Rigorous Analysis of Some Commutator Identities for Physical Observables

Abstract

This chapter serves as the primary “applications” sequel to Part III. Here, we connect the general “graph-density” sufficient condition for rigorous commutation theory with several equivalent conditions introduced by Kato [Kt 1] in his discussion of the canonical commutation relations. We indicate that generalizations of Kato’s conditions can be applied to a number of other commutation-theoretic matters that play an important role in mathematical physics: strong commutativity, dynamics of the harmonic oscillator, Cartan subalgebra methods (alias “raising and lowering operators”) in the study of the rotation group and Lorentz groups for two and three space dimensions, etc.