Abstract
The adiabatic regularization method was originally proposed by Parker and Fulling to renormalize the energy-momentum tensor of scalar fields in expanding universes. It can be extended to renormalize the electric current induced by quantized scalar fields in a time-varying electric background. This can be done in a way consistent with gravity if the vector potential is considered as a variable of adiabatic order one. Assuming this, we further extend the method to deal with Dirac fields in four spacetime dimensions. This requires a self-consistent ansatz for the adiabatic expansion, in presence of a prescribed time-dependent electric field, which is different from the conventional expansion used for scalar fields. Our proposal is consistent, in the massless limit, with the conformal anomaly. We also provide evidence that our proposed adiabatic expansion for the fermionic modes parallels the Schwinger-DeWitt adiabatic expansion of the two-point function. We give the renormalized expression of the electric current and analyze, using numerical and analytical tools, the pair production induced by a Sauter-type electric pulse. We also analyze the scaling properties of the current for a large field strength.
Highlights
The landmark work of Heisenberg and Euler [1], motivated by earlier work of Sauter [2], established the instability of the quantum vacuum under the influence of a prescribed electric field
III we introduce the basic ingredients of our ansatz to construct the adiabatic expansion of the four-dimensional fermionic modes coupled to a prescribed time-dependent electric field
To motivate the main idea of this work it is very convenient to present the status of the adiabatic regularization method for a charged 4-dimensional scalar field interacting with a classical, homogeneous, time-dependent electric background
Summary
The landmark work of Heisenberg and Euler [1], motivated by earlier work of Sauter [2], established the instability of the quantum vacuum under the influence of a prescribed (slowly varying) electric field. Due to the similarities with the gravitational case, it is a good strategy to readapt the adiabatic regularization scheme to the case in which the external background is an electric field This program was initiated in [32,33] to study backreaction problems when the matter field is a charged scalar field. It seems natural to extend the adiabatic regularization/renormalization method, with the assumption that Aμ is of adiabatic order 1, to Dirac fields in presence of an electric field background in four spacetime dimensions. This is the main aim of this work. We discuss in the Appendix B the connection between the adiabatic method and the Hadamard renormalization scheme for charged scalar fields
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