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

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

2004-04-23

arxiv.org/abs/gr-qc/0404100v4

Int.J.Mod.Phys. D14 (2005) 1131

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

30 pages. LaTeX with amsfonts, amsmath, amssymb. (v4: Typos corrected. Published version.)

Scientific paper

10.1142/S0218271805006894

We investigate refined algebraic quantisation 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^*R^{p+q} with p>0 and q>0, 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) is not (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 quantisation that imposes the constraints in the sense H_a 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 quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd.

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