arXiv subjects

Mathieu Beauvillain

Publications and source records attributed to Mathieu Beauvillain.

3 recordsLinked to original sources

arXiv2026

Quantum Mechanics on Lie Groups: II. Path Integrals

We continue our study of quantum dynamics on a Lie group $G$, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space $L^2(G)$. This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in $G$. We show that compactness can be handled through a sum over winding numbers in maximal tori of $G$, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.

quant-ph↗

arXiv2025

Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

quant-ph↗

arXiv2024

Berry Phases in the Bosonization of Nonlinear Edge Modes

We consider chiral, generally nonlinear density waves in one dimension, modelling the bosonized edge modes of a two-dimensional fermionic topological insulator. Using the coincidence between bosonization and Lie-Poisson dynamics on an affine U(1) group, we show that wave profiles which are periodic in time produce Berry phases accumulated by the underlying fermionic field. These phases can be evaluated in closed form for any Hamiltonian, and they serve as a diagnostic of nonlinearity. As an explicit example, we discuss the Korteweg-de Vries equation, viewed as a model of nonlinear quantum Hall edge modes.

cond-mat.mes-hall↗