Unitarily Inequivalent Representations in Algebraic Quantum Theory

Abstract

It has been maintained that the physical content of a model of a system is completely contained in the C∗-algebra of quasi-local observables \({\cal A}\) that is associated with the system. The reason given for this is that the unitarily inequivalent representations of \({\cal A}\) are physically equivalent. But, this view is dubious for at least two reasons. First, it is not clear why the physical content does not extend to the elements of the von Neumann algebras that are generated by representations of \({\cal A}\). It is shown here that although the unitarily inequivalent representations of \({\cal A}\) are physically equivalent, the extended representations are not. Second, this view detracts from special global features of physical systems such as temperature and chemical potential by effectively relegating them to the status of fixed parameters. It is desirable to characterize such observables theoretically as elements of the algebra that is associated with a system rather than as parameters, and thereby give a uniform treatment to all observables. This can be accomplished by going to larger algebras. One such algebra is the universal enveloping von Neumann algebra, which is generated by the universal representation of \({\cal A}\); another is the direct integral of factor representations that are associated with the set of values of the global features. Placing interpretive significance on the von Neumann algebras mentioned earlier sheds light on the significance of unitarily inequivalent representations of \({\cal A}\), and it serves to show the limitations of the notion of physical equivalence.