Group Averaging and Refined Algebraic Quantization

Refined Algebraic Quantization (RAQ) is an attempt (amongst others) to concretize Dirac’s program for the quantization of constrained systems within a generally applicable, well defined mathematical framework. It was first formulated as a general scheme in [1,9] and recently developed further in [3,4]. Here I wish to report on these recent developments. The method itself has already been used (partly implicitly) earlier in some successful applications to quantum gravity in specialized situations, like linearized gravity on symmetric backgrounds [6,7] or various minisuperspace models [10,11]. These suggested the program to find a general scheme of which these cases are just special cases. Any scheme that deals with constrained systems needs to interpret the phrase ‘solving the constraints’. Here the guiding idea of RAQ is to work from the onset within an auxiliary Hilbert space, Haux, by means of which a ∗-algebra of observables, Aobs, is constructed before the constraints are ‘solved’. The ∗-operation on Aobs derives from the adjoint-operation † on Haux, which allows to connect the auxiliary inner product with the inner product on the physical Hilbert space, Hphys, since the latter is required to support a ∗-representation of Aobs. A possibly non-trivial limitation derives from the fact that the constraint operators on Haux are required to be selfadjoint, which may not be possible in case they do not form a Lie-algebra (i.e. ‘close’ with structurefunctions only). This difficulty clearly does not arise if the constraints derive from the action of a Lie-group G. In the following we shall restrict attention to such cases. More precisely, we consider situations where a finite dimensional Lie group G acts by some unitary representation U on Haux.