We use the evolution operator method to find the one-loop effective action of scalar and spinor QED in electric field backgrounds in terms of the Bogoliubov coefficient between the ingoing and the outgoing vacua. We obtain the exact one-loop effective action for a Sauter-type electric field, E0 sech 2 (t/τ ), and show that the imaginary part correctly yields the vacuum persistence. The renormalized effective action shows the general relation between the vacuum persistence and the total mean number of created pairs for the constant and the Sauter-type electric field.

We propose a novel method for the effective action of spinor and scalar QED at finite temperature in time-dependent electric fields, where charged pairs evolve in a nonadiabatic way. The imaginary part of the effective action consists of thermal loops of the Fermi-Dirac or Bose-Einstein distribution for the initial thermal ensemble, weighted with factors of the Bogoliubov coefficients for quantum effects. And the real part of the effective action is determined by the mean number of produced pairs and vacuum polarization at zero temperature. In the weak-field limit, the mean number of produced pairs is shown twice the imaginary part. We explicitly find the finite temperature effective action in a constant electric field.

We find the Bogoliubov coefficient from the tunneling boundary condition on charged particles in a static electric field E0 sech 2 (z/L) and, using the regularization scheme in Phys. Rev. D 78, 105013 (2008), obtain the exact one-loop effective action in scalar and spinor QED. It is shown that the effective action satisfies the general relation between the vacuum persistence and the mean number of produced pairs. We advance an approximation method for general electric fields and show the duality between the space-dependent and time-dependent electric fields of the same form at the leading order of the effective actions. PACS numbers: 12.20.Ds, 11.15.Tk, I. INTRODUCTIONUnderstanding the vacuum structure of strong field backgrounds has been a challenging task in quantum field theory. Electromagnetic fields and spacetime curvatures provide a typical arena for strong field physics. The vacuum structure may be exploited by finding the effective action in these backgrounds. In quantum electrodynamics (QED), Sauter, Heisenberg and Euler, Weisskopf, and Schwinger obtained the effective action in a constant electromagnetic field several decades ago [1][2][3][4]. However, going beyond the constant electromagnetic field has been another long standing problem in QED, and the effective actions have been carried out only for certain field configurations (for a review and references, see Ref.[5] and for physical applications, see also Ref. [6]). For instance, there has been an attempt to compute the effective action in a pulsed electric field of the form E 0 sech 2 (t/τ ) in Refs. [7,8].The main purpose of this paper is to further develop the in-and out-state formalism based on the Bogoliubov transformation in Refs. [8] (hereafter referred to I) for a pulsed electric field and [9, 10] for a constant electric field to be applicable to the case of spatially localized electric fields. To quantize a charged particle in an electric field background is not trivial because the vacuum is unstable against pair production. Further, the boundary condition on the solution of the Klein-Gordon or Dirac equation distinguishes pulsed electric fields from spatially localized electric ones. In the former case of a pulsed electric field, the charged boson or fermion interacts for a finite period of time, and its positive frequency solution splits both into one branch of positive frequency solution and into another branch of negative frequency solution after completion of the interaction. In the second quantized field theory, the presence of negative frequency solution means that particle-antiparticle pairs of a given mode are created from the vacuum due to the external electric field.In the latter case of a spatially localized electric field, charged bosons or fermions experience a tunneling barrier from the Coulomb gauge potential. Nikishov elaborated the Feynman method to find the pair-production rate in the spatially localized electric field E 0 sech 2 (z/L) [11,12]. In fact, the quantum field confronts the Klein paradox from the tunneling b...

We use the evolution operator method to find the Schwinger pair-production rate at finite temperature in scalar and spinor QED by counting the vacuum production, the induced production and the stimulated annihilation from the initial ensemble. It is shown that the pair-production rate for each state is factorized into the mean number at zero temperature and the initial thermal distribution for bosons and fermions.

The asymptotic conformal invariance of some SU(2) model and Standard Model in curved space-time are investigated. We have examined the conditions for asymptotic conformal invariance for these models numerically.