcal{B}_{\phi}$ on $\mathcal{B}\times D$ defined by

[TABLE]

The bundle with fibre $D,$ associated to the principal bundle $\mathcal{B}_{\phi}\rightarrow\mathcal{B}\rightarrow\mathcal{B}/\mathcal{B}_{\phi},$ has total space $\mathcal{B}\times_{{\mathcal{B}}_{\phi}}D,$ which is defined to be the orbit space of $\mathcal{B}\times D$ under the action (50), and projection map

[TABLE]

where $[h,v]$ denotes the equivalence class of $(h,v)\in\mathcal{B}\times D$ under (50).

That $\pi^{-1}_{D}(h\mathcal{B}_{\phi})$ really is a copy $D_{h{\mathcal{B}}_{\phi}}$ of $D$ can be seen by noting the existence of the map

[TABLE]

which is a diffeomorphism for each $h\in\mathcal{B}.$

Therefore, for each (continuous unitary irreducible) representation $U$ of the ”little group” $L_{\phi}$ and for each character $\phi$ we have a bundle $D\rightarrow\mathcal{B}\times_{{\mathcal{B}}_{\phi}}D\rightarrow\mathcal{B}/\mathcal{B}_{\phi}$ with fibre $D,$ total space $\mathcal{B}\times_{{\mathcal{B}}_{\phi}}D,$ and base space $\mathcal{B}/\mathcal{B}_{\phi}.$ The bundle $D\rightarrow\mathcal{B}\times_{{\mathcal{B}}_{\phi}}D\rightarrow\mathcal{B}/\mathcal{B}_{\phi}$ depends, up to \mathcal{B}$$-isomorphism, only on the orbit $\mathcal{B}\phi$ and not on the choice of $\phi.$ There is a natural bijection

[TABLE]

$h\mathcal{B}_{\phi}\longrightarrow h\phi$ which will even be a homeomorphism Bour (Appendix I), under quite general conditions. We will often identify $\mathcal{B}/\mathcal{B}_{\phi}$ and $\mathcal{B}\phi$.

Since $A$ acts trivially on $\hat{A}$ the orbit $\mathcal{B}\phi$ is identical to the orbit $\mathcal{G}\phi$. Moreover, the map

[TABLE]

where $g\in\mathcal{G},$ and $h=(\alpha,g),\ \alpha\in A,$ is also a bijection. For this reason, we shall also identify the sets $\mathcal{G}/L_{\phi}$ and $\mathcal{B}\phi$. To conclude, the following bijections hold

[TABLE]

and, for this reason, we shall identify any two of the sets $\mathcal{B}/H_{\phi},\ \mathcal{B}\phi,\ \mathcal{G}/L_{\phi}$, and $\mathcal{G}\phi$.

It is well known (see e.g. Simms , p.45) that when $\mathcal{G}$ is locally compact and $L_{\phi}$ is a closed subgroup of $\mathcal{G}$, then there is a unique non$-$zero invariant measure class M on $\mathcal{G}/L_{\phi}$. For each measure $\mu$ in M and each $g\in\mathcal{G}$ the measure $\mu_{g}$:

[TABLE]

where $E$ is a Borel subset of $\mathcal{G}/L_{\phi}$, is also in $M$.

For each $\mu\in M$ let

[TABLE]

where $p\in{\mathcal{G}}/L_{\phi}$ and $<\psi(p),\psi(p)>$ denotes the inner product in the Hilbert space $\pi_{D}^{-1}(p)=D_{p}$ which is diffeomorphic to $D$. With the inner product

[TABLE]

and with sections identified if they differ only on a set of \mu$$-measure zero, $H_{\mu}$ is a Hilbert space.

Define an action of $\;\mathcal{B}\;$ on $\;H_{\mu}\;$ by

[TABLE]

where $h\in\mathcal{B},p\in\mathcal{B}/\mathcal{B}_{\phi},$ and $\;\mu_{h}(E)=\mu(h^{-1}E)\;$ for any Borel subset $\;E\;$ of $\;\mathcal{B}/\mathcal{B}_{\phi}.\;$ This action is a unitary action. The representation of $\mathcal{B}$ on $H_{\mu}$ given by (59) is called the representation of $\mathcal{B}$ induced by the representation $U$ of $L_{\phi}$. If $U$ and $W$ are equivalent representations of $L_{\phi}$ then they yield equivalent induced representations of $\mathcal{B}$.

If it is further assumed Mackey1 that $\mathcal{B}$ is a *regular *semidirect product then *all *IRS of $\mathcal{B}$ are obtained by inducing from the IRS of the “little groups” in the way described above. Regular means that the $\mathcal{B}-$orbit$-$space $\hat{A}$ is nice in a measure$-$theoretic sense: The $\mathcal{B}-$orbits can be enumerated in some way, namely, there is a Borel set in $\hat{A}$ that meets each $\mathcal{B}-$orbit exactly once.

The central results of induced representation theory are the following:

Given the topological restrictions on $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ (separability and local compactness), any representation of $\mathcal{B},$ constructed by the method above, is irreducible iff the representation $U$ of $L_{\phi}$ on $D$ is irreducible. An irreducible representation of $\mathcal{B}$ is obtained for each $\phi\in\hat{A}$ and each irreducible representation $U$ of $L_{\phi}$. 2. 2.

If $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ is a regular semidirect product then all of its IRS can be obtained in this way.

It follows from the previous discussion that in order to give the operators of the IRS of $B(2,1)$ explicitly it is necessary to give the following information

An irreducible unitary representation $U$ of $L_{\phi}$ on a Hilbert space $D$ for each $L_{\phi}.$ 2. 2.

A \mathcal{G}$$-quasi$-$invariant measure $\mu$ on each orbit $\mathcal{G}\phi\approx\mathcal{G}/L_{\phi};$ where $L_{\phi}$ denotes the ”little group” of the base point $\phi\in\hat{A}$ of the orbit $\mathcal{G}\phi.$

To find the IRS of $B(2,1)=L^{2}(P_{1}(R),\lambda,R)\rtimes_{T}G$ then, it is enough to provide the information cited in 1 and 2 for each of the orbit types.

5 Extension of Wigner$-$Mackey’s theory

In 1939 Wigner Wigner gave the first explicit treatment of infinite dimensional representations of the Poincare group $P$, a *ten$-$dimensional *Lie group. He constructed the IRS of $P$ by inducing from the IRS of some of its subgroups, the “little groups”. Mackey proved Mackey0 ; Mackey00 that Wigner’s inducing construction can be applied to any separable locally compact semidirect product $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ specified by the continuous action $T:\mathcal{G}\longrightarrow{\rm Aut}(A).$ Moreover Mackey proved Mackey0 ; Mackey00 that when the semidirect product is regular, then *all *the IRS of $\mathcal{B}$ can be obtained by this inducing construction.

It is noteworthy that both Wigner’s work and Mackey’s work are grounded on a classical theorem of Frobenius Frob according to which every “imprimitive” representation of a finite group is induced in a certain canonical fashion by a representation of one of its subgroups. The notion of “imprimitive” is not relevant here.

5.1 Need for an extension

What is relevant here is that Mackey initiates his study Mackey by assuming that $A$ is abelian, and that both the abelian group $A$ and the group $\mathcal{G}$ in the semidirect product $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ are locally compact. In the applications to Physics we are considering here, it is assumed that $A$ has additional structure, it is also a vector space, with vector addition being the group multiplication.

It is well known that every Hausdorff topological vector space is locally compact if and only if is finite. This implies in particular that a Hausdorff infinite dimensional topological vector space is never locally compact. Hausdorff is a desirable mathematical property which it is highly praised by Physicists since it allows to discern the points of the topological space under consideration in an way ideal for them. The same happens here; it is expected and desired both $A$ and its topological dual $A^{\prime}$ to be Hausdorff topological spaces.

In all the groups $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ we consider in the present paper, the abelian group $A$ is also a finite dimensional or an infinite dimensional topological vector space, which, in both cases, is Hausdorff in the whole class of topologies they come equipped with. In particular, on every Hausdorff finite dimensional topological vector space $A$ there is a unique topological vector space structure, whereas, in the case of infinite dimensional topological vector spaces $A$ there is a wide range of allowed topologies, stated in Section 6. In all these topologies $A$ is Hausdorff. We conclude, that for the groups $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ we consider in the present paper, Wigner$-$Mackey’s theory applies, as it is, when $A$ is a finite dimensional vector space.

The key new feature mac1 of the original BMS group $B$, as well as of all possible analogues of $B$, both real and in any signature, or complex, with all possible notions of asymptotic flatness “near infinity”, is the enlargement of the finite dimensional vector space of translations to the infinite dimensional vector space of supertranslations. This turns $B$, as well as of all possible analogues of $B$, into infinite dimensional Lie groups.

It is precisely this new feature which calls for an extension of Wigner$-$Mac$-$

key’s theory. The required extension needs to give answers to the following problems, emanating precisely from the infinite dimensions of the supertranslation space, which are not addressed by Wigner$-$Mackey’s theory.

*First problem. The “little groups” problem: Calculation of the “little groups” when the abelian subgroup $A$ of $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ is also an infinite dimensional vector space.

*In a nutshell Wigner$-$Mackey’s inducing construction allows to build up the representations of the whole group out of representations of some of its subgroups, the “little groups.” Wigner$-$Mackey’s theory applies to semidirect products $\mathcal{B},$ such that, when $A$ is also a vector space, it is a finite dimensional vector space. When $A$ is a finite dimensional vector space then its topological dual $A^{\prime}$ is also a finite dimensional vector space.

There is a natural bijection

[TABLE]

$g\mathcal{G}_{f}\longrightarrow gf,$ $f\in A^{\prime},$ between the coset space $\mathcal{G}/\mathcal{G}_{f}$ and the orbit $\mathcal{G}f$ of $\mathcal{G}$ in $A^{\prime}.$ $G_{f}$ is the “little group” of $\mathcal{G}$ at the point $f.$ A complete classification of the $\mathcal{G}$ orbits on $A^{\prime},$ when $A^{\prime}$ is a finite dimensional vector space, is a fairly easy task (see e.g. Barut , p. 516). This led Wigner and Mackey Wigner ; Mackey0 ; Mackey to search for the “little groups” and find them by classifying the $\mathcal{G}$ orbits in $A^{\prime}.$ The situation changes drastically when $A^{\prime}$ is an infinite dimensional vector space. Bijection (60) still holds, but a complete classification of the $\mathcal{G}$ orbits in $A^{\prime}$ is not generally feasible. Consequently the “little groups” cannot now be found by classification of the $\mathcal{G}$ orbits in $A^{\prime}$ and a different approach is required. 2. 2.

*Second problem. The irreducibility problem: When $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ is not locally compact it is not guaranteed that the representations of $\mathcal{B}$ obtained by Wigner$-$Mackey’s inducing construction are irreducible.

*When $A$ is an infinite dimensional vector space $\mathcal{B}$ is not locally compact, at least in the class of topologies $A,$ and subsequently $\mathcal{B},$ come equipped with in the present paper; these topologies are elaborated in Section 6. In Section 6 we concentrate on the “reasonable” topologies for $B(2,1)$. However similar topological structures are eligible for *any *BMS group $\mathcal{B}$, and for *any *of its analogues, in any number $d$, $d\geq 3$, of space$-$time dimensions. When $\mathcal{B}$ is not locally compact Mackey’s theorems 14.1 and 14.2 in Mackey0 , which prove the irreducibility of the representations of $\mathcal{B}$ derived by the inducing construction, do not no longer apply. 3. 3.

*Third problem. The exhaustivity problem: When the abelian group $A$ is also an infinite dimensional vector space, special care is needed in order to prove that the inducing construction is exhaustive.

*When $\mathcal{B}$ is a regular semidirect product Wigner$-$Mackey’s inducing construction is exhaustive, i.e., *all *IRS of $\mathcal{B}$ are induced from IRS of “little groups”. Regular semidirect product $\mathcal{B}$ means that the action of $\mathcal{G}$ on $A^{\prime}$ is not pathological, and as a result of this, the $\mathcal{G}-$orbit$-$space $A^{\prime}$ is nice in a measure$-$theoretic sense.

When the space of $\mathcal{G}-$orbits in $A^{\prime}$ is not nice, while a given $\mathcal{G}-$orbit and an irreducible representation of the ”little group” of a point on that orbit still define an irreducible unitary representation of $\mathcal{G},$ there will be still others not obtained by the inducing construction.

A sufficient condition which ensures the regularity of $\mathcal{B}$ (Mackey , p.43, p.141), is the existence of a a Borel cross section for the orbits in the action of $\mathcal{G}$ on $A^{\prime},$ i.e., the existence of a Borel set in $A^{\prime}$ that meets each $\mathcal{G}-$orbit exactly once.

The real purpose of this condition about the existence of a Borel cross section is that it implies that $A^{\prime}$ can have no $\mathcal{G}-$quasi$-$ invariant measures $\mu$ such that the action of $\mathcal{G}$ is strictly ergodic with respect to $\mu.$ Whenever such measures $\mu$ do exist, an irreducible representation of $\mathcal{B}$ may be associated with each that is not equivalent to any of the IRS of $\mathcal{B}$ induced from the IRS of the “little groups”. The nature of these representations is such that one can well despair of ever obtaining a classification for them as complete and satisfying as that available for regular semidirect products.

When the abelian group $A$ is also a finite dimensional vector space, then searching for a Borel cross section which intersects each $\mathcal{G}-$orbit in $A^{\prime}$ exactly once, is a fairly easy task (see e.g. Simms , p. 59). However, when the abelian group $A$ is also an infinite dimensional vector space, as it is actually the case for both the original BMS group $B$ mac3 and its generalizations mac1 , and the group $B(2,1)$ introduced here and its generalizations Mel8 , the situation is different. Then $A^{\prime}$ is also an infinite dimensional vector space. We note that, as we clarified in Subsection 5.1, at least in the class of topologies elaborated in Section 6, when the abelian group $A$ is also an infinite dimensional vector space, then both the BMS group $B$ in question and its analogues, in any number $d$ of space$-$time dimensions, $d\geq 3,$ is not locally compact and vice versa.

As a result, in this case, it is not generally feasible to classify completely mac4 ; Melas the orbits of the $\mathcal{G}$ action on $A^{\prime}$, let alone to search for a a Borel set in $A^{\prime}$ which meets each $\mathcal{G}-$orbit in $A^{\prime}$ exactly once. We conclude that when $A$ is also an infinite dimensional vector space, then the existence of a Borel cross section which meets each $\mathcal{G}-$orbit in $A^{\prime}$ exactly once, cannot be used as a sufficient condition, which can be practically implemented, for the regularity of the semidirect product $\mathcal{B}=A\rtimes_{T}\mathcal{G},$ and new, different sufficient conditions which ensure the regularity of the semidirect product $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ need to be introduced.

One more word of caution is in order here, word of caution brought to the fore again by the infinite dimensions of $A$, when $A$ is also an infinite dimensional vector space. The new sufficient conditions which need to be introduced in order to ensure the regularity of $\mathcal{B},$ involve the topologies of $A$ and of $A^{\prime}$.

In a nutshell there is an interplay between the class of functions in $A$, the class of functions in $A^{\prime},$ and the topologies of $A$ and $A^{\prime}:$ It is precisely the arbitrariness in the class of functions allowed in $A,$ expounded in Section 6, which permits a wide range of choices of “reasonable” topologies for $A$. The key point is that the type of functions in $A$ determines the type of functions in $A^{\prime}:$ Roughly speaking, the “smoother” the functions in $A$ are, the “rougher” the generalized functions in $A^{\prime}$ turn out to be. The functions in $A^{\prime}$ allow in turn a variety of choices of “reasonable” topologies for $A^{\prime}$. To conclude:

1. (a)

When the abelian group $A$ in the semidirect product $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ is also an infinite dimensional vector space new sufficient conditions are needed to insure the regularity of $\mathcal{B}.$ 2. (b)

The new sufficient conditions entail the topologies of $A$ and of its topological dual $A^{\prime}.$

The new feature of the original BMS group $B$, as well as of all possible analogues of $B$ mac1 , the enlargement of the finite dimensional vector space of translations to the infinite dimensional vector space of supertranslations, is also shared by $B(2,1)$ and its analogues Mel8 . In fact it is a common characteristic of *any *BMS group $\mathcal{B}$, and its analogues, in any number $d$, $d\geq 3$, of space$-$time dimensions. This turns $\mathcal{B}$, as well as of all possible analogues of $\mathcal{B}$, in any number $d$, $d\geq 3$, of space$-$time dimensions, into infinite dimensional Lie groups. Thus the required extension, which has been reduced above to the solution of three problems, is needed in order to construct the IRS of all these groups.

In Subsections 5.4 and 5.5 we find solutions to the three problems posed by the required extension of Wigner$-$Mackey’s theory. These solutions apply both to the original BMS group $B$ and also to $B(2,1).$ They also apply, possibly with appropriate modifications in each particular case, to *any *BMS group $\mathcal{B}$, and to *any *of its analogues, in any number $d$, $d\geq 3$, of space$-$time dimensions. In Subsection 5.4 we find a solution to the First problem posed by the needed extension of Wigner$-$Mackey’s theory. This solution applies to any topology $\mathcal{B}$ comes equipped with. The solution to the Second problem, given in Subsection 5.5, applies to the Hilbert and to the nuclear topologies. Finally, the solution to the Third problem, given in Subsection 5.5, applies to the Hilbert topology only. Thus solution to all three problems posed by the required extension of Wigner$-$Mackey’s theory are given in the Hilbert topology.

In the next Subsection we clarify which is the Hilbert topology the group $B(2,1)$ comes equipped with, and we point out that $B(2,1)$ is not locally compact in this topology.

5.2

$B(2,1)$  is  not  locally  compact  in  the  Hilbert  topology

$L^{2}(P_{1}(R),\lambda,R)$ is topologised Melas as a (pre) Hilbert space by using a natural measure $\lambda$ on $P_{1}(R)$ and by introducing scalar product (63) into $L^{2}(P_{1}(R),\lambda,R).$ The norm defined by this scalar product induces a metric in $L^{2}(P_{1}(R),\lambda,R).$ $L^{2}(P_{1}(R),\lambda,R)$ is endowed with the topology whose open sets are the balls determined by this metric, the Hilbert topology; the Hilbert topology is also called in the literature metric, norm, or, strong topology. If $R^{4}$ is endowed with the natural metric topology then the group $G=SL(2,R),$ considered as a subset of $R^{4},$ inherits the induced topology on $G.$

We name the product topology of $L^{2}(P_{1}(R),\lambda,R)\times G,$ Hilbert topology, to emphasize that $L^{2}(P_{1}(R),\lambda,R)$ is endowed with the topology induced by the scalar product (63) of the Hilbert space $L^{2}(P_{1}(R),\lambda,R).$ With the Hilbert topology, $B(2,1)$ becomes a topological group, in particular a non$-$locally compact group; the proof follows the corresponding proof Cantoni for the original BMS group $B$.

5.3 Orbits and “little groups”

Many symmetry groups in Physics, which are isometry groups of the underlying manifolds on which the dynamics unfolds, have the structure of a semidirect product

[TABLE]

where $A$ is a finite dimensional vector space, in particular $A=K^{d}$, $d\geq 2$, $K$ is either the field $R$ of real numbers, or $C$ of complex numbers; thus $A$ is the K$$-vector space of d$$-component column vectors with entries in $K$. $A$, being a vector space, is an abelian group which is assumed to be a normal subgroup of $\mathcal{B}$. Now let $s:A\times A\longrightarrow K$ be any scalar product (i.e. any non$-$degenerate symmetric K$$-bilinerar form) for $A$. $\mathcal{G}$ is the identity component of the “Lorentz group” of linear transformations preserving $s$. The remaining structure of the semidirect product is specified by the action $T$ of $\mathcal{G}$ on $A$ given by

[TABLE]

where $g\in\mathcal{G},\ \alpha\in A,$ and $g\cdot\alpha$ is matrix multiplication of $g\in\mathcal{G}$ from the left with $\alpha\in A$. Examples of symmetry groups with the semidirect product structure (61) are the connected Euclidean group and the connected Poincare group in any number of dimensions $d$, $d\geq 2$, and also, the connected Ultrahyperbolic group in any number of dimensions $d=2n$, $n=2,3...\ .$

In order to find the IRS of these groups, by applying Wigner$-$Mackey’s theory, the first step is to find the “little groups”. Determination of the “little groups” in turn typically requires (see e.g. Barut , p. 510, 516, 517) classification of the $\mathcal{G}$ orbits on the dual group $\hat{A}$ of $A$, or equivalently, due to the bijection (45), classification of the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$ of $A$. The key fact is that $A$ is a *finite *dimensional vector space and consequently its topological dual $A^{\prime}$ is a *finite *dimensional vector space isomorphic to $A^{\prime}$, and therefore, of the same dimension as $A$. This makes classification of the $\mathcal{G}$ orbits on $A^{\prime}$ a fairly easy task (see e.g. Barut , p. 516). In all cases the $\mathcal{G}$ orbits are parameterized by finite sets of parameters.

However, in the case of $B(2,1)$, $\mathcal{G}=G=SL(2,R)$, and $A={L^{2}}(P_{1}(R),\lambda,R),$ which is an *infinite *dimensional Hilbert space. As we pointed out in Subsection 3.3, Reisz$-$Fr$\acute{e}$chet theorem for Hilbert spaces states that the topological dual $A^{\prime}$ of $A$, is isomorphic to $A$. Therefore, $A^{\prime}\simeq{L^{2}}(P_{1}(R),\lambda,R)$, and $A^{\prime}$ is itself an *infinite *dimensional Hilbert space. Hereafter we shall identify $A^{\prime}$ with ${L^{2}}(P_{1}(R),\lambda,R)$ and we shall write $A^{\prime}={L^{2}}(P_{1}(R),\lambda,R),$ with the understanding that $A^{\prime}$ and $L^{2}(P_{1}(R),\lambda,R)$ are in fact isomorphic.

Infinite dimensional Hilbert spaces are too spacious to allow, in general, an easy classification of the orbits of a group action on them. When $A^{\prime}$ is an infinite dimensional Hilbert space the orbits of the $\mathcal{G}$ action on it, in general, are parameterized by countably infinite sets of parameters, and only a partial classification of them is feasible (see e.g. mac4 ; Melas ). This is precisely what happens Melas in the case of $B(2,1)$, classification of the $SL(2,R)$ orbits on $A^{\prime}={L^{2}}(P_{1}(R),\lambda,R)$ can only be partially achieved.

To conclude, determining the “little groups” by classifying the $\mathcal{G}$ orbits is appropriate when $A$ is a finite dimensional vector space; this is precisely what Wigner and Mackey did in their initial considerations Wigner ; Mackey0 ; Mackey . On the other hand, when $A$ is an infinite dimensional vector space, and only, in general, a partial classification of the $\mathcal{G}$ orbits can be attained, determining the “little groups” by classifying the $\mathcal{G}$ orbits is not appropriate, and a different method should be followed. This is precisely the problem which is addressed and solved by the *Programme *put forward in the next Subsection.

5.4 Determination of the “little groups”

Hereafter, we restrict attention to semidirect products $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ which share the new property of $B(2,1)$, namely, their largest proper normal abelian subgroup $A$, is also an infinite dimensional vector space. In fact it is assumed that $A$ is an infinite dimensional Hilbert space of real$-$valued functions defined on a locally compact space $\mathcal{Q}$, square integrable with respect to $\lambda$, $\lambda$ being a natural measure on $\mathcal{Q}$.

The subgroup $A$ may be topologized as a (pre) Hilbert space by introducing a scalar product into $A$ and by using a natural measure $\lambda$ on $\mathcal{Q}$. In the product topology of $A\times\mathcal{G}$, $\mathcal{B}$ becomes a topological group. In the case of $B(2,1)$, $A$ is topologized as a (pre)Hilbert space by introducing into $A$ the scalar product

[TABLE]

where $\lambda$ is chosen to be the standard normalized Lesbegue measure ${\rm d}\lambda=\frac{{\rm d}\theta}{2\pi}$ on the circle.

As the discussion in the previous Subsection suggests, in order to find the “little groups” when $A$ is an infinite dimensional vector space, we need to follow another route mac6 . Firstly a definition is in order:

Definition 1

For each closed subgroup $\mathcal{S}$ of $\mathcal{G}$, define its maximal invariant vector subspace $\mathcal{V}$($\mathcal{S}$) of $A^{\prime}$ by

[TABLE]

$\mathcal{V}$($\mathcal{S})$ is closed since the $\mathcal{G}$ action on $A^{\prime}$ is continuous. In the case of $B(2,1)$, $A^{\prime}={L^{2}}(P_{1}(R),\lambda,R),$ and consequently, $\mathcal{V}$($\mathcal{S})$ being a closed subspace of ${L^{2}}(P_{1}(R),\lambda,R),$ is itself a Hilbert space.

Now the following holds mac6 :

Theorem 5.1

Every “little group” of $\mathcal{B}$ is a closed Lie subgroup of $\mathcal{G}$. The “little groups” fall into three classes:

Discrete subgroups, 2. 2.

Connected Lie subgroups, 3. 3.

Non$-$connected non$-$discrete Lie subgroups.

The identity subgroup $e\in\mathcal{G}$ belongs both to 1 and 2, otherwise the three classes are non$-$overlapping.

When $A$ is infinite dimensional, the “little groups” of $\mathcal{B}$ may be found, in principle, by following the Programme proposed by McCarthy mac6 .

Programme

Determine, up to conjugacy, all subgroups $\mathcal{S}\subset\mathcal{G}$ belonging to the three classes 1, 2, and 3. 2. 2.

For each such $\mathcal{S}$ find the vector space $\mathcal{V}(\mathcal{S})$. 3. 3.

Find the subgroups $\mathcal{S}$ for which $\mathcal{V}(\mathcal{S})$ contains elements fixed under no properly larger group $\mathcal{S}^{{}^{\prime}}\supset\mathcal{S}$. These last subgroups, and only these, are the “little groups”.

The following remarks are now in order:

Regarding the first part of the *Programme *we note that as it was pointed out in the previous Subsection, it is not appropriate, when $A$ is an infinite dimensional vector space, to determine the “little groups” by classifying the $\mathcal{G}$ orbits. Instead, since “little groups” are subgroups of $\mathcal{G}$, in order to find the “little groups”, we start by finding, up to conjugacy, *all *subgroups $\mathcal{S}$ of $\mathcal{G}$ belonging to the three classes 1, 2, and 3. Up to conjugacy, because conjugate “little groups” lead to equivalent IRS of of $\mathcal{G}.$ Finding the subgroups $\mathcal{S}$ of $\mathcal{G}$ is a problem on its own right mac3 , which occasionally may be hard to solve Mel5 , or even does not have a complete solution mac6 with our current knowledge of affairs. 2. 2.

Regarding the second part of the *Programme *we note that for *each *subgroup $\mathcal{S}$ of $\mathcal{G}$, belonging to the three classes 1, 2, and 3, we search for non$-$zero $f\in A^{\prime}$ which are invariant under $\mathcal{S}.$ If there are such non$-$zero $f\in A^{\prime}$ which are invariant under a subgroup $\mathcal{S}$ of $\mathcal{G},$ then the set $\mathcal{V}(\mathcal{S})$ of all such non$-$zero $f$, as we already pointed out, is a vector space, in fact it is a Hilbert space, subspace of $A^{\prime}={L^{2}}(P_{1}(R),\lambda,R).$ It is appropriate now to draw attention to the following points:

1. (a)

An element $f$ is non$-$zero as an element of the Hilbert space $A^{\prime}.$ 2. (b)

It is easy to show that if $f_{1}$ and $f_{2}$ are two distinct elements of $\mathcal{V}(\mathcal{S})$ then the orbits $\mathcal{G}f_{1}$ and $\mathcal{G}f_{2}$ are also distinct. 3. (c)

If a subgroup $\mathcal{S}$ of $\mathcal{G}$ has non$-$zero invariant functionals $f\in A^{\prime},$ then such a subgroup $\mathcal{S}$ *does not *qualify yet as “little group.” Such a subgroup $\mathcal{S}$ is a *potential *”little group”.

$\mathcal{S}$ is a *potential *”little group” because *a priori *we cannot exclude the possibility that *all *the elements $f\in\mathcal{V}(\mathcal{S})$ are also invariant under some bigger group $\mathcal{S}^{*},$ where, $\mathcal{S}^{*}\subseteq\mathcal{G}.$ If this were the case then $\mathcal{S}$ would not qualify as an actual ”little group”. On the other hand, if for some group $\mathcal{S},$ there is at least one element $f\in\mathcal{V}(\mathcal{S}),$ which is invariant only under $\mathcal{S},$ and which has no higher invariance, then the potential ”little group” $\mathcal{S}$ does qualify as an actual ”little group”. 3. 3.

Regarding the third part of the *Programme *we note that, as we explained before, in order to find the (actual) ”little groups” we need to find those subgroups $\mathcal{S}$ of $\mathcal{G}$ which have at least one invariant functional $f\in\mathcal{V}(\mathcal{S})$ with no higher invariance. This is a problem on its own right. Usually, after having identified $\mathcal{V}(\mathcal{S}),$ one proceeds with brutal force macMel ; Mel6 ; Melas and constructs explicitly such an $f$; a task which is by no means trivial and can take considerable amount of time and effort macMel ; Mel6 ; Melas . If such an $f\in\mathcal{V}(\mathcal{S})$ is found then the potential ”little group” $\mathcal{S}$ does qualify as an actual ”little group”.

The implementation of each part of the Programme is by no means trivial, and it poses its own problems and challenges mac3 ; mac4 ; mac6 ; Mel5 ; Mel6 ; Mel7 ; Melas . In the case of $B(2,1)$ firstly we prove Melas that the “little groups” are compact in the employed Hilbert topology. Knowing that the “little groups” are compact facilitates subsequently the implementation of all parts of the Programme because search now is restricted to the compact subgroups $\mathcal{S}$ of $\mathcal{G}$ which belong to the three classes 1, 2, and 3. Moreover, the vector space $\mathcal{V}(\mathcal{S})$ is calculated only for the compact subgroups $\mathcal{S}$ of $\mathcal{G}$, and only among these groups we isolate those which contain elements fixed under no properly larger group.

5.5

Extension of Wigner$-$Mackey’s theory accomplished in the Hilbert topology

As we pointed out in Subsection 5.1 Wigner$-$Mackey’s theory needs to be extended to be applied to semidirect products $\mathcal{B}=A\rtimes_{T}\mathcal{G},$ when the Abelian group $A$ is also an infinite dimensional vector space. In order to be applied to this class of semidirect products Wigner$-$Mackey’s theory needs to be extended in such a way that solves the three problems stated in Subsection 5.1: The “little groups” problem, the irreducibility problem, and the exhaustivity problem.

We note that Wigner$-$Mackey’s theory is an enormously effective, systematic tool for analyzing representations of many different groups, has been used to good effect by many workers, and has been extended in different directions over the years. The extension given here is the appropriate one in order to study and construct the IRS of the BMS group $B$, and the IRS of its analogues, in any number $d$ of space$-$time dimensions, $d\geq 3$, and also in order to construct the IRS of their supersymmetric counterparts.

In Subsection 5.4 we solved the “little groups” problem. In order to solve the problem we restricted attention to semidirect products $\mathcal{B}=A\rtimes_{T}\mathcal{G},$ where $A={L^{2}}(\mathcal{Q},\lambda,R)$ is a separable infinite dimensional Hilbert space of real$-$valued functions defined on a locally compact space $\mathcal{Q}$, square integrable with respect to $\lambda$, $\lambda$ being a natural measure on $\mathcal{Q}$.

This is a large class of semidirect products which encompasses both the original BMS group $B$ mac3 ; mac8 ; mac4 ; mac5 ; mac6 and its generalizations mac1 and the group $B(2,1)$ and its generalizations Mel8 , when the functions which inhabit $A$ are square integrable with respect to the appropriate measure in each case.

For this class of semidirect products, when $A$ is endowed with the Hilbert topology, we can also solve the other two problems: The irreducibility problem, and the exhaustivity problem. It is noteworthy that the solution to the “little groups” problem, given in Section 5.4, applies also to all “reasonable” topologies $A$ can be endowed with. These topologies are explicated in the next section.

To conclude solutions to the three problems, stated in Section 5.1, when $\mathcal{B}$ is employed with the Hilbert topology, are obtained as follows:

*First problem. The “little groups” problem: *This problem was solved in Subsection 5.4. 2. 2.

*Second problem. The irreducibility problem: *When $\mathcal{B}$ is not locally compact Mackey’s theorems 14.1 and 14.2 in Mackey0 , which prove the irreducibility of the representations of $\mathcal{B}$ derived by the inducing construction, do not no longer apply. However, when $\mathcal{B}$ is employed with the Hilbert or nuclear topology, by using the results of mac5 and mac6 respectively, we can prove that the representations of $\mathcal{B}$ obtained by Wigner$-$Mackey’s inducing construction are irreducible despite the fact that $\mathcal{B}$ is not locally compact in the employed Hilbert or nuclear topology. In the case of $B(2,1)$, and for the Hilbert topology, we give the proof in Melas . 3. 3.

*Third problem. The exhaustivity problem: *As we pointed out in Subsection 5.1, when $A$ is an infinite dimensional vector space, and hence $\mathcal{B}$ is not locally compact, Mackey’s classic regularity condition, i.e., the existence of a Borel cross section which meets each $\mathcal{G}-$orbit in $A^{\prime}$ exactly once, which ensures the exhaustivity of Wigner$-$Mackey’s inducing construction, cannot be used as a sufficient condition, and new sufficient conditions which insure the exhaustivity of the IRS of $\mathcal{B}$ constructed by Wigner$-$Mackey’s “little group” method need to be introduced.

This was done in two remarkable papers by Piard Piard1 ; Piard2 in the case where $\mathcal{B}$ is endowed with the Hilbert topology. Piard followed what Mackey did Mackey0 ; Mackey00 , when $A$ is a finite dimensional vector space, and he appropriately adapted and modified it to the case where $A$ is an infinite dimensional vector space.

The sufficient conditions he introduced entail the strong topology of $A$ and the weak topology of $A^{\prime}$. Piard proved Piard2 that the original BMS group $B$ satisfies the sufficient conditions he introduced and so he proved *that in the case of $B$ Wigner$-$Mackey’s “little group” method gives all the IRS of $B.$ *This proves in particular that McCarthy in mac3 and mac4 obtained all the IRS of $B$ and this renders his results even more important.

It is not a trivial task to check if a BMS group $B$ in any number of $d\geq 3$ space$-$time dimensions satisfies Piard’s sufficient conditions. We undertake this task in Melas and we prove that the group $B(2,1)$ satisfies Piard’s sufficient conditions and therefore the IRS of $B(2,1)$ obtained here by applying Wigner$-$Mackey’s “little group” method are exhaustive.

The first one third of Piard’s paper deals also with the case where $A$ is endowed with the nuclear topology. It is an interesting and important open problem to complete Piard’s investigation, or apply a totally different approach, and prove or disprove that we obtain all IRS of $\mathcal{B}$, by applying Wigner$-$Mackey’s inducing construction, when $A$ is equipped with the nuclear topology.

This will inform us in particular if McCarthy in mac6 , where he equipped $A$ with the nuclear topology, obtained all IRS of the original BMS group $B$, and moreover, if we obtain all IRS of $B(2,1)$ by applying Wigner$-$Mackey’s inducing construction, if we endow $A$ with the nuclear topology. Similar remarks apply to any any BMS group $B$ in any number of $d\geq 3$ space$-$time dimensions.

It is possible and desirable, as it is explained in the next section, to endow $B(2,1)$ with other topologies, different from the Hilbert topology. It is not known if $B(2,1)$ is locally compact in these other topologies. Therefore, if $B(2,1)$ is not locally compact in these other topologies we should examine anew if the representations of $B(2,1)$ we obtain with Wigner$-$Mackey’s inducing construction are irreducible, and, if we obtain all of them.

6 Other choices of topologies

It is amusing that one can understand intuitively mac8 ; Melas that the “little groups” are compact in terms of light intensity distributions on distant “celestial” spheres. The dual action $T^{\prime}$ of $G$ on ${L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$ is given by equation (36):

[TABLE]

This action is similar to the transformation law for light intensity distributions on distant (“celestial”) 1$-$dim spheres, i.e. circles, under $G$. The latter action appears with a $\kappa_{g}^{-1}$ rather than $\kappa_{g}^{-2}$ factor (see e.g. Greiner , p.405, the discussion there refers to 4$-$dim space$-$time but it is easily adapted to the case of 3$-$dim space$-$time); but the analogy works nevertheless. The $\kappa_{g}$ factor corresponds physically to a blueshift factor. A subgroup $\mathcal{S}$ of $G$ is a ”little group” if it leaves a (non$-$zero) $f$ fixed under $T^{\prime}.$ We showed before that $\mathcal{S}$ is closed. Consequently, if $\mathcal{S}$ is non$-$compact, it must contain a sequence of pure boosts whose velocity parameter becomes arbitrary close to the speed of light. For a light intensity distribution $f$ to be invariant under $\mathcal{S}$, it must be such that, when subjected to arbitrary large blueshifts over a region of the circle with arbitrary small complement (the complement is redshifted), it remains the same. It is not surprising that there are no such non$-$zero $f^{\prime}s$, so that there cannot be any non$-$compact “little groups”.

In this paper we restrict attention to the case where the supertranslation space $F(P(N^{+}))$ is realized as a separable Hilbert space, namely $L^{2}(P_{1}(R),\lambda,R).$ We equip $L^{2}(P_{1}(R),\lambda,R)$ with the Hilbert topology and topologize $B(2,1)$ with the product topology of $L^{2}(P_{1}(R),\lambda,R)\times G.$ There is a wide range Crampin2 of “reasonable” topologies by which we can endow $B(2,1).$ This freedom we have in the choice of the topology for $B(2,1)$ stems from the infinite$-$dimensional additive supertranslation subgroup $F(P(N^{+}))$ of “arbitrary” real$-$valued functions defined on the $P(N^{+})\simeq S^{1}$. The range of choices available depends on the class of functions allowed in $F(P(N^{+})).$

Regarding the class of functions allowed in $F(P(N^{+}))$ we note the following: In his original derivation of the BMS group $B$ Sachs took supertranslations Sachs1 to be twice continuously differentiable functions, since they are coordinate transformations. Sach’s original derivation of $B$ was superseded by that of Penrose Pen1 ; Pen2 ; Pen3 , who gave a precise and intrinsic derivation of $B$ as that group of *exact *conformal motions of the future (or past) null boundary $\Im^{+}$ (or $\Im^{-}$) of (conformally compactified weakly asymptotically simple) space$-$times which preserve “null angles”. Since truly arbitrary supertranslation functions describe symmetry transformations in Penrose’s sense, supertranslations need not have some minimum degree of smoothness.

$B(2,1)$ can also be derived in Penrose’s sense, i.e., as as that group of *exact *conformal motions of the future (or past) null boundary $\Im^{+}$ (or $\Im^{-}$) of 3$-$dim (conformally compactified weakly asymptotically simple) space$-$times which preserve “null angles”. Consequently similar remarks for the available freedom in the choice of the degree of smoothness of the supertranslations apply also in the case of $B(2,1).$

We pointed out before that the dual action $T^{\prime}$ of $G$ on ${L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R)$ is similar to the transformation law for light intensity distributions on distant (“celestial”) 1$-$dim spheres under $G$, and this suggests, that there cannot exist any invariant, under non$-$compact “little group” $\mathcal{S}$, square integrable $f\in{L^{2}}^{{}^{\prime}}(P_{1}(R),\lambda,R).$ However, the same reasoning gives rise to the possibility that there may exist invariant distributional $f$ invariant under non$-$compact “little groups”.

Indeed this is what happens in the 4$-$dim case mac6 , when $F(P(N^{+})),$ $P(N^{+})\simeq S^{2}$ in the 4$-$dim case, is chosen to be the space $C^{\infty}(S^{2})$ of infinitely differential functions on the 2$-$dim sphere. If a sufficiently fine (actually nuclear) topology is chosen for $C^{\infty}(S^{2})$ then non$-$compact “little groups” arise associated with invariant distributions. We conjecture that the same is going to happen in the 3$-$dim case: If $F(P(N^{+}))$ is chosen to be the space $C^{\infty}(S^{1})$ of infinitely differential functions on the circle, and equip this space with the nuclear topology then non$-$compact “little groups” are going to arise associated with invariant distributions.

IRS both in the Hilbert topology and in the nuclear topology induced by compact “little groups” describe bounded sources, whereas, IRS in the nuclear topology induced by non$-$compact “little groups” describe scattering states mac6 ; Crampin2 . Another class of topologies for $F(P(N^{+}))$ are the Sobolev topologies Crampin2 , which depend strongly on smoothness properties (being defined by convergence in the mean of functions and derivatives). They are physically plausible in a gravitational context because they are adapted to initial$-$value problems for hyperbolic differential equations; in an ADM formulation of G.R. the evolution equations of the metric and the extrinsic curvature are hyperbolic equations.

7 Construction of the IRS of $B(2,1)$

7.1 Prerequisites

Let $\mathcal{B}=A\rtimes_{T}\mathcal{G}$ be the semidirect product of $A$ and $\mathcal{G},$ specified by the continuous action $T:\mathcal{G}\longrightarrow{\rm Aut}(A).$ In section 4 we pointed out that there is a bijective correspondence between the dual group $\hat{A}$ of $A$ and the topological dual $A^{\prime}$ of $A$. This bijection is also useful here so we recall the explicit form of the bijection between $\hat{A}$ and the $A^{\prime}$ and draw a conclusion not stated in Section 4. $\hat{A}$ and $A^{\prime}$ are isomorphic as real vector spaces, and in particular as abelian groups. The isomorphism $j$ between $\hat{A}$ and $A^{\prime}$ is given by equation (43):

[TABLE]

Therefore every element $j(f)=\phi\in\hat{A}$ is written as $\phi=e^{if}$, where $f$ is determined uniquely by $\phi$ and vice versa. We have (equation (41)):

[TABLE]

$\alpha\in A.$ The action $T$ of $\mathcal{G}$ on $\hat{A}$ is defined by (equation (44)):

[TABLE]

$b\in\mathcal{G},\ \phi\in\hat{A},\ \alpha\in A.$ The bijective correspondence $\hat{A}\leftrightarrow A^{\prime}$ is preserved pointwise by the $\mathcal{G}$ action if and only if the action of $\mathcal{G}$ on $A^{\prime}$ is defined by (equation (46)):

[TABLE]

$g\in\mathcal{G},\ f\in A^{\prime},\ \alpha\in A.$

In Section 4 we defined the “little group” $L_{\phi}$ of $\phi,$ $\phi\in\hat{A},$ to be the largest subgroup of $\mathcal{G}$ which leaves $\phi$ fixed. Similarly, we can define the “little group” $L_{f}$ of $f,$ $f\in A^{\prime},$ to be the largest subgroup of $\mathcal{G}$ which leaves $f$ fixed. It is an immediate consequence of equations (41), (44), and (46), that if $f$ is the unique element of $A^{\prime}$ which corresponds to the element $\phi$ of $\hat{A},$ then, $L_{f}=L_{\phi},$ i.e., we have:

[TABLE]

The last equality, not mentioned in Section 4, is going to be useful in the subsequent construction of the IRS of $B(2,1)$. One more remark is now in order before proceeding to the construction of the IRS of $B(2,1)$. As we stated in Section 4, the set of characters which can be reached from a character $\phi$ by the $\mathcal{G}$ action on $\hat{A}$ (equation (44)) is called the orbit of $\phi$, and is denoted by $\mathcal{G}\phi.$ There is a bijection between the orbit $\mathcal{G}\phi$ and the coset space $\mathcal{G}/L_{\phi}$, the set of left cosets of $L_{\phi}$ in $\mathcal{G},$ (equation (55))

[TABLE]

Similarly, the set of functionals which can be reached from a functional $f$ by the $\mathcal{G}$ action on $A^{\prime}$ (equation ((46)) is called the orbit of $f$, and is denoted by $\mathcal{G}f.$ There is a bijection between the orbit $\mathcal{G}f$ and the coset space $\mathcal{G}/L_{f}$, the set of left cosets of $L_{f}$ in $\mathcal{G},$

[TABLE]

Both bijections (66) and (67) will even be homeomorphisms Bour (Appendix I), under quite general conditions.

From equations (65), (66), and (67), we conclude that if $\phi$ is the unique element of $\hat{A}$ which corresponds to the element $f$ of $A^{\prime},$ then, there is a bijective correspondence between the orbits $\mathcal{G}\phi$ and $\mathcal{G}f$, i.e., we have

[TABLE]

Therefore we can identify, if needed, the sets $\mathcal{G}\phi$ and $Gf$. The previous comments also apply to $B(2,1)$, in which case we have, $A^{\prime}\simeq A={L^{2}}(P_{1}(R),\lambda,R)$, and, $\mathcal{G}=G=SL(2,R)$.

7.2 Necessary data for the construction of the IRS

To find explicitly the operators of the induced representations of $B(2,1)$, it suffices to provide the information stated in paragraphs 1 and 2 at the end of Section 4 for each of the orbit types. It is convenient, in order to construct the operators of the IRS of $B(2,1)$, to use the bijective correspondence between the dual group $\hat{A}$ of $A$ and the topological dual $A^{\prime}$ of $A$ and the consequences of this correspondence, i.e., to use equations (43), (65), and (68), and restate the information stated in paragraphs 1 and 2 at the end of Section 4 as follows: In order to give the operators of the IRS of $B(2,1)$ explicitly it is necessary to give the following information

An irreducible unitary representation $U$ of $L_{f}$ on a Hilbert space $D$ for each $L_{f}.$ 2. 2.

A \mathcal{G}$$-quasi$-$invariant measure $\mu$ on each orbit $\mathcal{G}f\approx\mathcal{G}/L_{f};$ where $L_{f}$ denotes the ”little group” of the base point $f\in A^{\prime}$ of the orbit $\mathcal{G}f.$

Thus for *each *orbit $Gf\in A^{\prime}$, with base point $f,$ and *each *irreducible unitary representation $U$ of $L_{f}$, we have an irreducible representation of $B(2,1),$ given by equation (59), which is unitary, provided the orbit $Gf$ comes equipped with a measure which is G$$-quasi$-$invariant. We conclude that in order to construct the operators of the IRS of $B(2,1)$ we need to give the following data in the prescribed order:

“Little groups” and their invariant functionals. 2. 2.

IRS of the “little groups” on a Hilbert space $D$. 3. 3.

A G$$-quasi$-$invariant measure on each orbit $Gf$.

We proceed now to carry out this task.

7.3 “Little groups” and their invariant functionals

The calculation of the “little groups” goes hand in hand with the calculation of their invariant functionals, and it is achieved by following the *Programme *stated in Subsection 5.4.

It is an important result that when $B(2,1)$ is employed with the Hilbert topology *all *“little groups” of $B(2,1)$ are compact. A formal proof is given in Melas . A heuristic, intuitive proof was given in Section 6.

The maximal compact subgroup of $G$, where $G=SL(2,R)$, is just the subgroup $SO(2).$ That is to say, if $K$ is a compact subgroup of $G,$ then some conjugate $gKg^{-1}$ of $K$ is a compact subgroup of $SO(2).$ In the representation theory of $B(2,1)$, the little groups are only significant up to conjugacy. So, after a possible conjugation, we may take every little group to be a compact (or, equivalently, closed) subgroup of $SO(2).$

This makes the Theorem which follows more intelligible. This Theorem, proved in Melas , describes in detail the “little groups” of $B(2,1).$

Theorem 7.1

The “little groups” $\;L_{f}\;$ for $B(2,1)$ are precisely the closed subgroups of $SO(2)$ which contain the element $-I$, $I$ being the identity element of $SL(2,R)$. These are (A) $SO(2)$ itself, and (B) the cyclic groups $C_{n}$ of even order $n$.

The proof of the preceding Theorem involves identifying $SO(2)$, and $C_{n},$ with $n$ even, as potential “little groups” and then proving that they do in fact occur as actual “little groups”. It is in this second part of the proof that we proceed with brutal force and actually construct in each case an invariant functional with no higher invariance. This construction, as we pointed out in Subsection 5.4, is by no means a trivial task, and can take in general macMel ; Mel6 , and in fact it does so here, considerable amount of time and effort, at least as much as required to identify the potential “little groups” at the first place.

As we pointed out in Section 5, for a given “little group” $\;L_{f}\;$ the elements $f\in A^{{}^{\prime}}$ which are invariant under $\;L_{f}\;$, i.e., the elements $f\in A^{{}^{\prime}}$ which satisfy

[TABLE]

$\theta\in P_{1}(R),$ form a (Hilbert) subspace of ${L^{2}}(P_{1}(R),\lambda,R)$. We denote this subspace by $L^{2}(L_{f})$. We have the following Theorem Melas :

Theorem 7.2

The Hilbert space $L^{2}(SO(2))$ of invariant vectors $f\in A^{{}^{\prime}}$ under $SO(2)$ is:

[TABLE]

So $L^{2}(SO(2))$ is just the one$-$dimensional space of constant real$-$valued functions defined on $P_{1}(R)$. The Hilbert space $L^{2}(C_{n})$ of invariant vectors $f\in A^{{}^{\prime}}$ under $C_{n}$ is:

[TABLE]

where $L^{2}(E_{n})$ is the Hilbert space of square integrable real$-$valued functions defined on

[TABLE]

It is important to bear in mind that, as we emphasized in Section 5, if $f_{1}$ and $f_{2}$ are two distinct elements of $L^{2}(SO(2))$, or of $L^{2}(C_{n})$, then the orbits $Gf_{1}$ and $Gf_{2}$, are also distinct.

Thus each $f\in L^{2}(SO(2)),$ and, each $f\in L^{2}(C_{n}),$ gives rise to a different orbit $Gf,$ and on each such orbit a continuous irreducible unitary representation of $B(2,1)$ can be built in the way summarized in paragraphs 1 and 2 of the previous Subsection. Representations built on distinct orbits are also distinct, i.e., not unitary equivalent.

Moreover, in Melas it is shown that the Wigner$-$Mackey’s inducing construction is exhaustive despite the fact that $B(2,1)$ is not locally compact in the employed Hilbert topology. This result is rather important because other group theoretical approaches to quantum gravity which invoke Wigner$-$Mackey’s inducing construction (see e.g. IshKak0 ; IshKak ) are typically plagued by the non$-$exhau$-$

stiveness of the inducing construction which results precisely from the fact that the group in question is not locally compact in the prescribed topology. Exhaustiveness is not just a mathematical nicety: If the inducing construction is not exhaustive one cannot know if the most interesting information or part of it is coded in the irreducibles which cannot be found by the Wigner$-$Mackey’s inducing procedure. These results, i.e. compactness of the “little groups” and exhaustiveness of the inducing construction, not only are significant for the group theoretical approach to quantum gravity advocated here, but also they have repercussions Melas for other approaches to quantum gravity.

7.4 IRS of the “little groups” on a Hilbert space $D$.

The “little groups” of $B(2,1)$ are given in Theorem 7.1 and are all abelian. All the irreducible representations of an abelian group are one$-$dimensional. We have in particular:

**(1). **    $L_{f}=SO(2)$.

The IRS $U$ of $SO(2)$ are parameterized by an integer $\nu$ which for distinct representations takes the values $\nu=...,-2,-1,0,1,2,...\ .$ Denoting these representations by $U^{(\nu)}$, they are given by multiplication in one complex dimension $D\approx C$ by

[TABLE]

where $R(\theta)=\begin{pmatrix}cos\theta&-sin\theta\\ sin\theta&cos\theta\end{pmatrix}\in SO(2)$.

**(2). **    $L_{f}=C_{n}$, $n$ is even.

The IRS $U$ of $C_{n}$ are parameterized by an integer $\lambda$ which for distinct representations takes the values $\lambda=0,1,2,...,n-1.$ Denoting these representations by $U^{(\lambda)}$, they are given by multiplication in one complex dimension $D\approx C$ by

[TABLE]

where $j$ parameterizes the elements of the group $C_{n}$.

7.5 A G$$-quasi$-$invariant measure on each orbit $Gf$.

A \;G\;$$-quasi$-$invariant measure $\;\mu\;$ on each orbit $\;Gf\approx G/L_{f}\;$ is required. However, a \;G\;$$-invariant measure $\;\mu\;$ on each orbit $\;Gf\approx G/L_{f}\;$ can be given in all cases.

7.5.1 Invariance and quasi$-$invariance

It is useful at this point to recall some basic facts from elementary measure theory Bourbaki . A measure $\mu$ on a measurable space $\mathcal{M}$ is called \mathcal{G}$$-invariant, or invariant with respect to $\mathcal{G}$, if for any measurable set $E\subset\mathcal{M}$, and $k\in\mathcal{G}$, the transformed measure $k\mu$, also denoted by $\mu_{k}$, defined by

[TABLE]

satisfies the relation

[TABLE]

for any measurable set $E\subset\mathcal{M}$.

Therefore a measure $\mu$ on a measurable space $\mathcal{M}$ is called \mathcal{G}$$-invariant, if

[TABLE]

for any $k\in\mathcal{G}$ and any $E\subset\mathcal{M}.$

The weaker condition of \mathcal{G}$$-quasi$-$invariant measure $\mu$, or measure $\mu$ quasi$-$

invariant with respect to $\mathcal{G}$, states that if $E$ has positive measure, then its image $k^{-1}E$ has also positive measure for any $k\in\mathcal{G}.$ Thus

[TABLE]

for any $k\in\mathcal{G}.$ The last condition is equivalent to the statement that the measures $\mu$ and $\mu_{k}$ have the same sets of measure zero, i.e.,

[TABLE]

Quasi$-$invariance is necessary and sufficient for the existence of the Radon$-$

Nikodym (R$-$N) derivative $\frac{\rm{d}\mu_{k}}{\rm{d}\mu}(p)$, $p\in\mathcal{M}$, for all group elements $k\in\mathcal{G}.$ In the case under consideration $\mathcal{G}=G=SL(2,R)$, and $\mathcal{M}=G/L_{f}.$ Note that it is precisely the quasi$-$invariance of $\mu$ with respect to $G$ that allows the square root of the R$-$N derivative to occur as a factor in (59), and this is exactly what is needed to make the representation of $\mathcal{B}$ in $H_{\mu}$, given in (59), unitary.

Construction of \mathcal{G}$$-quasi$-$invariant measures on the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$ of $A$ is a major mathematical problem in every group theoretical approach to quantum gravity which involves the construction of the IRS of a symmetry group of the form $\mathcal{B}=A\rtimes_{T}\mathcal{G}.$ In every approach this measure carries important information.

When $\mathcal{G}$ is a not locally compact group the construction of \mathcal{G}$$-quasi$-$invariant measures on the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$ of $A$ is an in general unsolved problem. This is the case in Isham’s approach (in Section 2 we adopted a different notation; we denoted $\mathcal{B}$ by $\mathcal{C}$, $A$ by $\mathcal{S}$, and, $\mathcal{G}$ by $G$). In fact Isham conjectured that there are not any \mathcal{G}$$-quasi$-$invariant measures on the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$ in his approach and that it is likely that such measures will of necessity be concentrated on genuine distributional elements in $A^{\prime}$. If this is true, Isham concluded, “the group representation theory determines by itself precisely what is to be understood by ‘distributional geometry’ ”. No matter how significant and insightful that is, it is still a conjecture.

The key fact in our approach which comes to our rescue is that $\mathcal{G}=G=SL(2,R).$ $SL(2,R)$ is locally compact and therefore has a Haar measure. This is true for any BMS group $B=A\rtimes_{T}\mathcal{G}$, and for any of its analogues, in any number $d\geq 3$ of space$-$time dimensions. Thus in all cases $\mathcal{G}$ is locally compact.

If this had not been the case we would have added as the fourth problem, in Subsection 5.1, the construction of \mathcal{G}$$-quasi$-$invariant measures on the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$, to the three problems stated there, namely, the “little groups” problem, the irreducibility problem, and the exhaustivity problem, the solutions to which are necessary in order to carry out the necessary extension of Wigner$-$Mackey’s theory.

In our approach the \mathcal{G}$$-quasi$-$invariant measures on the $\mathcal{G}$ orbits on the topological dual $A^{\prime}$ carries the information about how the outcomes of physical measurements distribute.

7.5.2 The Haar measure

The Haar measure is a generalization of the Lebesgue measure to the case where the measurable space $\mathcal{M}$ is a locally compact group $G$. The Haar measure shares the key property of the Lebesgue measure, namely it is invariant under translations. Since the measurable space is a group it is natural to distinguish left$-$invariance from right$-$invariance since the translation is realized now by group multiplication. In particular the following Theorem Haar holds:

Theorem 7.3

Let $G$ be a locally compact group. Then there exists a unique, up to a positive multiplicative constant, left$-$invariant measure, the Haar measure, on $G$. Similarly if “left$-$invariant” is replaced by“right$-$invariant”.

Let $\Omega$ be the class of Borel sets in $G.$ A left invariant (right invariant) Haar measure on $G$ is a measure $\mu$ such that

[TABLE]

($\mu(Ag^{-1})=\mu(A)$), for all $g\in G$ and $A\in\Omega.$ Note that according to the definition of invariant measures given in equation (77) a left invariant measure on $G$, is G$$-invariant.

Given a left Haar measure, one immediately obtains a right Haar measure, and vice versa. In particular, if $\mu$ is a left (right) invariant measure on $G$ then $\mu^{\prime}(A)=\mu(A^{-1})$ is a right (left) measure on $G$, where $A^{-1}$ contains the inverse elements of $A$.

In general, a left$-$invariant Haar measure on a locally compact group $G$, is not right$-$invariant. When on a group $G$ the Haar measure is both left and right invariant the group $G$ is called unimodular.

A subgroup of a locally compact group is locally compact. Therefore since $SL(2,R)$ is locally compact, its subgroups $SO(2)$, and $C_{n}$, where $n$ is even, are locally compact. As a matter of fact, the groups $SO(2)$, and $C_{n}$, where $n$ is even, are compact, and hence closed, subgroups of $SL(2,R)$. Compact groups are unimodular. Therefore the groups $SO(2)$, and $C_{n}$, where $n$ is even, are unimodular. Moreover, the group $SL(2,R)$ is also unimodular Lang .

The Haar measure on $SL(2,R)$ is given in Gel , p. 214$-$215, the Haar measure on $SO(2)$ is the usual Lebesgue measure on $SO(2)$, and the Haar measure on $C_{n}$, with $n$ even, is the counting measure on $C_{n}$ defined by:

[TABLE]

where $\mathcal{P}(C_{n})$ is the power set of $C_{n},$ $R_{\geq}$ is the set of non$-$negative real numbers, and $|E|$, is the order of the set $E.$

7.5.3 Projection of the Haar measure

Consider the canonical map

[TABLE]

where $G=SL(2,R)$, and $L_{f}=SO(2)$, or $L_{f}=C_{n}$, where $n$ is even. It is a beautiful, intuitively obvious, Theorem (Lang , p. 37, Theorem 1) that when $G$ and $L_{f}$ are unimodular, and $L_{f}$ is a closed subgroup of $G$, the unique, up to an arbitrary multiplicative positive constant, Haar measure on $G$ can be projected down uniquely, up to an arbitrary multiplicative positive constant, to a G$$-invariant Haar measure on the coset space $G/L_{f}$, and hence to a G$$-invariant Haar measure on the orbit $Gf.$

The Theorem asserts in particular that if $\mu_{G}$ and $\mu_{L_{f}}$ are the unique, up to an arbitrary multiplicative positive constant, Haar measures on the groups $G$ and $L_{f}$ respectively, then there is a unique, up to an arbitrary multiplicative positive constant, G$$-invariant Haar measure $\mu_{G/L_{f}}$ on the coset space $G/L_{f}.$ Thus there is one$-$to$-$one correspondence

[TABLE]

between the pairs of measures $(\mu_{G},\mu_{L_{f}})$ and the measures $\mu_{G/L_{f}}$ on the coset spaces $G/L_{f}.$ Moreover, the Theorem gives explicitly $\mu_{G/L_{f}}$, and it describes how $\mu_{G/L_{f}}$ is constructed from the measures $\mu_{G}$ and $\mu_{L_{f}}.$

We note that the projection from the Haar measure on $G$ to the G$$-invariant Haar measure on the orbit $Gf\simeq G/L_{f}$ is obvious when $G$ and $L_{f}$ are finite groups and the Theorem in question generalizes beautifully this obvious fact. The sought G$$-invariant measures on the orbits $Gf\simeq G/L_{f}$, where $G=SL(2,R)$, and $L_{f}=SO(2)$, or $L_{f}=C_{n}$, where $n$ is even, are obtained immediately by applying this Theorem. Details are given in Melas .

This completes the necessary information in order to construct the induced representations of $B(2,1)$. As already stated, it is shown in Melas that when $B(2,1)$ is equipped with the Hilbert topology the inducing construction is exhaustive notwithstanding the fact that $B(2,1)$ is not locally compact. To conclude, in this paper *all *IRS of $B(2,1)$ have been constructed in the Hilbert topology.