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

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.