Group theoretic quantization of punctured plane

TL;DR

This work applies Isham's group theoretic quantization to the punctured plane $X=\mathbb{R}^2-\{0\}$, whose phase space is $M=X\times\mathbb{R}^2$. It identifies the canonical group $\mathscr{G}=\mathbb{R}^2\rtimes(\mathrm{SO}(2)\times\mathbb{R}^+)$ and constructs a momentum map from the corresponding Lie algebra to $C^{\infty}(M)$, then realizes a quantization map via a weakly continuous irreducible unitary representation on $\mathscr{H}=L^2(X)$ with measure $\mathrm{d}\mu=\mathrm{d}\phi\mathrm{d}\rho/(2\pi\rho)$. For the punctured plane, the authors derive explicit Weyl-like relations and identify self-adjoint operators $\hat{c},\hat{s},\hat{\pi}_1,\hat{\pi}_2$ acting on $\mathscr{H}=L^2(S^1\times\mathbb{R})$, including a twisted representation parameter $\alpha$ arising from the universal cover, which yields inequivalent quantizations. The results demonstrate how nontrivial topology of the configuration space induces additional structure in the quantum theory, providing a concrete framework for quantizing multiply-connected systems within Isham's scheme.

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 $\dd μ= \dd φ\ddρ/(2πρ)$.

Figures (2)

• Figure 1: The fundamental vector fields, $\gamma$, are necessarily to be Hamiltonian vector fields, so that an isomorphism from the Lie algebra, $\mathfrak{g}$, into the Poisson algebra, $C^{\infty}(M)$, can be established.
• Figure 2: A schematic overview of Isham's group theoretic quantization procedure.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 1
• Theorem 2
• Theorem 3
• proof