QED in external field with space–time uniform invariants: Exact solutions

We study exact solutions of Dirac and Klein–Gordon equations and Green functions in $d$ -dimensional QED and in an external electromagnetic field with constant and homogeneous field invariants. The cases of even and odd dimensions are considered separately; they are essentially different. We direct attention to the asymmetry of the quasienergy spectrum, which appears in odd dimensions. The in and out classification of the exact solutions as well as the completeness and orthogonality relations is strictly proven. Different Green functions in the form of sums over the exact solutions are constructed. The Fock–Schwinger proper time integral representations of these Green functions are found. As physical applications, we consider the calculations of different quantum effects related to the vacuum instability in the external field. For example, we present mean values of particles created from the vacuum, the probability of the vacuum remaining a vacuum, the effective action, and the expectation values of the current and energy-momentum tensor.