4.3.2: The mathematical formulation of Heisenberg's uncertainty principle
4.3.2.1 Preparation In this section, we shall propose the mathematical formulation ( Theorem 4.15) of Heisenberg's uncertainty principle (Proposition 4.10). Consider the quantum basic structure: \begin{align} [{\mathcal C}(H) \subseteq B(H) \subseteq {B(H)} ] \end{align} Let $A_i$ $(i=1,2)$ be arbitrary self-adjoint operator on $H$. For example, it may satisfy that \begin{align} [A_1 , A_2](:=A_1 A_2 - A_2 A_1 ) =\hbar \sqrt{-1}I \end{align} Let ${\mathsf O}_{A_i}=({\mathbb R}, {\cal B}, F_{A_i} )$ be the spectral representation of $A_i$, i.e., $A_i=\int_{\mathbb R} \lambda F_{A_i}( d \lambda )$, which is regarded as the projective observable in $B(H)$. Let $\rho_0= |u\rangle \langle u |$ be a state, where $u \in H$ and $\|u\|=1$. Thus, we have two measurements:
 (B1): ${\mathsf{M}}_{B(H)} ({\mathsf{O}_{A_1}}{\; :=} ({\mathbb R}, {\cal B}, F_{A_1} ),$ $S_{[\rho_u]})$ $\qquad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}} \langle u, A_1 u \rangle$

(B2):${\mathsf{M}}_{B(H)} ({\mathsf{O}_{A_2}}{\; :=} ({\mathbb R}, {\cal B}, F_{A_2} ),$ $S_{[\rho_u]})$ $\qquad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}}\langle u, A_2 u \rangle$
\begin{align} (\forall \rho_u= |u\rangle \langle u | \in {\frak S}^p({\mathcal C}(H)^*)) \end{align}

However, since it is not always assumed that $A_1 A_2 - A_2A_1=0$, we can not expect the existence of the simultaneous observable ${\mathsf{O}_{A_1}}\times {\mathsf{O}_{A_2}}$, namely,

 $\bullet$ in general, two observables ${\mathsf{O}_{A_1}}$ and ${\mathsf{O}_{A_2}}$ can not be simultaneously measured
That is,
 $(B3):$ the measurement ${\mathsf{M}}_{B(H)} ({\mathsf{O}_{A_1}}\times {\mathsf{O}_{A_2}},$ $S_{[\rho_u]})$ is impossible, Thus, we have the question: \begin{align} \mbox{ Then, what should be done? } \end{align}

In what follows, we shall answer this. Let $K$ be another Hilbert space, and let $s$ be in $K$ such that $\| s \|=1$. Thus, we also have two observables ${\mathsf{O}_{A_1 \otimes I}}{\; :=} ({\mathbb R}, {\cal B}, F_{A_1} \otimes I )$ and ${\mathsf{O}_{A_2\otimes I}}{\; :=} ({\mathbb R}, {\cal B}, F_{A_2}\otimes I )$ in the tensor algebra $B(H \otimes K)$.

Put \begin{align} \mbox{ the tensor state ${\widehat \rho}_{us}=|u \otimes s \rangle \langle u \otimes s|$} \end{align} And we have the following two measurements:

 $(C_1):$ ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{A_1 \otimes I}},S_{[{\widehat \rho}_{us}]})$ $\qquad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}} \langle u, A_1 u \rangle$ $(C_2)$: ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{A_2 \otimes I}},S_{[{\widehat \rho}_{us}]})$ $\qquad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}} \langle u, A_1 u \rangle$
It is a matter of course that \begin{align} \mbox{(C$_1$)=(B$_1$) $\quad$ (C$_2$)=(B$_2$)} \end{align} and
$(C_3)$:${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{A_1\otimes I}}\times {\mathsf{O}_{A_2\otimes I}},$ $S_{[{\widehat{\rho}_{us}}]})$ is impossible.
Thus, overcoming this difficulty, we prepare the following idea:
Preparation 4.11 Let ${\widehat A}_i$ $(i=1,2)$ be arbitrary self-adjoint operator on the tensor Hilbert space $H \otimes K$, where it is assumed that \begin{align} [{\widehat A}_1, {\widehat A}_2](:= {\widehat A}_1{\widehat A}_2- {\widehat A}_2{\widehat A}_1)=0 \quad \mbox{(i.e., the commutativity)} \tag{4.21} \end{align} Let ${\mathsf O}_{{\widehat A}_i}=({\mathbb R}, {\cal B}, F_{{\widehat A}_i} )$ be the spectral representation of ${\widehat A}_i$, i.e.${\widehat A}_i=\int_{\mathbb R} \lambda F_{{\widehat A}_i} ( d \lambda )$, which is regarded as the projective observable in $B(H \otimes K)$. Thus, we have two measurements as follows:
 $(D_1):$ ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_1}},S_{[{\widehat \rho}_{us}]})$ $\quad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}}\langle u\otimes s,\widehat{A}_1( u\otimes s ) \rangle$ $(D_2)$: ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_2}},S_{[{\widehat \rho}_{us}]})$ $\quad \xrightarrow[\scriptsize{\mbox{ expectation}}]{\scriptsize{\mbox{ by (4.17)}}}\langle u\otimes s,\widehat{A}_2( u\otimes s ) \rangle$
Note, by the commutative condition (4.21), that the two can be measured by the simultaneous measurement ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_1}}\times{\mathsf{O}_{{\widehat A}_2}},S_{[{\widehat \rho}_{us}]})$, where ${\mathsf{O}_{{\widehat A}_1}}\times{\mathsf{O}_{{\widehat A}_2}}=({\mathbb R}^2, {\cal B}^2, F_{{\widehat A}_1} \times F_{{\widehat A}_2} )$. Again note that any relation between $A_i \otimes I$ and ${\widehat A}_i$ is not assumed. However,
 ${}$ we want to regard this simultaneous measurement as the substitute of the above two (C$_1$) and (C$_2$). That is, we want to regard (D$_1$) and (D$_2$) as the substitute of (C$_1$) and (C$_2$)
For this, we have to prepare Hypothesis 4.9 below.
Putting \begin{align} {\widehat N}_i := {\widehat A}_i -A_i \otimes I \quad (\mbox{and thus, } {\widehat A}_i={\widehat N}_i +A_i \otimes I) \tag{4.22} \end{align} we define the $\Delta_{\widehat{N}_i}^{{\widehat{\rho}_{us}}}$ and ${\overline \Delta}_{\widehat{N}_i}^{{\widehat{\rho}_{us}} }$ such that \begin{align} \Delta_{\widehat{N}_i}^{u \otimes s} =& \| {\widehat N}_i (u \otimes s) \| = \| ({\widehat A}_i -A_i \otimes I) (u \otimes s ) \| \tag{4.23} \\ {\overline \Delta}_{\widehat{N}_i}^{u \otimes s} =& \| ( {\widehat N}_i - \langle u \otimes s , {\widehat N}_i (u \otimes s)\rangle ) (u \otimes s) \| \nonumber \\ = & \| ( ({\widehat A}_i -A_i \otimes I) - \langle u \otimes s , ({\widehat A}_i -A_i \otimes I) (u \otimes s)\rangle ) (u \otimes s) \| \end{align} where the following inequality: \begin{align} \Delta_{\widehat{N}_i}^{{\widehat{\rho}_{us}}} \ge {\overline \Delta}_{\widehat{N}_i}^{{\widehat{\rho}_{us}}} \tag{4.24} \end{align} is common sense. By the commutative condition (4.21), (4.22) implies that \begin{align} [{\widehat N}_1,{\widehat N}_2] + [{\widehat N}_1, A_2 \otimes I]+[A_1 \otimes I ,{\widehat N}_2] = -[A_1 \otimes I, A_2 \otimes I] \tag{4.25} \end{align}

Here, we should note that the first term (or, precisely, $| \langle u \otimes s , [\mbox{the first term}] ( u \otimes s) \rangle |$ ) of (4.25) can be, by the Robertson uncertainty relation (Theorem 4.9), estimated as follows:

\begin{align} 2 {\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot {\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} \ge | \langle u \otimes s , [{\widehat N}_1,{\widehat N}_2] ( u \otimes s) \rangle | \tag{4.26} \end{align} 4.3.2.2: Average value coincidence conditions; approximately simultaneous measurement
However, it should be noted that \begin{align} \mbox{In the above, any relation between@$A_i \otimes I$@and@${\widehat A}_i$@is not assumed. } \end{align} Thus,@we think that@the following hypothesis is natural.
Hypothesis@4.12@[Average value coincidence conditions ]. @We assume that \begin{align} & \langle u \otimes s, {\widehat N}_i(u \otimes s) \rangle =0 \qquad ( \forall u \in H, i=1,2) \tag{4.27} \end{align} or equivalently, \begin{align} \langle u \otimes s, {\widehat A}_i(u \otimes s) \rangle = \langle u , {A}_i u \rangle \qquad ( \forall u \in H, i=1,2) \tag{4.28} \end{align} That is, \begin{align} &\mbox{the average measured value of ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_i}},S_{[{\widehat \rho}_{us}]})$} \\ = & \langle u \otimes s, {\widehat A}_i(u \otimes s) \rangle \\ = & \langle u , {A}_i u \rangle \\ = & \mbox{the average measured value of ${\mathsf{M}}_{B(H)} ({\mathsf{O}_{{A}_i}},S_{[{ \rho}_{u}]})$} \\ & \quad ( \forall u \in H, ||u||_H =1, i=1,2) \end{align}

Hence, we have the following definition.
Definition@4.13@[Approximately simultaneous measurement]@Let $A_1$ and@$A_2$ be (unbounded) self-adjoint operators@on a Hilbert space $H$.@The quartet $(K, s, \widehat{A}_1, \widehat{A}_2)$@is called @an approximately simultaneous observable@of@$A_1$ and $A_2$, if it satisfied that
 $(E_1):$ @$K$ is a Hilbert space.@$s \in K$, $\| s \|_K=1$,$\widehat{A}_1$ and $\widehat{A}_2$@are commutative self-adjoint operators on@a tensor Hilbert space $H \otimes K$@that satisfy@the average value coincidence condition i4.27),@that is, \begin{align} \langle u \otimes s, {\widehat A}_i(u \otimes s) \rangle = \langle u , {A}_i u \rangle \qquad ( \forall u \in H, i=1,2) \tag{4.29} \end{align}
Also,the measurement@${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_1}}\times{\mathsf{O}_{{\widehat A}_2}},S_{[{\widehat \rho}_{us}]})$@is called the approximately simultaneous measurement@of@${\mathsf{M}}_{B(H)} ({\mathsf{O}_{A_1}},$@$S_{[\rho_u]})$@and@${\mathsf{M}}_{B(H)} ({\mathsf{O}_{A_2}},$@$S_{[\rho_u]})$.
Thus, under the average coincidence condition, we regard
(D$_1$) and (D$_2$) as the substitute of (C$_1$) and (C$_2$)
And
 $(E_2):$ ${\Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}}$ $(= \| (\widehat{A}_1-A_1 \otimes I)(u \otimes s) \| )$ and ${ \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}}$ $(= \| (\widehat{A}_2-A_2 \otimes I)(u \otimes s) \| )$ are called errors of the approximate simultaneous measurement ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_1}}\times{\mathsf{O}_{{\widehat A}_2}},S_{[{\widehat \rho}_{us}]})$
Lemma 4.14 Let $A_1$ and $A_2$ be (unbounded) self-adjoint operators on a Hilbert space $H$. And let $(K, s, \widehat{A}_1, \widehat{A}_2)$ be an approximately simultaneous observable of $A_1$ and $A_2$. Then, it holds that \begin{align} & \Delta_{\widehat{N}_i}^{{\widehat{\rho}_{us}}}= {\overline \Delta}_{\widehat{N}_i}^{{\widehat{\rho}_{us}}} \tag{4.30} \\ & \langle u \otimes s, [{\widehat N}_1, A_2 \otimes I](u \otimes s) \rangle = 0 \qquad ( \forall u \in H) \tag{4.31} \\ & \langle u \otimes s, [A_1 \otimes I, {\widehat N}_2](u \otimes s) \rangle =0 \quad ( \forall u \in H) \tag{4.32} \end{align} \end{Lemma} The proof is easy, thus, we omit it.
Under the above preparations, we can easily get "Heisenberg's uncertainty principle" as follows. \begin{align} & {\Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot { \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} (= {\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot {\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} ) \ge \frac{1}{2} | \langle u , [A_1,A_2] u \rangle | \quad ( \forall u \in H \mbox{ such that } ||u||=1 ) \tag{4.33} \end{align} Summing up, we have the following theorem:
Theorem 4.15 [The mathematical formulation of Heisenberg's uncertainty principle] Let $A_1$ and $A_2$ be (unbounded) self-adjoint operators on a Hilbert space $H$. Then. we have the followings:
 $(i):$ There exists an approximately simultaneous observable $(K, s, \widehat{A}_1, \widehat{A}_2)$ of $A_1$ and $A_2$, that is, $s \in K$, $\| s \|_K=1$,$\widehat{A}_1$ and $\widehat{A}_2$ are commutative self-adjoint operators on a tensor Hilbert space $H \otimes K$ that satisfy the average value coincidence condition (4.27). Therefore, the approximately simultaneous measurement ${\mathsf{M}}_{B(H\otimes K)} ({\mathsf{O}_{{\widehat A}_1}}\times{\mathsf{O}_{{\widehat A}_2}},S_{[{\widehat \rho}_{us}]})$ exists. $(ii)$: And further, we have the following inequality (i.e., Heisenberg's uncertainty principle). \begin{align} {\Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot { \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} (= {\overline \Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot {\overline \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} ) & = \| (\widehat{A}_1-A_1 \otimes I)(u \otimes s) \| \cdot \| (\widehat{A}_2-A_2 \otimes I)(u \otimes s) \| \nonumber \\ & \ge \frac{1}{2} | \langle u , [A_1,A_2] u \rangle | \quad ( \forall u \in H \mbox{ such that } ||u||=1 ) \tag{4.34} \end{align} $(iii)$: In addition, if $A_1 A_2 - A_2 A_1 = \hbar \sqrt{-1}$, we see that \begin{align} {\Delta}_{\widehat{N}_1}^{{\widehat{\rho}_{us}}} \cdot { \Delta}_{\widehat{N}_2}^{{\widehat{\rho}_{us}}} \ge \hbar/2 \quad ( \forall u \in H \mbox{ such that } ||u||=1 ) \tag{4.35} \end{align}
For the proof of (i) and (ii), see
 $(\sharp):$ S. Ishikawa, Rep. Math. Phys. Vol.29(3), 1991, pp.257--273, doi: 10.1016/0034-4877(91)90046-P $(\sharp)$: S. Ishikawa, Mathematical Foundations of Measurement Theory,Keio University Press Inc. 2006, (335 pages)

Note that Theorem 4.15 says that
 $(\sharp):$ Heisenberg's indeterminacy principle had not been used as "scientific proposition" until 1991.
Therefore, we have the following question:
 $(\sharp):$ Had Heisenberg's indeterminacy principle been used as "scientific proposition"?
Now I have a clear answer for this question: