M ar 1 99 4 Adjointness relations as a criterion for choosing an inner product

1. Sufficient conditions for uniqueness In the quantisation of constrained systems it can happen that one obtains a representation of an algebra of quantum operators on a vector space without a preferred inner product. Since an inner product is necessary for the probabilistic interpretation of quantum theory, some way needs to be found of introducing an appropriate inner product on this vector space. If the classical system being quantised possesses some background structure then it may be possible to use this to fix an inner product. In the case of gravity, where no background structure is present, this is not an option. The algebra of quantum observables usually admits a preferred *-operation, often related to complex conjugation of functions on the classical phase space. It has been suggested by Ashtekar that a preferred inner product could be fixed by the requirement that this *-operation is mapped by the representation into the operation of taking the adjoint of an operator with respect to the inner product in question. Discussions of this proposal can be found in [1-3]. The purpose of the following is to discuss the circumstances under which this idea suffices to determine the inner product uniquely. Let A be an associative algebra with identity over the complex numbers. This is to be interpreted as the algebra of quantum observables. Suppose that a representation ρ of A on a complex vector space V is given. Suppose further that a *-operation a 7→ a is given on A. The defining properties of a *-operation are that it is conjugate linear ((λa + μb) = λ̄a + μ̄b), that (ab) = ba and that (a) = a. In this paper the origin of these various objects will not be discussed; information on that can be found in [1] and [2]. Instead we take this collection of objects as starting point. The condition which is supposed to characterise the inner product is that