206
N. Aizawa et al.
Proposition 2 Let Γ be an Hermitian operator anticommuting with Q. We assume
further that Γ 2 is the identity operator. Then
Q 1 = I 2 ⊗ I 4 ⊗ Q,
Q 2 = iσ 1 ⊗ γ 1 ⊗ QΓ,
Q 3 = iσ 1 ⊗ γ 3 ⊗ QΓ,
Q 4 = iσ 3 ⊗ γ 2 γ 4 ⊗ Q
(28)
realizes the Z 3
2 -SPA (25).
For completeness, we give the formula of the Hamiltonian and the central
elements:
H = I 8 ⊗ H,
Z 12 = σ 1 ⊗ γ 1 ⊗ H Γ,
Z 13 = σ 1 ⊗ γ 3 ⊗ H Γ,
Z 14 = iσ 3 ⊗ γ 2 γ 4 ⊗ H,
Z 23 = −iI 2 ⊗ γ 1 γ 3 ⊗ H,
Z 24 = −iσ 2 ⊗ γ 1 γ 2 γ 4 ⊗ H Γ,
Z 34 = iσ 2 ⊗ γ 2 γ 3 γ 4 ⊗ H Γ.
(29)
By (28) any model of (24) may be converted to the corresponding Z 3
2 -graded
version if the operator Γ exists. Such Γ would exist in many cases. The simplest
example is a single particle moving in one-dimensional space. In this case, Q
consists of σ 1 , σ 2 and differential operator, so that one may take Γ = σ 3 .
In the present work, we showed that Bruce–Duplij model of Z 2
2 -graded SQM
is easily extended to higher values of N and to superconformal setting. It was also
shown that N = 1 SQM is possible to generalize Z 3
2 -graded SQM. These were done
by finding a realization of a color superalgebra by an ordinary Lie superalgebra.
Therefore, we expect that it is possible to obtain models of Z n
2 -graded SQM and
SCM in a similar and a systematic way. If it is the case, then color superalgebras
would be a quite natural object in analysis of physical problems.
References
1. I. Ueba, Extended supersymmetry with central charges in Dirac action with curved extra
dimensions (2019). arXiv:1905.11673 [math-ph]
2. T. Okazaki, Superconformal quantum mechanics from M2-branes, Ph.D thesis (2015).
arXiv:1503.03906 [hep-th]
3. A.J. Bruce, On a Z n
2 -graded version of supersymmetry. Symmetry 11, 116 (2019). https://doi.
org/10.3390/sym11010116
4. A.J. Bruce, S. Duplij, Double-graded supersymmetric quantum mechanics (2019).
arXiv:1904.06975 [math-ph]
5. N. Aizawa, Z. Kuznetsova, H. Tanaka, F. Toppan, Z 2 × Z 2 -graded Lie Symmetries of the
Lévy-Leblond Equations. Prog. Theor. Exp. Phys. 2016, 123A01 (2016). https://doi.org/10.
1093/ptep/ptw176
6. V.N. Tolstoy, Once more on parastatistics. Phys. Part. Nucl. Lett. 11, 933 (2014). https://doi.
org/10.1134/S1547477114070449
N. Aizawa et al.
Proposition 2 Let Γ be an Hermitian operator anticommuting with Q. We assume
further that Γ 2 is the identity operator. Then
Q 1 = I 2 ⊗ I 4 ⊗ Q,
Q 2 = iσ 1 ⊗ γ 1 ⊗ QΓ,
Q 3 = iσ 1 ⊗ γ 3 ⊗ QΓ,
Q 4 = iσ 3 ⊗ γ 2 γ 4 ⊗ Q
(28)
realizes the Z 3
2 -SPA (25).
For completeness, we give the formula of the Hamiltonian and the central
elements:
H = I 8 ⊗ H,
Z 12 = σ 1 ⊗ γ 1 ⊗ H Γ,
Z 13 = σ 1 ⊗ γ 3 ⊗ H Γ,
Z 14 = iσ 3 ⊗ γ 2 γ 4 ⊗ H,
Z 23 = −iI 2 ⊗ γ 1 γ 3 ⊗ H,
Z 24 = −iσ 2 ⊗ γ 1 γ 2 γ 4 ⊗ H Γ,
Z 34 = iσ 2 ⊗ γ 2 γ 3 γ 4 ⊗ H Γ.
(29)
By (28) any model of (24) may be converted to the corresponding Z 3
2 -graded
version if the operator Γ exists. Such Γ would exist in many cases. The simplest
example is a single particle moving in one-dimensional space. In this case, Q
consists of σ 1 , σ 2 and differential operator, so that one may take Γ = σ 3 .
In the present work, we showed that Bruce–Duplij model of Z 2
2 -graded SQM
is easily extended to higher values of N and to superconformal setting. It was also
shown that N = 1 SQM is possible to generalize Z 3
2 -graded SQM. These were done
by finding a realization of a color superalgebra by an ordinary Lie superalgebra.
Therefore, we expect that it is possible to obtain models of Z n
2 -graded SQM and
SCM in a similar and a systematic way. If it is the case, then color superalgebras
would be a quite natural object in analysis of physical problems.
References
1. I. Ueba, Extended supersymmetry with central charges in Dirac action with curved extra
dimensions (2019). arXiv:1905.11673 [math-ph]
2. T. Okazaki, Superconformal quantum mechanics from M2-branes, Ph.D thesis (2015).
arXiv:1503.03906 [hep-th]
3. A.J. Bruce, On a Z n
2 -graded version of supersymmetry. Symmetry 11, 116 (2019). https://doi.
org/10.3390/sym11010116
4. A.J. Bruce, S. Duplij, Double-graded supersymmetric quantum mechanics (2019).
arXiv:1904.06975 [math-ph]
5. N. Aizawa, Z. Kuznetsova, H. Tanaka, F. Toppan, Z 2 × Z 2 -graded Lie Symmetries of the
Lévy-Leblond Equations. Prog. Theor. Exp. Phys. 2016, 123A01 (2016). https://doi.org/10.
1093/ptep/ptw176
6. V.N. Tolstoy, Once more on parastatistics. Phys. Part. Nucl. Lett. 11, 933 (2014). https://doi.
org/10.1134/S1547477114070449