$\mathbb{Z}_{2}^{3}$-Graded Extensions of Lie Superalgebras and Superconformal
Quantum Mechanics

$\boldsymbol{\mathbb{Z}_{2}^{3}}$-Graded Extensions of Lie Superalgebras
and Superconformal Quantum Mechanics

Shunya DOI and Naruhiko AIZAWA S. Doi and N. Aizawa Department of Physical
Science, Osaka Prefecture University,
Nakamozu Campus, Sakai, Osaka 599-8531, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received March 24, 2021, in final form July 14, 2021; Published online July
20, 2021

Quantum mechanical systems whose symmetry is given by
$\mathbb{Z}_{2}^{3}$-graded version of superconformal algebra are introduced.
This is done by finding a realization of a $\mathbb{Z}_{2}^{3}$-graded Lie
superalgebra in terms of a standard Lie superalgebra and the Clifford algebra.
The realization allows us to map many models of superconformal quantum
mechanics (SCQM) to their $\mathbb{Z}_{2}^{3}$-graded extensions. It is
observed that for the simplest SCQM with $\mathfrak{osp}(1|2)$ symmetry there
exist two inequivalent $\mathbb{Z}_{2}^{3}$-graded extensions. Applying the
standard prescription of conformal quantum mechanics, spectrum of the SCQMs
with the $\mathbb{Z}_{2}^{3}$-graded $\mathfrak{osp}(1|2)$ symmetry is
analyzed. It is shown that many models of SCQM can be extended to
$\mathbb{Z}_{2}^{n}$-graded setting.

graded Lie superalgebras; superconformal mechanics

17B75; 17B81; 81R12

##  1 Introduction

In the recent works [2, 3, 14], $\mathbb{Z}_{2}^{n}$-graded extensions of
supersymmetric quantum mechanics (SQM) were introduced and their properties
were investigated ($\mathbb{Z}_{2}^{n}$ denotes the direct product of $n$
copies of the Abelian group $\mathbb{Z}_{2}$). They are a quantum mechanical
realization of $\mathbb{Z}_{2}^{n}$-graded version of supersymmetry algebra
introduced by Bruce [12] (see also [18]), i.e., the Hamiltonian is a matrix
differential operator acting on a $\mathbb{Z}_{2}^{n}$-graded Hilbert space
and the symmetry is given by a $\mathbb{Z}_{2}^{n}$-graded Lie superalgebra.
As a $\mathbb{Z}_{2}^{n}$-graded Lie superalgebra (see Appendix for
definition) is an extension of Lie superalgebra to more complex grading
structure [23, 24, 25, 26], the $\mathbb{Z}_{2}^{n}$-graded SQM is a natural
generalization of the standard SQM. It is observed in [2, 3] that the
$\mathbb{Z}_{2}^{n}$-graded SQM is constructed by a combination of the
standard SQM and Clifford algebras. In fact, it is known that a tensor product
of a Clifford algebra and a standard Lie superalgebra realizes a
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebra [1, 25]. Such realization is not
unique since for a given Lie superalgebra there exist some distinct ways of
tensoring Clifford algebras. Usually, the distinct tensoring produces
inequivalent $\mathbb{Z}_{2}^{n}$-graded extensions of the Lie superalgebra.
However, it can happen that those $\mathbb{Z}_{2}^{n}$-graded extensions are
identical and the different tensoring produces inequivalent representations of
a single $\mathbb{Z}_{2}^{n}$-graded Lie superalgebra. This is the case of
$\mathbb{Z}_{2}^{n}$-graded SQM studied in [2] where tensor product of a
standard SQM and a sequence of Clifford algebras gives inequivalent
representations of the $\mathbb{Z}_{2}^{n}$-graded version of supersymmetry
algebra.

The realizations in [2] are restricted to the standard SQM and it is not clear
that it can be applicable to other Lie superalgebras. On the other hand, the
realization presented in [3] is applied to a larger class of Lie superalgebras
though it produce only $\mathbb{Z}_{2}^{2}$-graded extensions. Thus one can
use it to define a $\mathbb{Z}_{2}^{2}$-graded extension of superconformal
quantum mechanics (SCQM). It is shown that by this realization many models of
the standard SCQM are mapped to their $\mathbb{Z}_{2}^{2}$-graded extension.
The simplest case, $\mathbb{Z}_{2}^{2}$-graded $\mathfrak{osp}(1|2)$ SCQM, is
investigated in some detail [3] and an abstract representation theory of the
$\mathbb{Z}_{2}^{2}$-graded $\mathfrak{osp}(1|2)$ is developed in [10] where
the richness of irreducible representations of the $\mathbb{Z}_{2}^{2}$-graded
$\mathfrak{osp}(1|2)$ is observed.

As a continuation of the works on quantum mechanical realizations of
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebras, in the present work we explore
$\mathbb{Z}_{2}^{n}$-graded version of SCQM and present models of
$\mathbb{Z}_{2}^{3}$-graded SCQM explicitly. Although we focus on
$\mathbb{Z}_{2}^{3}$-graded SCQM, models of $\mathbb{Z}_{2}^{n}$-graded SCQM
for any $n$ are also introduced. In fact, our result is more general since we
start with a new way of mapping a standard Lie superalgebra to its higher
graded version. This means that higher graded extensions of physically
relevant algebras such as super-Poincaré, super-Schrödinger etc are also
obtained in our formalism.

The present work is motivated by the recent renewed interest in
$\mathbb{Z}_{2}^{n}$-graded superalgebras in physics and mathematics. In
physics side, they give a new symmetry different from the ones generated by
Lie algebras and superalgebras. Here we mention only some of them. It was
found that symmetries of some differential equations such as Lévy-Leblond
equation (non-relativistic Dirac equation) are generated by
$\mathbb{Z}_{2}^{2}$-graded Lie superalgebras [5, 6]. Some supersymmetric
classical theories are extended to $\mathbb{Z}_{2}^{2}$-graded setting [8, 9,
13]. It is shown that non-trivial physics can be detected in the multiparticle
sector of the $\mathbb{Z}_{2}^{2}$-graded SQM [28].
$\mathbb{Z}_{2}^{2}$-Graded version of spacetime symmetries are proposed by
several authors, e.g., [27]. In mathematics side, $\mathbb{Z}_{2}^{n}$-graded
supergeometry which is an extension of supergeometry on supermanifolds, is
studied extensively, see, e.g., [22]. More exhaustive list of references of
physical and mathematical aspects of $\mathbb{Z}_{2}^{n}$-graded Lie
superalgebras is found in [10].

We organise this paper as follows: We start Section 2 with the definition of
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebra. Then we review briefly the
results of [2] on $\mathbb{Z}_{2}^{n}$-graded SQM. An emphasis is put on the
fact that there exists a sequence of inequivalent models of
$\mathbb{Z}_{2}^{n}$-graded SQM for a given standard SQM because we also
consider the sequence of $\mathbb{Z}_{2}^{3}$-graded SCQM in this work. In
Section 3 the first member of the sequence (there are three members), ${\rm
Cl}(4)$ model, is presented. We give a realization of a
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebra in terms of the Clifford algebra
${\rm Cl}(4)$ and a standard Lie superalgebra. This realization is applied to
$\mathfrak{osp}(1|2)$ SCQM, then we obtain its $\mathbb{Z}_{2}^{3}$-graded
extension. The spectrum of $\mathbb{Z}_{2}^{3}$-graded $\mathfrak{osp}(1|2)$
SCQM is investigated by employing the standard procedure of conformal quantum
mechanics. In Section 4 other two members of the sequence, for which ${\rm
Cl}(6)$ is used, are considered and it will be shown that one of them is
irrelevant as it does not give an irreducible representation of
$\mathbb{Z}_{2}^{3}$-graded $\mathfrak{osp}(1|2)$. For the relevant one which
defines an another $\mathbb{Z}_{2}^{3}$-graded extension of
$\mathfrak{osp}(1|2)$, the same analysis as ${\rm Cl}(4)$ case is repeated. We
close the paper with some remarks in Section 5.

##  2 Preliminaries

###  2.1 $\boldsymbol{\mathbb{Z}_{2}^{3}}$-graded Lie superalgebras

We define the $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra according to [24,
25]. Let $\vec{a}=(a_{1},a_{2},a_{3})$, $\vec{b}=(b_{1},b_{2},b_{3})$ be
elements of $\mathbb{Z}_{2}^{3}$. Here we regard an element of
$\mathbb{Z}_{2}^{3}$ as a three-dimensional vector and their sum and inner
product are computed in modulus 2

$\vec{a}+\vec{b}=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3}),\qquad\vec{a}\cdot\vec{b}=\sum_{k=1}^{3}a_{k}b_{k}.$

We also introduce the parity of $\vec{a}$ defined by

$|\vec{a}|:=\sum_{k=1}^{3}a_{k}\mod 2.$

Consider a complex vector space $\mathfrak{g}$ consisting of eight subspaces
each of which is labelled by an element of $\mathbb{Z}_{2}^{3}$:

$\mathfrak{g}=\mathfrak{g}_{(0,0,0)}\oplus\mathfrak{g}_{(0,0,1)}\oplus\mathfrak{g}_{(0,1,0)}\oplus\mathfrak{g}_{(1,0,0)}\oplus\mathfrak{g}_{(0,1,1)}\oplus\mathfrak{g}_{(1,0,1)}\oplus\mathfrak{g}_{(1,1,0)}\oplus\mathfrak{g}_{(1,1,1)}.$

The vector space $\mathfrak{g}$ is refereed to as a
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebra if its elements are closed in
commutator or anticommutator which is chosen according to the following rule

$\llbracket
X_{\vec{a}},X_{\vec{b}}\rrbracket:=\begin{cases}[X_{\vec{a}},X_{\vec{b}}],&\vec{a}\cdot\vec{b}=0,\vspace{1mm}\\\
\\{X_{\vec{a}},X_{\vec{b}}\\},&\vec{a}\cdot\vec{b}=1,\end{cases}\qquad
X_{\vec{a}}\in\mathfrak{g}_{\vec{a}},\qquad
X_{\vec{b}}\in\mathfrak{g}_{\vec{b}}.$ (2.1)

We use $\llbracket\cdot,\cdot\rrbracket$ as a unified notation of commutator
and anticommutator. See appendix for more rigorous definition of
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebras.

###  2.2 A sequence of $\boldsymbol{\mathbb{Z}_{2}^{n}}$-graded SQM

We review the results of [2] briefly as the present work is an algebraic
generalization of them. The $\mathbb{Z}_{2}^{n}$-graded SQM is defined as a
realization of $\mathbb{Z}_{2}^{n}$-graded supersymmetry algebra in a
$\mathbb{Z}_{2}^{n}$-graded Hilbert space. The $\mathbb{Z}_{2}^{n}$-graded
supersymmetry algebra consists of one Hamiltonian, $2^{n-1}$ supercharges of
parity 1 and $2^{n-2}\big{(}2^{n-1}-1\big{)}$ central elements of parity 0.

It was shown that a tensor product of ${\mathcal{N}}=1$ standard SQM and a
complex irreducible representation (irrep) of the Clifford algebra ${\rm
Cl}(2m)$ can give the realization which define a model of the
$\mathbb{Z}_{2}^{n}$-graded SQM. The ${\mathcal{N}}=1$ standard SQM is
generated by one supercharge $Q$ and its defining relations are given by

$\\{Q,Q\\}=2H,\qquad[H,Q]=0.$

Both $Q$ and $H$ are $2\times 2$ matrix differential operators acting on
$\mathbb{Z}_{2}$-graded Hilbert space.

The Clifford algebra ${\rm Cl}(2m)$ is generated by $\gamma_{j}$
$(j=1,2,\dots,2m)$ subject to the conditions

$\\{\gamma_{j},\gamma_{k}\\}=2\delta_{jk}.$

The Hermitian complex irrep of ${\rm Cl}(2m)$ is $2^{m}$-dimensional and given
explicitly as follows [15, 20]

$\displaystyle\gamma_{1}$ $\displaystyle=\sigma_{1}^{\otimes
m},\qquad\gamma_{j}=\sigma_{1}^{\otimes(m-j+1)}\otimes\sigma_{3}\otimes\mathbb{I}_{2}^{\otimes(j-2)},\qquad
2\leq j\leq m,$ $\displaystyle\tilde{\gamma}_{j}$
$\displaystyle:=\gamma_{j+m}=\sigma_{1}^{\otimes(m-j)}\otimes\sigma_{2}\otimes\mathbb{I}_{2}^{\otimes(j-1)},\qquad
1\leq j\leq m,$ (2.2)

where $\sigma_{k}$ is the Pauli matrix and $\mathbb{I}_{2}$ denotes the
$2\times 2$ identity matrix. Therefore, a model of $\mathbb{Z}_{2}^{n}$-graded
SQM is a set of $2^{m+1}$-dimensional matrix differential operators.

For a fixed value of $n$, one may have a sequence of inequivalent models of
$\mathbb{Z}_{2}^{n}$-graded SQM by tensoring the standard SQM and the
following sequence of the Clifford algebra

${\rm Cl}(2(n-1)),\quad{\rm Cl}(2n),\quad{\rm Cl}(2(n+1)),\quad\dots,\quad{\rm
Cl}(2^{n}-2).$ (2.3)

For instance, we have five distinct models of $\mathbb{Z}_{2}^{4}$-graded SQM
from the Clifford algebras

${\rm Cl}(6),\quad{\rm Cl}(8),\quad{\rm Cl}(10),\quad{\rm Cl}(12),\quad{\rm
Cl}(14).$

The difference in the models is the number of linearly independent central
elements. The $\mathbb{Z}_{2}^{n}$-graded supersymmetry algebra has a lot of
central elements. Some of the central elements are realized as dependent
operators unless the Clifford algebra of the maximal dimension in the above
sequence is used. Lower the dimension of the Clifford algebra, more central
elements are realized as dependent operators.

In the next two sections, we show the existence of a sequence of realizations
of $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra and by which one may introduce
models of $\mathbb{Z}_{2}^{3}$-graded SCQM.

##  3 $\boldsymbol{{\rm Cl}(4)}$ model of
$\boldsymbol{\mathbb{Z}_{2}^{3}}$-graded SCQM

In this and the following sections, we deal with $\mathbb{Z}_{2}^{3}$-graded
Lie superalgebras and $\mathbb{Z}_{2}^{3}$-graded SCQM. Setting $n=3$ in
(2.3), we see that the sequence has only two Clifford algebras: ${\rm
Cl}(2(n-1))={\rm Cl}(4)$, ${\rm Cl}(2n)={\rm Cl}(2^{n}-2)={\rm Cl}(6)$.
However, realizations for ${\rm Cl}(2n)$ and ${\rm Cl}(2^{n}-2)$ considered in
[2] are not identical. We thus explore three cases, one for ${\rm Cl}(4)$ and
two for ${\rm Cl}(6)$. It will then turn out that, contrary to
$\mathbb{Z}_{2}^{n}$-graded SQM, we have two inequivalent
$\mathbb{Z}_{2}^{3}$-graded extensions of $\mathfrak{osp}(1|2)$. In this
section, we focus on ${\rm Cl}(4)$.

In the sequel, we denote a standard ($\mathbb{Z}_{2}$-graded) Lie superalgebra
by $\mathfrak{s}$ and its even and odd subspaces by $\mathfrak{s}_{0}$ and
$\mathfrak{s}_{1}$, respectively. We use a Hermitian representation of
$\mathfrak{s}$ to realize a $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra.
Recalling that $|\vec{a}|=0$ or $1$ for $\vec{a}\in\mathbb{Z}_{2}^{3}$, we
denote a Hermitian matrix representing an element of
$\mathfrak{s}_{|\vec{a}|}$ by $X_{|\vec{a}|}$ and suppose its size is
$2m\times 2m$.

###  3.1 $\boldsymbol{{\rm Cl}(4)}$ realization of
$\boldsymbol{\mathbb{Z}_{2}^{3}}$-graded Lie superalgebra

The irrep (2.2) for ${\rm Cl}(4)$ is given by

$\displaystyle\gamma_{1}=\sigma_{1}\otimes\sigma_{1},\qquad\gamma_{2}=\sigma_{1}\otimes\sigma_{3},\qquad\gamma_{3}=\sigma_{1}\otimes\sigma_{2},\qquad\gamma_{4}=\sigma_{2}\otimes\mathbb{I}_{2}.$

Let $\Gamma$ be a matrix subject to

$[X_{0},\Gamma]=0,\qquad\\{X_{1},\Gamma\\}=0,\qquad\Gamma^{2}=\mathbb{I}_{2m},\qquad\forall\,X_{|\vec{a}|}\in\mathfrak{s}_{|\vec{a}|}.$
(3.1)

Then the matrices defined by

$\displaystyle\boldsymbol{X}_{\vec{a}}={\rm
i}^{f(\vec{a})}\gamma_{1}^{a_{1}}\gamma_{2}^{a_{2}}\otimes
X_{|\vec{a}|}\Gamma^{a_{1}+a_{2}},\qquad
f(\vec{a}):=a_{1}a_{2}+|\vec{a}|(a_{1}+a_{2})\mod 2$ (3.2)

are Hermitian and define a $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra. More
explicitly, $\boldsymbol{X}_{\vec{a}}$ is given by

$\displaystyle\boldsymbol{X}_{(0,0,0)}=\mathbb{I}_{4}\otimes X_{0},\qquad$
$\displaystyle\boldsymbol{X}_{(1,0,0)}={\rm i}\gamma_{1}\otimes
X_{1}\Gamma,\qquad$ $\displaystyle\boldsymbol{X}_{(0,1,0)}={\rm
i}\gamma_{2}\otimes X_{1}\Gamma,$
$\displaystyle\boldsymbol{X}_{(0,0,1)}=\mathbb{I}_{4}\otimes X_{1},\qquad$
$\displaystyle\boldsymbol{X}_{(1,1,1)}={\rm i}\gamma_{1}\gamma_{2}\otimes
X_{1},\qquad$ $\displaystyle\boldsymbol{X}_{(1,1,0)}={\rm
i}\gamma_{1}\gamma_{2}\otimes X_{0},$
$\displaystyle\boldsymbol{X}_{(1,0,1)}=\gamma_{1}\otimes X_{0}\Gamma,\qquad$
$\displaystyle\boldsymbol{X}_{(0,1,1)}=\gamma_{2}\otimes X_{0}\Gamma.\qquad$

It is immediate to see $\boldsymbol{X}_{\vec{a}}$ is Hermitian

$\displaystyle(\boldsymbol{X}_{\vec{a}})^{\dagger}=(-{\rm
i})^{f(\vec{a})}\gamma_{2}^{a_{2}}\gamma_{1}^{a_{1}}\otimes\Gamma^{a_{1}+a_{2}}X_{|\vec{a}|}=(-1)^{f(\vec{a})+a_{1}a_{2}+|\vec{a}|(a_{1}+a_{2})}\boldsymbol{X}_{\vec{a}}=\boldsymbol{X}_{\vec{a}}.$

To verify the $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra structure, we need
to prove the closure in (anti)commutator and graded Jacobi relations (A.1).
This will be done by showing that the $\mathbb{Z}_{2}^{3}$-graded commutators
and Jacobi relations are reduced to those for the Lie superalgebra
$\mathfrak{s}$. It is not difficult to see the (anti)commutators (see (A.2))
are computed as

$\displaystyle\llbracket\boldsymbol{X}_{\vec{a}},\boldsymbol{X}_{\vec{b}}\rrbracket$
$\displaystyle=X_{\vec{a}}X_{\vec{b}}-(-1)^{\vec{a}\cdot\vec{b}}X_{\vec{b}}X_{\vec{a}}$
$\displaystyle=(-1)^{a_{2}b_{1}+(a_{1}+a_{2})|\vec{b}|}{\rm
i}^{f(\vec{a})+f(\vec{b})}\gamma_{1}^{a_{1}+b_{1}}\gamma_{2}^{a_{2}+b_{2}}\otimes\langle
X_{|\vec{a}|},X_{|\vec{b}|}\rangle\Gamma^{a_{1}+a_{2}+b_{1}+b_{2}},$ (3.3)

where

$\langle
X_{|\vec{a}|},X_{|\vec{b}|}\rangle:=X_{|\vec{a}|}X_{|\vec{b}|}-(-1)^{|\vec{a}||\vec{b}|}X_{|\vec{b}|}X_{|\vec{a}|}$

is the (anti)commutator of the Lie superalgebra $\mathfrak{s}$. Writing the
(anti)commutation relations of $\mathfrak{s}$ in the form

$\langle X_{|\vec{a}|},X_{|\vec{b}|}\rangle={\rm
i}^{1-|\vec{a}||\vec{b}|}X_{|\vec{a}|+|\vec{b}|}={\rm
i}^{1-|\vec{a}||\vec{b}|}X_{|\vec{a}+\vec{b}|},$

(3.3) yields

$\llbracket\boldsymbol{X}_{\vec{a}},\boldsymbol{X}_{\vec{b}}\rrbracket=(-1)^{a_{2}b_{1}+(a_{1}+a_{2})|\vec{b}|}{\rm
i}^{1+f(\vec{a})+f(\vec{b})-f(\vec{a}+\vec{b})-|\vec{a}||\vec{b}|}\boldsymbol{X}_{\vec{a}+\vec{b}}$

with

$\boldsymbol{X}_{\vec{a}+\vec{b}}={\rm
i}^{f(\vec{a}+\vec{b})}\gamma_{1}^{a_{1}+b_{1}}\gamma_{2}^{a_{2}+b_{2}}\otimes
X_{|\vec{a}+\vec{b}|}\Gamma^{a_{1}+a_{2}+b_{1}+b_{2}}.$

Therefore,
$\llbracket\boldsymbol{X}_{\vec{a}},\boldsymbol{X}_{\vec{b}}\rrbracket\in\mathfrak{g}_{\vec{a}+\vec{b}}$,
i.e., closure of $\mathbb{Z}_{2}^{3}$-graded (anti)commutator has been proved.

By the similar computation one may see

$\displaystyle(-1)^{\vec{a}\cdot\vec{c}}\llbracket\boldsymbol{X}_{\vec{a}},\llbracket\boldsymbol{X}_{\vec{b}},\boldsymbol{X}_{\vec{c}}\rrbracket\rrbracket=$
$\displaystyle(-1)^{\kappa}{\rm
i}^{f(\vec{a})+f(\vec{b})+f(\vec{c})}\gamma_{1}^{a_{1}+b_{1}+c_{1}}\gamma_{2}^{a_{2}+b_{2}+c_{2}}$
$\displaystyle{}\otimes(-1)^{|\vec{a}||\vec{c}|}\langle X_{|\vec{a}|},\langle
X_{|\vec{b}|},X_{|\vec{c}|}\rangle\rangle\Gamma^{\sum_{k=1}^{2}(a_{k}+b_{k}+c_{k})},$

where

$\displaystyle\kappa:=\sum_{k=1}^{2}(a_{k}b_{k}+b_{k}c_{k}+c_{k}a_{k})+a_{1}b_{2}+b_{1}c_{2}+c_{1}a_{2}$
$\displaystyle\hphantom{\kappa:=}{}+a_{1}b_{3}+b_{1}c_{3}+c_{1}a_{3}+a_{2}b_{3}+b_{2}c_{3}+c_{2}a_{3}.$

$\kappa$ is invariant under the cyclic permutation of $a$, $b$, $c$. This
shows that the graded Jacobi relations are reduced to those for
$\mathfrak{s}$. It follows that the graded Jacobi identity holds true and the
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebra structure has been proved.

The realization (3.2) is able to generalize to a realization of
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebras by $\mathfrak{s}$ and ${\rm
Cl}(2(n-1))$:

$\displaystyle\boldsymbol{X}_{\vec{a}}={\rm
i}^{f(\vec{a})}\prod_{j=1}^{n-1}\gamma_{j}^{a_{j}}\otimes
X_{|\vec{a}|}\Gamma^{\sum_{k=1}^{n-1}a_{k}},$ (3.4) $\displaystyle
f(\vec{a})=\sum_{k=1}^{n-2}a_{k}\prod_{l=k+1}^{n-1}a_{l}+|\vec{a}|\sum_{l=1}^{n-1}a_{l}\mod
2.$

One may prove this in the same way as $\mathbb{Z}_{2}^{3}$-graded Lie
superalgebras so we do not present the proof.

###  3.2 $\boldsymbol{{\rm Cl}(4)}$ model of
$\boldsymbol{\mathbb{Z}_{2}^{3}}$-graded $\boldsymbol{\mathfrak{osp}(1|2)}$
SCQM

As shown in Section 3.1, any Lie superalgebra satisfying the condition (3.1)
can be promoted to a $\mathbb{Z}_{2}^{3}$-graded superalgebra. If one starts
with a matrix differential operator realization of a superconformal algebra,
i.e., a model of SCQM, then one may obtain its $\mathbb{Z}_{2}^{3}$-graded
version. Many models of SCQM have been obtained so far (see, e.g., [17, 19,
21]). Some of the models, e.g., the ones in [4, 7], satisfy the condition
(3.1) so that we may have the $\mathbb{Z}_{2}^{3}$-graded SCQM of
${\mathcal{N}}=2,4,8$ and so on.

Here we analyse the simplest example of $\mathbb{Z}_{2}^{3}$-graded SCQM
obtained from the $\mathfrak{osp}(1|2)$ superconformal algebra. Let us
consider the following realization of $\mathfrak{osp}(1|2)$ which is a
${\mathcal{N}}=1$ SCQM:

$\displaystyle
Q=\frac{1}{\sqrt{2}}\left(\sigma_{1}p-\sigma_{2}\frac{\beta}{x}\right),\qquad$
$\displaystyle S=\frac{x}{\sqrt{2}}\sigma_{1},\qquad$ $\displaystyle
H=\frac{1}{2}\left(p^{2}+\frac{\beta^{2}}{x^{2}}\right)\mathbb{I}_{2}+\frac{\beta}{2x^{2}}\sigma_{3},\qquad$
$\displaystyle D=-\frac{1}{4}\\{x,p\\}\mathbb{I}_{2},\qquad$ $\displaystyle
K=\frac{x^{2}}{2}\mathbb{I}_{2},$ (3.5)

where $\beta$ is a coupling constant. The non-vanishing relations of
$\mathfrak{osp}(1|2)$ read as follows

$\displaystyle[D,K]={\rm i}K,\qquad$ $\displaystyle[H,K]=2{\rm i}D,\qquad$
$\displaystyle[D,H]=-{\rm i}H,$ $\displaystyle\\{Q,Q\\}=2H,\qquad$
$\displaystyle\\{S,S\\}=2K,\qquad$ $\displaystyle\\{Q,S\\}=-2D,$
$\displaystyle[D,Q]=-\frac{{\rm i}}{2}Q,\qquad$ $\displaystyle[D,S]=\frac{{\rm
i}}{2}S,\qquad$ $\displaystyle[Q,K]=-{\rm i}S,$ $\displaystyle[S,H]={\rm i}Q.$

One may immediately see that $\Gamma=\sigma_{3}$ satisfies the condition
(3.1). Thus by (3.2) we obtain twenty operators: the diagonal degree $(0,0,0)$
operators are given by

$\boldsymbol{H}_{000}=\mathbb{I}_{4}\otimes
H,\qquad\boldsymbol{D}_{000}=\mathbb{I}_{4}\otimes
D,\qquad\boldsymbol{K}_{000}=\mathbb{I}_{4}\otimes K.$ (3.6)

Here and in the following sections we use a simplified notation
$\boldsymbol{X}_{a_{1}a_{2}a_{3}}:=\boldsymbol{X}_{(a_{1},a_{2},a_{3})}.$ The
operator $\boldsymbol{H}_{000}$ is the Hamiltonian of the model and these
three operators form the one-dimensional conformal algebra
$\mathfrak{so}(1,2)$. The other parity even operators, which are not diagonal,
are given by

$\displaystyle\boldsymbol{X}_{110}=i\gamma_{1}\gamma_{2}\otimes
X,\qquad\boldsymbol{X}_{101}=\gamma_{1}\otimes
X\sigma_{3},\qquad\boldsymbol{X}_{011}=\gamma_{2}\otimes X\sigma_{3},\qquad
X=H,\,D,\,K,$

and the parity odd ones are given by

$\displaystyle\boldsymbol{Q}_{100}={\rm i}\gamma_{1}\otimes
Q\sigma_{3},\qquad$ $\displaystyle\boldsymbol{S}_{100}={\rm
i}\gamma_{1}\otimes S\sigma_{3},$ $\displaystyle\boldsymbol{Q}_{010}={\rm
i}\gamma_{2}\otimes Q\sigma_{3},\qquad$
$\displaystyle\boldsymbol{S}_{010}={\rm i}\gamma_{2}\otimes S\sigma_{3},$
$\displaystyle\boldsymbol{Q}_{001}=\mathbb{I}_{4}\otimes Q,\qquad$
$\displaystyle\boldsymbol{S}_{001}=\mathbb{I}_{4}\otimes S,$
$\displaystyle\boldsymbol{Q}_{111}={\rm i}\gamma_{1}\gamma_{2}\otimes
Q,\qquad$ $\displaystyle\boldsymbol{S}_{111}={\rm
i}\gamma_{1}\gamma_{2}\otimes S.$ (3.7)

(Anti)commutator of these operators are closed and define a
$\mathbb{Z}_{2}^{3}$-graded extension of $\mathfrak{osp}(1|2)$ which we denote
simply by ${\mathcal{G}}_{1}$. We note that $\dim{\mathcal{G}}_{1}=20$.

For the range of $\beta$ where the potential is repulsive, the Hamiltonian $H$
in (3.5) has continuous spectrum. It is known that the eigenfunctions of $H$
with the positive eigenvalue are plane wave normalizable, however, the zero
energy state is not even plane wave normalizable [16]. This property is
inherited to the Hamiltonian $\boldsymbol{H}_{000}$ of the
$\mathbb{Z}_{2}^{3}$-graded SCQM (3.7). In order to analyse the syetem (3.7)
we follow the standard prescription of conformal mechanics. That is, the
eigenspace of $\boldsymbol{H}_{000}$ is not taken as the Hilbert space of the
theory. Instead, the eigenspace of an operator which is a linear combination
of $\boldsymbol{H}_{000}$ and $\boldsymbol{K}_{000}$ is chosen as the Hilbert
space. We thus introduce the following operators

$\displaystyle\boldsymbol{R}_{\vec{a}}=\boldsymbol{H}_{\vec{a}}+\boldsymbol{K}_{\vec{a}},\qquad$
$\displaystyle\boldsymbol{L}_{\vec{a}}^{\pm}=\frac{1}{2}(\boldsymbol{K}_{\vec{a}}-\boldsymbol{H}_{\vec{a}})\pm{\rm
i}\boldsymbol{D}_{\vec{a}},$
$\displaystyle\boldsymbol{a}_{\vec{a}}=\boldsymbol{S}_{\vec{a}}+{\rm
i}\boldsymbol{Q}_{\vec{a}},\qquad$
$\displaystyle\boldsymbol{a}_{\vec{a}}^{\dagger}=\boldsymbol{S}_{\vec{a}}-{\rm
i}\boldsymbol{Q}_{\vec{a}}.$

The diagonal operator $\boldsymbol{R}_{000}$ is the new Hamiltonian and it has
discrete eigenvalues due to the oscillator potential (see (3.5) and (3.6)).
The eigenspace of $\boldsymbol{R}_{000}$ is
$\mathscr{H}=L^{2}(\mathbb{R})\otimes\mathbb{C}^{8}$ which is taken to be the
Hilbert space of the model. The space $\mathscr{H}$ has a vector space
decomposition according to the $\mathbb{Z}_{2}^{3}$-degree

$\mathscr{H}=\bigoplus_{\vec{a}\in\mathbb{Z}_{2}^{3}}\mathscr{H}_{\vec{a}}.$

The operators $\boldsymbol{a}_{\vec{a}}$,
$\boldsymbol{a}_{\vec{a}}^{\dagger}$, $\boldsymbol{L}_{\vec{a}}^{\pm}$
generate the spectrum of $\boldsymbol{R}_{000}$

$\displaystyle[\boldsymbol{R}_{000},\boldsymbol{a}_{\vec{a}}]=-\boldsymbol{a}_{\vec{a}},\qquad\big{[}\boldsymbol{R}_{000},\boldsymbol{a}_{\vec{a}}^{\dagger}\big{]}=-\boldsymbol{a}_{\vec{a}}^{\dagger},\qquad\big{[}\boldsymbol{R}_{000},\boldsymbol{L}_{\vec{a}}^{\pm}\big{]}=\pm
2\boldsymbol{L}_{\vec{a}}^{\pm},$ (3.8)
$\displaystyle\big{\\{}\boldsymbol{a}_{\vec{a}},\boldsymbol{a}_{\vec{a}}^{\dagger}\big{\\}}=2\boldsymbol{R}_{000},$

and the operators $\boldsymbol{a}_{\vec{a}}$,
$\boldsymbol{a}_{\vec{a}}^{\dagger}$, together with $F$ defined below, form a
Klein deformed oscillator algebra

$\displaystyle\big{[}\boldsymbol{a}_{\vec{a}},\boldsymbol{a}_{\vec{a}}^{\dagger}\big{]}=\mathbb{I}_{8}-2\beta
F,\qquad$ $\displaystyle F:=\mathbb{I}_{4}\otimes\sigma_{3},$ $\displaystyle
F^{2}=\mathbb{I}_{8},\qquad$
$\displaystyle\\{F,\boldsymbol{a}_{\vec{a}}\\}=\big{\\{}F,\boldsymbol{a}_{\vec{a}}^{\dagger}\big{\\}}=0.$

It follows that

$\boldsymbol{R}_{000}=\boldsymbol{a}_{\vec{a}}^{\dagger}\boldsymbol{a}_{\vec{a}}+\frac{1}{2}(\mathbb{I}_{8}-2\beta
F).$

Thus the ground state is obtained by

$\boldsymbol{a}_{\vec{a}}\Psi(x)=0,$

where

$\Psi(x)=(\psi_{000}(x),\psi_{001}(x),\psi_{110}(x),\psi_{111}(x),\psi_{011}(x),\psi_{010}(x),\psi_{101}(x),\psi_{100}(x))^{\rm
T}\in\mathscr{H}.$

For all $\boldsymbol{a}_{\vec{a}}$ this condition is reduced to the set of
relations for the components of $\Psi(x)$

$\displaystyle\left(\partial_{x}+x-\frac{\beta}{x}\right)\psi_{\vec{a}}(x)=0,\qquad$
$\displaystyle\vec{a}=(0,0,1),(1,1,1),(0,1,0),(1,0,0),$ (3.9)
$\displaystyle\left(\partial_{x}+x+\frac{\beta}{x}\right)\psi_{\vec{a}}(x)=0,\qquad$
$\displaystyle\vec{a}=(0,0,0),(1,1,0),(0,1,1),(1,0,1).$ (3.10)

The solution of these equations are given by
$\psi_{\vec{a}}(x)=x^{\pm\beta}{\rm e}^{-x^{2}/2}$ and the normalizability of
the functions are studied in detail in [4]. For $\beta>1$ (repulsive
potential) only one of them is normalizable so that the ground state is either

$x^{\beta}{\rm e}^{-x^{2}/2}(0,C_{1},0,C_{2},0,C_{3},0,C_{4})^{\rm T}$

with the energy $\frac{1}{2}(1+2\beta)$ or

$x^{-\beta}{\rm e}^{-x^{2}/2}(C_{1},0,C_{2},0,C_{3},0,C_{4},0)^{\rm T}$

with the energy $\frac{1}{2}(1-2\beta)$ where $C_{i}$ is a constant. Thus the
ground state is four-fold degenerate and belongs to either parity odd or even
subspaces of $\mathscr{H}$.

The excited states with various $\mathbb{Z}_{2}^{3}$-degree are obtained by
repeated application of $\boldsymbol{a}_{\vec{a}}^{\dagger}$ on the ground
state and one may see from (3.8) that the operator $\boldsymbol{R}_{(0,0,0)}$
has equally spaced spectrum. We remark that no need to consider the action of
$\boldsymbol{L}_{\vec{a}}^{+}$ because of the relation
$\big{\\{}\boldsymbol{a}_{\vec{a}}^{\dagger},\boldsymbol{a}_{\vec{a}}^{\dagger}\big{\\}}=4\boldsymbol{L}^{+}_{\vec{a}}$.
The excited state is also four-fold degenerate. This is verified as follows.
Let $\phi_{\vec{a}}\in\mathscr{H}_{\vec{a}}$ and
$\phi_{\vec{b}}\in\mathscr{H}_{\vec{b}}$ be eigenfunctions of
$\boldsymbol{R}_{000}$ with the same eigenvalue. Then
$\boldsymbol{a}_{\vec{c}}\phi_{\vec{a}}$ equals to
$\boldsymbol{a}_{\vec{d}}\phi_{\vec{b}}$ up to a constant multiple if
$\vec{a}+\vec{c}=\vec{b}+\vec{d}.$ For instance, it is not difficult to see
the following two functions in $\mathscr{H}_{(1,0,1)}$ are identical up to a
constant

$\displaystyle\boldsymbol{a}_{100}\psi_{001},\qquad$
$\displaystyle\psi_{001}=x^{\beta}{\rm e}^{-x^{2}/2}(0,C_{1},0,0,0,0,0,0),$
$\displaystyle\boldsymbol{a}_{010}\psi_{111},\qquad$
$\displaystyle\psi_{111}=x^{\beta}{\rm e}^{-x^{2}/2}(0,0,0,C_{2},0,0,0,0).$

This ${\rm Cl}(4)$ model of $\mathbb{Z}_{2}^{3}$-graded SCQM gives irreps of
${\mathcal{G}}_{1}$. Recalling that the order of $\mathbb{Z}_{2}^{3}$ is
$\big{|}\mathbb{Z}_{2}^{3}\big{|}=8$, the minimal dimension of non-trivial
irreps of ${\mathcal{G}}_{1}$ in a $\mathbb{Z}_{2}^{3}$-graded vector space is
also eight which is the dimension of the ${\rm Cl}(4)$ model.

As mentioned at the end of Section 3.1, we have a realization of
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebras by $\mathfrak{s}$ and ${\rm
Cl}(2(n-1))$ where the condition (3.1) is required. The condition is satisfied
for $\mathfrak{osp}(1|2)$ SCQM (3.5) as we have seen. It is also satisfied for
other models of SCQM. For instance,
$\Gamma=\sigma_{3},\sigma_{3}\otimes\mathbb{I}_{2}$ and
$\left(\begin{smallmatrix}\mathbb{I}_{8}&0\\\
0&\mathbb{I}_{8}\end{smallmatrix}\right)$ for the $\mathfrak{osp}(2|2)$,
$D(2,1;\alpha)$ and $F(4)$ models, respectively (see [4, 7] for the models).
These models are promoted to their $\mathbb{Z}_{2}^{n}$-graded version by
using (3.4). Thus, there exists various ${\rm Cl}(2(n-1))$ models of
$\mathbb{Z}_{2}^{n}$-graded SCQM and we may analyze its properties in a manner
similar to this section.

##  4 $\boldsymbol{{\rm Cl}(6)}$ models of
$\boldsymbol{\mathbb{Z}_{2}^{3}}$-graded SCQM

In this section, we explore the models of $\mathbb{Z}_{2}^{3}$-graded SCQM
obtained via ${\rm Cl}(6)$ in a way similar to the ones via ${\rm Cl}(4)$. As
already mentioned, we investigate two realizations of
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebras via ${\rm Cl}(2n)$ and ${\rm
Cl}(2^{n}-2)$. These two Clifford algebras are degenerate for $n=3$, however
the way of realizing the $\mathbb{Z}_{2}^{3}$-graded Lie superalgebras are not
identical.

The irrep (2.2) for ${\rm Cl}(6)$, which is common for ${\rm Cl}(2n)$ and
${\rm Cl}(2^{n}-2)$, is given by

$\displaystyle\gamma_{1}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1},\qquad$
$\displaystyle\gamma_{2}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{3},\qquad$
$\displaystyle\gamma_{3}=\sigma_{1}\otimes\sigma_{3}\otimes\mathbb{I}_{2},$
$\displaystyle\gamma_{4}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{2},\qquad$
$\displaystyle\gamma_{5}=\sigma_{1}\otimes\sigma_{2}\otimes\mathbb{I}_{2},\qquad$
$\displaystyle\gamma_{6}=\sigma_{2}\otimes\mathbb{I}_{2}\otimes\mathbb{I}_{2}.$

###  4.1 $\boldsymbol{{\rm Cl}(2n)}$ model

The realization of $\mathbb{Z}_{2}^{n}$-graded Lie superalgebra in terms of
${\rm Cl}(2n)$ and a ordinary Lie superalgebra $\mathfrak{s}$ is given in [1].
Thus we are able to use the result to investigate a model of
$\mathbb{Z}_{2}^{3}$-graded SCQM. For $n=3$, the realization of
$\mathbb{Z}_{2}^{3}$-graded Lie superalgebra by $X_{a}\in\mathfrak{s}_{a}$
reads as follows

$\displaystyle\boldsymbol{X}_{000}=\mathbb{I}_{8}\otimes X_{0},$
$\displaystyle\boldsymbol{X}_{100}=\gamma_{1}\otimes X_{1},\qquad$
$\displaystyle\boldsymbol{X}_{010}=\gamma_{2}\otimes X_{1},\qquad$
$\displaystyle\boldsymbol{X}_{001}=\gamma_{3}\otimes X_{1},\qquad$
$\displaystyle\boldsymbol{X}_{111}={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes X_{1},$
$\displaystyle\boldsymbol{X}_{110}={\rm i}\gamma_{1}\gamma_{2}\otimes
X_{0},\qquad\\!$ $\displaystyle\boldsymbol{X}_{101}={\rm
i}\gamma_{1}\gamma_{3}\otimes X_{0},\qquad\\!$
$\displaystyle\boldsymbol{X}_{011}={\rm i}\gamma_{2}\gamma_{3}\otimes X_{0}.$

Contrast to ${\rm Cl}(4)$, there is no condition like (3.1) so that any models
of SCQM can be extended to $\mathbb{Z}_{2}^{3}$-grading by this realization.
We consider again the $\mathfrak{osp}(1|2)$ model (3.5) as the simplest
example. The $\mathbb{Z}_{2}^{3}$-graded SCQM so obtained is the set of matrix
differential operators

$\displaystyle\boldsymbol{X}_{000}=\mathbb{I}_{8}\otimes X,\qquad$
$\displaystyle\boldsymbol{Q}_{100}=\gamma_{1}\otimes Q,\qquad$
$\displaystyle\boldsymbol{S}_{100}=\gamma_{1}\otimes S,\qquad$
$\displaystyle\boldsymbol{Q}_{010}=\gamma_{2}\otimes Q,\qquad$
$\displaystyle\boldsymbol{S}_{010}=\gamma_{2}\otimes S,\qquad$
$\displaystyle\boldsymbol{Q}_{001}=\gamma_{3}\otimes Q,\qquad$
$\displaystyle\boldsymbol{S}_{001}=\gamma_{3}\otimes S,\qquad$
$\displaystyle\boldsymbol{Q}_{111}={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes Q,\qquad$
$\displaystyle\boldsymbol{S}_{111}={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes S,\qquad$
$\displaystyle\boldsymbol{X}_{110}={\rm i}\gamma_{1}\gamma_{2}\otimes
X,\qquad$ $\displaystyle\boldsymbol{X}_{101}={\rm
i}\gamma_{1}\gamma_{3}\otimes X,\qquad$
$\displaystyle\boldsymbol{X}_{011}={\rm i}\gamma_{2}\gamma_{3}\otimes X$ (4.1)

with $X=H,D,K$.

It is not difficult to see that these twenty operators form an closed algebra
whose (anti)commutation relations are identical to the ones for ${\rm Cl}(4)$
model (3.7). Namely, the operators in (4.1) give 16-dimensional representation
of ${\mathcal{G}}_{1}$. However, this is a reducible representation of
${\mathcal{G}}_{1}$. To see this, let $F(\mathbb{R})$ be a space of complex
valued functions on a real line and
$\mathscr{H}=F(\mathbb{R})\otimes\mathbb{C}^{16}$. The operators (4.1) act on
$\mathscr{H}$ and it is readily seen from the explicit form of the operators
that the following subspaces $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ are
invariant under the action of (4.1):

$\displaystyle\mathscr{H}=\mathscr{H}_{1}\oplus\mathscr{H}_{2},$
$\displaystyle\mathscr{H}_{1}=(\psi_{000},0,\psi_{110},0,\psi_{011},0,\psi_{101},0,0,\psi_{001},0,\psi_{111},0,\psi_{010},0,\psi_{100})^{\rm
T},$
$\displaystyle\mathscr{H}_{2}=(0,\psi_{000},0,\psi_{110},0,\psi_{011},0,\psi_{101},\psi_{001},0,\psi_{111},0,\psi_{010},0,\psi_{100},0)^{\rm
T}.$

This shows that the operators (4.1) are a reducible representation of
${\mathcal{G}}_{1}$.

Although the combination of ${\rm Cl}(2n)$ and $\mathfrak{osp}(1|2)$ are not
physically relevant for $n=3$. This does not implies other models such as
$D(2,1;\alpha),F(4)$ are irrelevant, either. At least, one may see the
existence of various models of $\mathbb{Z}_{2}^{n}$-graded SCQM. Because the
realization given in [1] does not require any further conditions like (3.1) so
than any models of standard SCQM can be promoted to their
$\mathbb{Z}_{2}^{n}$-graded counterparts. Physical relevance of those models
should be studied case by case.

As already mentioned, some inequivalent realizations of
$\mathbb{Z}_{2}^{n}$-graded Lie superalgebras in terms of the Clifford algebra
and a standard Lie superalgebra are known. However, the present example
elucidates not all such realizations are suitable for physical applications.
We need to find an appropriate one to discuss physical problems. Therefore,
general study of realizations of $\mathbb{Z}_{2}^{n}$-graded Lie superalgebra
by ordinary superalgebras is an important research problem.

###  4.2 $\boldsymbol{{\rm Cl}(2^{n}-2)}$ model

A non-trivial ${\rm Cl}(6)$ model is obtained by an analogy of
$\mathbb{Z}_{2}^{n}$-extension of SQM considered in [2]. First, we introduce
the following ordering into the parity odd elements of $\mathbb{Z}_{2}^{3}$:

$\displaystyle\vec{a}_{0}=(1,1,1),\qquad\vec{a}_{1}=(1,0,0),\qquad\vec{a}_{2}=(0,1,0),\qquad\vec{a}_{3}=(0,0,1).$

Define the following Hermitian matrices

$\displaystyle\boldsymbol{X}_{(0,0,0)}=\mathbb{I}_{8}\otimes X_{0},$
$\displaystyle\boldsymbol{X}_{\mu}=\gamma_{\mu}\otimes X_{1},$
$\displaystyle\boldsymbol{X}_{\mu\nu}={\rm
i}^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{\mu}\gamma_{\nu}\otimes
X_{0},\qquad\mu<\nu,$ $\displaystyle\boldsymbol{X}_{\mu\nu\rho}={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes X_{1},\qquad\mu<\nu<\rho,$
$\displaystyle\boldsymbol{X}_{0123}={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes X_{0},$ (4.2)

where the Greek indices run from 0 to 3 and $\gamma_{0}=\mathbb{I}_{8}.$ The
suffix $(0,0,0)$ of $\boldsymbol{X}_{(0,0,0)}$ denotes its
$\mathbb{Z}_{2}^{3}$-degree where the original notation is restored to avoid
confusion. The $\mathbb{Z}_{2}^{3}$-degree of the matrices with the Greek
indices is determined as follows

$\displaystyle\deg(\boldsymbol{X}_{\mu})=\vec{a}_{\mu},\qquad$
$\displaystyle\deg(\boldsymbol{X}_{\mu\nu})=\vec{a}_{\mu}+\vec{a}_{\nu},$
$\displaystyle\deg(\boldsymbol{X}_{\mu\nu\rho})=\vec{a}_{\mu}+\vec{a}_{\nu}+\vec{a}_{\rho},\qquad$
$\displaystyle\deg(\boldsymbol{X}_{0123})=(0,0,0).$

With this assignment of $\mathbb{Z}_{2}^{3}$-degree the matrix operators of
(4.2) define a $\mathbb{Z}_{2}^{3}$-graded Lie superalgebra. This is verified
by observing that the $\mathbb{Z}_{2}^{3}$-graded (anti)commutators are
reduced to those for a standard superalgebra. In order to see this we write
the relations of the superalgebra $\mathfrak{s}$ as follows

$[X_{0},Y_{0}]={\rm i}Z_{0},\qquad[X_{0},Y_{1}]={\rm
i}Z_{1},\qquad\\{X_{1},Y_{1}\\}=W_{0}.$

By definition
$\llbracket\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}\rrbracket=[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}]$
for any $\boldsymbol{Y}$. One may compute the commutator as follows

$\displaystyle[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}_{(0,0,0)}]=\mathbb{I}_{8}\otimes[X_{0},Y_{0}]={\rm
i}\boldsymbol{Z}_{(0,0,0)},$
$\displaystyle[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}_{\mu}]=\gamma_{\mu}\otimes[X_{0},Y_{1}]={\rm
i}\boldsymbol{Z}_{\mu},$
$\displaystyle[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}_{\mu\nu}]={\rm
i}^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{\mu}\gamma_{\nu}\otimes[X_{0},Y_{0}]={\rm
i}\boldsymbol{Z}_{\mu\nu},$
$\displaystyle[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}_{\mu\nu\rho}]={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes[X_{0},Y_{1}]={\rm
i}\boldsymbol{Z}_{\mu\nu\rho},$
$\displaystyle[\boldsymbol{X}_{(0,0,0)},\boldsymbol{Y}_{0123}]={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes[X_{0},Y_{0}]={\rm
i}\boldsymbol{Z}_{0123}.$

Similarly for $\boldsymbol{X}_{0123}$

$\displaystyle[\boldsymbol{X}_{0123},\boldsymbol{Y}_{\mu}]={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{\mu}\otimes[X_{0},Y_{1}]=\sum_{\nu,\rho,\sigma}f_{\mu\nu\rho\sigma}\boldsymbol{Z}_{\nu\rho\sigma},$
$\displaystyle[\boldsymbol{X}_{0123},\boldsymbol{Y}_{\mu\nu}]=-{\rm
i}^{-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{\mu}\gamma_{\nu}\otimes[X_{0},Y_{0}]={\rm
i}\sum_{\rho,\sigma}g_{\mu\nu\rho\sigma}\boldsymbol{Z}_{\rho\sigma},$
$\displaystyle[\boldsymbol{X}_{0123},\boldsymbol{Y}_{\mu\nu\rho}]=-\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes[X_{0},Y_{1}]={\rm
i}\sum_{\sigma}h_{\mu\nu\rho\sigma}\boldsymbol{Z}_{\sigma},$
$\displaystyle[\boldsymbol{X}_{0123},\boldsymbol{Y}_{0123}]=\mathbb{I}_{8}\otimes[X_{0},Y_{0}]={\rm
i}\boldsymbol{Z}_{(0,0,0)},$

where the structure constants are given as follows

$\displaystyle f_{0123}=f_{1023}=f_{3012}=1,\qquad$ $\displaystyle
f_{2013}=-1,$ $\displaystyle g_{0123}=g_{0312}=g_{1203}=g_{2301}=1,\qquad$
$\displaystyle g_{0213}=g_{1302}=-1,$ $\displaystyle
h_{0123}=h_{0231}=h_{1230}=1,\qquad$ $\displaystyle h_{0132}=-1,$

and the others are zero.

Other (anti)commutation relations are more involved ((2.1), see also (3.3)):

$\displaystyle\llbracket\boldsymbol{X}_{\mu},\boldsymbol{Y}_{\nu}\rrbracket=\boldsymbol{X}_{\mu}\boldsymbol{Y}_{\nu}-(-1)^{\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\boldsymbol{Y}_{\nu}\boldsymbol{X}_{\mu}=\gamma_{\mu}\gamma_{\nu}\otimes\\{X_{1},Y_{1}\\}={\rm
i}^{-1+\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\boldsymbol{W}_{\mu\nu},$
$\displaystyle\llbracket\boldsymbol{X}_{\mu},\boldsymbol{Y}_{\nu\rho}\rrbracket=\boldsymbol{X}_{\mu}\boldsymbol{Y}_{\nu\rho}-(-1)^{\vec{a}_{\mu}\cdot(\vec{a}_{\nu}+\vec{a}_{\rho})}\boldsymbol{Y}_{\nu\rho}\boldsymbol{X}_{\mu}={\rm
i}^{1-\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes[X_{1},Y_{0}]={\rm
i}^{1-\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\boldsymbol{W}_{\mu\nu\rho},$
$\displaystyle\llbracket\boldsymbol{X}_{\mu},\boldsymbol{Y}_{\nu\rho\sigma}\rrbracket=\boldsymbol{X}_{\mu}\boldsymbol{Y}_{\nu\rho\sigma}-(-1)^{\vec{a}_{\mu}\cdot(\vec{a}_{\nu}+\vec{a}_{\rho}+\vec{a}_{\sigma})}\boldsymbol{Y}_{\nu\rho\sigma}\boldsymbol{X}_{\mu}$
$\displaystyle\hphantom{\llbracket\boldsymbol{X}_{\mu},\boldsymbol{Y}_{\nu\rho\sigma}\rrbracket}{}={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\otimes\\{X_{1},Y_{1}\\}=\boldsymbol{W}_{\mu\nu\rho\sigma},$
(4.3)

where we introduced $W_{1}\in\mathfrak{s}_{1}$ by $[X_{1},Y_{0}]={\rm i}W_{1}$
and $\boldsymbol{W}_{\mu\nu\rho}={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes W_{1}$. The indices of
$\boldsymbol{W}$ on the right-hand side of (4.3) do not always respect the
restriction given in (4.2). Such $\boldsymbol{W}$ is converted to the one in
(4.2) by the following relations:

$\displaystyle\boldsymbol{W}_{\mu\mu}=\boldsymbol{W}_{(0,0,0)},\qquad$
$\displaystyle\boldsymbol{W}_{\mu\nu}=(-1)^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\boldsymbol{W}_{\nu\mu},\quad\mu\neq\nu,$
$\displaystyle\boldsymbol{W}_{\mu\mu\rho}={\rm i}\boldsymbol{W}_{\rho},\qquad$
$\displaystyle\boldsymbol{W}_{\mu\nu\mu}=-{\rm i}\boldsymbol{W}_{\nu},$
$\displaystyle\boldsymbol{W}_{\mu\mu\nu\rho}=\boldsymbol{W}_{\mu\nu\rho\mu}=(-{\rm
i})^{\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\boldsymbol{W}_{\nu\rho},\qquad$
$\displaystyle\boldsymbol{W}_{\mu\nu\mu\rho}=-(-{\rm
i})^{\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\boldsymbol{W}_{\nu\rho},$
$\displaystyle\boldsymbol{W}_{\nu\mu\mu\rho}={\rm
i}^{\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\boldsymbol{W}_{\nu\rho},\qquad$
$\displaystyle\boldsymbol{W}_{\nu\mu\rho\mu}=-{\rm
i}^{\vec{a}_{\nu}\cdot\vec{a}_{\rho}}\boldsymbol{W}_{\nu\rho},$

and if all the indices are different value, then

$\displaystyle\boldsymbol{W}_{\mu\nu\rho}=(-1)^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\boldsymbol{W}_{\nu\mu\rho}=(-1)^{\vec{a}_{\mu}\cdot(\vec{a}_{\nu}+\vec{a}_{\rho})}\boldsymbol{W}_{\nu\rho\mu},$
$\displaystyle\boldsymbol{W}_{\mu\nu\rho\sigma}=(-1)^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\boldsymbol{W}_{\nu\mu\rho\sigma}=(-1)^{\vec{a}_{\mu}\cdot(\vec{a}_{\nu}+\vec{a}_{\rho})}\boldsymbol{W}_{\nu\rho\mu\sigma}=-(-1)^{\vec{a}_{\mu}\cdot(\vec{a}_{\nu}+\vec{a}_{\rho}+\vec{a}_{\sigma})}\boldsymbol{W}_{\nu\rho\sigma\mu}.$
(4.4)

We further need to check the closure of multi-index matrices

$\displaystyle\llbracket\boldsymbol{X}_{\mu\nu},\boldsymbol{Y}_{\rho\sigma}\rrbracket$
$\displaystyle=\boldsymbol{X}_{\mu\nu}\boldsymbol{Y}_{\rho\sigma}-(-1)^{(\vec{a}_{\mu}+\vec{a}_{\nu})\cdot(\vec{a}_{\rho}+\vec{a}_{\sigma})}\boldsymbol{Y}_{\rho\sigma}\boldsymbol{X}_{\mu\nu}$
$\displaystyle=-{\rm
i}^{-\vec{a}_{\mu}\cdot\vec{a}_{\nu}-\vec{a}_{\rho}\cdot\vec{a}_{\sigma}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\otimes[X_{0},Y_{0}]=-{\rm
i}^{-\vec{a}_{\mu}\cdot\vec{a}_{\nu}-\vec{a}_{\rho}\cdot\vec{a}_{\sigma}}\boldsymbol{Z}_{\mu\nu\rho\sigma},$
(4.5)

where $\boldsymbol{Z}_{\mu\nu\rho\sigma}$ is understood as in (4.4) and (4.5),

$\displaystyle\llbracket\boldsymbol{X}_{\mu\nu},\boldsymbol{Y}_{\rho\sigma\tau}\rrbracket$
$\displaystyle=\boldsymbol{X}_{\mu\nu}\boldsymbol{Y}_{\rho\sigma\tau}-(-1)^{(\vec{a}_{\mu}+\vec{a}_{\nu})\cdot(\vec{a}_{\rho}+\vec{a}_{\sigma}+\vec{a}_{\tau})}\boldsymbol{Y}_{\rho\sigma\tau}\boldsymbol{X}_{\mu\nu}$
$\displaystyle=-{\rm
i}^{-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\otimes[X_{0},Y_{1}]=-{\rm
i}^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\otimes
Z_{0}.$ (4.6)

There are five gamma matrices in this case so that one or two pairs of
identical gamma matrices exist. When there exist one pair of identical
matrices, say $\gamma_{\nu}=\gamma_{\tau}$, (4.6) equals to
$\boldsymbol{Z}_{\mu\rho\sigma}$ up to a constant factor. When there exist two
pair of identical matrices, say $\gamma_{\mu}=\gamma_{\rho}$ and
$\gamma_{\nu}=\gamma_{\sigma}$, (4.6) equals to $\boldsymbol{Z}_{\tau}$ up to
a constant multiple. In this way, we see the closure of (4.6). Similarly,
there exist identical gamma matrices in the following (anti)commutator

$\displaystyle\llbracket\boldsymbol{X}_{\mu\nu\rho},\boldsymbol{Y}_{\sigma\tau\lambda}\rrbracket$
$\displaystyle=\boldsymbol{X}_{\mu\nu\rho}\boldsymbol{Y}_{\sigma\tau\lambda}-(-1)^{(\vec{a}_{\mu}+\vec{a}_{\nu}+\vec{a}_{\rho})\cdot(\vec{a}_{\sigma}+\vec{a}_{\tau}+\vec{a}_{\lambda})}\boldsymbol{Y}_{\sigma\tau\lambda}\boldsymbol{X}_{\mu\nu\rho}$
$\displaystyle=-\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\gamma_{\lambda}\otimes\\{X_{1},Y_{1}\\}=-\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\gamma_{\tau}\gamma_{\lambda}\otimes
W_{0}.$ (4.7)

A special subcase of this is three pairs of identical gamma matrices

$\\{\boldsymbol{X}_{\mu\nu\rho},\boldsymbol{X}_{\mu\nu\rho}\\}=\mathbb{I}_{8}\otimes
W_{0}=\boldsymbol{W}_{(0,0,0)}.$

Except the special case, there always exist two pairs of identical gamma
matrices. Therefore, (4.7) equals to $\boldsymbol{W}_{\mu\nu}$ up to a
constant factor. We thus have proved the closure of (anti)commutators.

We observed that the $\mathbb{Z}_{2}^{3}$-graded (anti)commutator is reduced
to the one of superalgebra. It follows from this fact that a
$\mathbb{Z}_{2}^{3}$-graded Jacobi relation is also reduced to the one of
superalgebra. Therefore, it is straightforward to verify that (4.2) satisfies
the $\mathbb{Z}_{2}^{3}$-graded Jacobi relations.

Now we are able to use (4.2) to construct a ${\rm Cl}(6)$ model of
$\mathbb{Z}_{2}^{3}$-graded SCQM. Taking $\mathfrak{s}$ as any model of SCQM,
(4.2) produces a corresponding model of $\mathbb{Z}_{2}^{3}$-graded SCQM. As a
simplest example, here we take $\mathfrak{s}=\mathfrak{osp}(1|2)$ given in
(3.5). Then (4.2) gives us 40 operators which close in a
$\mathbb{Z}_{2}^{3}$-graded extension of $\mathfrak{osp}(1|2)$. We denote this
algebra simply by ${\mathcal{G}}_{2}$:

$\displaystyle\boldsymbol{X}_{(0,0,0)}=\mathbb{I}_{8}\otimes X,$
$\displaystyle\boldsymbol{Q}_{\mu}=\gamma_{\mu}\otimes
Q,\qquad\boldsymbol{S}_{\mu}=\gamma_{\mu}\otimes S,$
$\displaystyle\boldsymbol{X}_{\mu\nu}={\rm
i}^{1-\vec{a}_{\mu}\cdot\vec{a}_{\nu}}\gamma_{\mu}\gamma_{\nu}\otimes
X,\qquad\mu<\nu,$ $\displaystyle\boldsymbol{Q}_{\mu\nu\rho}={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes
Q,\qquad\boldsymbol{S}_{\mu\nu\rho}={\rm
i}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\otimes S,\qquad\mu<\nu<\rho,$
$\displaystyle\boldsymbol{X}_{0123}={\rm
i}\gamma_{1}\gamma_{2}\gamma_{3}\otimes X,\qquad X=H,D,K.$ (4.8)

The number of operator is double of the ${\rm Cl}(4)$ model discussed in
Section 3.2 where we have 20 operators. The reason of this difference is same
as the $\mathbb{Z}_{2}^{n}$-graded SQM considered in [2] and it is best seen
in the next example:

$\displaystyle[\boldsymbol{Q}_{1},\boldsymbol{Q}_{2}]=2\gamma_{1}\gamma_{2}\otimes
H=-2{\rm
i}\boldsymbol{H}_{12},\qquad\\{\boldsymbol{Q}_{0},\boldsymbol{Q}_{3}\\}=2\gamma_{3}\otimes
H=2\boldsymbol{H}_{03}.$

These are the relations of ${\rm Cl}(6)$ model and
$\deg(\boldsymbol{H}_{12})=\deg(\boldsymbol{H}_{03})=(1,1,0)$. As
$\gamma_{1}\gamma_{2}\neq\gamma_{3}$, $\boldsymbol{H}_{12}$ and
$~{}\boldsymbol{H}_{03}$ are linearly independent. The corresponding relations
in the ${\rm Cl}(4)$ model are

$\displaystyle[\boldsymbol{Q}_{100},\boldsymbol{Q}_{010}]=2\gamma_{1}\gamma_{2}\otimes
H,\qquad\\{\boldsymbol{Q}_{111},\boldsymbol{Q}_{001}\\}=2{\rm
i}\gamma_{1}\gamma_{2}\otimes H.$

Obviously, the operators on the right hand side are not linearly independent.
Namely, in the ${\rm Cl}(4)$ model degeneracy of operators, which are linearly
independent operators in ${\rm Cl}(6)$ model, happens and the number of the
operators are reduced.

By using the explicit form of (4.8), it is not difficult to see that the space
$\mathscr{H}=F(\mathbb{R})\otimes\mathbb{C}^{16}$ is not decoupled into two
subspaces by the action of ${\mathcal{G}}_{2}$. This is a sharp contrast to
the model in Section 4.1 and suggests the $\mathbb{Z}_{2}^{3}$-graded SCQM
(4.8) gives an irreducible representation of ${\mathcal{G}}_{2}$. More precise
analysis of irreducible representation of ${\mathcal{G}}_{2}$ will be done in
a way similar to [10] but it is beyond the scope of the present work.

Let us briefly analyse the spectrum of the ${\rm Cl}(6)$ model by employing
the standard prescription of conformal mechanics. That is, we define the
operators

$\displaystyle\boldsymbol{R}_{(0,0,0)}=\boldsymbol{H}_{(0,0,0)}+\boldsymbol{K}_{(0,0,0)},\qquad$
$\displaystyle\boldsymbol{L}_{(0,0,0)}^{\pm}=\frac{1}{2}(\boldsymbol{K}_{(0,0,0)}-\boldsymbol{H}_{(0,0,0)})\pm{\rm
i}\boldsymbol{D}_{(0,0,0)},$
$\displaystyle\boldsymbol{a}_{\mu}=\boldsymbol{S}_{\mu}+{\rm
i}\boldsymbol{Q}_{\mu},\qquad$
$\displaystyle\boldsymbol{a}_{\mu}^{\dagger}=\boldsymbol{S}_{\mu}-{\rm
i}\boldsymbol{Q}_{\mu},$
$\displaystyle\boldsymbol{a}_{\mu\nu\rho}=\boldsymbol{S}_{\mu\nu\rho}+{\rm
i}\boldsymbol{Q}_{\mu\nu\rho},\qquad$
$\displaystyle\boldsymbol{a}_{\mu\nu\rho}^{\dagger}=\boldsymbol{S}_{\mu\nu\rho}-{\rm
i}\boldsymbol{Q}_{\mu\nu\rho},$

and take the eigenspace of $\boldsymbol{R}_{(0,0,0)}$, which is
$L^{2}(\mathbb{R})\otimes\mathbb{C}^{16}$, as the Hilbert space of the theory.
There exist twice many creation and annihilation operators than ${\rm Cl}(4)$
model

$\displaystyle\big{[}\boldsymbol{R}_{(0,0,0)},\boldsymbol{a}_{\mu}\big{]}=-\boldsymbol{a}_{\mu},\qquad$
$\displaystyle\big{[}\boldsymbol{R}_{(0,0,0)},\boldsymbol{a}^{\dagger}_{\mu}\big{]}=-\boldsymbol{a}^{\dagger}_{\mu},$
$\displaystyle\big{[}\boldsymbol{R}_{(0,0,0)},\boldsymbol{a}_{\mu\nu\rho}\big{]}=-\boldsymbol{a}_{\mu\nu\rho},\qquad$
$\displaystyle\big{[}\boldsymbol{R}_{(0,0,0)},\boldsymbol{a}^{\dagger}_{\mu\nu\rho}\big{]}=-\boldsymbol{a}^{\dagger}_{\mu\nu\rho},$
$\displaystyle\big{[}\boldsymbol{R}_{(0,0,0)},\boldsymbol{L}_{(0,0,0)}^{\pm}\big{]}=\pm
2\boldsymbol{L}_{(0,0,0)}^{\pm}.\qquad$

These creation and annihilation operators satisfy the relations similar to the
${\rm Cl}(4)$ model

$\displaystyle\big{\\{}\boldsymbol{a}_{\mu},\boldsymbol{a}_{\mu}^{\dagger}\big{\\}}=\big{\\{}\boldsymbol{a}_{\mu\nu\rho},\boldsymbol{a}_{\mu\nu\rho}^{\dagger}\big{\\}}=2\boldsymbol{R}_{(0,0,0)},$
$\displaystyle\big{\\{}\boldsymbol{a}_{\mu}^{\dagger},\boldsymbol{a}_{\mu}^{\dagger}\big{\\}}=\big{\\{}\boldsymbol{a}_{\mu\nu\rho}^{\dagger},\boldsymbol{a}_{\mu\nu\rho}^{\dagger}\big{\\}}=2\boldsymbol{L}^{+}_{(0,0,0)},\qquad\\{\boldsymbol{a}_{\mu},\boldsymbol{a}_{\mu}\\}=\\{\boldsymbol{a}_{\mu\nu\rho},\boldsymbol{a}_{\mu\nu\rho}\\}=2\boldsymbol{L}^{-}_{(0,0,0)}.$

Furthermore, $\boldsymbol{a}_{\mu}$ and $\boldsymbol{a}_{\mu}^{\dagger}$
satisfy a Klein deformed oscillator algebra

$\displaystyle\big{[}\boldsymbol{a}_{\mu},\boldsymbol{a}_{\mu}^{\dagger}\big{]}=\big{[}\boldsymbol{a}_{\mu\nu\rho},\boldsymbol{a}_{\mu\nu\rho}^{\dagger}\big{]}=\mathbb{I}_{16}-2\beta
F,\qquad F:=\mathbb{I}_{8}\otimes\sigma_{3},$
$\displaystyle\\{F,\boldsymbol{a}_{\mu}\\}=\big{\\{}F,\boldsymbol{a}_{\mu}^{\dagger}\big{\\}}=\\{F,\boldsymbol{a}_{\mu\nu\rho}\\}=\big{\\{}F,\boldsymbol{a}_{\mu\nu\rho}^{\dagger}\big{\\}}=0,\qquad
F^{2}=\mathbb{I}_{16}.$

With these relations one may have

$\displaystyle\boldsymbol{R}_{(0,0,0)}=\boldsymbol{a}_{\mu}^{\dagger}\boldsymbol{a}_{\mu}+\frac{1}{2}(\mathbb{I}_{16}-2\beta
F)=\boldsymbol{a}_{\mu\nu\rho}^{\dagger}\boldsymbol{a}_{\mu\nu\rho}+\frac{1}{2}(\mathbb{I}_{16}-2\beta
F).$

Thus the ground state $\Psi(x)$ or $\boldsymbol{R}_{(0,0,0)}$ is defined by

$\boldsymbol{a}_{\mu}\Psi(x)=\boldsymbol{a}_{\mu\nu\rho}\Psi(x)=0.$ (4.9)

We write $\Psi(x)$ in components

$\displaystyle\Psi(x)=(\psi_{000},\psi_{111},\psi_{110},\psi_{001},\psi_{011},\psi_{100},\psi_{101},\psi_{010},\psi_{110},\psi_{001},\psi_{000},$
$\displaystyle\hphantom{\Psi(x)=(}{}\psi_{111},\psi_{101},\psi_{010},\psi_{011},\psi_{100})^{\rm
T}.$

Then the condition (4.9) gives the same relations as ${\rm Cl}(4)$ model (3.9)
and (3.10). It follows that the ground state of ${\rm Cl}(6)$ model is eight-
fold degenerate. The excited states are obtained by repeated application of
$\boldsymbol{a}_{\mu}^{\dagger}$ and $\boldsymbol{a}_{\mu\nu\rho}^{\dagger}$
on the ground state. Repeating the argument same as the ${\rm Cl}(4)$ model,
one may see that the spectrum of $\boldsymbol{R}_{(0,0,0)}$ is equally spacing
and the excited state has eight-fold degeneracy.

As an abstract Lie algebra we regard ${\mathcal{G}}_{2}$ is inequivalent to
${\mathcal{G}}_{1}$ as $\dim{\mathcal{G}}_{2}=40>\dim{\mathcal{G}}_{1}$.

##  5 Concluding remarks

We showed that many models of SCQM are able to extend to
$\mathbb{Z}_{2}^{n}$-graded setting by the use of the Clifford algebras ${\rm
Cl}(2(n-1))$ and ${\rm Cl}(2n)$. It was also shown the existence of a sequence
of models of $\mathbb{Z}_{2}^{3}$-graded $\mathfrak{osp}(1|2)$ SCQM and we
analyzed the spectrum of the models. Most likely, for a given model of
standard SCQM there would exists a sequence of models of
$\mathbb{Z}_{2}^{n}$-graded SCQM produced via the sequence of the Clifford
algebras (2.3). However, full analysis of $\mathbb{Z}_{2}^{n}$-graded SCQM
will require lengthy computation and invention of better notations (especially
for the models via higher-dimensional Clifford algebras) which make the
presentation simpler and more readable. Therefore, we are planning to present
them in a separate publication. We convince that the present analysis of
$\mathbb{Z}_{2}^{3}$-graded SCQM provides all the essentials of
$\mathbb{Z}_{2}^{n}$-graded extensions.

Although the existence of $\mathbb{Z}_{2}^{n}$-graded SCQM has been
established, its physical implications and how much it differs from the
standard SCQM are not clear yet. To have better understanding of
$\mathbb{Z}_{2}^{3}$ and higher graded SCQM, there would be some more works to
be done. For example, one may consider multiparticle extensions of the models
presented in this paper. As showed in $\mathbb{Z}_{2}^{2}$-graded SQM,
difference from the standard SQM becomes clear when a multiparticle model is
considered. A multiparticle extension may be done in a way similar to [28].

The second example is classical theories of $\mathbb{Z}_{2}^{3}$-graded SCQM
which reproduce the models of this work upon quantization. Such classical
theories will shed a new light on $\mathbb{Z}_{2}^{3}$-graded SCQM and they
have their own interest. For the simpler grading by $\mathbb{Z}_{2}^{2}$,
$D$-module presentation and superfield approach to the classical theory of
$\mathbb{Z}_{2}^{2}$-graded SQM are discussed in the literature [8, 13]. It is
a very interesting but challenging problem to generalize these to
$\mathbb{Z}_{2}^{n}$-graded $(n\geq 3)$ setting since integration on
$\mathbb{Z}_{2}^{n}$-graded $(n\geq 3)$ supermanifolds has not been
established yet [22]. Nonlinear realization is a widely used approach to
superconformal mechanics, see, e.g., [11, 17]. $\mathbb{Z}_{2}^{3}$-graded
extension of nonlinear realization will be possible and it will give some
geometrical understanding of $\mathbb{Z}_{2}^{3}$-graded SCQM.

##  Appendix A Definition of $\boldsymbol{\mathbb{Z}_{2}^{n}}$-graded Lie
superalgebra

In this appendix we give a rigorous definition of $\mathbb{Z}_{2}^{n}$-graded
Lie superalgebra [23, 24, 25]. Let $\mathfrak{g}$ be a vector space over
$\mathbb{R}$ or $\mathbb{C}$ and $\vec{a}=(a_{1},a_{2},\dots,a_{n})$ an
element of $\mathbb{Z}_{2}^{n}$. Suppose that $\mathfrak{g}$ is a direct sum
of graded subspaces labelled by $\vec{a}$

$\mathfrak{g}=\bigoplus_{\vec{a}}\mathfrak{g}_{\vec{a}}.$

Homogeneous elements of $\mathfrak{g}_{\vec{a}}$ are denoted by
$X_{\vec{a}},Y_{\vec{a}},\dots$. If $\mathfrak{g}$ admits a bilinear operation
(the general Lie bracket), denoted by $\llbracket\cdot,\cdot\rrbracket$,
satisfying the identities

$\displaystyle\llbracket
X_{\vec{a}},Y_{\vec{b}}\rrbracket\in\mathfrak{g}_{\vec{a}+\vec{b}},\qquad\llbracket
X_{\vec{a}},Y_{\vec{b}}\rrbracket=-(-1)^{\vec{a}\cdot\vec{b}}\llbracket
Y_{\vec{b}},X_{\vec{a}}\rrbracket,$
$\displaystyle(-1)^{\vec{a}\cdot\vec{c}}\llbracket X_{\vec{a}},\llbracket
Y_{\vec{b}},Z_{\vec{c}}\rrbracket\rrbracket+(-1)^{\vec{b}\cdot\vec{a}}\llbracket
Y_{\vec{b}},\llbracket
Z_{\vec{c}},X_{\vec{a}}\rrbracket\rrbracket+(-1)^{\vec{c}\cdot\vec{b}}\llbracket
Z_{\vec{c}},\llbracket X_{\vec{a}},Y_{\vec{b}}\rrbracket\rrbracket=0,$ (A.1)

where

$\vec{a}+\vec{b}=(a_{1}+b_{1},a_{2}+b_{2},\dots)\in\mathbb{Z}_{2}^{n},\qquad\vec{a}\cdot\vec{b}=\sum_{k=1}^{n}a_{k}b_{k},$

then $\mathfrak{g}$ is referred to as a $\mathbb{Z}_{2}^{n}$-graded Lie
superalgebra. The relation (A.1) is called the $\mathbb{Z}_{2}^{n}$-graded
Jacobi relation.

We take $\mathfrak{g}$ to be contained in its enveloping algebra, via the
identification

$\llbracket
X_{\vec{a}},Y_{\vec{b}}\rrbracket=X_{\vec{a}}Y_{\vec{b}}-(-1)^{\vec{a}\cdot\vec{b}}Y_{\vec{b}}X_{\vec{a}},$
(A.2)

where an expression such as $X_{\vec{a}}Y_{\vec{b}}$ is understood to denote
the associative product on the enveloping algebra. In other words, by
definition, in the enveloping algebra the general Lie bracket
$\llbracket\cdot,\cdot\rrbracket$ for homogeneous elements coincides with
either a commutator or anticommutator.

This is a natural generalization of Lie superalgebra which is defined on
$\mathbb{Z}_{2}$-grading structure. Namely, the vector $\vec{a}$ is one-
dimensional

$\mathfrak{g}=\mathfrak{g}_{(0)}\oplus\mathfrak{g}_{(1)}$

with $\vec{a}+\vec{b}=(a+b)$, $\vec{a}\cdot\vec{b}=ab$.

### Acknowledgements

The authors would like to thank the referees for the valuable comments for
improvement of this paper.

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