arXiv subjects

N. M. Muller

Publications and source records attributed to N. M. Muller.

4 recordsLinked to original sources

arXiv2002

Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity

Starting with the Chern-Simons formulation of (2+1)-dimensional gravity we show that the gravitational interactions deform the Poincare symmetry of flat space-time 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.

hep-th↗

arXiv1998

Topological field theory and the quantum double of SU(2)

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.

hep-th↗

arXiv1998

Tensor product representations of the quantum double of a compact group

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.

q-alg↗

arXiv1996

Quantum double of a (locally) compact group

We generalise the quantum double construction of Drinfel'd to the case of the (Hopf) algebra of suitable functions on a compact or locally compact group. We will concentrate on the *-algebra structure of the quantum double. If the conjugacy classes in the group are countably separated, then we classify the irreducible *-representations by using the connection with so-called transformation group algebras. For finite groups, we will compare our description to the result of Dijkgraaf, Pasquier and Roche. Finally we will work out the explicit examples of SU(2) and SL(2,R).

q-alg↗