Consequences of Dirac Theory of the Positron
W. Heisenberg, H. Euler
TL;DR
The paper investigates how Dirac's positron theory implies nonlinear corrections to vacuum electrodynamics through vacuum polarization, framing the analysis with the Dirac density matrix and the auxiliary R_S operator to compute the vacuum energy density $U(\mathfrak E,\mathfrak B)$ for slowly varying fields. By subtracting the singular vacuum contribution and expressing the result as a Lagrangian depending on the invariants ${\mathfrak E}^2-{\mathfrak B}^2$ and $({\mathfrak EB})^2$, the work reveals Euler-Kockel-type corrections that reproduce known light-by-light scattering in the weak-field limit. The study discusses how virtual processes lead to divergences in perturbation theory for the quantum wave-field, and argues that the fourth-order terms are physically meaningful while higher orders and real pair creation at the critical field limit the applicability of the results. Ultimately, the findings provide a foundational, albeit provisional, link between the positron theory and nonlinear vacuum electrodynamics, highlighting both the potential corrections to Maxwell's equations and the limitations imposed by high-field phenomena and the ongoing development of quantum electrodynamics.
Abstract
According to Dirac's theory of the positron, an electromagnetic field tends to create pairs of particles which leads to a change of Maxwell's equations in the vacuum. These changes are calculated in the special case that no real electrons or positrons are present and the field varies little over a Compton wavelength.