Group theoretic quantization of punctured plane

Abstract

We quantize punctured plane, $X=\mathbb{R}^2-\{0\}$, employing Isham's group theoretic quantization procedure. After sketching out a brief review of group theoretic quantization procedure, we apply the quantization scheme to the phase space, $M=X \times \R^2$, corresponding to the punctured plane, $X$. Particularly, we find the canonical Lie group, $\mathscr{G}$, corresponding to the phase space, $M=X \times \R^2$, to be $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$. We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, and the smooth functions, $f\in C^{\infty}(M)$, in the phase space, $M=X \times \R^2$. Making use of this homomorphism and unitary representation of the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, we deduce a quantization map that maps a subspace of classical observables, $f\in C^{\infty}(M)$, to self-adjoint operators on the Hilbert space, $\mathscr{H}$, which is the space of all square integrable functions on $X=\mathbb{R}^2-\{0\}$ with respect to the measure $\dd \mu = \dd \phi\dd\rho/(2\pi\rho)$.