Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations

François Golse, Nikolai Leopold, Norbert J. Mauser, Jakob Möller, Chiara Saffirio

TL;DR

The paper rigorously derives the semiclassical limit $\hbar\to0$ for the Dirac equation under time-dependent external electromagnetic fields, showing that the electron/positron phase-space densities converge to solutions of the relativistic Vlasov equations with Lorentz force. It employs matrix-valued Wigner transforms and a Lagrange-multiplier framework to pass to the limit in the full Dirac-Wigner equation without projecting first, allowing for rough time-dependent potentials. A key result is that antimatter and spin persist as intrinsically relativistic effects at the classical level, encoded in a matrix Liouville equation for the Wigner measure with a explicit Y-term; projecting yields Berry-phase and Poisson-curvature corrections in the electron/positron transport. The analysis provides a rigorous bridge from quantum Dirac dynamics to classical relativistic kinetic theory, with precise convergence of charge and current densities under weak assumptions on the external fields and mixed-state initial data.

Abstract

We prove the mathematically rigorous (semi-)classical limit $\hbar \to 0$ of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. Gérard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform $L^2$ bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data.

The Semi-Classical Limit from the Dirac Equation with Time-Dependent External Electromagnetic Field to Relativistic Vlasov Equations

TL;DR

The paper rigorously derives the semiclassical limit for the Dirac equation under time-dependent external electromagnetic fields, showing that the electron/positron phase-space densities converge to solutions of the relativistic Vlasov equations with Lorentz force. It employs matrix-valued Wigner transforms and a Lagrange-multiplier framework to pass to the limit in the full Dirac-Wigner equation without projecting first, allowing for rough time-dependent potentials. A key result is that antimatter and spin persist as intrinsically relativistic effects at the classical level, encoded in a matrix Liouville equation for the Wigner measure with a explicit Y-term; projecting yields Berry-phase and Poisson-curvature corrections in the electron/positron transport. The analysis provides a rigorous bridge from quantum Dirac dynamics to classical relativistic kinetic theory, with precise convergence of charge and current densities under weak assumptions on the external fields and mixed-state initial data.

Abstract

We prove the mathematically rigorous (semi-)classical limit of the Dirac equation with time-dependent external electromagnetic field to relativistic Vlasov equations with Lorentz force for electrons and positrons. In this limit antimatter and spin remain as intrinsically relativistic effects on a classical level. Our global-in-time results use Wigner transforms and a Lagrange multiplier viewpoint of the matrix-valued Wigner equation. In particular, we pass to the limit in the ''full" Wigner matrix equation without projecting on the eigenspaces of the matrix-valued symbol of the Dirac operator. In the limit, the Lagrange multiplier maintains the constraint that the Wigner measure and the symbol of the Dirac operator commute and vanishes when projected on the electron or positron eigenspace. This is a different approach to the problem as discussed in [P. Gérard, P. Markowich, N.J. Mauser, F. Poupaud: Comm. Pure Appl. Math. 50(4):323--379, 1997], where the limit is taken in the projected Wigner equation. By explicit calculation of the remainder term in the expansion of the Moyal product we are able to generalize to time-dependent potentials with much less regularity. We use uniform bounds for the Wigner transform, which are only possible for a special class of mixed states as initial data.

Paper Structure

This paper contains 19 sections, 18 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. The relativistic Vlasov equation
  3. The Dirac-von Neumann equation
  4. The Dirac-Wigner equation
  5. Analogy with Navier-Stokes
  6. Main results
  7. Notation
  8. Proof of the main theorems
  9. Outline of the proof
  10. Preliminaries
  11. Proof of Theorem \ref{['thm:main1']}
  12. Proof of Theorem \ref{['thm:main2']}
  13. Derivation of the Dirac-Wigner equation
  14. Explicit calculations
  15. Dirac matrices
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\varepsilon = \hbar$ be a small parameter. Then the matrix-valued Wigner transform $W^{\varepsilon}$ of a mixed state density matrix $R^{\varepsilon}$ – whose eigenvalues ("occupation probabilities") $\lambda_j^{\varepsilon}$ satisfy and which solves the von Neumann equation associated with the Dirac equation eq:dirac equation spinor – converges weakly$^{\ast}$ in $L^{\infty}(\mathbb{R}_t,L^

Theorems & Definitions (36)

  • Theorem
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • proof
  • Corollary 1
  • ...and 26 more