Exponential Hilbert Space: Fock Space Revisited

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Abstract
An exponential Hilbert space, which is an abstraction of the familiar Fock space for bosons, provides a natural framework to discuss a wide class of field-operator representations. This framework is especially convenient when wide invariance groups, such as a unique translationally invariant state, are involved. In this paper, we develop the theory of exponential Hilbert spaces in a functional fashion suitable to discuss representations of field operators enjoying such invariance features. Representations of both current algebras and canonical field operators are discussed, and it is shown that these representations are natural generalizations of those characterizing infinitely divisible random processes. Questions of reducibility and equivalence are treated, and we prove that our construction gives rise to infinitely many unitarily inequivalent representations. Nevertheless, an extremely simple expression, bilinear in annihilation and creation operators, abstractly characterizes the operators of both the current algebras and canonical fields. Dynamical applications to quantum field theory will be treated in subsequent papers.