Velocity Averaging for the Wigner Equation with Electromagnetic Fields
Abstract
We discuss the application of L2-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the L2-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables (x,v=ξ−A(t,x)), which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in L2 but no gain in regularity.
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BibTeX
@misc{golse2026velocity,
title = {{Velocity Averaging for the Wigner Equation with Electromagnetic Fields}},
author = {François Golse and Jakob Möller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21885},
eprint = {2609.21885},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21885},
}
RIS
TY - UNPB
TI - Velocity Averaging for the Wigner Equation with Electromagnetic Fields
AU - François Golse
AU - Jakob Möller
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21885
UR - https://arxiv.org/abs/2609.21885
AB - We discuss the application of L^2-based velocity averaging to the Wigner kinetic equation governing the quantum evolution of the Wigner transform of a density operator of a particle subjected to an external electromagnetic field. This extends the L^2-based averaging lemma from [F. Golse, J. Möller: Commun. Math. Sci. (2026)] to electromagnetic fields. We make use of the magnetic or gauge-invariant Wigner transform due to Stratonovich [Sov. Phys. D., (1)414-418, 1956] which differs from the usual definition by a phase factor involving the circulation of the magnetic vector potential along a line segment. The magnetic Wigner transform obeys the Wigner equation in the generalized phase space variables (x,v=ξ−A(t,x)), which is a more natural setting for the application of velocity averaging. Since the magnetic Wigner equation contains a first order derivative in the spatial variable, we obtain compactness in L^2 but no gain in regularity.
ER -