UNIQUE DETERMINATION OF AN INNER-PRODUCT BY ADJOINTNESS RELATIONS IN THE ALGEBRA OF QUANTUM OBSERVABLES

It is shown that if a representation of a -algebra on a vector space V is an irreducible -representation with respect to some inner produc t on V, then under appropriate technical conditions this property dete rmines the inner product uniquely up to a constant factor. Ashtekar ha s suggested using the condition that a given representation of the alg ebra of quantum observables is a -representation to fix the inner pro duct on the space of physical states. This idea is of particular inter est for the quantization of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show r igorously that Ashtekar's criterion does suffice to determine the inne r product in very general circumstances. Two versions of the result ar e proved: a simpler one which only applies to representations by bound ed operators and a more general one which allows for unbounded operato rs. Some concrete examples are worked out in order to illustrate the m eaning and range of applicability of the general theorems.