On local representations of rotations on discrete configuration spaces

Abstract

A quantum mechanical spin-half system may be characterized abstractly as having a set of two-valued observables which generate infinitesimal rotations in three dimensions. We consider a concrete realization of such a two-level system within the formalism of ensembles on configuration space, an approach which is not only capable of describing quantum mechanical systems but allows also for theories that are generalizations of quantum theory. Such a spin-half system may be called a rotational bit or robit, to distinguish it from the standard quantum qubit. After reviewing ensembles on configuration space and examining the example of constructing representations of the Galilean Lie algebra for the free particle, we construct probabilistic models for ensembles that consist of one and two spin-half systems. In the case of a single spin-half system, there are two main requirements: the configuration space must be a discrete set, labeling the outcomes of two-valued spin observables, and these observables must provide an algebraic representation of so(3). The case of a pair of spin-half systems is more complicated, in that additional physical requirements concerning locality and subsystem independence must also be taken into account and now the observables must provide an algebraic representation of so(3) ⊕ so(3). We compare the resulting theories to the corresponding quantum mechanical systems.