Unique determination of an inner product by adjointness relations in the algebra of quantum observables

TL;DR

This paper proves that, under certain conditions, the inner product in a quantum observable algebra's representation is uniquely determined by the adjointness relations, supporting Ashtekar's approach in quantum gravity.

Contribution

It establishes that the *-representation property uniquely determines the inner product, providing rigorous support for Ashtekar's criterion in quantum gravity quantization.

Findings

Inner product is uniquely determined by *-representation under technical conditions

Results apply to both bounded and unbounded operator cases

Concrete examples illustrate the theorems' applicability

Abstract

It is shown that if a representation of a *-algebra on a vector space $V$ is an irreducible *-representation with respect to some inner product on $V$ then under appropriate technical conditions this property determines the inner product uniquely up to a constant factor. Ashtekar has suggested using the condition that a given representation of the algebra of quantum observables is a *-representation to fix the inner product on the space of physical states. This idea is of particular interest for the quantisation of gravity where an obvious prescription for defining an inner product is lacking. The results of this paper show rigorously that Ashtekar's criterion does suffice to determine the inner product in very general circumstances. Two versions of the result are proved: a simpler one which only applies to representations by bounded operators and a more general one which allows for unbounded operators. Some concrete examples are worked out in order to illustrate the meaning and range of applicability of the general theorems.