[Paper Review] Four classes of modified relativistic symmetry transformations

J. Lukierski, Anatol Nowicki|ArXiv.org|Oct 11, 2002

Crystallography and Radiation Phenomena3 citations

TL;DR

This paper introduces and classifies four classes of modified relativistic symmetry transformations based on nonlinear realizations of the Poincaré algebra, extending Doubly Special Relativity (DSR) theories. It proposes DSR1 (bounded momentum), DSR2 (bounded energy and momentum), DSR3 (bounded energy), and Smoothly Modified Special Relativity (SMSR, unbounded in both) through nonlinear momentum and energy transformations, with explicit boost transformations and deformed dispersion relations derived for each framework.

ABSTRACT

We discuss the nonlinear transformations of standard Poincaré symmetry in the context of recently introduced Doubly Special Relativity (DSR) theories. We introduce four classes of modified relativistic theories with three of them describing various DSR frameworks. We consider four examples of modified relativistic symmetries, which illustrate each of the considered class.

Motivation & Objective

• To classify and systematically describe four distinct classes of modified relativistic symmetries beyond standard Poincaré invariance.
• To extend Doubly Special Relativity (DSR) theories by introducing new frameworks, including DSR3 (bounded energy) and SMSR (unbounded, smooth modifications).
• To analyze how nonlinear basis transformations in the Poincaré algebra affect covariance relations, dispersion relations, and boost transformations.
• To provide explicit examples of nonlinear momentum and energy mappings that preserve the Lorentz algebra but deform the boost sector.
• To establish connections between these modified symmetries and quantum group structures via coproduct deformations.

Proposed method

• Introduces nonlinear transformations of momentum and energy variables: $\vec{\cal{P}} = \vec{P} g(E/\kappa c^2, \vec{P}^2/\kappa^2 c^2)$, $\cal{E} = \kappa c^2 f(E/\kappa c^2, \vec{P}^2/\kappa^2 c^2)$, preserving the classical Lorentz algebra.
• Derives modified covariance relations for $[N_i, P_j]$, $[N_i, E]$, and $[M_i, E]$ using the new nonlinear basis, showing dependence on functions $f$ and $g$.
• Constructs deformed mass Casimir invariants (dispersion relations) for each class, such as $C_2 = (2\kappa \sinh(E/2\kappa c^2))^2 - \vec{P}^2 e^{E/\kappa c^2}$ for DSR1.
• Derives nonlinear boost transformations via the standard rapidity formulas (4)-(5), adapted to the new variables, yielding nontrivial $E(\alpha)$ and $\vec{P}(\alpha)$ dependencies.
• Analyzes coproduct structures for the nonlinear generators, showing symmetric coproducts for $\Delta E$ and $\Delta \vec{P}$ in terms of the classical $\Delta \cal{P}_\mu$.
• Relates the framework to quantum group structures by considering Drinfeld twist deformations of primitive coproducts, leading to nonprimitive coproducts in quantum Poincaré algebras.

Experimental results

Research questions

• RQ1 What are the distinct classes of modified relativistic symmetries that generalize Doubly Special Relativity (DSR) theories?
• RQ2 How do nonlinear basis transformations in the Poincaré algebra affect the boost generators and covariance relations?
• RQ3 What are the resulting dispersion relations and how do they differ across DSR1, DSR2, DSR3, and SMSR frameworks?
• RQ4 Can unbounded energy and momentum be consistently described in a modified relativistic framework with two invariant scales?
• RQ5 How do the coproduct structures of the nonlinear generators relate to quantum group deformations and deformed addition laws for momenta?

Key findings

• Four classes of modified relativistic symmetries are classified: DSR1 (bounded momentum), DSR2 (bounded energy and momentum), DSR3 (bounded energy), and SMSR (unbounded, smooth modifications).
• For DSR1, the energy is unbounded while momentum is bounded by $\kappa c$, with the dispersion relation $C_2 = (2\kappa \sinh(E/2\kappa c^2))^2 - \vec{P}^2 e^{E/\kappa c^2} = M^2$.
• In DSR2, both energy and momentum are bounded: $E \leq \kappa c^2$, $|\vec{P}| \leq \kappa c$, with a symmetric dispersion relation derived from the nonlinear transformation.
• For DSR3, energy is bounded ($E \leq \kappa c^2$) while momentum is unbounded, with the dispersion relation $C_2 = E^2(1 + \vec{P}^2/\kappa^2 c^2) - c^2 \vec{P}^2 = M^2 c^4$.
• In SMSR, both energy and momentum are unbounded, with the transformation $\cal{E} = 2\kappa c^2 \sinh(E/2\kappa c^2)$, and the dispersion relation $C_2 = (2\kappa c^2 \sinh(E/2\kappa c^2))^2 - c^2 \vec{P}^2 = M^2 c^4$.
• The nonlinear boost transformations for each class are explicitly derived, showing nontrivial dependence on rapidity and the new variables, with $E(\alpha)$ and $\vec{P}(\alpha)$ expressed in terms of hyperbolic functions and scaling factors.