Scalar Field Theory on Non-commutative Snyder Space-Time

TL;DR

This paper develops a scalar field theory on Snyder non-commutative space-time, addressing non-associativity issues and providing explicit first-order corrections, thus advancing the understanding of quantum geometry and field interactions.

Contribution

It introduces a realization-based approach to construct a consistent scalar field theory on Snyder space-time, overcoming non-associativity challenges and explicitly computing correction terms.

Findings

Constructed a co-algebraic framework for Snyder geometry.

Defined a self-interacting scalar field theory on Snyder space.

Computed first-order correction terms in the Lagrangian.

Abstract

We construct a scalar field theory on the Snyder non-commutative space-time. The symmetry underlying the Snyder geometry is deformed at the co-algebraic level only, while its Poincar\'e algebra is undeformed. The Lorentz sector is undeformed at both algebraic and co-algebraic level, but the co-product for momenta (defining the star-product) is non-co-associative. The Snyder-deformed Poincar\'e group is described by a non-co-associative Hopf algebra. The definition of the interacting theory in terms of a non-associative star-product is thus questionable. We avoid the non-associativity by the use of a space-time picture based on the concept of realization of a non-commutative geometry. The two main results we obtain are: (i) the generic (namely for any realization) construction of the co-algebraic sector underlying the Snyder geometry and (ii) the definition of a non-ambiguous self interacting scalar field theory on this space-time. The first order correction terms of the corresponding Lagrangian are explicitly computed. The possibility to derive Noether charges for the Snyder space-time is also discussed.