2.10.1: de Broglie paradox in $B({\mathbb C}^2)$

Axiom1 (measurement in $\S$2.7) includes the paradox ( that is, so called de Broglie paradox :"there is something faster than light"). In what follows, we shall explain de Broglie paradox in $B({\mathbb C}^2)$, though the original idea is mentioned in $B(L^2({\mathbb R}))$ (cf. $\S$11.2). Also, it should be noted that the argument below is essentially the same as the Stern=Gerlach experiment.

Example 2.37 [de Broglie paradox in $B({\mathbb C}^2)$ ] Let $H$ be a two dimensional Hilbert space, i.e., $H={\mathbb C}^2$. Consider the quantum basic structure:

\begin{align*} [B({\mathbb C}^2)\subseteq B({\mathbb C}^2 ) \subseteq {B({\mathbb C}^2 )}] \end{align*} Now consider the situation in the following Figure 2.11.
Let us explain this figure in what follows. Let $f_1, f_2 \in H$ such that \begin{align*} f_1 = \left[\begin{array}{l} 1 \\ 0 \end{array}\right] \in {\mathbb C}^2 , \qquad f_2 = \left[\begin{array}{l} 0 \\ 1 \end{array}\right] \in {\mathbb C}^2 \end{align*} Put \begin{align*} u=\frac{f_1 +f_2}{{\sqrt 2}} \end{align*}

Thus, we have the state $\rho = |u \rangle \langle u |$ $(\in {\frak S}^p(B({\mathbb C}^2)))$. Let $U (\in B({\mathbb C}^2 ))$ be an unitary operator such that

\begin{align*} U = \left[\begin{array}{ll} 1 & 0 \\ 0 & e^{i\pi/2 } \end{array}\right] \quad \end{align*} and let $\Phi: B({\mathbb C}^2) \to B({\mathbb C}^2)$ be the homomorphism such that \begin{align*} \Phi(F) = U^* F U \qquad (\forall F \in B({\mathbb C}^2) ) \end{align*}

Consider the observable ${\mathsf O}_f=(\{1,2\}, 2^{\{1,2\}}, F)$ in $B({\mathbb C}^2 )$ such that

\begin{align*} & F(\{1\}) = |f_1 \rangle \langle f_1 | , \quad F(\{ 2 \}) = |f_2 \rangle \langle f_2 | \end{align*} and thus, define the observable $\Phi {\mathsf O}_f=(\{1,2\}, 2^{\{1,2\}}, \Phi F)$ by \begin{align*} \Phi F (\Xi) = U^* F(\Xi ) U \qquad (\forall \Xi \subseteq \{1, 2\} ) \end{align*}

Let us explain Figure 2.38. The photon P with the state $u=\frac{1}{\sqrt{2}}( f_1+f_2 )$ ( precisely, $|u \rangle \langle u |$ ) rushed into the half-mirror 1

 $(A_1):$ the $f_1$ part in $u$ passes through half-mirror 1, and goes along course 1 to photon detector $D_1$. $(A_2):$ the $f_2$ part in $u$ rebounds on half-mirror 1 (and strictly saying, the $f_2$ changes to ${\sqrt{-1}}f_2$, we are not concerned with it ), and goes along course 2 to photon detector $D_2$.
Thus, we have the measurement: \begin{align} {\mathsf M}_{B({\mathbb C}^2)} ( \Phi{\mathsf O}_f, S_{[\rho]} ) \tag{2.77} \end{align} And thus, we see:
 $(B):$ The probability that a $\left[\begin{array} \mbox{\rm measured\;\; value }1 \\ \mbox{measured value }2 \end{array}\right]$ is obtained by the measurement ${\mathsf M}_{B({\mathbb C}^2)} ( \Phi{\mathsf O}_f, S_{[\rho]} )$@is given by \begin{align*} \left[\begin{array}{l} \mbox{Tr}(\rho \cdot \Phi F(\{1\}) ) \\ \mbox{Tr}(\rho \cdot \Phi F(\{2\}) ) \end{array}\right] = \left[\begin{array}{l} \langle u, \Phi F(\{1\})u \rangle \\ \langle u, \Phi F(\{2\}) u \rangle \end{array}\right] = \left[\begin{array}{l} \langle Uu, F(\{1\}) Uu \rangle \\ \langle Uu, F(\{2\}) U u \rangle \end{array}\right] = \left[\begin{array}{l} | \langle u, f_1 \rangle|^2 \\ | \langle u, f_2 \rangle|^2 \end{array}\right] = \left[\begin{array}{l} \frac{1}{2} \\ \frac{1}{2} \end{array}\right] \end{align*}
This is easy, but it is deep in the following sense.
 $(C):$ @Assume that Detector $D_1$ and Detector $D_2$ are very far apart. And assume that photon P is discovered at detector $D_1$.@Then, we are troubled if photon P is also discovered at detector $D_2$.@Thus, in order to avoid this difficulty,@photon P (discovered at detector $D_1$)@has to eliminate the wave function $\frac{\sqrt{-1}}{\sqrt{2}}f_2$@in an instant.@In this sense, the (B) implies that \begin{align*} \mbox{ there may be something faster than light } \end{align*}

 $\qquad \qquad$Fig. 1.1; history of world-descriptions
This is the de Broglie paradox. From the view point of quantum language,@we give up to solve the paradox, that is, we ( as well as Mermin ) declare

Stop being bothered!

Or

If you want to be bothered,
be bothered in physics D in Figure 1.1 ( in $\S$1.1) !

$\fbox{Note 2.8}$The de Broglie paradox@(i.e.,@there may be something faster than light@)@always appears in quantum mechanics.@For example,@the readers should confirm that@it appears in Example 2.36@(Schtern-Gerlach experiment).@I think that
 $\bullet$ the de Broglie paradox is the only paradox in quantum mechanics
The readers will find that the other paradoxes ( see "paradox" in the index of this book ) in quantum mechanics are solved in this book.