# Quantum Mechanics of the Extended Snyder Model

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## Abstract

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## 1. Introduction

## 2. The Snyder Model

## 3. The Harmonic Oscillator

## 4. Noncovariant Formalism

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Snyder, H.S. Quantized space-time. Phys. Rev.
**1947**, 71, 38. [Google Scholar] [CrossRef] - Hellund, E.J.; Tanaka, K. Quantized space-time. Phys. Rev.
**1954**, 94, 192. [Google Scholar] [CrossRef] - Gol’fand, Y.A. On the introduction of an “elementary length” in the relativistic theory of elementary particles. Sov. Phys. JETP
**1960**, 10, 356. [Google Scholar] - Gol’fand, Y.A. Quantum field theory in constant curvature p-space. Sov. Phys. JETP
**1963**, 16, 184. [Google Scholar] - Kadyshevskii, V.G. On the Theory of Quantization of Space-Time. Sov. Phys. JETP
**1963**, 14, 1340. [Google Scholar] - Tamm, I.E. The Evolution of Quantum Theory. In Selected Papers; Bolotovskii, B.M., Frenkel, V.Y., Peierls, R., Eds.; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar]
- Jaroszkiewicz, G. A dynamical model for the origin of Snyder’s quantized spacetime algebra. J. Phys.
**1995**, A28, L343. [Google Scholar] [CrossRef] - Romero, J.M.; Zamora, A. Adolfo Zamora. Phys. Rev.
**2004**, D70, 105006. [Google Scholar] - Banerjee, R.; Kulkarni, S.; Samanta, S. Deformed Symmetry in Snyder Space and Relativistic Particle Dynamics. arXiv
**2006**, arXiv:hep-th/0602151. [Google Scholar] [CrossRef] - Guo, H.Y.; Huang, C.G.; Tian, Y.; Wu, H.T.; Xu, Z.; Zhou, B. From the Anti-Yang Model to the Anti-Snyder Model and Anti-De Sitter Special Relativity. Class. Quantum Grav.
**2007**, 24, 4009. [Google Scholar] [CrossRef] - Doplicher, S.; Fredenhagen, K.; Roberts, J.E. Spacetime Quantization Induced by Classical Gravity. Phys. Lett.
**1994**, B331, 39. [Google Scholar] [CrossRef][Green Version] - Lukierski, J.; Ruegg, H.; Novicki, A.; Tolstoi, V.N. Jerzy Lukierski. Phys. Lett.
**1991**, B264, 331. [Google Scholar] [CrossRef] - Majid, S. Foundations of Quantum Group Theory; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Battisti, M.V.; Meljanac, S. Modification of Heisenberg uncertainty relations in noncommutative Snyder space-time geometry. Phys. Rev.
**2009**, D79, 067505. [Google Scholar] [CrossRef][Green Version] - Battisti, M.V.; Meljanac, S. Scalar field theory on noncommutative Snyder spacetime. Phys. Rev.
**2010**, D82, 024028. [Google Scholar] [CrossRef][Green Version] - Girelli, F.; Livine, E. Scalar field theory in Snyder space-time: Alternatives. J. High Energy Phys.
**2011**, 1103, 132. [Google Scholar] [CrossRef][Green Version] - Meljanac, S.; Mignemi, S. Associative realizations of the extended Snyder model. Phys. Rev.
**2020**, D102, 126011. [Google Scholar] [CrossRef] - Meljanac, S.; Mignemi, S. Unification of κ-Minkowski and extended Snyder spaces. Phys. Lett.
**2021**, B814, 136117. [Google Scholar] [CrossRef] - Meljanac, S.; Mignemi, S. Associative realizations of κ-deformed extended Snyder model. Phys. Rev.
**2021**, D104, 086006. [Google Scholar] [CrossRef] - Meljanac, S.; Meljanac, D.; Samsarov, A.; Stojić, M. Kappa Snyder deformations of Minkowski spacetime, realizations, and Hopf algebra. Phys. Rev.
**2011**, D83, 065009. [Google Scholar] [CrossRef][Green Version] - Amorim, R. Tensor Operators in Noncommutative Quantum Mechanics. Phys. Rev. Lett.
**2008**, 101, 081602. [Google Scholar] [CrossRef][Green Version] - Chang, L.N.; Minic, D.; Okamura, N.; Takeuchi, T. Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem. Phys. Rev.
**2002**, D65, 125027. [Google Scholar] [CrossRef][Green Version] - Mignemi, S. Classical and quantum mechanics of the nonrelativistic Snyder model. Phys. Rev.
**2011**, D84, 025021. [Google Scholar] [CrossRef][Green Version] - Gubitosi, G.; Mignemi, S. Diffeomorphisms in momentum space: Physical implications of different choices of momentum coordinates in the Galilean Snyder model. Universe
**2022**, 8, 108. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Bianco, S.; Rosati, G. Planck-scale-deformed relativistic symmetries and diffeomorphisms in momentum space. Phys. Rev.
**2020**, D101, 026018. [Google Scholar] [CrossRef][Green Version] - Franchino-Viñas, S.; Mignemi, S. Snyder-de Sitter meets the Grosse-Wulkenhaar model. Nucl. Phys.
**2020**, B959, 115152. [Google Scholar] [CrossRef] - Shen, L.T.; Yang, J.W.; Zhong, Z.R.; Yang, Z.B.; Zheng, S.B. Quantum phase transition and quench dynamics in the two-mode Rabi model. Phys. Rev.
**2021**, A104, 063703. [Google Scholar] [CrossRef] - Daszkiewicz, M.; Walczyk, C.J. Classical mechanics of many particles defined on canonically deformed nonrelativistic spacetime. Mod. Phys. Lett.
**2011**, A26, 819. [Google Scholar] [CrossRef][Green Version] - Gnatenko, K.P.; Tkachuk, V.M. Composite system in rotationally invariant noncommutative phase space. Int. J. Mod. Phys.
**2018**, A33, 1850037. [Google Scholar] [CrossRef][Green Version] - Pramanik, S.; Ghosh, S.; Pal, P. Conformal invariance in noncommutative geometry and mutually interacting Snyder particles. Phys. Rev.
**2014**, D90, 105027. [Google Scholar] [CrossRef][Green Version] - Amorim, R.; Abreu, E.M.C.; Ramirez, W.G. Walberto Guzmán. Noncommutative relativistic particles. Phys. Rev.
**2010**, D81, 105005. [Google Scholar] - Mir-Kasimov, R.M. Focusing singularity in p-space of constant curvature. Sov. Phys. JETP
**1966**, 22, 629. [Google Scholar] - Mir-Kasimov, R.M. The Snyder space-time quantization, q-deformations, and ultraviolet divergences. Phys. Lett.
**1996**, B378, 181. [Google Scholar] [CrossRef] - Breckenridge, J.C.; Steele, T.G.; Elias, V. Massless scalar field theory in a quantized spacetime. Class. Quantum Gravity
**1995**, 12, 637. [Google Scholar] [CrossRef] - Meljanac, S.; Mignemi, S.; Trampetić, J.; You, J. UV-IR mixing in nonassociative Snyder ϕ4 theory. Phys. Rev.
**2017**, D96, 045021. [Google Scholar] [CrossRef][Green Version]

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**MDPI and ACS Style**

Meljanac, S.; Mignemi, S.
Quantum Mechanics of the Extended Snyder Model. *Symmetry* **2023**, *15*, 1373.
https://doi.org/10.3390/sym15071373

**AMA Style**

Meljanac S, Mignemi S.
Quantum Mechanics of the Extended Snyder Model. *Symmetry*. 2023; 15(7):1373.
https://doi.org/10.3390/sym15071373

**Chicago/Turabian Style**

Meljanac, Stjepan, and Salvatore Mignemi.
2023. "Quantum Mechanics of the Extended Snyder Model" *Symmetry* 15, no. 7: 1373.
https://doi.org/10.3390/sym15071373