# Poincaré Symmetry from Heisenberg’s Uncertainty Relations

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## Abstract

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## 1. Introduction

## 2. Sp(2) Symmetry for the Single-Variable Uncertainty Relation

## 3. Two-Oscillator System

## 4. Contraction of SO(3, 2) to ISO(3, 1)

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Dirac, P.A.M. The Quantum Theory of the Emission and Absorption of Radiation. Proc. Roy. Soc.
**1927**, A114, 243–265. [Google Scholar] [CrossRef] - Dirac, P.A.M. Unitary Representations of the Lorentz Group. Proc. Roy. Soc.
**1945**, A183, 284–295. [Google Scholar] - Dirac, P.A.M. Forms of Relativistic Dynamics. Rev. Mod. Phys.
**1949**, 21, 392–399. [Google Scholar] [CrossRef][Green Version] - Dirac, P.A.M. A Remarkable Representation of the 3 + 2 de Sitter Group. J. Math. Phys.
**1963**, 4, 901–909. [Google Scholar] [CrossRef] - Yuen, H.P. Two-photon coherent states of the radiation field. Phys. Rev. A
**1976**, 13, 2226–2243. [Google Scholar] [CrossRef] - Yurke, B.S.; McCall, B.L.; Klauder, J.R. SU(2) and SU(1, 1) interferometers. Phys. Rev. A
**1986**, 33, 4033–4054. [Google Scholar] - Dirac, P.A.M. The Conditions for a Quantum Field Theory to be Relativistic. Rev. Mod. Phys.
**1962**, 34, 592–696. [Google Scholar] [CrossRef] - Wigner, E. On unitary representations of the inhomogeneous Lorentz group. Ann. Math.
**1939**, 40, 149–204. [Google Scholar] [CrossRef] - Arnold, V.I. Mathematical Methods of Classical Mechanics; Springer: Heidelberg, Germany, 1978. [Google Scholar]
- Guillemin, V.; Sternberg, S. Symplectic Techniques in Physics; Cambridge University Press: Cambridge, UK, 1984. [Google Scholar]
- Han, D.; Kim, Y.S.; Noz, M.E. Linear canonical transformations of coherent and squeezed states in the Wigner phase space. Phys. Rev. A
**1988**, 37, 807–814. [Google Scholar] [CrossRef] - Kim, Y.S.; Wigner, E.P. Canonical transformation in quantum mechanics. Am. J. Phys.
**1990**, 58, 439–447. [Google Scholar] [CrossRef] - Kim, Y.S.; Noz, M.E. Phase Space Picture of Quantum Mechanics; World Scientific Publishing Company: Singapore, 1991. [Google Scholar]
- Han, D.; Kim, Y.S.; Noz, M.E. SO(3, 3)-like Symmetries of Coupled Harmonic Oscillators. J. Math. Phys.
**1995**, 36, 3940–3954. [Google Scholar] [CrossRef] - Abraham, R.; Marsden, J.E. Foundations of Mechanics, 2nd ed.; Benjamin Cummings: Reading, MA, USA, 1978. [Google Scholar]
- Goldstein, H. Classical Mechanics, 2nd ed.; Addison-Wesley: Reading, MA, USA, 1980. [Google Scholar]
- Inönü, E.; Wigner, E.P. On the Contraction of Groups and their Representations. Proc. Natl. Acad. Sci. USA
**1953**, 39, 510–524. [Google Scholar] [CrossRef] [PubMed] - Kim, Y.S.; Wigner, E.P. Cylindrical group and massless particles. J. Math. Phys.
**1987**, 28, 1175–1179. [Google Scholar] [CrossRef] - Kim, Y.S.; Wigner, E.P. Space-time geometry of relativistic-particles. J. Math. Phys.
**1990**, 31, 55–60. [Google Scholar] [CrossRef] - Yukawa, H. Structure and Mass Spectrum of Elementary Particles. I. General Considerations. Phys. Rev.
**1953**, 91, 415–416. [Google Scholar] [CrossRef] - Feynman, R.P.; Kislinger, M.; Ravndal, F. Current matrix elements from a relativistic quark model. Phys. Rev. D
**1971**, 3, 2706–2732. [Google Scholar] [CrossRef] - Loewe, M.; Magnollay, P.; Mukunda, N.; Drechsler, W.; Korny, S.R. Relativistic rotator. I. Quantum observables and constrained Hamiltonian mechanics. Phys. Rev. D
**1983**, 28, 3020–3031. [Google Scholar] - Bars, I. Relativistic oscillator revisted. Phys. Rev. D
**2009**, 79, 045009. [Google Scholar] [CrossRef] - Gilmore, R. Lie Groups, Lie Algebras, and Some of Their Applications; Wiley: New York, NY, USA, 1974. [Google Scholar]
- Bohm, A.; Loewe, M.; Magnollay, P.; Tarlini, M.; Aldinger, R.R.; Kielanowski, P. New Relativistic Generalization of the Heisenberg Commutation Relations. Pys. Rev. Lett.
**1984**, 53, 2292–2295. [Google Scholar] [CrossRef] - Bohm, A.; Loewe, M.; Magnollay, P.; Tarlini, M.; Aldinger, R.R.; Biedenharn, L.C.; van Dam, H. Quantum relativistic oscillator. III. Contraction between the algebras of SO(3,2) and the three-dimensional harmonic oscillator. Phys. Rev. D
**1985**, 32, 2828–2834. [Google Scholar] [CrossRef]

**Figure 1.**Contraction of the $SO(3,2)$ group to the Poincaré group. The time-like s coordinate is contracted with respect to the space-like x variable, and with respect to the time-like variable t.

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**MDPI and ACS Style**

Başkal, S.; Kim, Y.S.; Noz, M.E.
Poincaré Symmetry from Heisenberg’s Uncertainty Relations. *Symmetry* **2019**, *11*, 409.
https://doi.org/10.3390/sym11030409

**AMA Style**

Başkal S, Kim YS, Noz ME.
Poincaré Symmetry from Heisenberg’s Uncertainty Relations. *Symmetry*. 2019; 11(3):409.
https://doi.org/10.3390/sym11030409

**Chicago/Turabian Style**

Başkal, Sibel, Young S. Kim, and Marilyn E. Noz.
2019. "Poincaré Symmetry from Heisenberg’s Uncertainty Relations" *Symmetry* 11, no. 3: 409.
https://doi.org/10.3390/sym11030409