Groupoids and the tomographic picture of quantum mechanics

TL;DR

This paper explores the connection between groupoid algebra structures and the tomographic representation of quantum states, providing new constructions and conditions for their mathematical and physical interpretation.

Contribution

It introduces a novel framework linking groupoid convolution algebras with quantum tomograms, including explicit constructions for qudit, Fock states, and symplectic tomography.

Findings

Constructed realizations of groupoid algebras for various quantum states

Established conditions linking convolution products with star-products

Derived explicit intertwining kernels for symplectic tomograms

Abstract

The existing relation between the tomographic description of quantum states and the convolution algebra of certain discrete groupoids represented on Hilbert spaces will be discussed. The realizations of groupoid algebras based on qudit, photon-number (Fock) states and symplectic tomography quantizers and dequantizers will be constructed. Conditions for identifying the convolution product of groupoid functions and the star--product arising from a quantization--dequantization scheme will be given. A tomographic approach to construct quasi--distributions out of suitable immersions of groupoids into Hilbert spaces will be formulated and, finally, intertwining kernels for such generalized symplectic tomograms will be evaluated explicitly.