Localization for Dirac fermions

TL;DR

This paper develops a formalism for localizing Dirac fermions in 1+1 dimensional quantum field theory, creating localized particle states and operators, and analyzing their relation to standard global Fock space representations.

Contribution

It introduces a new localized excitation formalism for Dirac fields, extending scalar field methods, and explores the implications for particle localization in QFT.

Findings

Localized operators are well-defined in Fock space despite inequivalence.

The global vacuum state affects particle localization.

A method to achieve localization consistent with QFT principles.

Abstract

This work is devoted to incorporating into QFT the notion that particles and hence the particle states should be localizable in space. It focuses on the case of the Dirac field in 1+1 dimensional flat spacetime, generalizing a recently developed formalism for scalar fields. This is achieved exploiting again the non-uniqueness of quantization process. Instead of elementary excitations carrying definite amounts of energy and momentum, we construct the elementary excitations of the field localized (at some instant) in a definite region of space. This construction not only leads to a natural notion of localized quanta, but also provides a local algebra of operators. Once constructed, the new representation is confronted to the standard global (Fock) construction. In spite of being unitarily inequivalent representations, the localized operators are well defined in the conventional Fock space. By using them we dig up the issues of localization in QFT showing how the globality of the vacuum state is responsible for the lack of a "common sense localization" notion for particles in the standard Fock representation of QFT and propose a method to meet its requirements.