# Spacetime and Deformations of Special Relativistic Kinematics

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## Abstract

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## 1. Introduction

- “... quantum field theory ... is the only way to reconcile the principles of quantum mechanics (including the cluster decomposition property) with those of special relativity.” Steven Weinberg, The Quantum Theory of Fields, Preface to Volume I.
- “It is one of the fundamental principles of physics (indeed, of all science) that experiments that are sufficiently separated in space have unrelated results... In S-matrix theory, the cluster decomposition principle states that if multiparticle processes .. are studied in ... very distant laboratories, then the S-matrix element for the overall process factorizes.” Steven Weinberg, The Quantum Theory of Fields, Volume I, pg. 177.

## 2. Deformation of SR Kinematics

## 3. Single-Interaction Process

## 4. Multi-Interaction Process

## 5. Observable Effects in the Propagation of a Particle over Very Large Distances

- (1)
- Model for the propagation based on a DDR. The simplest model for the propagation of a particle is based on neglecting the interaction term in the action (Equation (1)): the extrema of the action correspond to the solutions of$${\dot{p}}_{\mu}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{{\dot{x}}^{\mu}}{N}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{\partial C(p)}{\partial {p}_{\mu}}.$$The velocity ($\overrightarrow{v}$) of propagation of a particle with a given momentum $\overrightarrow{p}$ is$${v}^{i}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{\partial C(p)/\partial {p}_{i}}{\partial C(p)/\partial {p}_{0}},$$
- (2)
- Model for the propagation based on a DDR and a physical spacetime defined by the DCL. A less trivial model for the propagation of a particle is based on the observation that spacetime coordinates should be defined by the crossing of worldlines in the interaction of particles. This, together with the decoupling of particles at very large distances due to the cluster decomposition principle, leads to identify the linear combinations of the canonical spacetime coordinates with coefficients depending on momentum variables$${\tilde{x}}^{\alpha}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{x}^{\mu}{\phi}_{\mu}^{\alpha}(p)$$$${\tilde{v}}^{i}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{\dot{\tilde{x}}}^{i}}{{\dot{\tilde{x}}}^{0}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{\dot{x}}^{\mu}{\phi}_{\mu}^{i}(p)}{{\dot{x}}^{\nu}{\phi}_{\nu}^{0}(p)}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{(\partial C(p)/\partial {p}_{\mu}){\phi}_{\mu}^{i}(p)}{(\partial C(p)/\partial {p}_{\nu}){\phi}_{\nu}^{0}(p)}.$$The dependence of the velocity on the momentum is determined by the DDR (through $C(p)$) and also by the DCL (through ${\phi}_{\mu}^{\alpha}(p)$, which is determined by the DCL). Both ingredients of the deformation of SR kinematics have to be considered in the propagation of a particle.The relation between the DCL and ${\phi}_{\mu}^{\alpha}(p)$ has been determined to be [35,48]$${\phi}_{\mu}^{\alpha}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\underset{q\to 0}{lim}\frac{\partial {(q\oplus p)}_{\mu}}{\partial {q}_{\alpha}}.$$An example of a DDR and DCL leading to a relativistic deformed kinematics is [34]$$C(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\Lambda}^{2}\left({e}^{{p}_{0}/\Lambda}+{e}^{-{p}_{0}/\Lambda}-2\right)-{e}^{{p}_{0}/\Lambda}{\overrightarrow{p}}^{2},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{(q\oplus p)}_{0}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{q}_{0}+{p}_{0},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{(q\oplus q)}_{i}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{q}_{i}+{e}^{-{q}_{0}/\Lambda}{p}_{i},$$$$\begin{array}{l}\frac{\partial C(p)}{\partial {p}_{0}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\Lambda \left({e}^{{p}_{0}/\Lambda}-{e}^{-{p}_{0}/\Lambda}\right)-{e}^{{p}_{0}/\Lambda}{\overrightarrow{p}}^{2}/\Lambda ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\frac{\partial C(p)}{\partial {p}_{i}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-2{e}^{{p}_{0}/\Lambda}{p}_{i},\hfill \\ {\phi}_{0}^{0}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\phi}_{0}^{i}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\phi}_{i}^{0}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-{p}_{i}/\Lambda ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\phi}_{j}^{i}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\delta}_{j}^{i},\hfill \end{array}$$$${\tilde{v}}^{i}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{-2{e}^{{p}_{0}/\Lambda}{p}_{i}}{\Lambda \left({e}^{{p}_{0}/\Lambda}-{e}^{-{p}_{0}/\Lambda}\right)+{e}^{-{p}_{0}/\Lambda}{\overrightarrow{p}}^{2}/\Lambda}.$$The dispersion relation $C(p)={m}^{2}$ gives$${e}^{{p}_{0}/\Lambda}{\overrightarrow{p}}^{2}/{\Lambda}^{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{e}^{{p}_{0}/\Lambda}+{e}^{-{p}_{0}/\Lambda}-2-{m}^{2}/{\Lambda}^{2},$$$$1-{\overrightarrow{\tilde{v}}}^{2}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{m}^{2}\left(1+{m}^{2}/4{\Lambda}^{2}\right)}{{\left[\Lambda \left({e}^{{p}_{0}/\Lambda}-1\right)-{m}^{2}/2\Lambda \right]}^{2}},$$$$1-{\overrightarrow{\tilde{v}}}^{2}\approx \frac{{m}^{2}}{{p}_{0}^{2}}\left(1-\frac{{p}_{0}}{\Lambda}\right),$$We have not considered, in all the discussion of the model for the effects of a deformation of the kinematics in the propagation of a particle, the arbitrariness in the starting point corresponding to the choice of canonical coordinates in phase space. In fact, if one considers new momentum coordinates ${p}_{\mu}^{\prime}$ related nonlinearly to ${p}_{\nu}$, then one will have a new dispersion relation defined by a function ${C}^{\prime}$ and a new modified composition law ${\oplus}^{\prime}$ which are related to the function C and ⊕ by$$C(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{C}^{\prime}({p}^{\prime}),\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{({q}^{\prime}{\oplus}^{\prime}{p}^{\prime})}_{\mu}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{(q\oplus p)}_{\mu}^{\prime}\phantom{\rule{0.166667em}{0ex}}.$$Then, we have$$\begin{array}{ll}\hfill {\phi}_{\mu}^{\prime \alpha}({p}^{\prime})& \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\underset{{q}^{\prime}\to 0}{lim}\frac{\partial {({q}^{\prime}{\oplus}^{\prime}{p}^{\prime})}_{\mu}}{\partial {q}_{\alpha}^{\prime}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\underset{{q}^{\prime}\to 0}{lim}\frac{\partial {(q\oplus p)}_{\mu}^{\prime}}{\partial {q}_{\alpha}^{\prime}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\underset{q\to 0}{lim}\frac{\partial {q}_{\beta}}{\partial {q}_{\alpha}^{\prime}}\frac{\partial {(q\oplus p)}_{\mu}^{\prime}}{\partial {q}_{\beta}}\hfill \\ & \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\underset{q\to 0}{lim}\frac{\partial {(q\oplus p)}_{\mu}^{\prime}}{\partial {(q\oplus p)}_{\nu}}\frac{\partial {(q\oplus p)}_{\nu}}{\partial {q}_{\alpha}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{\partial {p}_{\mu}^{\prime}}{\partial {p}_{\nu}}{\phi}_{\nu}^{\alpha}(p).\hfill \end{array}$$On the other hand, the nonlinear change of momentum variables $p\to {p}^{\prime}$ defines a canonical change of coordinates in phase space with$${x}^{\prime \mu}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{x}^{\rho}\frac{\partial {p}_{\rho}}{\partial {p}_{\mu}^{\prime}}\phantom{\rule{0.166667em}{0ex}},$$$${x}^{\prime \mu}{\phi}_{\mu}^{\prime \alpha}({p}^{\prime})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{x}^{\rho}\frac{\partial {p}_{\rho}}{\partial {p}_{\mu}^{\prime}}{\phi}_{\mu}^{\prime \alpha}({p}^{\prime})\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{x}^{\rho}\frac{\partial {p}_{\rho}}{\partial {p}_{\mu}^{\prime}}\frac{\partial {p}_{\mu}^{\prime}}{\partial {p}_{\nu}}{\phi}_{\nu}^{\alpha}(p)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{x}^{\nu}{\phi}_{\nu}^{\alpha}(p).$$This means that ${\tilde{x}}^{\prime \alpha}={\tilde{x}}^{\alpha}$ and then one has ${\tilde{v}}^{\prime i}({p}_{0}^{\prime})={\tilde{v}}^{i}({p}_{0})$. Every relativistic deformed kinematics obtained from Equation (23) by a canonical transformation of coordinates of the form $p\to {p}^{\prime}=f(p)$ (what is usually called a change of “basis” in momentum space) will not show any effect of the deformation in the propagation of photons, and the modified energy dependence of the velocity for a massive particle can be obtained directly from Equation (27) by using the nonlinear change of momentum variables together with the dispersion relation to express ${p}_{0}$ in terms of ${p}_{0}^{\prime}$.The absence of an energy dependence for the velocity of propagation of photons in the model for the propagation of a particle with a relativistic deformed kinematics has not taken into account the expansion of the Universe. However, this cannot be neglected in the propagation over astrophysical distances required to have a possible observable effect. It is an open question how to combine consistently a deformation of SR kinematics with the cosmological model based on a curved spacetime. Although some effects have been considered in the propagation of particles in a curved $\kappa $-Minkowski spacetime [49], there are different approaches to include a curvature in spacetime in a deformed relativistic kinematics. This has been done, in the DSR framework, using Finsler spacetimes for curved spacetimes [50]. A different approach was used in Ref. [51], where the modification is carried out by Hamiltonian geometry (see also [52]). In this case, the metric is momentum dependent (the Hamiltonian version of a Lagrange space). The starting point in all of them is a modified dispersion relation, but a key ingredient of a modified relativistic kinematics is a deformed composition law, which does not appear in the previous works. Another approach is considered in Ref. [53], where the authors tried to combine a curvature in momentum space and in spacetime including a modified composition law, generalizing the original relative locality action introducing what they called non-local variables. In any case, it remains to be seen whether the absence of signals of the deformed kinematics in timing measurements applies after the proper inclusion of the expansion of the universe.
- (3)
- Model for the propagation based on a DDR, including the interactions at production and detection of the particle that determine the initial and final points of the trajectory through the DCL. The study of the two-interaction process in the previous section suggests to use the results for the trajectory (${y}^{\mu}(\tau )$) of the particle between the two interactions as a third alternative model for the effects of a deformation of relativistic kinematics on the propagation of a particle identifying the two interactions in the process with the production and detection of the particle. This model has in common with the first simplest model based on neglecting the interaction term in the action (Equation (1)) that the velocity of propagation of the particle ${v}^{i}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\dot{y}}^{i}/{\dot{y}}^{0}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}(\partial C(q)/\partial {q}_{i})/(\partial C(q)/\partial {q}_{0})$ will have a momentum dependence determined by the DDR. However, to determine the time of flight, one has to consider that the end points of the trajectory of the particles (${y}^{\mu}({\tau}_{1})$, ${y}^{\mu}({\tau}_{2})$), and then the length of the path of the particle, depend also on the momentum of the particle and the momenta of other particles participating in the production and detection of the particles. One will have an spectral and timing distribution of gamma-rays from a short GRB which differs from the expectation in SR but one does not have a definite prediction on the effect of the deformation of SR kinematics due to its dependence on details of the production and detection interactions to which we do not have access. On top of these problems, this third model for the effects of a deformed kinematics on the propagation of a particle based on a process with two interactions presents doubts on its consistency. One has to justify why one can consider the process with just two interactions. One can ask about the interactions in the production of each of the three particles in the in-state and the detection of each of the three particles in the out-state. The argument to neglect these interactions is that they are separated by very large distances from the two interactions in the process but if we want to use the model to study the effect of a deformation of SR kinematics on the propagation of a particle over large distances then the two interactions are already separated by a very large distance and then, according with the implementation of the cluster decomposition principle at the classical level, we should treat the process with two interactions as two independent single-interaction processes. The conclusion of this discussion is that the third model for the propagation of a particle is disfavoured with respect to the other two models.

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Carmona, J.M.; Cortés, J.L.; Relancio, J.J.
Spacetime and Deformations of Special Relativistic Kinematics. *Symmetry* **2019**, *11*, 1401.
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Carmona JM, Cortés JL, Relancio JJ.
Spacetime and Deformations of Special Relativistic Kinematics. *Symmetry*. 2019; 11(11):1401.
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Carmona, José Manuel, José Luis Cortés, and José Javier Relancio.
2019. "Spacetime and Deformations of Special Relativistic Kinematics" *Symmetry* 11, no. 11: 1401.
https://doi.org/10.3390/sym11111401