Weyl Quantization of Exponential Lie Groups for Square Integrable Representations
Stine Marie Berge, Simon Halvdansson
TL;DR
This work develops a general Weyl quantization for square-integrable functions on connected exponential Lie groups by composing the Fourier–Wigner and Fourier–Kirillov transforms, yielding a unitary isometry $A:L^2_r(G) o ext{S}^2(\\mathcal{H})$ with a dequantization $a= ext{A}^{-1}$. The construction preserves translations and adjoints via $ fg x^* A_f fg x = A_{R_{x^{-1}}f}$ and $A_f^* = A_{ar{f}}$, and endows the phase-space with an $H^*$-algebra structure through twisted products satisfying $A_{f at g}=A_f A_g$. Wigner distributions are defined as $W(oldsymbol{ullet},oldsymbol{ullet})=a_{oldsymbol{ullet}oldsymbol{ullet}}$, with key properties including sesquilinearity, covariance under the group action, and a trace/orthogonality relation that extends classical Moyal-type identities. The framework integrates quantum harmonic analysis by showing operator convolutions correspond to group convolutions of Weyl symbols, enabling generalization of spectrogram-Wigner relations to wider groups and providing tools for phase retrieval, Wigner-approximation, and $L^p$-bounds in this generalized setting. Overall, the paper extends Weyl quantization beyond the Weyl–Heisenberg and affine groups to broad classes of exponential Lie groups, leveraging co-adjoint-orbit geometry to unify time–frequency analysis, representation theory, and operator theory with potential impact on analysis and quantum signal processing on nontraditional phase spaces.
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.