# Curved Momentum Space, Locality, and Generalized Space-Time

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Deformed Relativistic Invariance

- a representation of the Lorentz transformations (J) in momentum space$${J}_{\omega}:\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mathcal{M}\to \mathcal{M}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{k}_{\mu}^{\prime}={\left[{J}_{\omega}\left(k\right)\right]}_{\mu}\doteq {J}_{\mu}(\omega ,k),$$
- another representation of the Lorentz group of transformations (${J}^{\left(2\right)}$) in the system of two particles such that, when the total momentum is $\mathcal{P}=p\oplus q$,$${J}_{\omega}^{\left(2\right)}:\phantom{\rule{1.em}{0ex}}\mathcal{M}\otimes \mathcal{M}\to \mathcal{M}\otimes \mathcal{M}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{J}_{\omega}^{\left(2\right)}(p,q)=({p}^{\prime},\overline{q}),$$$${p}_{\mu}^{\prime}={J}_{\mu}(\omega ,p)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\overline{q}}_{\mu}={J}_{\mu}^{\left(2\right)}(\omega ,p,q),$$$${(p\oplus q)}^{\prime}={p}^{\prime}\oplus \overline{q}.$$

## 3. Derivation of an RDK from the Momentum Space Geometry

#### 3.1. Construction of a Relativistic Kinematics at the Two-Particle Level

#### 3.2. Isotropic Relativistic Deformed Kinematics: The $\kappa $-Poincaré Example

## 4. Generalized Space-Time and Locality of Interactions with an RDK

#### 4.1. Generalized Space-Time Coordinates

#### 4.2. Propagation of a Massless Particle in Space-Time

## 5. Complementarity of the Algebraic, Geometric, and Locality Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Carmona, J.M.; Cortés, J.L.; Relancio, J.J.
Curved Momentum Space, Locality, and Generalized Space-Time. *Universe* **2021**, *7*, 99.
https://doi.org/10.3390/universe7040099

**AMA Style**

Carmona JM, Cortés JL, Relancio JJ.
Curved Momentum Space, Locality, and Generalized Space-Time. *Universe*. 2021; 7(4):99.
https://doi.org/10.3390/universe7040099

**Chicago/Turabian Style**

Carmona, José Manuel, José Luis Cortés, and José Javier Relancio.
2021. "Curved Momentum Space, Locality, and Generalized Space-Time" *Universe* 7, no. 4: 99.
https://doi.org/10.3390/universe7040099