Comments on the Weyl-Wigner calculus for lattice models

TL;DR

This paper critically examines a recent Weyl-Wigner calculus for lattice models, highlighting its unphysical assumptions, mathematical inconsistencies, and failure to accurately represent quantum physics of discrete systems.

Contribution

It clarifies the physical and mathematical shortcomings of the recent formalism, emphasizing the importance of proper phase space and Fourier transform properties.

Findings

The formalism uses unphysical continuous momentum space.

It lacks a bijective Fourier transform, undermining the uncertainty principle.

Fails to correctly model quantum systems like qubits.

Abstract

Here, we clarify the physical aspects between the discrete Weyl-Wigner (W-W) formalism, well developed in condensed matter physics, and the so-called 'precise Weyl-Wigner calculus for lattice models' recently appearing in the literature. We point out that the use of compact continuous momentum space for a discrete lattice model is unphysically founded. It has an incommensurate phase space, highly unphysical, lacks the finite fields aspects, as exemplified by the Born-von Karman boundary condition of compactified Bravais lattice of solid-state physics, and leads to several ambiguities. This new W-W formalism simply lacks bijective Fourier transformation, which is well-known to support the uncertainty principle of canonical conjugate dynamical variables of quantum physics. Moreover, this new W-W formalism for lattice models failed to handle the quantum physics of qubits, representing two discrete lattice sites.