On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation |
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Authors: | Asao Arai |
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Institution: | (1) Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan |
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Abstract: | Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e−itH
D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation .
This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science
(JSPS). |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 81Q10 47N50 |
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