Color Algebraic Extension of
Supersymmetric Quantum Mechanics
Naruhiko Aizawa, Kosuke Amakawa, and Shunya Doi
Abstract In the recent paper, Bruce and Duplij introduced a Z 2
2 -graded version of
supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic
nature of N = 1 SQM to a Z 2
2 -graded color superalgebra. We present three
extensions of the result of Bruce and Duplij. Namely, Z 2
2 -graded SQM with higher
values of N , Z 2
2 -graded version of superconformal mechanics, and Z 3
2 -graded SQM.
All these were done by realizations of color superalgebra in terms of ordinary Lie
superalgebra.
Keywords Z 2
2 -graded Lie algebras · Supersymmetric quantum mechanics ·
Superconformal mechanics
1 Introduction
Supersymmetric and superconformal quantum mechanics have been discussed in
surprisingly wide variety of problems in physics. Even in some of modern problems
such as curved extra dimension or M-theory they play fundamental and important
roles, see, for instance [1, 2] and references therein. It is, therefore, natural that
there exist many considerations on possible extensions of supersymmetric quantum
mechanics. Supersymmetric quantum mechanics (SQM) is a quantum mechanical
realization of the super-Poincaré algebra in (0 + 1)-dimensional spacetime. Thus
many extensions of SQM discuss possible replacement of Lie superalgebraic nature
of super-Poincaré algebra with more general setting (also in connection with the
no-go theorem of Coleman and Mandula).
One of the most recent works in this direction is due to Bruce [3] where Z 2 -
grading of the super-Poincaré algebra in (3 + 1) dimensional Minkowski space
are replaced with Z n
2 -grading. Z n
2 denotes a direct product of n copies of the
N. Aizawa () · K. Amakawa · S. Doi
Department of Physical Science, Osaka Prefecture University, Sakai, Osaka, Japan
e-mail: aizawa@p.s.osakafu-u.ac.jp
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_18
199
Supersymmetric Quantum Mechanics
Naruhiko Aizawa, Kosuke Amakawa, and Shunya Doi
Abstract In the recent paper, Bruce and Duplij introduced a Z 2
2 -graded version of
supersymmetric quantum mechanics (SQM). It is an extension of Lie superalgebraic
nature of N = 1 SQM to a Z 2
2 -graded color superalgebra. We present three
extensions of the result of Bruce and Duplij. Namely, Z 2
2 -graded SQM with higher
values of N , Z 2
2 -graded version of superconformal mechanics, and Z 3
2 -graded SQM.
All these were done by realizations of color superalgebra in terms of ordinary Lie
superalgebra.
Keywords Z 2
2 -graded Lie algebras · Supersymmetric quantum mechanics ·
Superconformal mechanics
1 Introduction
Supersymmetric and superconformal quantum mechanics have been discussed in
surprisingly wide variety of problems in physics. Even in some of modern problems
such as curved extra dimension or M-theory they play fundamental and important
roles, see, for instance [1, 2] and references therein. It is, therefore, natural that
there exist many considerations on possible extensions of supersymmetric quantum
mechanics. Supersymmetric quantum mechanics (SQM) is a quantum mechanical
realization of the super-Poincaré algebra in (0 + 1)-dimensional spacetime. Thus
many extensions of SQM discuss possible replacement of Lie superalgebraic nature
of super-Poincaré algebra with more general setting (also in connection with the
no-go theorem of Coleman and Mandula).
One of the most recent works in this direction is due to Bruce [3] where Z 2 -
grading of the super-Poincaré algebra in (3 + 1) dimensional Minkowski space
are replaced with Z n
2 -grading. Z n
2 denotes a direct product of n copies of the
N. Aizawa () · K. Amakawa · S. Doi
Department of Physical Science, Osaka Prefecture University, Sakai, Osaka, Japan
e-mail: aizawa@p.s.osakafu-u.ac.jp
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_18
199