On the Uniqueness of Weak Weyl Representations of the Canonical

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 spaceH satisfying the weakWeyl relation: For all t 2 R (the set of real numbers), e¡itHD(T) D(T) (i is the imaginary unit and D(T) denotes the domain of T) and Te¡itHψ = e¡itH(T + t)ψ, 8t 2 R,8ψ 2 D(T). 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 H be separable. Assume that H is bounded below with ε0: = inf σ(H) and σ(T) = fz 2 CjIm z ¸ 0g, where C is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Suppose that fT, T ¤,Hg (T is the closure of T) is irreducible. Then (T,H) is unitarily equivalent to the weak Weyl representation (¡pε0,+, qε0,+) on the Hilbert space L2((ε0,1)), where qε0,+ is the multiplication operator by the variable λ 2 (ε0,1) and pε0,+: = ¡id/dλ with D(d/dλ) = C10 ((ε0,1)). Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation (T,H)