10.2.2: Simple example---Finite causal operator is represented by matrix

Example 10.7 [Deterministic causal operator, deterministic dual causal operator, deterministic causal map ]

Define the two states space $\Omega_1$ and $\Omega_2$ such that $\Omega_1=\Omega_2={\mathbb R}$ with the Lebesgue measure $\nu$. Thus we have the classical basic structures:

\begin{align} [C_0(\Omega_k)\subseteq L^\infty (\Omega_k, \nu) \subseteq B( L^2 (\Omega_k, \nu ) ) ] \quad(k=1,2) \end{align}

Define the deterministic causal map $\phi_{1,2} : \Omega_1 \to \Omega_2$ such that

\begin{align} \omega_2 = \phi_{1,2}(\omega_1 )= 3 (\omega_1)^2 +2 \qquad (\forall \omega_1 \in \Omega_1 ={\mathbb R}) \end{align}

Then, by (10.11), we get the deterministic dual causal operator ${\Phi}^*_{1,2}:{\cal M}_{}(\Omega_1) \to {\cal M}_{}(\Omega_2)$ such that

\begin{align} {\Phi}^*_{1,2} \delta_{\omega_1} = \delta_{3 (\omega_1)^2 +2} \qquad (\forall \omega_1 \in \Omega_1 ) \end{align}

where $\delta_{(\cdot)}$ is the point measure. Also, the deterministic causal operator $\Phi_{1,2}:L^\infty(\Omega_2) \to L^\infty (\Omega_1)$ is defined by

\begin{align} [\Phi_{1,2} (f_2)](\omega_1 ) = f_2(3 (\omega_1)^2 +2) \qquad (\forall f_2 \in C_0(\Omega_2), \forall \omega_1 \in \Omega_1 ) \end{align}

Example 10.8 [Dual causal operator, causal operator] Recall Remark 2.13, that is, if $\Omega$ $(=\{1,2,...,n \})$ is finite set ( with the discrete metric $d_D$ and the counting measure $\nu$,), we can consider that

\begin{align} C_0(\Omega)=L^\infty(\Omega,\nu)= {\mathbb C}^n, \qquad {\mathcal M}(\Omega ) = L^1(\Omega, \nu )= {\mathbb C}^n, \qquad {\mathcal M}_{+1}(\Omega ) = L^1_{+1}(\Omega, \nu ) \end{align}

For example, put $\Omega_1=\{ \omega_1^1, \omega_1^2, \omega_1^3 \}$ and $\Omega_2=\{ \omega_2^1 , \omega_2^2\}$. And define $\rho_1 (\in {\cal M}_{+1}(\Omega_1))$ such that

\begin{align} \rho_1= a_1 \delta_{\omega_1^1} +a_2 \delta_{\omega_1^2} +a_3 \delta_{\omega_1^3} \quad (0{{\; \leqq \;}}a_1,a_2 ,a_3 {{\; \leqq \;}}1, a_1+a_2 +a_3=1) \end{align}

Then, the dual causal operator ${\Phi}^*_{1,2}: {\cal M}_{+1}(\Omega_1) \to {\cal M}_{+1}(\Omega_2)$ is represented by

\begin{align} {\Phi}^*_{1,2}(\rho_1 ) = & (c_{11}a_1+c_{12}a_2+c_{13}a_3)\delta_{\omega_2^1} + (c_{21}a_1+c_{22}a_2+c_{23}a_3)\delta_{\omega_2^2} \\ \; & (0 {{\; \leqq \;}}c_{ij} {{\; \leqq \;}}1, \sum\limits_{i=1}^2 c_{ij} =1) \end{align}

and, consider the identification:${\cal M}(\Omega_1) {\; \approx \;} {\mathbb C}^3$, ${\cal M}(\Omega_2) {\; \approx \;} {\mathbb C}^2$, That is,

\begin{align} & {\cal M}(\Omega_1) \ni \alpha_1 \delta_{\omega_1^1} +\alpha_2 \delta_{\omega_1^2} + \alpha_3 \delta_{\omega_1^3} \underset{\scriptsize{\mbox{ (identification)}}}{\longleftrightarrow} \left[\begin{array}{l} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \end{array}\right] \in {\mathbb C}^3 \\ & {\cal M}(\Omega_2) \ni \beta_1 \delta_{\omega_2^1} +\beta_2 \delta_{\omega_2^2} \underset{\scriptsize{\mbox{ (identification)}}}{\longleftrightarrow} \left[\begin{array}{l} \beta_1 \\ \beta_2 \\ \end{array}\right] \in {\mathbb C}^2 \end{align}

Then, putting

\begin{align} & {\Phi}^*_{1,2}(\rho_1 ) = \beta_1\delta_{\omega_2^1} + \beta_2\delta_{\omega_2^1} = \left[\begin{array}{l} \beta_1 \\ \beta_2 \\ \end{array}\right], \\ & \rho_1 = \alpha_1 \delta_{\omega_1^1} +\alpha_2 \delta_{\omega_1^2} + \alpha_3 \delta_{\omega_1^3} = \left[\begin{array}{l} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \end{array}\right] \end{align}

write, by matrix representation, as follows.

\begin{align} {\Phi}^*_{1,2} (\rho_1 ) = \left[\begin{array}{l} \beta_1 \\ \beta_2 \\ \end{array}\right] = \left[\begin{array}{lll} c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ \end{array}\right] \left[\begin{array}{l} \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \end{array}\right] \end{align}

Next, from this dual causal operator ${\Phi}^*_{1,2}: {\cal M}_{}(\Omega_1) \to {\cal M}_{}(\Omega_2)$, we will construct a causal operator $\Phi_{1,2}: C_0(\Omega_2) \to C_0(\Omega_1)$. Consider the identification:${C_0}(\Omega_1) {\; \approx \;} {\mathbb C}^3$, ${C_0}(\Omega_2) {\; \approx \;} {\mathbb C}^2$, that is,

\begin{align} {C_0}(\Omega_1) \ni f_1 \underset{\scriptsize{\mbox{(identification)}}}{\longleftrightarrow} \left[\begin{array}{l} f_1 ( \omega_1^1) \\ f_1 ( \omega_1^2) \\ f_1 ( \omega_1^3) \\ \end{array}\right] \in {\mathbb C}^3, \qquad {C_0}(\Omega_2) \ni f_2 \underset{ \scriptsize{\mbox{(identification)}} }{\longleftrightarrow} \left[\begin{array}{l} f_2 ( \omega_2^1) \\ f_2 ( \omega_2^2) \\ \end{array}\right] \in {\mathbb C}^2 \end{align}

Let $f_2 \in C_0(\Omega_2)$, $f_1 = \Phi_{1,2} f_2$. Then, we see

\begin{align} \left[\begin{array}{l} f_1(\omega_1^1) \\ f_1(\omega_1^2) \\ f_1(\omega_1^3) \\ \end{array}\right] = f_1 = \Phi_{1,2} (f_2 ) = \left[\begin{array}{ll} c_{11} & c_{21} \\ c_{12} & c_{22} \\ c_{13} & c_{23} & \\ \end{array}\right] \left[\begin{array}{l} f_2(\omega_2^1) \\ f_2(\omega_2^2) \\ \end{array}\right] \end{align}

Therefore, the relation between the dual causal operator ${\Phi}^*_{1,2}$ and causal operator $\Phi_{1,2}$ is represented as the the transposed matrix.

$\square \quad$

Example 10.9[Deterministic dual causal operator, deterministic causal map, deterministic causal operator] Consider the case that dual causal operator ${\Phi}^*_{1,2}: {\cal M}(\Omega_1) ({\approx} {\mathbb C}^3) \to {\cal M}(\Omega_2) ({\approx} {\mathbb C}^2)$ has the matrix representation such that

\begin{align} {\Phi}^*_{1,2}(\rho_1 ) = \left[\begin{array}{l} b_1 \\ b_2 \\ \end{array}\right] = \left[\begin{array}{lll} 0 & 1& 1 \\ 1 & 0& 0 \\ \end{array}\right] \left[\begin{array}{l} a_1 \\ a_2 \\ a_3 \\ \end{array}\right] \end{align}

In this case, it is the deterministic dual {{causal operator}}. This deterministic causal operator $\Phi_{1,2}: C_0(\Omega_2) \to C_0(\Omega_1)$ is represented by

\begin{align} \left[\begin{array}{l} f_1(\omega_1^1) \\ f_1(\omega_1^2) \\ f_1(\omega_1^3) \\ \end{array}\right] = f_1 = \Phi_{1,2} (f_2 ) = \left[\begin{array}{ll} 0 & 1 \\ 1& 0 \\ 1 & 0 & \\ \end{array}\right] \left[\begin{array}{l} f_2(\omega_2^1) \\ f_2(\omega_2^2) \\ \end{array}\right] \end{align}

with the deterministic causal map $\phi_{1,2}: \Omega_1 \to \Omega_2$ such that

\begin{align} \phi_{1,2}(\omega_1^1)= \omega_2^2, \quad \phi_{1,2}(\omega_1^2)= \omega_2^1, \quad \phi_{1,2}(\omega_1^3)= \omega_2^1 \quad \end{align} 10.2.3: Sequential causal operator---A chain of causalities

Let $(T,\le)$ be a finite tree (in Chapter 14, we discuss the infinite case), i.e., a tree like semi-ordered finite set such that "$t_1 \le t_3$ and $t_2 \le t_3$" implies "$t_1 \le t_2$ or $t_2 \le t_1$"

Assume that there exists an element $t_0 \in T$, called the root of $T$, such that $t_0 \le t$ ($\forall t \in T$) holds.

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. Since we usually consider the subtree $T_{t_0}$ $(\subseteq T)$ with the root $t_0$, we assume that the tree has a root. In this chapter, assume, for simplicity, that $T$ is finite (though it is sometimes infinite in applications).

For simplicity, assume that $T$ is finite, or a finite subtree of a whole tree. Let $T$ $(= \{ 0, 1, ..., N \})$ be a tree with the root $0$. Define the parent map $\pi{}: T \setminus \{ 0 \} \to T$ such that $\pi (t) = \max \{ s \in T{}: s < t \}$. It is clear that the tree $(T\equiv \{ 0,1,..., N\} , \le)$ can be identified with the pair $(T\equiv \{ 0,1,..., N\} , \pi: T \setminus \{ 0 \} \to T)$. Also, note that, for any $t \in T \setminus \{ 0 \}$, there uniquely exists a natural number $h(t)$ $($called the height of $t$ $)$ such that $\pi^{h(t)} (t) = 0$. Here, $\pi^2 (t) = \pi (\pi (t))$, $\pi^3 (t) = \pi (\pi^2 (t))$, etc. Also, put $\{ 0,1,..., N \}^2_{{}_{\le}}$ $=$ $\{ (m,n) \; | \; 0 \le m \le n \le N \}$. In Fig.10.2, see the root $t_0$, the parent map: $\pi(t_3)=\pi(t_4)=t_2$, $\pi(t_2)=\pi(t_5)=t_1$, $\pi(t_1)=\pi(t_6)=\pi(t_7)=t_0$

Figure 10.2:Tree: $(T=\{t_0, t_1, ...,t_7\}, \pi: T \setminus \{t_0\} \to T )$

Definition 10.10 [Sequential causal operator; Heisenberg picture of causality] The family $\{ \Phi_{t_1,t_2}{}:$ $\overline{\mathcal A}_{t_2} \to \overline{\mathcal A}_{t_1} \}_{(t_1,t_2) \in T^2_{\leqq}}$ $\Big($ or, $\{$ $\overline{\mathcal A}_{t_2} \overset{\Phi_{t_1,t_2}}\to \overline{\mathcal A}_{t_1} \}_{(t_1,t_2) \in T^2_{\leqq}}$ $\Big)$ is called a sequential causal operator, if it satisfies that

 (i): For each $t \;(\in T)$, a basic structure $[{\mathcal A}_{t} \subseteq \overline{\mathcal A}_{t} \subseteq B(H_t)]$ is determined. (ii): For each $(t_1,t_2) \in T^2_{\leqq}$, a {{causal operator}} $\Phi_{t_1,t_2}{}: \overline{\mathcal A}_{t_2} \to \overline{\mathcal A}_{t_1}$ is defined such as $\Phi_{t_1,t_2} \Phi_{t_2,t_3} = \Phi_{t_1,t_3}$ $(\forall (t_1,t_2)$, $\forall (t_2,t_3) \in T^2_{\leqq})$. Here, $\Phi_{t,t} : \overline{\mathcal A}_t \to \overline{\mathcal A}_t$ is the identity operator.

Definition 10.11

(i): [pre-dual sequential causal operator : Schrödinger picture of causality ] The sequence $\{ ({\Phi}_{t_1,t_2})_*{}:$ $(\overline{\mathcal A}_{t_1})_* \to (\overline{\mathcal A}_{t_1})_* \}_{(t_1,t_2) \in T^2_{\leqq}}$ is called a pre-dual sequential causal operator of $\{ {\Phi_{t_1,t_2}}{}:$ ${ \overline{\mathcal A}_{t_2}} \to {\overline{\mathcal A}{t_1}} \}_{(t_1,t_2) \in T^2_{\leqq}}$

(ii): [Dual sequential causal operator : Schrödinger picture of causality ] A sequence $\{ {\Phi}^*_{t_1,t_2}{}:$ ${\mathcal A}^\ast_{t_1} \to {\mathcal A}^\ast_{t_1} \}_{(t_1,t_2) \in T^2_{\leqq}}$ is called a dual sequential causal operator of $\{ \Phi_{t_1,t_2}{}:$ ${ \overline{\mathcal A}_{t_2}} \to {\overline{\mathcal A}_{t_1}} \}_{(t_1,t_2) \in T^2_{\leqq}}$.

Figure 10.4: Schrödinger picture} ( dual sequential causal operator)

Remark 10.12 [The Heisenberg picture is formal; the Schrödinger picture is makeshift]

The Schrödinger picture is intuitive and handy. Consider the Schrödinger picture$\{ {\Phi}^*_{t_1,t_2}{}:$ ${\mathcal A}^\ast_{t_1} \to {\mathcal A}^\ast_{t_1} \}_{(t_1,t_2) \in T^2_{\leqq}}$. For $C^*$-mixed state $\rho_{t_1} (\in {\frak S}^m({\mathcal A}_{t_1}^*)$ (i.e., a state at time $t_1$),

 $\bullet$ $C^*$-mixed state $\rho_{t_2} (\in {\frak S}^m({\mathcal A}_{t_2}^*))$ (at time $t_2 (\ge t_1)$) is defined by \begin{align} \rho_{t_2}=\Phi_{t_1, t_2}^* \rho_{t_1} \end{align}

However, the linguistic interpretation says "state does not move", and thus, we consider that

 $\bulllet$ $\left\{\begin{array}{ll} \mbox{ the Heisenberg picture is formal} \\ \\ \mbox{ the Schrödinger picture is makeshift } \end{array}\right.$