The Locally Covariant Dirac Field

Abstract

We describe the free Dirac field in a four dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case. *

Quantum Field Theories on Categories Fibered in Groupoids

“…Examples of quantum field theories defined on π : SLoc → Loc include Dirac quantum fields, see e.g. [Ver01,DHP09,San10]. (In order to include also fermionic quantum field theories, Definition 2.1 has to be generalized in the usual way to Z 2 -graded algebras, see e.g.…”

“…For example, Dirac quantum fields are typically defined on the category of globally hyperbolic Lorentzian spin manifolds (cf. [Ver01,DHP09,San10]) and charged quantum fields in the presence of background gauge fields are defined on a category of principal bundles with connections over spacetimes (cf. [Zah14,SZ16]).…”

Quantum estimation of parameters of classical spacetimes

“…The above relative Cauchy evolution equation has also been shown to hold for spin-1 2 and spin-1 fields [16][17][18], with an appropriately defined stress-energy tensor. For example, for the electromagnetic field, which we consider in Sec.…”

“…For any free field in a (globally hyperbolic) spacetime, there exist zero-mean-field reference states, called Hadamard states, characterized by welldefined two-point correlators [16,19,25]. Such a reference state allows us to perform normal ordering via the point-splitting approach.…”

“…The aim of algebraic quantum field theory is to put quantum field theory on rigorous mathematical footing, while the aim of what might be termed pragmatic quantum field theory is to make experimental predictions [14,15]. Strides toward connecting the two have been made recently [5,[16][17][18][19], and this progress makes the current work possible.…”