Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

Abstract

Dirac, Fock, and Podolsky devised a relativistic model in 1932 in which a fixed number of N Dirac electrons interact through a second-quantized electromagnetic field. It is formulated with the help of a multitime wave function ψ(t1,x1,…,tN,xN) that generalizes the Schrödinger multiparticle wave function to allow for a manifestly relativistic formulation of wave mechanics. The dynamics is given in terms of N evolution equations that have to be solved simultaneously. Integrability imposes a rather strict constraint on the possible forms of interaction between the N particles and makes the rigorous construction of interacting dynamics a long-standing problem, also present in the modern formulation of quantum field theory. For a simplified version of the multitime model, in our case describing N Dirac electrons that interact through a relativistic scalar field, we prove well-posedness of the corresponding multitime initial value problem and discuss the mechanism and type of interaction between the charges. For the sake of mathematical rigor, we are forced to employ an ultraviolet cutoff in the scalar field. Although this again breaks the desired relativistic invariance, this violation occurs only on the arbitrarily small but finite length-scale of this cutoff. In view of recent progress in this field, the main mathematical challenges faced in this work are, on the one hand, the unboundedness from below of the free Dirac Hamiltonians and the unbounded, time-dependent interaction terms, and on the other hand, the necessity of pointwise control of the multitime wave function.