Deformation quantization and unitary representations of the Euclidean motion group

Abstract

As an autonomous quantization procedure, deformation quantization describes quantum mechanics as a deformed classical mechanics by introducing a noncommutative but associative ⋆-product on the space of C^∞-functions on a symplectic manifold. In this paper, we demonstrate to construct and classify unitary irreducible representations of a Lie group, in particular, the Euclidean motion group M(2) via deformation quantization. These representations uniquely correspond with infinite cylinders, generated by the coadjoint action of M(2) on the dual of its Lie algebra. Via the chart defined by the cylindrical coordinate system, an M(2)-covariant Moyal ⋆-product gives rise to representations of the Lie algebra of M(2). The exponentiation of these representations are exactly the desired unitary representations of M(2).