# 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

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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