Revisiting the Poincaré and Little Groups with Physical Applications
Abstract
1. Introduction
2. The Lorentz Group
Orbits of the Lorentz Group
- (a)
- For time-like , we have . The little group is, thus, the rotation group.
- (b)
- For light-like , this four-vector is invariant under rotations around the z-axis.
- (c)
- For, again, light-like , the rotation matrix is the same.
- (d)
- For space-like , this four-vector remains invariant under rotations around the z-axis and boosts along either the x- or y-axis. Together, these form the three-dimensional Lorentz group , satisfying the required condition.
- (e)
- For , the entire Lorentz group leaves this zero momentum invariant, where the origin is the orbit.
3. The Covering Group of the Lorentz Group: SL(2, c)
4. Subgroups of the Lorentz Group
The Squeeze-Rotation and the Shear-Squeeze Representations of the Sp(2) Group
5. Poincaré Group and Wigner’s Little Groups
5.1. Poincaré Group
5.2. Wigner’s Little Groups
5.3. Wigner Four-Momentum-Matrices
6. Examples
6.1. Applications to Quantum Mechanics: Lorentz-Covariant Harmonic Oscillators and Entangled Excited States
6.1.1. Lorentz-Covariant Harmonic Oscillators
6.1.2. Entangled Excited States
6.2. Applications to High Energy Physics: Proton Form Factor and Feynman’s Parton Model
6.2.1. The Proton Form Factor
6.2.2. The Parton Picture
- (a)
- The picture is only valid if the hadrons are moving at close to light speed.
- (b)
- The partons behave as independent free particles, and the interaction time of the quarks becomes dilated between the quarks.
- (c)
- The hadron appears to have a widespread momentum distribution of partons.
- (d)
- The parton number appears to be much greater than that of quarks or even infinite.
6.3. Application to Classical Optics: Laser Cavity
6.4. Applications to Quantum Optics: Shear States
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Exponentiation | Two-by-Two | Four-by-Four |
---|---|---|
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Particle Mass | Wigner Four-Vector | Wigner Transformation Matrix |
---|---|---|
Massive | ||
Massless | ||
Imaginary-mass |
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Başkal, S.; Kim, Y.S.; Noz, M.E. Revisiting the Poincaré and Little Groups with Physical Applications. Symmetry 2025, 17, 1003. https://doi.org/10.3390/sym17071003
Başkal S, Kim YS, Noz ME. Revisiting the Poincaré and Little Groups with Physical Applications. Symmetry. 2025; 17(7):1003. https://doi.org/10.3390/sym17071003
Chicago/Turabian StyleBaşkal, Sibel, Young S. Kim, and Marilyn E. Noz. 2025. "Revisiting the Poincaré and Little Groups with Physical Applications" Symmetry 17, no. 7: 1003. https://doi.org/10.3390/sym17071003
APA StyleBaşkal, S., Kim, Y. S., & Noz, M. E. (2025). Revisiting the Poincaré and Little Groups with Physical Applications. Symmetry, 17(7), 1003. https://doi.org/10.3390/sym17071003