$ \mathcal{N} = 2 $ double graded supersymmetric quantum mechanics via dimensional reduction

Naruhiko Aizawa; Ren Ito; Toshiya Tanaka; Naruhiko Aizawa; Ren Ito; Toshiya Tanaka

doi:10.3934/math.2024513

AIMS Mathematics

2024, Volume 9, Issue 5: 10494-10510. doi: 10.3934/math.2024513

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$ \mathcal{N} = 2 $ double graded supersymmetric quantum mechanics via dimensional reduction

Department of Physics, Graduate School of Science, Osaka Metropolitan University, Nakamozu Campus, Sakai, Osaka 5998531, Japan

Received: 11 January 2024 Revised: 02 March 2024 Accepted: 11 March 2024 Published: 18 March 2024
MSC : 81Q60, 81Q80, 81S08

We presented a novel $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-graded supersymmetric quantum mechanics ($ {\mathbb{Z}_2^2} $-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-supersymmetry algebra. The $ {\mathbb{Z}_2^2} $-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $ {\mathbb{Z}_2^2} $-supersymmetric Lagrangian of $ \mathcal{N} = 1 $, which we constructed in our previous work. The ground states of the $ {\mathbb{Z}_2^2} $-SQM were also investigated.
- $ {\mathbb{Z}_2^2} $-supersymmetry,
- constrained system,
- Dirac-Bergmann method,
- quantization,
- quantum mechanics
Citation: Naruhiko Aizawa, Ren Ito, Toshiya Tanaka. $ \mathcal{N} = 2 $ double graded supersymmetric quantum mechanics via dimensional reduction[J]. AIMS Mathematics, 2024, 9(5): 10494-10510. doi: 10.3934/math.2024513

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Abstract

We presented a novel $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-graded supersymmetric quantum mechanics ($ {\mathbb{Z}_2^2} $-SQM) which has different features from those introduced so far. It is a two-dimensional (two-particle) system and was the first example of the quantum mechanical realization of an eight-dimensional irreducible representation (irrep) of the $ \mathcal{N} = 2 $ $ \mathbb{Z}_2^2 $-supersymmetry algebra. The $ {\mathbb{Z}_2^2} $-SQM was obtained by quantizing the one-dimensional classical system derived by dimensional reduction from the two-dimensional $ {\mathbb{Z}_2^2} $-supersymmetric Lagrangian of $ \mathcal{N} = 1 $, which we constructed in our previous work. The ground states of the $ {\mathbb{Z}_2^2} $-SQM were also investigated.

References

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