# Multi-Time Dynamics of the Dirac-Fock-Podolsky Model of QED

## TL;DR

This paper proves the well-posedness of a simplified relativistic multi-time model of QED involving Dirac electrons and a scalar field, addressing mathematical challenges like unbounded operators and interaction control.

## Contribution

It establishes rigorous mathematical results for a simplified multi-time Dirac-Fock-Podolsky model, including well-posedness and interaction analysis, with a focus on handling unbounded operators.

## Key findings

- Proved well-posedness of the multi-time initial value problem.
- Analyzed the interaction mechanism between charges.
- Addressed mathematical challenges like unbounded Hamiltonians.

## Abstract

Dirac, Fock, and Podolsky [Ref. 1] 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 multi-time wave function $\psi(t_1,\mathbf{x}_1,...,t_N,\mathbf{x}_N)$ that generalizes the Schr\"odinger multi-particle 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 multi-time model, in our case describing $N$ Dirac electrons that interact through a relativistic scalar field, we prove well-posedness of the corresponding multi-time 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 cut-off in the scalar field. Although this again breaks the desired relativistic invariance, this violation occurs only on the arbitrary small but finite length-scale of this cut-off. 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 multi-time wave function.