Representations of the Heisenberg algebra on holomorphic functions and Krein structures

Abstract

Representations of CCR algebras in spaces of entire functions are classified on the basis of isomorphisms between the Heisenberg CCR algebra A_H and star algebras of holomorphic operators. To each representations of such algebras, satisfying a regularity and a reality condition, one can associate isomorphisms and inner products so that they become Krein star representations of A_H, with the gauge transformations implemented by a continuous U(1) group of Krein isometries. Conversely, any holomorphic Krein representation of A_H, having the gauge transformations implemented as before and no null subrepresentation, is shown to be contained in a direct sum of the above representations. The analysis is extended to infinite dimensional CCR algebras, under a spectral condition for the implementers of the gauge transformations.