Group theoretic quantization of punctured plane

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 R², corresponding to the punctured plane, X. Particularly, we find the canonical Lie group, mathscr{G}, corresponding to the phase space, M=X times R², to be mathscr{G} = R² rtimes (SO(2)times R^+). We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, mathscr{G} = R² rtimes (SO(2)times R^+), and the smooth functions, fin C^infty(M), in the phase space, M=X times R². Making use of this homomorphism and unitary representation of the canonical group, mathscr{G} = R² rtimes (SO(2)times R^+), we deduce a quantization map that maps a subspace of classical observables, fin 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 μ= dd φddρ/(2πρ).