The Deformation of Poincar\'e Subgroups Concerning Very Special Relativity

TL;DR

This paper explores how semi-product subgroups of the Poincaré group deform within the framework of Very Special Relativity, revealing new scale transformations affecting rotations and boosts, which constrain invariant metrics and field theories.

Contribution

It systematically classifies all possible deformations of Poincaré subgroups in Very Special Relativity and analyzes their natural representations and geometric implications.

Findings

Rotation operations may include additional scale transformations upon deformation.

Boost operations acquire additional scale transformations in all deformation cases.

Additional scale transformations impose constraints on invariant metrics and field theories.

Abstract

We investigate here various kinds of semi-product subgroups of Poincar\'e group in the scheme of Cohen-Glashow's very special relativity along the deformation approach by Gibbons- Gomis-Pope. For each proper Poincar\'e subgroup which is a semi-product of proper lorentz group with the spacetime translation group T(4), we investigate all possible deformations and obtain all the possible natural representations which inherit from the $5-d$ representation of Poincar\'e group. We find from the obtained natural representation that rotation operation may have additional accompanied scale transformation in the case of the original Lorentz subgroup is deformed and the boost operation get the additional accompanied scale transformation in all the deformation cases. The additional accompanied scale transformation has strong constrain on the possible invariant metric function of the corresponding geometry and the field theories in the spacetime with the corresponding geometry.