Group averaging in the (p,q) oscillator representation of SL(2,R)

Abstract

We investigate refined algebraic quantisation with group averaging in a finite-dimensional constrained Hamiltonian system that provides a simplified model of general relativity. The classical theory has gauge group SL(2,R) and a distinguished o(p,q) observable algebra. The gauge group of the quantum theory is the double cover of SL(2,R), and its representation on the auxiliary Hilbert space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial representation of the o(p,q) quantum observable algebra. For p=q=1, the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form.Comment: 34 pages. LaTeX with amsfonts, amsmath, amssymb. (References added; minor typos corrected.

Group averaging, positive definiteness and superselection sectors

“…In quantisation of constrained systems, one approach to finding gauge invariant states is to first build an unconstrained quantum theory and then to average states in this theory over the action of the gauge group [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. When the gauge group is a compact Lie group with a unitary action on the unconstrained Hilbert space, the mathematical setting is well-understood: the averaging converges and yields a projection operator to the Hilbert subspace of gauge invariant states.…”

Ostrogradski approach for the Regge-Teitelboim type cosmology

“…Important to mention is the fact that the total Hamiltonian (29) leads us directly to the right equations of motion (13) through the conventional Ostrogradski approach for higher-order derivative systems [17]. We also note that under the symplectic structure (31), the constraints (27) and (28) result to be in involution, {C 1 , C 2 } = 0. According to the Dirac program for constrained systems, both C 1 and C 2 must be preserved by the evolution which demands the existence of the secondary constraints…”

“…we will work on the assumption that the commutators of these quantum constraints form a closed Lie algebra which will be also isomorphic to the algebra g. In fact, the classical first-class constraints are isomorphic to the algebra g associated to the lower triangular subgroup G of SL(2, R) (see the Appendix). Quantization of the lower triangular subgroup of SL(2, R) by algebraic methods was extensively studied in [27] (see also [28] for comparison). Now we explore the rather different senses in which the quantum constraints can be used to define appropriate physical states.…”

The $${\varvec{SL}}(2,\mathbb {R})$$ S L ( 2 , R ) totally constrained model: three quantization approaches

“…[17]. Besides, a considerable number of publications [18][19][20] has appeared within the Algebraic Quantization, and also when testing the Master Constraint Programme [13]. In both schemes, the constraints can be imposed simultaneously at the quantum level, but the final physical Hilbert space requires additional inputs for achieving a semiclassical limit compatible with the classical theory.…”

“…In both schemes, the constraints can be imposed simultaneously at the quantum level, but the final physical Hilbert space requires additional inputs for achieving a semiclassical limit compatible with the classical theory. More specifically, in Algebraic Quantization approach [16,[18][19][20] the group averaging technique cannot be suitably applied since the symmetry group is non-amenable. Therefore, in these cases, one has to appeal to the reality conditions of a given family of Dirac observables in order to determine the inner product of the physical Hilbert space.…”