Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions

Abstract

Abstract. We consider the deformation of the Poincaré group in 2+1 dimensions into the quantum double of the Lorentz group and construct Lorentz-covariant momentum-space formulations of the irreducible representations describing massive particles with spin 0, 1 2 and 1 in the deformed theory. We discuss ways of obtaining non-commutative versions of relativistic wave equations like the Klein-Gordon, Dirac and Proca equations in 2+1 dimensions by applying a suitably defined Fourier transform, and point out the relation between non-commutative Dirac equations and the exponentiated Dirac operator considered by Atiyah and Moore.

Non-commutative waves for gravitational anyons

“…For our purposes it is more convenient to first consider the double cover SU(1, 1) because it allows for an easy transition to the universal cover. In order to translate results from [3], the reader will need to apply the unitary matrix…”

“…The basis t a , a = 0, 1, 2, of sl(2, R) used in [3] is related to our basis of su(1, 1) via s a = h −1 t a h. An arbitrary matrix M ∈ SU(1, 1) can be parametrised in terms of two complex numbers a, b which satisfy |a| 2 − |b| 2 = 1 via…”

“…We will need to understand the conjugacy classes of SU(1, 1) for various applications in this paper. These are obtained from the well-known conjugacy classes of SL(2, R) (see [3], for example) by conjugation with h and therefore determined by the trace (which, for determinant one, fixes the eigenvalues up to ordering) plus additional data. Elements u ∈ SU(1, 1) with absolute value of the trace less than 2 are called elliptic elements.…”

“…with SU(1, 1) acting on translations via the co-adjoint action Ad * . More precisely, the product of elements (g 1 , a 1 ), (g 2 , a 2 ) ∈P 3 in the conventions of [3,2] is (g 1 , a 1 )(g 2 , a 2 ) = (g 1 g 2 , Ad * g 2 a 1 + a 2 ). (2.15) The motivation for the interpretation (2.14) of the double cover of the Poincaré group comes from the formulation of 3d gravity as a Chern-Simons theory with the Poincaré group as a gauge group.…”

“…2, we introduce our notation and review the definition and parametrisation of the universal cover SU(1, 1) as well as the discrete series UIRs. We also revisit the UIRs of the Poincaré group and briefly summarise the covariantisation of the UIRs and their Fourier transform, following [3]. In Sect.…”