A novel approach to particle production in an uniform electric field

Abstract

We outline a different method of describing scalar field particle production: in a uniform electric field. In the standard approach, the (analytically continued) harmonic oscillator paradigm is important in describing particle production. In the gauges normally considered, in which the four vector potential depends only on the time or space coordinate, the system reduces to a non-relativistic effective Schri:idinger equation with an inverted oscillator potential. The Bogolubov coefficients are determined by tunnelling in this potential. In the Schwinger proper time method of determining the effective Lagrangian, the analytically continued propagator for the usual oscillator system is regarded as the correct propagator for the inverted oscillator system and is used to obtain the gauge invariant result. However, there is another gauge in which the particle production process has striking similarities with the one used to describe Hawking radiation in black holes. This gauge we use to describe the electric field in is the lightcone gauge, so named because the mode functions for a scalar field are found to be singular on the lightcone. We use these modes in evaluating the effective Lagrangian using the proper time technique. The key feature of this analysis is that these modes can be explicitly "normalized" by using the criterion that they reduce to the usual flat space modes in the limit of the electric field tending to zero. This normalization procedure allows one to determine the Schwinger proper time kernel without using the analytical continuation of the harmonic oscillator kernel that is resorted to in the standard analysis. We find that the proper time kernel is not the same as the analytically continued oscillator kernel though the effective Lagrangian is the standard result as it should be. We also consider an example of a confined electric field system using the lightcone gauge modes that has several features of interest. In particular, our analysis indicates that the Bogolubov coefficients, in taking the limit to the uniform electric field case, are multiplied by energy dependent boundary factors that have not been taken into account before.