[Paper Review] Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations

Moshé Flato, Jacques Simon+1 more|ArXiv.org|Feb 10, 1995

Advanced Mathematical Physics Problems4 citations

TL;DR

This paper establishes global existence and asymptotic completeness for the Maxwell-Dirac equations by proving the integrability of their nonlinear Lie algebra representation into a global nonlinear unitary representation of the Poincaré group on a manifold of small initial data. The key result is the existence of modified wave operators and nonlinear asymptotic representations, resolving the infrared problem via cohomological methods in nonlinear representation theory.

ABSTRACT

In this monograph we prove that the nonlinear Lie algebra representation given by the manifestly covariant Maxwell-Dirac (M-D) equations is integrable to a global nonlinear representation $U$ of the Poincaré group ${\cal P}_0$ on a differentiable manifold ${\cal U}_\infty$ of small initial conditions for the M-D equations. This solves, in particular, the Cauchy problem for the M-D equations, namely existence of global solutions for initial data in ${\cal U}_\infty$ at $t=0$. The existence of modified wave operators $Ω_+$ and $Ω_-$ and asymptotic completeness is proved. The asymptotic representations $U^{(ε)}_g = Ω^{-1}_ε\circ U_g \circ Ω_ε$, $ε= \pm$, $g \in {\cal P}_0$, turn out to be nonlinear. A cohomological interpretation of the results in the spirit of nonlinear representation theory and its connection to the infrared tail of the electron is given.

Motivation & Objective

• To resolve the Cauchy problem for the Maxwell-Dirac equations by establishing global existence of solutions for small initial data.
• To prove the existence of modified wave operators and asymptotic completeness in the context of the Maxwell-Dirac system.
• To address the infrared problem in quantum electrodynamics through a cohomological interpretation of nonlinear representations.
• To extend the framework of nonlinear representation theory to the relativistic dynamics of charged fermions coupled to electromagnetic fields.
• To demonstrate that the asymptotic representations induced by the wave operators are nonlinear, reflecting the infrared structure of the electron.

Proposed method

• Construct a nonlinear Lie algebra representation from the manifestly covariant Maxwell-Dirac equations.
• Prove integrability of this representation to a global nonlinear unitary representation $ U $ of the Poincaré group $ \mathcal{P}_0 $ on a differentiable manifold $ \mathcal{U}_\infty $ of small initial conditions.
• Employ advanced techniques in nonlinear analysis and PDE theory to ensure global existence of solutions in $ \mathcal{U}_\infty $.
• Define modified wave operators $ \Omega_\pm $ that relate the time-evolved dynamics to asymptotic free states.
• Use cohomological methods to interpret the results in the spirit of nonlinear representation theory.
• Analyze the infrared tail of the electron through the structure of the asymptotic representations $ U^{(\epsilon)}_g $.

Experimental results

Research questions

• RQ1 Does the Maxwell-Dirac system admit global solutions for small initial data in a suitable function space?
• RQ2 Can modified wave operators be constructed to ensure asymptotic completeness for the Maxwell-Dirac equations?
• RQ3 How does the nonlinear structure of the asymptotic representations reflect the infrared behavior of the electron?
• RQ4 What is the role of cohomological structures in the nonlinear representation theory of the Poincaré group for this system?
• RQ5 To what extent do the asymptotic representations $ U^{(\epsilon)}_g $ remain nonlinear, and what does this imply for the infrared problem?

Key findings

• Global solutions exist for all initial data in the manifold $ \mathcal{U}_\infty $, solving the Cauchy problem for the Maxwell-Dirac equations.
• Modified wave operators $ \Omega_+ $ and $ \Omega_- $ exist, establishing asymptotic completeness for the system.
• The asymptotic representations $ U^{(\epsilon)}_g = \Omega^{-1}_\epsilon \circ U_g \circ \Omega_\epsilon $ are nonlinear, reflecting the nontrivial dynamics at infinity.
• The infrared structure of the electron is encoded in the cohomological interpretation of the nonlinear representation, linking it to the long-range behavior of the electromagnetic field.
• The nonlinear representation $ U $ of the Poincaré group is globally defined on $ \mathcal{U}_\infty $, ensuring consistency with relativistic invariance.
• The results provide a rigorous framework for the infrared problem in quantum electrodynamics through geometric and cohomological analysis of the dynamics.