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