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International Journal of Theoretical Physics

On the Relation Between Time Representations and Inner Product Spaces

Heinrich Saller

Published Oct 1, 2003• 2 citations
Publisher
DOI
10.1023/b:ijtp.0000005954.51726.b3

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It is shown that if ℍ is a Hilbert space for a representation of a group G, then there are triplets of spaces (FH,H,FH), in which FH is a space of coherent state or vector coherent state wave functions and FH is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps FH→H→FH which facilitates the construction of the corresponding inner products. After completion if necessary, the spaces (FH,H,FH) become isomorphic Hilbert spaces. It is shown that the inner product for ℍ is often easier to evaluate in FH than in FH. Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of K-matrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.