The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea

TL;DR

This paper revisits the causal perturbation expansion for the Dirac sea, introduces a rescaling to obtain a proper projector, and explores the implications for the fermionic projector and light-cone expansion.

Contribution

It provides a modified construction of the fermionic projector using rescaling, ensuring it is a true projector and clarifies its relation to the free case.

Findings

The rescaled operator is generally not idempotent.

A rescaling procedure yields a genuine projector.

The fermionic projector with interaction is unitarily related to the free projector.

Abstract

The causal perturbation expansion defines the Dirac sea in the presence of a time-dependent external field. It yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces. After giving a self-contained introduction to the ideas and techniques, we show that this operator is in general not idempotent. We modify the standard construction by a rescaling procedure giving a projector on the generalized negative-energy subspace. The resulting rescaled causal perturbation expansion uniquely defines the fermionic projector in terms of a series of distributional solutions of the Dirac equation. The technical core of the paper is to work out the combinatorics of the expansion in detail. It is also shown that the fermionic projector with interaction can be obtained from the free projector by a unitary transformation. We finally analyze the consequences of the rescaling procedure on the light-cone expansion.