Refined algebraic quantization in the oscillator representation of SL(2, ℝ)

We investigate refined algebraic quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space $T^{*}{\mathbb{R}}^4$ and gauge group SL(2, R). The reduced phase space M is connected and contains four mutually disconnected “regular” sectors with topology $\mathbb{R}{\mathrm{\times S}}^1,$ but these sectors are connected to each other through an exceptional set, where M is not a manifold and where M has non-Hausdorff topology. The RAQ physical Hilbert space ${\mathcal{H}}_{\mathrm{phys}}$ decomposes as ${\mathcal{H}}_{\mathrm{phys}}\mathrm{\simeq \oplus }{\mathcal{H}}_i,$ where the four subspaces ${\mathcal{H}}_i$ naturally correspond to the four regular sectors of M. The RAQ observable algebra ${\mathcal{A}}_{\mathrm{obs}},$ represented on ${\mathcal{H}}_{\mathrm{phys}},$ contains natural subalgebras represented on each ${\mathcal{H}}_i.$ The group averaging takes place in the oscillator representation of SL(2, R) on $L^2({\mathbb{R}}^{\text{2,2}}\mathrm{),}$ and ensuring convergence requires a subtle choice for the test state space: the classical analog of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space ${\mathcal{H}}_{\mathrm{phys}}$ and a finitely generated observable subalgebra of ${\mathcal{A}}_{\mathrm{obs}}$ is recovered through both Ashtekar’s algebraic quantization and Isham’s group theoretic quantization.