Lie Group Contractions and Relativity Symmetries

TL;DR

This paper systematically explores Lie algebra contractions of $SO(m,n)$ symmetries to derive new relativity symmetries, including Galilean-type symmetries, with potential implications for quantum spacetime physics.

Contribution

It introduces a systematic classification of Lie algebra contractions of $SO(m,n)$ symmetries, extending previous work and providing explicit examples relevant to quantum spacetime.

Findings

Five different contractions of G(m,n) symmetries preserving same type at reduced dimension

Explicit coset space representations for physical models

Contractions of SO(2,4) relevant to quantum spacetime physics

Abstract

With a more relaxed perspective on what constitutes a relativity symmetry mathematically, we revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions. We focus on the contractions of an $SO(m,n)$ symmetry as a relativity symmetry on an $m+n$ dimension geometric arena, which generalizes the notion of spacetime, and discuss systematically contractions that reduce the dimension one at a one, aiming at going one step beyond what has been discussed in the literature. Our key results are five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension $m+n-1$, e.g. a G(m,n-1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for "quantum spacetime". The contractions from G(1,3) may be relevant to real physics.