14.1: Infinite realized causal observable in classical systems

In what follows, we will generalize the argument ( concerning the finite realized causal observable in Chapter 12) to infinite case. In the case of infinite trees, it is impossible to discuss quantum system deeply. thus, in this chapter,

\begin{align} \mbox{ we devote ourselves to classical systems } \end{align}

Let $(T,\le)$ be an infinite tree, i.e., an infinite tree like semi-ordered set such that \begin{align} \mbox{ "$t_1 {{\; \leqq \;}}t_3$ and $t_2 {{\; \leqq \;}}t_3$" $\Longrightarrow$ "$t_1 {{\; \leqq \;}}t_2$ or $t_2 {{\; \leqq \;}}t_1$" } \end{align}

Put $T^2_\le = \{ (t_1,t_2) \in T^2{}: t_1 \le t_2 \}$. An element $t_0 \in T$ is called a root if $t_0 \le t$ ($\forall t \in T$) holds. If $T$ has the root $t_0$, we sometimes denote $T$ by $T(t_0)$. $T' (\subseteq T )$ is called lower bounded if there exists an element $t_i (\in T)$ such that $t_i {{\; \leqq \;}}t$ $(\forall t \in T')$.

Therefore, if $T$ has the root, any $T' (\subseteq T )$ is lower bounded. We always assume that $T$ is complete, that is, for any $T' (\subseteq T )$ which is lower bounded, there exists an element ${\rm Inf}_T ( T' ) (\in T )$ that satisfies the following (i) and (ii):

(i): ${\rm Inf}_T ( T' ) {{\; \leqq \;}}t \qquad ( \forall t \in T' )$
(ii): If $s {{\; \leqq \;}}t \; \; ( \forall t \in T' )$, then it holds that $s {{\; \leqq \;}}{\rm Inf}_T ( T' )$

As typical examples, it suffices to consider the finite tree $T$ (mentioned in $\S$6) such that or, non-negative real numbers $T(0)=\{ t \in {\mathbb R} \;|\; t {\; \geqq \;}0 \}$ or natual numbers $T(1)=\{1,2,\ldots \}$

Let $(T{(t_0)}, {{\; \leqq \;}})$ be an infinite tree with the root $t_0$. For each $t \in T$, consider the classical basic structure:

\begin{align} [C_0(\Omega_t ) \subseteq L^\infty (\Omega_t, \nu_t ) \subseteq B(L^2 (\Omega_t, \nu_t ) ) ] \end{align}

Also, for each $t \in T$, define the separable complete metric space $X_t$, and the Borel field ${\cal B}_{X_t}$, and further, define the observable ${\mathsf O}_t {{=}} (X_t, {\cal F}_t, F_t)$ in ${L^\infty(\Omega_t, \nu_t)}$. That is, we have a sequential causal observable:

\begin{align} [{}{\mathbb O}_{T(t_0)}{}] =[{}\{ {\mathsf O}_t \}_{ t \in T} , \{ \Phi_{t_1,t_2}{}: L^\infty(\Omega_{t_2}, \nu_{t_2}) \to L^\infty(\Omega_{t_1}, \nu_{t_1}) \}_{(t_1,t_2) \in T^2_{\leqq}} ] \end{align}

Now let us construct the realized causal observable in what follows:

Here, define, $\overline{\cal P}_0(T)$ $( = \overline{\cal P}_0(T(t_0)) \subseteq {\cal P}(T) )$ such that

\begin{align} & \overline{\cal P}_0(T(t_0)) \nonumber \\ {{=}} & \{ {T'} \subseteq T \;|\; {T'} \mbox{ is finite}, t_0 \in T' \mbox{ and satisfies } {\rm Inf}_{T'} S = {\rm Inf}_T S \;\; (\forall S \subseteq T') \} \tag{14.1} \end{align}

Let ${T'{(t_0)}} \in \overline{\cal P}_0(T{(t_0)} )$. Since $(T'(t_0), {{\; \leqq \;}})$ is finite, we can put $({T'} {{=}} \{ t_0, t_1,\ldots , t_N \},$ $\pi{}: {T'} \setminus \{t_0\} \to T')$, where $\pi$ is a parent map.

Review 14.1 [The review of Definition 12.4]
Finite case: The realization of asequential causal observable is easy as follows: Let $T' (= T'(t_0) ) \in \overline{\mathcal P}_0(T)$. Consider the sequential causal observable $[{}\{ {\mathsf O}_t \}_{ t \in {T'}} , \{ \Phi_{\pi(t), t }{}:$ $L^\infty(\Omega_t, \nu_t) \to L^\infty(\Omega_{\pi(t)}, \nu_{\pi(t)}) \}_{ t \in {T'} \setminus \{t_0 \} }$ $]$. For each $s$ $(\in {T'})$, putting $T_s =\{ t \in T' \;|\; t {\; \geqq \;}s \}$, define the observable $\widehat{\mathsf O}_s {{=}} (\times_{t \in T_s } X_t,$ $\times_{t \in T_s } {\cal F}_t, {\widehat F}_s)$ in $L^\infty(\Omega_t, \nu_t)$ such that

\begin{align} \widehat{\mathsf O}_s = \left\{\begin{array}{ll} {\mathsf O}_s \quad & \mbox{($s \in {T'} \setminus \pi ({T'})$\; \mbox{ and })} \\ \\ {\mathsf O}_s {\times} ({ \underset{{t \in \pi^{-1} (\{ s \})}}{\times} } \Phi_{ \pi(t), t} \widehat {\mathsf O}_t) \quad & \mbox{($s \in \pi ({T'})${}\; \mbox{ and })} \end{array}\right. \tag{14.2} \end{align}

And further, iteratively, we get $\widehat{\mathsf O}_{t_0}{{=}} (\times_{t \in T' } X_t,$ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{t_0})$, which is also denoted by $\widehat{\mathsf O}_{{T'}}{{=}} (\times_{t \in T' } X_t,$ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{T'})$.
$\Big($ In classical cases, the existence is guaranteed by Definition 12.4 $\Big)$

Ininite case:

For any subsets $T_{1} \subseteq T_{2}( \subseteq {{T}})$, define the natural map $\pi_{T_{1},T_{2}}: \times_{{t} \in T_{2}} X_{{t}} \longrightarrow \times_{{t} \in T_{1}} X_{{t}}$ by

\begin{align} \mathop{\Large{\mbox{ $\times$}}}_{t\in T_{2}}X_{t} \ni (x_t )_{t \in T_2 } \mapsto (x_t )_{t \in T_1} \in \mathop{\Large{\mbox{ $\times$}}}_{t\in T_{1}}X_{t} \tag{14.3} \end{align}

It is clear that the observables $\bigl\{$ $\widehat{\mathsf O}_{T'}{{=}} (\times_{t \in T' } X_t,$ $\times_{t \in T' } {\cal F}_t, {\widehat F}_{T'})$ $~|~$ ${T'}\in{\overline{\cal P}_0}({{T}})$ $\bigr\}$ in $L^\infty(\Omega_{t_0}, \nu_{t_0})$ satisfy the following consistency condition , that is,

 $\bullet$ for any $T_1, T_2$ ($\in$ ${\overline{\cal P}_0}({ {T}})$) such that $T_1 \subseteq T_2$, it holds that
\begin{align} {\widehat F}_{T_2} \bigl( \pi_{T_{1},T_{2}}^{-1}({\Xi}_{T_1}) \bigr) = {\widehat F}_{T_1} \bigl({\Xi}_{T_1} \bigr) \quad (\forall {\Xi}_{T_1} \in \times_{{t} \in T_1} {\cal F}_{{t}}) \tag{14.4} \end{align}

Then, by Theorem 4.1[ Kolmogorov extension theorem in measurement theory ], there uniquely exists the observable ${\widehat{\mathsf O}}_{{{T}}}$ ${{=}}$ $\bigl(\mathop{\Large{\mbox{$\times$}}}_{{t} \in {T}}X_{{t}},$ $\mathop{\mbox{$\boxtimes$}}_{{t}\in{{T}}} {\cal F}_{{t}},$ ${\widehat F}_{{T}} \bigr)$ in $L^\infty(\Omega_{t_0}, \nu_{t_0})$ such that:

\begin{align} {\widehat F}_{{T}} \bigl( \pi_{{{T'}, T}}^{-1}({\Xi}_{{{T'}}}) \bigr) = {\widehat F}_{{{T'}}} \bigl({\Xi}_{{{T'}}} \bigr) \quad (\forall {\Xi}_{{{T'}}} \in \mathop{\mbox{$\boxtimes$}}_{{t}\in{{T'}}} {\cal F}_{{t}},~ \forall{{T'}}\in{\overline{\cal P}_0}({{T}})) \tag{14.5} \end{align}

This observable ${\widehat{\mathsf O}}_{{{T}}}$ ${{=}}$ $(\mathop{\Large{\mbox{$\times$}}}_{{t} \in {T}}X_{{t}},$ $\mathop{\mbox{$\boxtimes$}}_{{t}\in{{T}}} {\cal F}_{{t}},$ ${\widehat F}_{{T}} )$ is called the realization of the sequential causal observable $[{}{\mathbb O}_{T(t_0)}{}]$ $=$ $[{}\{ {\mathsf O}_t \}_{ t \in T} , \{ \Phi_{t_1,t_2}{}:$ $L^\infty(\Omega_{t_2}, \nu_{t_2})$ $\to L^\infty(\Omega_{t_1}, \nu_{t_1}) \}_{(t_1,t_2) \in T^2_{\leqq}}$ $]$.

Summing up the above argument, we have the following theorem in classical systems. This is the infinite version of Definition 12.4.

Theorem 14.2 [The existence theorem of an infinite realized causal observable in classical systems]

Let $T$ be an infinite tree with the root $t_0$. For each $t \in T$, consider the basic structure:

\begin{align} [C_0(\Omega_t ) \subseteq L^\infty (\Omega_t ,\nu_t ) \subseteq B( L^2 (\Omega_t ,\nu_t ) )] \end{align}

Also, for each ${t}\in{{T}}$, define the separable complete metric space $X_{t}$, the Borel field $(X_{t} , {\cal F}_{{t}})$ and an observable ${\mathsf O}_t {{=}} (X_t, {\cal F}_t, F_t)$ in $L^\infty (\Omega_t ,\nu_t )$. And, consider the sequential causal observable$[{}{\mathsf O}_{T(t_0)}{}]$ $=$ $[{}\{ {\mathsf O}_t \}_{ t \in T} , \{ \Phi_{t_1,t_2}{}:$ $L^\infty (\Omega_{t_2} ,\nu_{t_2} ) \to L^\infty (\Omega_{t_1} ,\nu_{t_1} ) \}_{(t_1,t_2) \in T^2_{\leqq}}$ $]$. Then, there uniquely exists the realized causal observable ${\widehat{\mathsf O}}_{{{T}}}$ ${{=}}$ $\bigl(\times_{{t} \in {T}}X_{{t}},$ $\boxtimes_{{t}\in{{T}}} {\cal F}_{{t}},$ ${\widehat F}_{{T}} \bigr)$ in $L^\infty (\Omega_{t_0} ,\nu_{t_0} )$, that is, it satisfies that

\begin{align} {\widehat F}_{{T}} \bigl( \pi_{{{T'}, T}}^{-1}({\Xi}_{{{T'}}}) \bigr) = {\widehat F}_{{{T'}}} \bigl({\Xi}_{{{T'}}} \bigr) \quad (\forall {\Xi}_{{{T'}}} \in \boxtimes_{{t}\in{{T'}}} {\cal F}_{{t}},~ \forall{{T'}}\in{\overline{\cal P}_0}({{T}})) \tag{14.6} \end{align}