= <\Phi_{0},P(t) \Psi_{0}>\; ;
\Phi_{0},\Psi_{0} \in \dom$,
\item
$t \to <\Phi_{0},P(t) \Psi_{0}>$ is continuous,
\item
$P(t) \Omega = \Omega\; ; \Omega =[\id]$.
\end{enumerate}
\end{prop}
\begin{rem}
In the case of the stochastic process defined by photon field, $U(t)$
commutes with $\Gamma$, thus $\{ P(t) \} _{t \geq 0}$ form a strongly
continuous semi - group of contractions on the Hilbert space
completion
of $\dom$ and $P(t) = e^{-tH}$ for some positive self - adjoint
operator
$H$ \cite{K}.
\end{rem}
\subsection*{\bf c. Quantum algebra.}
Now we consider the representation of the algebra
$L^{\infty}(Q,\Sigma_{0},\mu)$ on the space $\kr$. For $F_{0}
\in L^{\infty}(Q,\Sigma_{0},\mu)$ define
$$\pi(F_{0})[G] = [F_{0} G]\;;\; G\in L^{2}(Q,\Sigma_{+},\mu) \eqno
(3.5)$$
$\pi(F_{0})$ is well defined on $\dom$ since
\begin{eqnarray*}
** \geq 0$$
Thus, we obtain the following:
\begin{tw}
Starting from the stochastic process corresponding to the photon
field, we construct the algebraic quantum system on the Krein
space consisting of
\begin{enumerate}
\item
the algebra $\balg$ of bounded operators on $\kr$ which is
closed with respect to the Krein space involution,
\item
the group $\{ \alpha_{t} \}$ of automorphisms of $\balg$ representing
time evolution,
\item
the vector $\Omega$ which is cyclic for $\balg$ i.e. the set
$\dom_{+}$ is dense in $\kr$,
\item
the state $\omega(B) = <\Omega,B \Omega>$ on $\balg$ which is
Krein - positive
and $\alpha_{t}$ - invariant.
\end{enumerate}
\end{tw}
\begin{rem}
It is worth to stress, that the Krein space of states obtained from
the
pair $(\balg, \omega)$ by generalized GNS construction \cite{J2} is
unitarily equivalent to the Krein space $\kr$.
\end{rem}
\section*{\sf IV Markov property}
The stochastic process defined by $S^{(2)}$ satisfies the Markov
property \cite{N}. Now we examine the consequences of Markov property
for the construction of quantum system. Let $E_{+},\, E_{-}$ and
$E_{0}$
be the conditional expectations with respect to $\Sigma_{+},\,
\Sigma_{-}$
\linebreak
( $\sigma$ - algebra generated by $\bigcup_{t \leq 0} U(t)
\Sigma_{0}$ ) and $\Sigma_{0}$. The
Markov property of $\varphi$ can be
formulated as follows \cite{K}
$$E_{+} E_{-} = E_{+} E_{0} E_{-} \eqno (4.1)$$
In the case of photon field, the stochastic process is not reflection
positive, so instead of reflection property \cite{N}, we have
$$E_{0} \Theta = \Theta E_{0} \Gamma \eqno (4.2)$$
Thus for every $F,G \in L^{2}(Q,\Sigma_{+},\mu)$
\begin{eqnarray*}
=(\Theta F,G)_{L^{2}}&=& (\Theta E_{+} F,E_{+} G)_{L^{2}} =\\
(E_{+} \Theta E_{+} F,G)_{L^{2}}&=& (E_{+} E_{-} \Theta E_{+}
F,G)_{L^{2}}=\\
(E_{+} E_{0} E_{-} \Theta E_{+} F,G)_{L^{2}}&=& (E_{+} E_{0} \Theta
E_{+} F,G)_{L^{2}}=\\
(\Gamma E_{0} F,G)_{L^{2}}&=& (E_{0}F, \Gamma E_{0} G)_{L^{2}}
\end{eqnarray*}
and similarly
$$_{\Gamma} =(E_{0} F,E_{0} G)_{L^{2}}$$
So we arrive at the following conclusion:
\begin{prop}
\begin{enumerate}
\item
The Krein space $\kr$ can be identified with the space
$L^{2}(Q,\Sigma_{0},\mu)$ and the indefinite inner product is given by
$$ = (F,\Gamma G)_{L^{2}}$$
\item
Since $\alg \simeq L^{\infty}(Q,\Sigma_{0},\mu)$ and $P(t) \simeq
E_{0}U(t) E_{0},\, \Omega$ is cyclic also for $\alg$.
\end{enumerate}
\end{prop}
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\end{document}
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