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

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 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) ≠ (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 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 = 1 (mod 2). © World Scientific Publishing Company.
  • 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%2bq 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) ≠ (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 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 %2b q = 1 (mod 2). © World Scientific Publishing Company.

publication date

  • 2005-01-01