Abstract Harmonic Analysis on Spacetime

TL;DR

This paper develops a Fourier transform framework for the Poincaré group in Minkowski spacetime, establishing a Plancherel theorem to facilitate harmonic analysis in relativistic physics.

Contribution

It introduces a Fourier transform on the Poincaré group and proves the Plancherel theorem, bridging harmonic analysis with spacetime symmetries.

Findings

Defined the Fourier transform on the Poincaré group

Established the Plancherel theorem for spacetime

Provides mathematical tools for harmonic analysis in relativity

Abstract

In this paper, we consider the Poincare group (space time). In mathematics, the Poincar\'e group of spacetime, named after Henri Poincar\'e, is the group of isometries of Minkowski spacetime, introduced by Hermann Minkowski. It is a non-abelian Lie group with ten generators. Spacetime, in physical science, single concept that recognizes the union of space and time, posited by Albert Einstein in the theories of relativity. One of the interesting problems for Mathematicians and Physicists is. Can we do the Fourier analysis on space time. The purpose of this paper is to define the Fourier transform the Poincar\'e group, and then we establish the Plancherel theorem for spacetime