08-27 Asao Arai

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation (225K, Latex2e) Feb 12, 08

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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 $\mathscr{H}$ satisfying the weak Weyl relation: For all $t\in \R$ (the set of real numbers), $e^{-itH}D(T)\subset D(T)$ ($i$ is the imaginary unit and $D(T)$ denotes the domain of $T$) and $Te^{-itH}\psi=e^{-itH}(T+t)\psi, \ \forall t\in \R, \forall \psi \in D(T)$. 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 $\mathscr{H}$ be separable. Assume that $H$ is bounded below with $\varepsilon_0:=\inf\sigma(H)$ and $\sigma(T)=\{z \in \C| \Im z\geq 0\}$, where $\C$ is the set of complex numbers and, for a linear operator $A$ on a Hilbert space, $\sigma(A)$ 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 $p_{\varepsilon_0,+}:=-id/d\lambda$ with $D(d/d\lambda)=C_0^{\infty}((\varepsilon_0,\infty))$. Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation $(\overline{T},H)$.

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