REFINED ALGEBRAIC QUANTIZATION WITH THE TRIANGULAR SUBGROUP OFSL(2, ℝ)

Abstract

We investigate refined algebraic quantization with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL (2, ℝ). The unreduced phase space is T*ℝp+qwith p≥1 and q≥1, and the system has a distinguished classical [Formula: see text] observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O (p, q). The representation is nontrivial iff (p, q) ≠ (1, 1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantization that imposes the constraints in the sense ĤaΨ = 0 and postulates self-adjointness of the [Formula: see text] observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantization gives no quantum theory when (p, q) = (1, 2) or (2, 1), or when p≥2, q≥2 and p+q ≡1 ( mod 2).