# Non-commutative Fourier transform for the Lorentz group via the Duflo map

## TL;DR

This paper introduces a non-commutative Fourier transform for the Lorentz group using the Duflo map, establishing a new algebraic framework for quantum systems with Lorentz symmetry.

## Contribution

It develops a first-principles construction of a non-commutative Fourier transform for the Lorentz group based on the Duflo quantization map, linking classical and quantum structures.

## Key findings

- Defined a non-commutative algebra representation for Lorentz group quantum systems
- Constructed a non-commutative Fourier transform ensuring unitarity
- Derived all structures from the Duflo quantization map

## Abstract

We defined a non-commutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the non-commutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.