Representations of CCR Algebras in Krein Spaces of Entire Functions

Representations of CCR algebras in spaces of entire functions are classified on the basis of isomorphisms between the Heisenberg CCR algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{A}_H$$ \end{document} and star algebras of holomorphic operators. To each representation 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{A}_H$$ \end{document}, with the gauge transformations implemented by a continuous U(1) group of Krein space isometries. Conversely, any holomorphic Krein representation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{A}_H$$ \end{document}, having the gauge transformations implemented as before and no null subrepresentation, are shown to be contained in a direct sum of the above representations. The analysis is extended to CCR algebras with [ai, aj*]=δi jηi, ηi=±1, i=1,...,M, the infinite-dimensional case included, under a spectral condition for the implementers of the gauge transformations.