Weyl Quantization of Exponential Lie Groups for Square Integrable Representations

TL;DR

This paper develops a new Weyl quantization method for exponential Lie groups with maximal co-adjoint orbits, linking quantum harmonic analysis and operator convolutions, and extending Weyl quantization results.

Contribution

It introduces a general quantization framework for exponential Lie groups using Fourier-Kirillov transforms, broadening the applicability of Weyl quantization techniques.

Findings

Quantization respects function translations and yields a well-behaved Wigner distribution.

Convolution relations in quantum harmonic analysis are expressed as group convolutions.

Wavelet transform squared modulus can be represented as a convolution of Wigner distributions.

Abstract

We construct a general quantization procedure for square integrable functions on well-behaved connected exponential Lie groups. The Lie groups in question should admit at least one co-adjoint orbit of maximal possible dimension. The construction is based on composing the Fourier-Wigner transform with another Fourier transform we call the Fourier-Kirillov transform. This quantization has many desirable properties including respecting function translations and inducing a well-behaved Wigner distribution. Moreover, we investigate the connection to the operator convolutions of quantum harmonic analysis. This is intricately connected to Weyl quantization in the Weyl-Heisenberg setting. We find that convolution relations in quantum harmonic analysis can be written as group convolutions of Weyl quantizations. This implies that the squared modulus of the wavelet transform of the representation can be written as a convolution between two Wigner distributions. Lastly, we look at how we can extend known results based on Weyl quantization to wider classes of groups using our quantization procedure.