The current density in quantum electrodynamics in external potentials

Abstract

We review different definitions of the current density for quantized fermions in the presence of an external electromagnetic field. Several deficiencies in the popular prescription due to Schwinger and the mode sum formula for static external potentials are pointed out. We argue that Dirac's method, which is the analog of the Hadamard pointsplitting employed in quantum field theory in curved space-times, is conceptually the most satisfactory. As a concrete example, we discuss vacuum polarization and the stress-energy tensor for massless fermions in 1+1 dimension. Also a general formula for the vacuum polarization in static external potentials in 3+1 dimensions is derived.

The current density in quantum electrodynamics in time-dependent external potentials and the Schwinger effect

“…However, both these integrals are not computed correctly. Regarding the vanishing of the time integral over (20), Serber first introduces the variable s = √ t 2 − r 2 and correctly notes that [35,Sect. 13.6] the step from the first to the second line being the result of a deformation of the integration path to the imaginary axis.…”

“…Numerical integration shows that integrals in the first and the second line do not coincide, so the path deformation seems to have been performed incorrectly. Performing the integral in the first line by computer algebra, one finds that I (19) (r, ψ) = I (20) (r, ψ). Hence, no supplementary term localized on the boundary of the light cone is present.…”

“…the singular part of the two-point function ψ(x)ψ(y) , the so-called Hadamard parametrix H(x, y), cf. [20] for a detailed discussion. This method is conceptually appealing as it is explicitly local and gauge covariant.…”

“…This may be appropriate in the 1+1 dimensional setting considered there, as the remainder is continuous (though not continuously differentiable) for equal times. However, it easily follows from the results in [20], that in 3+1 dimensions the remainder would not be continuous. In particular, the logarithmic divergence that is responsible for the charge renormalization ambiguity is not canceled, so one expects the problems with greatly enhanced values at intermediate times to persist.…”

Semiclassical energy of open Nambu-Goto strings

“…This method has been reliably used for the computation of Casimir energies and vacuum polarization, cf. [16,27,28] for example. For our purposes, it is advantageous to perform the limit of coinciding points from the time direction, i.e., we take x ¼ ðτ; σÞ, x 0 ¼ ðτ þ t; σÞ, and t → þ0.…”