Killing symmetries of generalized Minkowski spaces. 1-Algebraic-infinitesimal structure of space-time rotation groups

TL;DR

This paper introduces the algebraic structure of space-time rotation groups in N-dimensional generalized Minkowski spaces with non-diagonal, coordinate-dependent metrics, focusing on their Killing symmetries and infinitesimal generators.

Contribution

It characterizes the maximal Killing group as a product of generalized Lorentz and translation groups, deriving explicit generators and algebra for these spaces.

Findings

Maximal Killing group is a product of generalized Lorentz and translation groups.

Explicit form of generalized Lorentz generators derived.

Specialization to 4D deformed Minkowski space with energy-dependent metric coefficients.

Abstract

In this paper, we introduce the concept of N-dimensional generalized Minkowski space, i.e. a space endowed with a (in general non-diagonal) metric tensor, whose coefficients do depend on a set of non-metrical coodinates. This is the first of a series of papers devoted to the investigation of the Killing symmetries of generalized Minkowski spaces. In particular, we discuss here the infinitesimal-algebraic structure of the space-time rotations in such spaces. It is shown that the maximal Killing group of these spaces is the direct product of a generalized Lorentz group and a generalized translation group. We derive the explicit form of the generators of the generalized Lorentz group in the self-representation and their related, generalized Lorentz algebra. The results obtained are specialized to the case of a 4-dimensional, ''deformed'' Minkowski space $% \widetilde{M_{4}}$, i.e. a pseudoeuclidean space with metric coefficients depending on energy.