The role of Heisenberg group in Harmonic analysis

Abstract

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics.

we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space .

we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties.

The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group.

Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .