# Momentum Gauge Fields and Non-Commutative Space–Time

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## Abstract

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## 1. Gauge Theory in Momentum Space

## 2. Connection to Non-Commutative Space–Time

#### 2.1. Constant Non-Commutativity Parameter

#### 2.2. Variable Non-Commutativity Parameter

- In the vacuum (${J}^{\nu}=0$), (14) has the solution ${A}^{\nu}\propto {e}^{i(px-Et)}{\epsilon}^{\nu}\delta ({p}^{2}-{E}^{2}/{c}^{2})$, where the $\delta $-function enforces the mass shell condition $\frac{{E}^{2}}{{p}^{2}}={c}^{2}$ and ${\epsilon}^{\nu}$ is the polarization vector.
- For a point charge at rest, one has the current ${J}^{\nu}=(q{\delta}^{3}\left(r\right),0,0,0)$, which has the solution ${A}^{0}=\frac{q}{r}$ and $\overrightarrow{A}=0$, since ${\nabla}_{x}^{2}\left(\frac{1}{r}\right)=4\pi \delta \left(r\right)$.

- In the vacuum (${\mathcal{J}}^{\nu}=0$), (15) has solution ${C}^{\nu}\propto {e}^{i(px-Et)}{\epsilon}^{\nu}\delta ({x}^{2}-{c}^{2}{t}^{2})$, where the $\delta $-function enforces the light cone condition $\frac{{x}^{2}}{{t}^{2}}={c}^{2}$ and ${\epsilon}^{\nu}$ is the polarization vector.
- The momentum gauge equivalent of the charge at rest is given by ${\mathcal{J}}^{\nu}=(g{\delta}^{3}\left(p\right),0,0,0)$, with ${C}^{0}=\frac{g}{p}$ and $\overrightarrow{C}=0$ since ${\nabla}_{p}^{2}\left(\frac{1}{p}\right)=4\pi \delta \left(p\right)$.

## 3. Generalized Landau Levels

## 4. Momentum Dependent Non-Commutativity Parameter

#### 4.1. Capacitor-Type Momentum Electric Field Configuration

**same**“surface charge”, $\Sigma $. This same “surface charge” setup leads to a momentum “electric” field in the ${p}_{z}$ direction given by

**inverse**of the normal capacitor is due to the connection between the non-commutativity parameter, ${\Theta}_{\mu \nu}$, and the momentum gauge field tensor, ${G}_{\mu \nu}$, as given Equations (11) and (12) i.e., ${\Theta}_{\mu \nu}=g{G}_{\mu \nu}$. We want to have a normal position–position commutator (i.e., $[{X}_{\mu},{X}_{\nu}]=0$) for momenta near zero (i.e., for $-{p}_{a}\le {p}_{z}\le {p}_{a}$), but we want non-commutative space–time effects for large momenta, i.e., we want ${\Theta}_{\mu \nu}\propto {G}_{\mu \nu}\ne 0$ for large momenta, $|{p}_{a}|\le |{p}_{z}|$. This is different from the usual non-commutative space–time approach, where the non-commutative parameter is “turned on” for all momenta. Here, the non-commutativity, at least for the ${\Theta}_{0i}$ components, is turned on only for the z-momentum magnitude satisfying $|{p}_{a}|<|{p}_{z}|$.

#### 4.2. Current Sheet-Type Momentum Magnetic Field

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- O’Raifeartaigh, L.; Straumann, N. Gauge theory: Historical origins and some modern developments. Rev. Mod. Phys.
**2000**, 72, 1. [Google Scholar] [CrossRef][Green Version] - Abers, E.S.; Lee, B.W. Gauge Theories. Phys. Rep.
**1973**, 9, 1. [Google Scholar] [CrossRef] - Glashow, S.L. Partial Symmetries of Weak Interactions. Nucl. Phys.
**1961**, 22, 579. [Google Scholar] [CrossRef] - Weinberg, S. A Model of Leptons. Phys. Rev. Lett.
**1967**, 19, 1264. [Google Scholar] [CrossRef] - Salam, A.; Svartholm, N. (Eds.) “Elementary Particle Physics: Relativistic Groups and Analyticity” Eighth Nobel Symposium; Almquvist and Wiksell: Stockholm, Sweden, 1968. [Google Scholar]
- Guendelman, E.; Owen, D. Relativistic Quantum Mechanics and Related Topics; World Scientific: Singapore, 2022. [Google Scholar] [CrossRef]
- Lasenby, A.; Doran, C.; Phil, S.G. Gravity, gauge theories and geometric algebra. Trans. R. Soc. A
**1998**, 356, 487. [Google Scholar] [CrossRef][Green Version] - Weinberg, S. A New Light Boson? Phys. Rev. Lett.
**1978**, 40, 223. [Google Scholar] [CrossRef] - Weinberg, S. Supersymmetry at Ordinary Energies. 1. Masses and Conservation Laws. Phys. Rev. D
**1982**, 26, 287. [Google Scholar] [CrossRef] - Weinberg, S. The Cosmological Constant Problem. Rev. Mod. Phys.
**1989**, 61, 1. [Google Scholar] [CrossRef] - Berry, M.V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A
**1984**, 392, 45. [Google Scholar] - Dirac, P.A.M. Quantised singularities in the electromagnetic field. Proc. R. Soc. Lond. A
**1931**, 133, 60. [Google Scholar] - Zee, A. Quantum Field theory in a Nutshell, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Born, M. Reciprocity Theory of Elementary Particles. Rev. Mod. Phys.
**1949**, 21, 463. [Google Scholar] [CrossRef] - Douglas, M.R.; Nekrasov, N.A. Noncommutative field theory. Rev. Mod. Phys.
**2001**, 73, 977. [Google Scholar] [CrossRef][Green Version] - Chaichian, M.; Sheikh-Jabbari, M.M.; Tureanu, A. Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED. Phys. Rev. Lett.
**2001**, 86, 2716. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nicolini, P. Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review. Int. J. Mod. Phys. A
**2009**, 24, 1229. [Google Scholar] [CrossRef] - Gamboa, J.; Loewe, M.; Rojas, J.C. Noncommutative quantum mechanics. Phys. Rev. D
**2001**, 64, 067901. [Google Scholar] [CrossRef][Green Version] - Abers, E.S. Quantum Mechanics; Pearson Education Inc.: Upper Saddle River, NJ, USA, 2004. [Google Scholar]
- Singh, R.; Kothawala, D. Covariant formulation of the generalized uncertainty principle. Phys. Rev. D
**2022**, 105, L101501. [Google Scholar] [CrossRef] - Guendelman, E.I.; Wagner, F. Momentum Gauge Fields from Curved Momentum Space through Kaluza-Klein Reduction. arXiv
**2022**, arXiv:2208.00409. [Google Scholar] - Singleton, D.; Kato, A.; Yoshida, A. Alternative Gauge Procedure with Fields of Various Ranks. Phys. Lett. A
**2004**, 330, 326. [Google Scholar] [CrossRef][Green Version] - Guendelman, E.I.; Singleton, D. Scalar Gauge Fields. J. High Energy Phys.
**2014**, 5, 96. [Google Scholar] [CrossRef][Green Version] - Guendelman, E.I. Gauge invariance and mass without spontaneous symmetry breaking. Phys. Rev. Lett.
**1979**, 43, 543. [Google Scholar] [CrossRef] - Kato, A.; Singleton, D. Gauging dual symmetry. Int. J. Theor. Phys.
**2002**, 41, 1563. [Google Scholar] [CrossRef] - Curtright, T. Generalized gauge fields. Phys. Lett. B
**1985**, 165, 304. [Google Scholar] [CrossRef] - Copinger, P.; Morales, P. Emergent Spacetime from a Momentum Gauge and Electromagnetism. arXiv
**2022**, arXiv:2211.14165. [Google Scholar]

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**MDPI and ACS Style**

Guendelman, E.; Singleton, D.
Momentum Gauge Fields and Non-Commutative Space–Time. *Symmetry* **2023**, *15*, 126.
https://doi.org/10.3390/sym15010126

**AMA Style**

Guendelman E, Singleton D.
Momentum Gauge Fields and Non-Commutative Space–Time. *Symmetry*. 2023; 15(1):126.
https://doi.org/10.3390/sym15010126

**Chicago/Turabian Style**

Guendelman, Eduardo, and Douglas Singleton.
2023. "Momentum Gauge Fields and Non-Commutative Space–Time" *Symmetry* 15, no. 1: 126.
https://doi.org/10.3390/sym15010126