Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we show that the gravitational interactions deform the Poincaré symmetry of flat spacetime to a quantum group symmetry. The relevant quantum group is the quantum double of the universal cover of the (2+1)-dimensional Lorentz group, or Lorentz double for short. We construct the Hilbert space of two gravitating particles and use the universal R-matrix of the Lorentz double to derive a general expression for the scattering cross section of gravitating particles with spin. In appropriate limits our formula reproduces the semi-classical scattering formulae found by 't Hooft, Deser, Jackiw and de Sousa Gerbert.
We study the quantum mechanics of a system of topologically interacting particles in 2+1 dimensions, which is described by coupling the particles to a Chern-Simons gauge field of an inhomogeneous group. Analysis of the phase space shows that for the particular case of ISO(3) Chern-Simons theory the underlying symmetry is that of the quantum double D(SU(2)), based on the homogeneous part of the gauge group. This in contrast to the usual q-deformed gauge group itself, which occurs in the case of a homogeneous gauge group. Subsequently, we describe the structure of the quantum double of a continuous group and the classification of its unitary irreducible representations. The comultiplication and the R-element of the quantum double allow for a natural description of the fusion properties and the nonabelian braid statistics of the particles. These typically manifest themselves in generalised Aharonov-Bohm scattering processes, for which we compute the differential cross sections. Finally, we briefly describe the structure of D(SO(2,1)), the underlying quantum double symmetry of (2+1)-dimensional quantum gravity.Comment: 48 pages, 3 figures, LaTeX2e; two remarks and a reference added, typos corrected; to appear in Nucl.Phys.
We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible * -representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible * -representations. The example of D(SU (2)) is treated in detail, with explicit formulas for direct integral decomposition ('Clebsch-Gordan series') and Clebsch-Gordan coefficients. We point out possible physical applications. * email: thk@wins.uva.nl † email: bais@phys.uva.nl ‡ email: nmuller@phys.uva.nl rigorous definitions for the quantum double or its dual have been proposed, see in particular Majid [17] and Podles' and Woronowicz' [20].An important mathematical application of the Drinfel'd double is a rather simple construction of the 'ordinary' quasi-triangular quantum groups (i.e. q-deformations of universal enveloping algebras of semisimple Lie algebras and of algebras of functions on the corresponding groups), see for example [8] and [17].In physics the quantum double has shown up in various places: in integrable field theories [6], in algebraic quantum field theory [18], and in lattice quantum field theories. For a short summary of these applications, see [12]. Another interesting application lies in orbifold models of rational conformal field theory, where the physical sectors in the theory correspond to irreducible unitary representations of the quantum double of a finite group. This has been constructed by Dijkgraaf, Pasquier and Roche in [9]. Directly related to the latter are the models of topological interactions between defects in spontaneously broken gauge theories in 2+1 dimensions. In [2] Bais, Van Driel and De Wild Propitius show that the non-trivial fusion and braiding properties of the excited states in broken gauge theories can be fully described by the representation theory of the quantum double of a finite group. For a detailed treatment see [23].Both from a mathematical and a physical point of view it is interesting to consider the quantum double D(G) of the Hopf * -algebra of functions on a (locally) compact group G, and to study its representation theory. For G a finite group, D(G) can be realized as the linear space of all complex-valued functions on G × G. Its Hopf * -algebra structure, which rigorously follows from Drinfel'd's definition, can be given explicitly. In [16] and in the present paper we take the following approach to D(G) for G (locally) compact: We realize D(G) as a linear space in the form C c (G×G), the space of complex valued, continuous functions of compact support on G × G. Then the Hopf * -algebra operations for G finite can be formally carried over to operations on C c (G × G) for G non-finite (formally because of the occurrence of Dirac delta's). Finally it can be shown that these operations formally satisfy the axioms of a Hopf * -algebra.In [16], we focussed on the * -algebra structure of D(G), and we derived a classification of the irreducible * -representat...
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