A free Lie algebra approach to curvature corrections to flat space-time

TL;DR

This paper develops a systematic algebraic framework using free Lie algebras to incorporate curvature corrections into flat space-time symmetries, leading to an infinite-dimensional algebra that captures (A)dS curvature effects at all orders.

Contribution

It introduces a novel free Lie algebra extension of the Poincaré algebra to encode curvature corrections order-by-order, and constructs a particle action reflecting these corrections.

Findings

The extended algebra, Poincaré_infinity, includes (A)dS as a quotient.

A non-linear realization of the algebra yields a geodesic equation with curvature corrections.

The approach systematically encodes curvature effects in symmetry algebra.

Abstract

We investigate a systematic approach to include curvature corrections to the isometry algebra of flat space-time order-by-order in the curvature scale. The Poincar\'e algebra is extended to a free Lie algebra, with generalised boosts and translations that no longer commute. The additional generators satisfy a level-ordering and encode the curvature corrections at that order. This eventually results in an infinite-dimensional algebra that we refer to as Poincar\'e${}_\infty$, and we show that it contains among others an (A)dS quotient. We discuss a non-linear realisation of this infinite-dimensional algebra, and construct a particle action based on it. The latter yields a geodesic equation that includes (A)dS curvature corrections at every order.