# 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).

*Proceedings of the Samahang Pisika ng Pilipinas*

**36**, SPP-2018-1D-05 (2018). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2018-1D-05.