Abstract
Representations of CCR algebras in spaces of entire functions are classified on the basis of isomorphisms between the Heisenberg CCR algebra \(\mathcal{A}_H\) 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 \(\mathcal{A}_H\), with the gauge transformations implemented by a continuous U(1) group of Krein space isometries. Conversely, any holomorphic Krein representation of \(\mathcal{A}_H\), 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 [a i , a j *]=δ i j η i , η i =±1, i=1,...,M, the infinite-dimensional case included, under a spectral condition for the implementers of the gauge transformations.
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Mnatsakanova, M., Morchio, G., Strocchi, F. et al. Representations of CCR Algebras in Krein Spaces of Entire Functions. Letters in Mathematical Physics 65, 159–172 (2003). https://doi.org/10.1023/B:MATH.0000010715.93852.84
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DOI: https://doi.org/10.1023/B:MATH.0000010715.93852.84