Noncommutative Spacetime in Very Special Relativity

Very Special Relativity (VSR) framework, proposed by Cohen and Glashow [1], demonstrated that a proper subgroup of the Poincar\'e group, (in particular ISIM(2)), is sufficient to describe the spacetime symmetries of the so far observed physical phenomena. Subsequently a deformation of the latter, $DISIM_b(2)$, was suggested by Gibbons, Gomis and Pope [2]. In the present work, we introduce a novel Non-Commutative (NC) spacetime structure, underlying the $DISIM_b(2)$. This allows us to construct explicitly the $DISIM_b(2)$ generators, consisting of a sector of Lorentz rotation generators and the translation generators. Exploiting the Darboux map technique, we construct a point particle Lagrangian that lives in the NC phase space proposed by us and satisfies the modified dispersion relation proposed by Gibbons et. al. [2]. It is interesting to note that in our formulation the momentum algebra becomes non-commutative.