Time evolution of the QED vacuum in a uniform electric field: Complete analytic solution by spinorial decomposition

Exact analytical solutions are presented for the time evolution of the density of pairs produced in the QED vacuum by a time-independent, uniform electric field. The mathematical tool used here to describe the pair production is the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44, 1825 (1991)]. The initial value problem for this function is solved by decomposing the solution into a product of spinors. The equations for spinors are much simpler and are solved analytically. These calculations are nonperturbative since pair production is due to quantum-mechanical tunneling and the explicit solutions clearly exhibit their nonanalytic behavior.

Figure 1

(Color online) The pair density $n(q,\tau )$ as a function of the dimensionless momentum $q$ plotted at the values 10, 20, 30, and 40 of the dimensionless time $\tau$. The dimensionless field strength $\mathcal{E}$ is set to one.