Scalar field theory in Snyder space-time: alternatives

TL;DR

This paper develops two scalar field theories on Snyder space-time, exploring non-associative deformations and extra-dimensional approaches, revealing implications for Poincaré symmetries in non-commutative geometry.

Contribution

It introduces two novel scalar field theories on Snyder space-time, analyzing their symmetry properties and connections to larger non-commutative spaces.

Findings

Non-associative deformation of Poincaré symmetries uncovered.

Extra-dimensional scalar field theories can be restricted to Snyder space.

Results extend to any dimension and signature.

Abstract

We construct two types of scalar field theory on Snyder space-time. The first one is based on the natural momenta addition inherent to the coset momentum space. This construction uncovers a non-associative deformation of the Poincar\'e symmetries. The second one considers Snyder space-time as a subspace of a larger non-commutative space. We discuss different possibilities to restrict the extra-dimensional scalar field theory to a theory living only on Sndyer space-time and present the consequences of these restrictions on the Poincar\'e symmetries. We show moreover how the non-associative approach and the Doplicher-Fredenhagen-Roberts space can be seen as specific approximations of the extra-dimensional theory. These results are obtained for the 3d Euclidian Snyder space-time constructed from $\SO(3,1)/\SO(3)$, but our results extend to any dimension and signature.