On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

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 ${\mathcal{H}}$ satisfying the weak Weyl relation: for all ${t \in \mathbb{R}}$ (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 ${T{\rm e}^{-itH}\psi = {\rm e}^{-itH}(T+t)\psi, \forall t \in \mathbb{R}, \forall\psi \in D(T)}$ . 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 ${\mathcal{H}}$ be separable. Assume that H is bounded below with ${\varepsilon_0 := \inf \sigma(H)}$ and ${\sigma(T)=\{z \in \mathbb{C}|{\rm Im} z \ge 0\}}$ , where ${\mathbb{C}}$ is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ${(\overline{T}, H)}$ (${\overline{T}}$ is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation ${(-\overline{p}_{\varepsilon_0,+}, q_{\varepsilon_0,+})}$ on the Hilbert space ${L^2((\varepsilon_0,\infty))}$ , where ${q_{\varepsilon_0,+}}$ is the multiplication operator by the variable ${\lambda \in (\varepsilon_0,\infty)}$ and ${p_{\varepsilon_0,+} :=-i{\rm d}/{\rm d}\lambda}$ with ${D({\rm d}/{\rm d}\lambda)=C_0^{\infty}((\varepsilon_0,\infty))}$ . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation ${(\overline{T}, H)}$.