Localization for Dirac fermions

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.