Dual DSR

TL;DR

This paper introduces dual DSR, a framework based on dual kappa Poincare algebra, showing its equivalence to de Sitter relativity, and explores its implications for invariant scales, scalar fields, and a minimum momentum.

Contribution

It establishes the isomorphism between dual kappa Poincare algebra and de Sitter algebra, deriving key physical quantities and proposing a dual generalized uncertainty principle.

Findings

Dual kappa Poincare algebra is isomorphic to de Sitter algebra.

Derived the Casimir invariant and scalar field equations in dual DSR.

Identified an observer-independent minimum momentum and its implications.

Abstract

We develop the physics of dual kappa Poincare algebra, which we will call dual DSR. First, we show that the dual kappa Poincare algebra is isomorphic to de Sitter algebra and its spactime is essentially de Sitter spacetime. Second, we show how to derive the coproduct rules for Beltrami and conformal coordinates of de Sitter spacetime. It follows from the current literature on de Sitter relativity that the speed of light c and the de Sitter length are the two invariant scales of the physics of dual kappa Poincare algebra. Third, we derive the Casimir invariant of the dual kappa Popincare algebra and use this to derive an expression for the speed of light, our fourth result. Fifth, the field equation for the scalar field is derived from the Casimir invariant. The results for the coordinate speed of light and the scalar field theory are the same as in de Sitter theory in the planar coordinate basis. Thus, we have shown that the physics of dual kappa Poincare algebra (in the dual bicrossproduct basis), which can be apprpriately called dual DSR, is essentially de Sitter relativity. Sixth, we argue the existence of an observer-independent minimum momentum. Seventh, we argue heuristically that the existence of minimum momentum will lead to a dual generalized uncertainty principle. Finally, we note that dual DSR is not a quantum theory of spacetime but a quantum theory of momenta.