Refined algebraic quantization with the triangular subgroup of SL(2, R)

摘要

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+q, with p >= 1 and q >= 1, and the system has a distinguished classical o(p,q) 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)not equal(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 (H) over cap (alpha)Psi=0 and postulates self-adjointness of the o(p,q) 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 equivalent to 1 (mod 2).