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, $M=X\times {\mathbb{R}}^2$, corresponding to the punctured plane, $X$. Particularly, we find the canonical Lie group, $\mathcal{G}$, corresponding to the phase space, $M=X\times {\mathbb{R}}^2$, to be $\mathcal{G}={\mathbb{R}}^2⋊(SO(2)\times {\mathbb{R}}^{+})$. We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, $\mathcal{G}={\mathbb{R}}^2⋊(SO(2)\times {\mathbb{R}}^{+})$, and the smooth functions, $f\in C^{\infty }(M)$, in the phase space, $M=X\times {\mathbb{R}}^2$. Making use of this homomorphism and unitary representation of the canonical group, $\mathcal{G}={\mathbb{R}}^2⋊(SO(2)\times {\mathbb{R}}^{+})$, 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, $\mathcal{H}$, which is the space of all square integrable functions on $X={\mathbb{R}}^2-\{0\}$ with respect to the measure $\dd μ= \dd ϕ\ddρ/(2πρ)$.

17 pages, 2 figures