The fermionic projector in a time-dependent external potential: Mass oscillation property and Hadamard states

Abstract

Abstract. We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of Hadamard form, provided that the time integral of the spatial supnorm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state.

The fermionic signature operator in de Sitter spacetime

“…On account of Theorem 2.5 and of Lemma 2.6, we can construct a distinguished state which is invariant under the action of all background isometries, characterized by the fact that its two-point distribution coincides with P (see [17,Theorem 1.4] and the constructions in [17, Section 6]):…”

“…Hadamard states have been thoroughly investigated for all free fields, see [31] for a recent review and, in the special case of spinors, see [30,38]. Most notably several results have been proven both in cosmological spacetimes [9,26] and in the framework of fermionic projectors [17,21].…”

“…The price to pay for such generality is the impossibility to conclude a priori that the FP state is of Hadamard form. Several analyses of this issue have been made, and it is now clear that, although one cannot expect the Hadamard condition to hold true generically, see [12], it is nonetheless verified in many interesting scenarios, see in particular [16,17,21]. We remark that the construction of the fermionic signature operator goes back to [14], where the FP state was constructed perturbatively in Minkowski space in the presence of an external potential.…”

“…This is closely linked to the fermionic signature operator, which is a symmetric operator acting on the space of solutions of the massive Dirac equation on a globally hyperbolic spacetime. It was introduced in [19,20] and it has the advantage of producing a distinguished quasi-free, pure state for the C * -algebra of Dirac quantum fields, provided that a suitable condition, known as the strong mass oscillation property, holds true [17]. For a related analysis, containing a weaker but non-canonical condition, refer to [11].…”

The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

“…The fermionic signature operator introduced in [12,13] gives a general setting for spectral geometry in Lorentzian signature [9] and is useful for constructing quasifree Dirac states in globally hyperbolic space-times [4,10]. In the present paper, the fermionic signature operator is constructed for the first time in a black hole geometry, namely the exterior Schwarzschild geometry.…”