Asao Arai
On the Uniqueness of Weak Weyl Representations of the Canonical Commutation 
Relation
(225K, Latex2e)

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 $(\overline{T},H)$.