Disentangling: Definitions, Properties and Elementary Examples

A rigorous approach to Feynman’s operational calculus, the formation of functions of several noncommuting operators, is introduced, using Feynman’s heuristic “rules” as a guide. Two equivalent (i.e., isomorphic) commutative Banach algebras are constructed, in particular the disentangling algebra, which enable Feynman’s rules to be applied in a mathematically rigorous way. Once the necessary time-ordering calculations (i.e. the disentangling) have been carried out in the disentangling algebra, the resulting expression is mapped via the disentangling map to the noncommutative environment of...

A rigorous approach to Feynman’s operational calculus, the formation of functions of several noncommuting operators, is introduced, using Feynman’s heuristic “rules” as a guide. Two equivalent (i.e., isomorphic) commutative Banach algebras are constructed, in particular the disentangling algebra, which enable Feynman’s rules to be applied in a mathematically rigorous way. Once the necessary time-ordering calculations (i.e. the disentangling) have been carried out in the disentangling algebra, the resulting expression is mapped via the disentangling map to the noncommutative environment of L(X), the Banach algebra of linear operators on the Banach space X. The basic properties of the disentangling map are discussed. Simple (and detailed) examples of the disentangling process are presented in order to familiarize the reader with disentangling techniques. At the end of this chapter, there is a discussion of the behavior of the operational calculus under the assumption that one or more of the operators involved commute with all of the others.