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{\large \bf Cubic Matrix, Generalized Spin Algebra and Uncertainty Relation}


\vspace{1cm}
 
{Yoshiharu Kawamura}\footnote{E-mail:
haru@azusa.shinshu-u.ac.jp} 

\vspace{0.7cm}
Department of Physics, Shinshu University,
Matsumoto 390-8621, Japan\\
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\begin{abstract}
We propose a generalization of spin algebra using three-index objects.
There is a possibility that a triple commutation relation among three-index objects
implies a kind of uncertainty relation among their expectation values.
\end{abstract}
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\section{Introduction}

Matrix is an instrument indispensable to mathematics and physics.
One of the main reason is as follows.
Physical systems are often described by many variables, some of which are
treated on an equal footing, e.g. coordinates of space.
The systems have symmetries under some transformations for variables
and such symmetry transformations, in many cases, form a group, e.g., 
rotations for coordinates of space form a rotational group.
Use of matrices makes the analysis with many variables simple and systematic
because an action for group elements can be represented by a matrix.
In fact, the group theoretical analysis has been applied to a wide range of systems, e.g.,
the classification of elementary particles
and the determination of interactions among them.\cite{G}
Considering its success, it is natural to ask the following questions:
\begin{enumerate}
\item Is there a generalization of matrix?

\item If there is a generalization, what advantages and applications for mathematics and physics?
\end{enumerate}

For the first question, one might hit upon a many-index object such as $A_{m_1 m_2 ...m_n}$
since a matrix is regarded as a two-index object.
For the second one, 
a possible answer is that a new mechanics has been proposed 
based on many-index objects\cite{YK}
and its basic structure has been studied from the algebraic point of view\cite{YK2,YK3}.
This mechanics has a counterpart of the canonical structure in classical mechanics
or Nambu mechanics\cite{Nambu},
and is interpreted as its $\lq$quantum' or $\lq$discretized' version.
It is also regarded as a generalization of Heisenberg's matrix mechanics
because a generalization of the Ritz rule is taken as a guiding principle,
This mechanics has such interesting properties, but
it is not clear whether it is applicable to a real physical system
and what physical meanings many-index objects possess.
There is also the possibility of examining an algebraic structure 
of many-index objects independent of its dynamics to provide information 
on their physical reality and implications.
This is a main motivation of our work.

In this paper, we propose a generalization of spin algebra using three-index objects and 
point out a possibility that a triple commutation relation leads to
a kind of uncertainty relation.
% paying our attention to the algebraic structure of three-index object.

This paper is organized as follows.
In the next section, we explain the definition of three-index object.
We study a generalization of spin algebra in $\S$3. 
In section 4, we discuss a connection between a triple commutation relation among three-index objects
and an uncertainty among their expectation values.
Section 5 is devoted to conclusions and discussion.


\section{Cubic matrix}

Here we state our definition of a cubic matrix\footnote{
Awata, Li, Minic and Yoneya introduced many-index objects
to realize the quantum version of Nambu bracket.\cite{ALMY}
They refer to three-index object as $\lq$cubic matrix' and we use
the naming in this paper 
although the definition of triple product is different each other.}
and its related terminology.
A cubic matrix is an object with three indices, $A_{lmn}$, 
which is a generalization
of a usual matrix, such as $B_{mn}$.
We refer to a cubic matrix whose elements possess cyclic symmetry, i.e., $A_{lmn} = A_{mnl} = A_{nlm}$,
as a cyclic cubic matrix.
We define the hermiticity of a cubic matrix by 
$A_{l'm'n'} = A_{lmn}^{*}$ for odd permutations among indices
and refer to a cubic matrix possessing hermiticity as a hermitian cubic matrix.
Here, the asterisk indicates complex conjugation.
A hermitian cubic matrix is a special type of cyclic cubic matrix, because it obeys the relations
$A_{lmn} = A_{mln}^{*} = A_{mnl} = A_{nml}^{*} = A_{nlm} = A_{lnm}^{*}$.
We refer to the following form of a cubic matrix as a normal form or a normal cubic matrix:
\begin{eqnarray}
A_{lmn} = \delta_{lm} a_{mn} + \delta_{mn} a_{nl} + \delta_{nl} a_{lm}  .
\label{AN} 
\end{eqnarray}
A normal cubic matrix is also a special type of cyclic cubic matrix.
The elements of a cubic matrix are treated as $c$-numbers throughout this paper.

We define the triple product among cubic matrices $A_{lmn}$, 
$B_{lmn}$ and $C_{lmn}$ by
\begin{eqnarray}
(ABC)_{lmn} \equiv
\sum_k A_{lmk} B_{lkn} C_{kmn} .
\label{cubicproduct}
\end{eqnarray}
The resultant three-index object, $(A B C)_{lmn}$,
does not always have cyclic symmetry, even if $A_{lmn}$, $B_{lmn}$ and $C_{lmn}$
are cyclic cubic matrices.
Note that this product is, in general, neither commutative nor associative; that is,
$(ABC)_{lmn} \neq (BAC)_{lmn}$ and
$(AB(CDE))_{lmn} \neq (A(BCD)E)_{lmn} \neq ((ABC)DE)_{lmn}$.
The triple-commutator is defined by
\begin{eqnarray}
&~& [A, B, C]_{lmn} \equiv (ABC + BCA 
+ CAB  - BAC - ACB - CBA)_{lmn} .
\label{T-comm}
\end{eqnarray}
The triple-anticommutator is defined by
\begin{eqnarray}
&~& \{A, B, C\}_{lmn} \equiv (ABC + BCA 
+ CAB + BAC + ACB + CBA)_{lmn} .
\label{T-anticomm}
\end{eqnarray}
If $A_{lmn}$, $B_{lmn}$ and $C_{lmn}$ are hermitian cubic matrices,
$i[A, B, C]_{lmn}$ and $\{A, B, C\}_{lmn}$ are also hermitian cubic matrices. 
%If $G^{(N)}_1$ and $G^{(N)}_2$ are normal cubic matrices 
%and $\widetilde{(G^{(N)}_1 G^{(N)}_2)}_{lmn}$ is a 3-cocycle,
%the fundamental identity holds so that
%\begin{eqnarray}
%&~& [[A, B, C], G^{(N)}_1, G^{(N)}_2]_{lmn} =  [[A, G^{(N)}_1, G^{(N)}_2], B, C]_{lmn} \nonumber \\
%&~& ~~~~~~~~~~~~~~~~ + [A, [B, G^{(N)}_1, G^{(N)}_2], C]_{lmn}  + [A, B, [C, G^{(N)}_1, G^{(N)}_2]]_{lmn} .
%\label{C-fund}
%\end{eqnarray}

Further we define the product between cubic matrices $A_{lmn}$ and $B_{lmn}$ by
\begin{eqnarray}
(AB)_{lm} \equiv
\sum_k A_{lmk} B_{klm} .
\label{product}
\end{eqnarray}
If $A_{lmn}$ and $B_{lmn}$
are hermitian cubic matrices,
the two-index object, $(AB)_{lm}$,
possess a hermiticity, i.e., $(AB)_{lm} = (AB)^{*}_{ml}$.
If $A_{lmn}$ and $B_{lmn}$
are cyclic cubic matrices,
they commute with respect to this product, i.e. ,
$(AB)_{lm} = (BA)_{lm}$.


\section{Generalized spin algebra}

\subsection{Spin algebra}

Here we review the spin algebra ${\it su}(2)$.
The algebra is defined by
\begin{eqnarray}
[J^a, J^b] = i \hbar \varepsilon^{abc} J^c ,
\label{spin-alg}
\end{eqnarray}
where $J^a$s are spin variables $(a = 1, 2, 3)$, $\hbar$ is the reduced Planck constant
and $\varepsilon^{abc}$ is the Levi-Civita symbol.
Matrices on the adjoint representation are written by $3 \times 3$ matrices such as
\begin{eqnarray}
(J^a)_{mn} = -i \hbar \varepsilon^{amn} ,
\label{adj}
\end{eqnarray}
where each of the indices $m$ and $n$ runs from 1 to 3.
Further matrices on the spinor representation are written by $2 \times 2$ matrices such as
\begin{eqnarray}
(J^a)_{mn} =  \frac{\hbar}{2} (\sigma^a)_{mn} ,
\label{spinor}
\end{eqnarray}
where $\sigma^a$s are Pauli matrices and each of the indices $m$ and $n$ runs from 1 to 2.

In general, matrices on the spin $j$ representation are given by $N \times N$ matrices:
\begin{eqnarray}
&~& (J^1)_{mn} = \frac{\hbar}{2} \Bigl(\sqrt{n(N-n)} \delta_{m n+1} 
+ \sqrt{m(N-m)} \delta_{m n-1}\Bigr) , \nonumber \\
&~& (J^2)_{mn} = \frac{\hbar}{2i} \Bigl(\sqrt{n(N-n)} \delta_{m n+1} 
- \sqrt{m(N-m)} \delta_{m n-1}\Bigr) , \nonumber \\
&~& (J^3)_{mn} = \frac{\hbar}{2} (2m-N-1) \delta_{mn} ,
\label{N*N}
\end{eqnarray}
where each of the indices $m$ and $n$ runs from 1 to $N = 2j +1$.
The Casimir operator $\vec{J}^2$ is given by
\begin{eqnarray}
(\vec{J}^2)_{mn} \equiv (J^1)^2_{mn} + (J^2)^2_{mn} + (J^3)^2_{mn} 
 = \hbar^2 j (j + 1) \delta_{mn} .
\label{casimir}
\end{eqnarray}
The spinor representation matrices (\ref{spinor}) are obtained from (\ref{N*N}) with $j = \frac{1}{2}$, and
the adjoint representation matrices (\ref{adj}) are obtained from (\ref{N*N}) with $j = 1$
after making a suitable unitary transformation.
 

\subsection{Generalization}

Let us generalize the spin algebra (\ref{spin-alg}) using hermitian cubic matrices.
We consider the following $4 \times 4 \times 4$ matrices as a counterpart of (\ref{adj}),
\begin{eqnarray}
(J^a)_{lmn} = -i \hbar_C \varepsilon^{almn} , ~~
(K^a)_{lmn} = \hbar_C |\varepsilon^{almn}|  ,
\label{adj-cubic}
\end{eqnarray}
where each of the indices $a$, $l$, $m$ and $n$ runs from 1 to 4
and $\hbar_C$ is a new physical constant.
The $(J^a)_{lmn}$ and $(K^a)_{lmn}$ satisfy the following algebra,
\begin{eqnarray}
&~& [J^a, J^b, J^c] = - i \hbar_C^2 \varepsilon^{abcd} K^d , 
~~~~ [J^a, J^b, K^c] = - i \hbar_C^2 \varepsilon^{abcd} J^d ,  \nonumber \\
&~& [J^a, K^b, K^c] =  i \hbar_C^2 \varepsilon^{abcd} K^d ,  
~~~~ [K^a, K^b, K^c] =  i \hbar_C^2 \varepsilon^{abcd} J^d ,
\label{cubic-spin-alg}
\end{eqnarray}
where the indices $l$, $m$ and $n$ are omit.
There exists a subalgebra in (\ref{cubic-spin-alg})
whose elements consist of $J^a$, $J^b$, $J^c$ and $K^d$ (or 
$K^a$, $K^b$, $K^c$ and $J^d$).
Here $a$, $b$, $c$ and $d$ are different numbers among them.
For example, $G^a = (J^1, J^2, J^3, K^4)$ satisfy the algebra:
\begin{eqnarray}
&~& [G^a, G^b, G^c] = - i \hbar_C^2 \varepsilon^{abcd} G^d .
\label{cubic-spin-alg-G}
\end{eqnarray}
We refer to the above algebra (\ref{cubic-spin-alg-G}) as a $\lq$cubic spin algebra'.
The $G^a$s satisfy the so-called $\lq$fundamental indentity':
\begin{eqnarray}
&~& [[G^a, G^b, G^c], G^d, G^e] = [[G^a, G^d, G^e], G^b, G^c] \nonumber\\
&~& ~~~~~~~~~~~~ + [G^a, [G^b, G^d, G^e], G^c] + [G^a, G^b, [G^c, G^d, G^e]] .
\label{fund-id}
\end{eqnarray}

 As a counterpart of spinor representation matrices (\ref{spinor}),
we define four kinds of hermitian $3 \times 3 \times 3$ matrices:
\begin{eqnarray}
&~& (S^1)_{lmn} \equiv \frac{\hbar_C}{\sqrt{2}} |\varepsilon_{lmn}| , 
~~~~ (S^2)_{lmn} \equiv \frac{\hbar_C}{i\sqrt{2}} \varepsilon_{lmn} , 
\nonumber \\
&~& (S^3)_{lmn} \equiv \frac{\hbar_C}{\sqrt{2}} \Bigl(\delta_{lm} \xi_{mn} + \delta_{mn} \xi_{nl}
 + \delta_{nl} \xi_{lm}\Bigr) ,
\nonumber \\
&~& (S^4)_{lmn} \equiv \frac{\hbar_C}{\sqrt{2}} \Bigl(\delta_{lm} \zeta_{mn} 
+ \delta_{mn} \zeta_{nl} + \delta_{nl} \zeta_{lm}\Bigr) ,
\label{S3*3*3}
\end{eqnarray} 
where each of the indices $l$, $m$ and $n$ runs from 1 to 3, 
$\xi_{mn} \equiv (\delta_{m1} - \delta_{m2}) \delta_{n3}$ and  
$\zeta_{mn} \equiv \delta_{m1} \delta_{n2} +  \delta_{m2} \delta_{n1}$.
It is shown that the variables $(S^a)_{lmn}$ yield the cubic spin algebra (\ref{cubic-spin-alg-G}).\footnote{
This choice is not unique, but it belongs to $(S^a)_{lmn}$ with $\xi_{mn} = \epsilon_m \varepsilon_{12n}$
and $\zeta_{mn} = \epsilon_m \varepsilon_{mn3}$. 
Here $\epsilon_m$ takes 1 or $-1$.}

For $(N+1) \times (N+1) \times (N+1)$ matrices ($N \geq 3$) which satisfy the cubic spin algebra
(\ref{cubic-spin-alg-G}), we can find the example:
\begin{eqnarray}
&~& (G^1)_{lmn} = \frac{\hbar_C}{8^{1/4}} \Bigl( (\delta_{lm-1} + \delta_{lm+1} \Bigr.
+ \delta_{lm-N+1} + \delta_{lm+N-1}) \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \cdot (1-\delta_{lN+1})(1-\delta_{mN+1})\delta_{nN+1}  \nonumber \\
&~& ~~~~~~~~~ + (\delta_{mn-1} + \delta_{mn+1}
+ \delta_{mn-N+1} + \delta_{mn+N-1})(1-\delta_{mN+1})(1-\delta_{nN+1})\delta_{lN+1} \nonumber \\
&~& ~~~~~~~~~ + (\delta_{nl-1} + \delta_{nl+1}
\Bigl. + \delta_{nl-N+1} + \delta_{nl+N-1})(1-\delta_{nN+1})(1-\delta_{lN+1})\delta_{mN+1} \Bigr) , \nonumber \\
&~& (G^2)_{lmn} = \frac{\hbar_C}{8^{1/4}i} \Bigl( (\delta_{lm-1} - \delta_{lm+1} \Bigr.
+ \delta_{lm-N+1} - \delta_{lm+N-1}) \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \cdot (1-\delta_{lN+1})(1-\delta_{mN+1})\delta_{nN+1}  \nonumber \\
&~& ~~~~~~~~~ + (\delta_{mn-1} - \delta_{mn+1}
+ \delta_{mn-N+1} - \delta_{mn+N-1})(1-\delta_{mN+1})(1-\delta_{nN+1})\delta_{lN+1} \nonumber \\
&~& ~~~~~~~~~ + (\delta_{nl-1} - \delta_{nl+1}
\Bigl. + \delta_{nl-N+1} - \delta_{nl+N-1})(1-\delta_{nN+1})(1-\delta_{lN+1})\delta_{mN+1} \Bigr) , \nonumber \\
&~& (G^3)_{lmn} = \frac{\hbar_C}{2^{1/4}} \Bigl( \delta_{lm} g^3_{mn}
+ \delta_{mn} g^3_{nl} + \delta_{nl} g^3_{lm} \Bigr) , \nonumber \\
&~& (G^4)_{lmn} = \frac{\hbar_C}{8^{1/4}} \Bigl( \delta_{lm} g^4_{mn}
+ \delta_{mn} g^4_{nl} + \delta_{nl} g^4_{lm} \Bigr) ,
\label{(N+1)*(N+1)*(N+1)}
\end{eqnarray}
where each of the indices $l$, $m$ and $n$ runs from 1 to $N = 2j +1$, and
$g^3_{mn}$ and $g^4_{mn}$ are defined by
\begin{eqnarray}
g^3_{mn} \equiv \epsilon_m (1 - \delta_{m N+1}) \delta_{n N+1} ~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{g3}
\end{eqnarray}
and
\begin{eqnarray}
&~& g^4_{mn} \equiv \epsilon_m (\delta_{mn-1} - \delta_{mn+1} + \delta_{mn-N+1} - \delta_{mn+N-1}) \nonumber \\
&~&  ~~~~~~~~~~~~~~~~~~~~ \cdot (1-\delta_{mN+1})(1-\delta_{nN+1}) ,
\label{g4}
\end{eqnarray}
respectively.
Here $\epsilon_m$ takes 1 or $-1$.

As a comment,
we can find $N \times N$ matrices $(M^a)_{mn}$ which satisfy the algebra
%\begin{eqnarray}
$[M^a, M^b, M^c]_{mn} = - i \hbar^2 \varepsilon^{abcd} (M^d)_{mn}$.
%\label{cubic-spin-alg-M}
%\end{eqnarray}
Here the triple commutator $[M^a, M^b, M^c]_{mn}$ is defined by
\begin{eqnarray}
&~& [M^a, M^b, M^c]_{mn} \equiv (M^aM^bM^c + M^bM^cM^a + M^cM^aM^b \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~  - M^bM^aM^c - M^aM^cM^b - M^cM^bM^a)_{mn} \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~ =  ([M^a, M^b] M^c)_{mn} + ([M^b, M^c] M^a)_{mn} \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + ([M^c, M^a] M^b)_{mn} 
\label{T-comm-matrix}
\end{eqnarray}
with a usual definition of the triple product among matrices:
\begin{eqnarray}
(M^aM^bM^c)_{mn} \equiv \sum_{k,l} (M^a)_{mk} (M^b)_{kl} (M^c)_{ln} . 
\label{T-pro-matrix}
\end{eqnarray}
An example is as follows,
\begin{eqnarray}
(M^a)_{mn} = \frac{1}{(j(j+1))^{1/4}} (J^a)_{mn} , ~~~~~ (M^4)_{mn} = - (j(j+1))^{1/4} \hbar \delta_{mn} ,
\label{Na}
\end{eqnarray}
where $(J^a)_{mn}$ is given in (\ref{N*N}).
%But the following relation, in general, does not hold,
%\begin{eqnarray}
%\delta N^a  \delta N^b \delta N^c \geq \frac{1}{6} |\langle [N^a, N^b, N^c] \rangle| 
%= \frac{1}{6} |\langle N^d \rangle| .
%\label{G-uncertainty3}
%\end{eqnarray}

\section{Uncertainty relation}

\subsection{Uncertainty relation in quantum mechanics}

The uncertainty relation in quantum mechanics is expressed by
\begin{eqnarray}
\delta x \delta p \geq \frac{\hbar}{2}
\label{H-uncertainty}
\end{eqnarray}
where $\delta x$ and $\delta p$ represent an uncertainty of position 
and its canonical momentum, respectively.
%This relation was obtained by several kinds of thought experiments \cite{Heisenberg},
This relation and a generalization are nicely formulated.\cite{Uncertainty}
For any observable $A$, there is a correspondence to a hermitian operator
$\hat{A}$.
The expectation value of $A$ is defined by
\begin{eqnarray}
\langle A \rangle \equiv \int \psi^*(\vec{r}) A \psi(\vec{r}) d^3r , 
\label{<A>}
\end{eqnarray}
where $\psi(\vec{r})$ is a wave function and describes a state of the system.
The uncertainty of the value of measurement for $A$ is defined by
\begin{eqnarray}
\delta A  \equiv \sqrt{\langle (A - \langle A \rangle)^2 \rangle} 
= \sqrt{\langle A^2 \rangle - \langle A \rangle^2} .
\label{deltaA}
\end{eqnarray}
Here $\delta A$ is a standard deviation which  
represents the magnitude of fluctuation about the mean value.
For any observables $A$ and $B$, the following uncertainty relation holds;
\begin{eqnarray}
\delta A  \delta B \geq \frac{1}{2} |\langle [A, B] \rangle| .
\label{H-uncertainty2}
\end{eqnarray}
Let us derive the above relation in the matrix formalism for a later convenience.
The following relationship exists between
the matrix $A_{mn}$ in Heisenberg's matrix mechanics 
and the hermitian operator $\hat{A}$,
\begin{eqnarray}
A_{mn} = \int \phi_m^*(\vec{r}) \hat{A} \phi_n(\vec{r}) d^3r ,
\label{Amn}
\end{eqnarray}
where $\phi_n(\vec{r})$s constitute a complete set of orthonormal functions.\footnote{
Here we treat a case with a discrete spectrum for simplicity, but
it is straightforward to extend a case with a continuous one.} From 
(\ref{<A>}) and (\ref{Amn}), the $\langle A \rangle$ is written down as
\begin{eqnarray}
\langle A \rangle = \sum_{m,n} a_m^* a_n A_{mn}  
\label{<A>-matrix}
\end{eqnarray}
for the wave function $\psi(\vec{r}) = \sum_{n} a_n \phi_n(\vec{r})$. 
In the same way, the expectation value of $\hat{A}\hat{B}$ is written by
\begin{eqnarray}
\langle A B \rangle  = \sum_{m,n}\sum_{k} a_m^* a_n A_{mk} B_{kn} \equiv (\vec{\cal A}, \vec{\cal B}) .
\label{<AB>}
\end{eqnarray}
Here 
%$(\vec{\cal A}, \vec{\cal B})$ stands for an inner product if
$\vec{\cal B}$ stands for a complex vector whose component is $\sum_{n} B_{kn} a_n$.
Then the uncertainty relation (\ref{H-uncertainty2}) is shown by use of
the Schwarz inequality $|\vec{\cal A}|^2 |\vec{\cal B}|^2 \geq |(\vec{\cal A}, \vec{\cal B})|^2$
and the relation $\hat{A}\hat{B} = \frac{1}{2} [\hat{A}, \hat{B}] + \frac{1}{2} \{\hat{A}, \hat{B}\}$.
Hence the uncertainty relation is understood as a consequence of
algebraic relation between physical variables.
% and has a kinematical origin.
If the expectation value $\langle [A, B] \rangle$ does not vanish,
it is not possible to measure the values of $A$ and $B$ simultaneously
as precise as possible.
The uncertainty relation (\ref{H-uncertainty}) is derived
from the commutation relation for $\hat{x}$ and $\hat{p}$, i.e., $[\hat{x}, \hat{p}] = i\hbar$.

\subsection{Generalized uncertainty relation}

We have seen that the uncertainty relation 
$\delta A  \delta B \geq \frac{1}{2} |\langle C \rangle|$
is derived from the commutation relation such as $[A, B]_{mn} = i C_{mn}$ in quantum mechanics.
 Then, we have a question whether there is an uncertainty relation originated from 
the triple commutation relation such as $[A, B, C]_{lmn} = i D_{lmn}$.
(The typical triple commutation relation is the cubic spin algebra discussed in the previous section.)
In the following, we will find that an inequality such as
%\begin{eqnarray}
$\delta A  \delta B \delta C \geq \frac{1}{6} |\langle D \rangle|$
%\label{G-uncertainty}
%\end{eqnarray}
is derived with certain types of definitions for expectation values of many-index objects.
We define the expectation value for a cubic matrix $A_{lmn}$ by
%\footnote{The existence of operator formalism has not known yet.}
\begin{eqnarray}
\langle A \rangle_c \equiv  \sum_{l,m,n} |a_l a_m a_n| 
e^{i(\theta_{lm} + \theta_{mn} + \theta_{nl})} A_{lmn} , 
\label{cubic<A>}
\end{eqnarray}
where $a_l$ is a complex number and
$\theta_{lm}$ is a real antisymmetric object, $\theta_{lm} = - \theta_{ml}$.
Then the expectation value of $(A B C)_{lmn}$ is written by
\begin{eqnarray}
&~& \langle A B C \rangle_c  = \sum_{l,m,n} |a_l a_m a_n| 
e^{i(\theta_{lm} + \theta_{mn} + \theta_{nl})} \sum_{k} A_{lmk} B_{lkn} C_{kmn} \nonumber \\
&~& ~~~~~~~~~~ \equiv \sum_{l,m,n} \sum_k {\cal{A}}^{(k)}_{lm} {\cal{B}}^{(k)}_{nl} {\cal{C}}^{(k)}_{mn} 
= \sum_{k} Tr({\cal{A}}^{(k)} {\cal{C}}^{(k)} {\cal{B}}^{(k)}), 
\label{<ABC>}
\end{eqnarray}
where ${\cal{A}}^{(k)}_{lm} \equiv |a_l a_m|^{1/2} e^{i \theta_{lm}} A_{lmk}$.
Further we define the expectation value for a two-index object $B_{lm}$ by
%, which is constructed from the product (\ref{product}), by
\begin{eqnarray}
\langle B \rangle_s \equiv  \sum_{l, m} |a_l a_m| 
e^{i(\theta_{lm} - \theta_{ml})} B_{lm} . 
\label{square<B>}
\end{eqnarray}
The expectation value of $(A^2)_{lm}$ is written by
\begin{eqnarray}
&~& \langle A^2 \rangle_s  = \sum_{l,m} |a_l a_m|
e^{i(\theta_{lm} - \theta_{ml})} \sum_{k} A_{lmk} A_{klm} \nonumber \\
&~& = \sum_{l,m}\sum_k {\cal{A}}^{(k)}_{lm} {\cal{A}}^{(k)}_{lm}
= \sum_{l,m}\sum_k {\cal{A}}^{(k)}_{lm} {\cal{A}}^{*(k)}_{ml} \equiv |{\cal{A}}^{(k)}|^2.  \label{<A2>}
\end{eqnarray}
In this way, two kinds of expectation values\footnote{
In case that $\theta_{lm} = \frac{1}{2}(\beta_m - \beta_l)$,
the expectation values $\langle A \rangle_c$ and $\langle B \rangle_s$ are reduced to
%\begin{eqnarray}
$\langle A \rangle_c \equiv  \sum_{l,m,n} |a_l a_m a_n|  A_{lmn}$ and
$\langle B \rangle_s \equiv  \sum_{l,m} a_l^* a_m  B_{lm}$
%\label{<A><B>}
%\end{eqnarray}
where $a_m = |a_m| e^{i\beta_m}$.}
are introduced without any guiding principle.
It is an important subject to pursue their physical implication.
By use of the inequality:
\begin{eqnarray}
(\sum_{k_1} |{\cal{A}}^{(k_1)}|^2) (\sum_{k_2} |{\cal{B}}^{(k_2)}|^2) (\sum_{k_3} |{\cal{C}}^{(k_3)}|^2)
\geq |\sum_{k} Tr({\cal{A}}^{(k)} {\cal{C}}^{(k)} {\cal{B}}^{(k)})|^2
\label{G-Schwarz}
\end{eqnarray}
and the relation:
\begin{eqnarray}
\langle A B C \rangle_c  = \frac{1}{6} 
(\langle [A, B, C] \rangle_c + \langle \{A, B, C\} \rangle_c) ,
\label{ABC-rel}
\end{eqnarray}
the following uncertainty relation is derived,
\begin{eqnarray}
\delta A  \delta B \delta C \geq \frac{1}{6} |\langle [A, B, C] \rangle_c| 
= \frac{1}{6} |\langle D \rangle_c| ,
\label{G-uncertainty}
\end{eqnarray}
where the uncertainty $\delta A$ is defined by
\begin{eqnarray}
\delta A  \equiv \sqrt{\langle (A - \langle A \Delta \rangle_s \Delta)^2 \rangle_s} .
\label{deltaA-cubic}
\end{eqnarray}
Here $\Delta_{lmn} = \delta_{lm} \delta_{mn}$ and 
hence $\langle A \Delta \rangle_s = \sum_{m} |a_m|^2 A_{mmm}$.
Note that there is the identity $[A, B, \Delta]_{lmn} =0$
for arbitrary cyclic cubic matrices $A_{lmn}$ and $B_{lmn}$.

Finally we discuss a physical implication on the uncertainty relation
(\ref{G-uncertainty}).
Let us assume that the 4-dimensional space-time coordinates 
are described by cubic matrices $(X^{\mu})_{lmn}$ ($\mu = 0,1,2,3$) which satisfy the following relation,
\begin{eqnarray}
[X^1, X^2, X^3]_{lmn} = - i l_{P}^2 (X^0)_{lmn} ,
\label{space-time-T-comm}
\end{eqnarray}
where $l_P$ is the Planck length defined by $l_P \equiv \sqrt{2 G \hbar/c^3}$.
Here $G$ is the Newton constant and $c$ is a speed of light. From
the above argument, the following uncertainty relation can be derived,
\begin{eqnarray}
\delta X^1 \delta X^2 \delta X^3 \geq \frac{l_P^2}{6} |\langle X^0 \rangle| .
\label{space-time-uncertainty}
\end{eqnarray}
Many people have discussed uncertainty relations
concerning the measurement of space-time distances
based on various kinds of thought experiments.\cite{Review,K,NvD,A-C,NS,Y}
Among them, the relation\cite{K,NvD,A-C,NS} like 
$(\delta r)^3$ \protect\raisebox{-0.5ex}{$\stackrel{\scriptstyle >}{\sim}$} 
$l_P^2 r \sim l_P^2 c \delta t$
is deeply related to ours (\ref{space-time-uncertainty}).
Here $\delta r$ and $\delta t$ are an uncertainty of space distance $r$ and a time period
of observer, respectively.
The Lorentz covariant form has been proposed in Ref.\cite{NS};
\begin{eqnarray}
|\varepsilon_{\mu\nu\rho\sigma} n^{\mu} \delta x_{i_1}^{\nu} \delta x_{i_2}^{\rho} \delta x_{i_3}^{\sigma}|~
\protect\raisebox{-0.5ex}{$\stackrel{\scriptstyle >}{\sim}$}~ l_P^2 \delta x_{i_4}^{\mu} n_{\mu} ,
\label{space-time-uncertainty2}
\end{eqnarray}
where $\delta x_{i}^{\mu}$s are the four vectors to define a space-time volume and
$n_{\mu}$ is any four vector which represents the velocity vector of an observer.
There is a possibility that the space-time uncertainty relation
(\ref{space-time-uncertainty2}) has the origin from the algebraic relation such as
\begin{eqnarray}
[X^{\mu}, X^{\nu}, X^{\rho}]  = - i l_{P}^2 \varepsilon^{\mu\nu\rho\sigma} X_{\sigma} .
\label{X-rel}
\end{eqnarray}
%where $X^{\mu}$s are cubic matrices representing space-time cooridinates.

\section{Conclusions and discussion}

We have studied a generalization of spin algebra using cubic matrices and 
pointed out a possibility that a triple commutation relation among cubic matrices implies
a kind of uncertainty relation among their expectation values.
As a physical implication, we suggest that the space-time uncertainty relation
can be connected to the triple commtation relation
such as $[X^1, X^2, X^3]_{lmn} = - i l_{P}^2 (X^0)_{lmn}$.

The physical meanings of cubic matrices have not been completely understood and
there still exist several questions.
The $(J^a)_{mn}$s are representation matrices that operate a representation space called spin space.
We have a question whether $(G^a)_{lmn}$ also act as generators or not and
what kind of representation space exists.
In quantum mechanics, the matrix element $A_{mn}$ is interpreted 
as a probability amplitude between the state 
described by $\phi_m$ and that described by $\phi_n$.
However the meaning of the cubic matrix element $A_{lmn}$ has not been known yet.
Further the derivation of uncertainty relation (\ref{G-uncertainty}) seems tricky

because the definition of expectation values looks ad hoc.
We need to find the meaning of $a_l$ and $\theta_{lm}$.
The physical implication on expectation values could be a key to
find a way out.
It is also an interesting subject to find the relationship 
between the space-time uncertainty relations derived
from string/M theories \cite{Y} and ours (\ref{space-time-uncertainty}).

\section*{Acknowledgements}
We would like to thank Professor S. Odake for useful discussions. 


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\end{thebibliography}


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