Pure quasifree states of the Dirac field from the fermionic projector

TL;DR

This paper constructs and analyzes pure quasifree states for the free Dirac field on specific spacetimes, exploring their properties, including conditions under which they satisfy the Hadamard condition.

Contribution

It introduces a method to derive gauge invariant pure quasifree states from the fermionic projector and modifies the construction to ensure Hadamard states.

Findings

The constructed states are generally non-Hadamard.

A modified construction yields Hadamard states.

Hadamard condition relates to fluctuations of Wick polynomials.

Abstract

We consider the quantised free Dirac field on oriented and globally hyperbolic ultrastatic slab spacetimes with compact spatial section and demonstrate how a gauge invariant, pure and quasifree state on the C*-completion of the self-dual CAR-algebra can be extracted from the fermionic projector construction of Finster and Reintjes. This state is analogous to the "SJ-state" of the free scalar field recently discussed in the literature. We prove that this state generically fails to be Hadamard. However, we also show how a modified version of the construction, inspired by work of Brum and Fredenhagen, yields states which are Hadamard. We also relate the Hadamard condition to the finiteness of fluctuations of Wick polynomials.