The fermionic signature operator in de Sitter spacetime

Abstract

The fermionic projector state is a distinguished quasi-free state for the algebra of Dirac fields in a globally hyperbolic spacetime. We construct and analyze it in the four-dimensional de Sitter spacetime, both in the closed and in the flat slicing. In the latter case we show that the mass oscillation properties do not hold due to boundary effects. This is taken into account in a so-called mass decomposition. The involved fermionic signature operator defines a fermionic projector state. In the case of a closed slicing, we construct the fermionic signature operator and show that the ensuing state is maximally symmetric and of Hadamard form, thus coinciding with the counterpart for spinors of the Bunch-Davies state.

The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

“…We finally remark that a similar connection between boundary terms and double mass integrals involving a principal value is found in cosmological De Sitter spacetime [2].…”

“…We denote the corresponding spinor bundle by SM. Its fibres S x M are endowed with an inner product ≺.|.≻ x of signature (2,2), referred to as the spin scalar product. Clifford multiplication is described by a mapping γ which satisfies the anti-commutation relations,…”

“…This is based on the observation that spectral subspaces of S m are distinguished subspaces of H m characterized purely geometrically independent of observers. These distinguished subspaces can be used to construct distinguished quasi-free Dirac states, referred to as the fermionic projector state or generalized fermionic projector states (see [12,4] and [13,14,11,10,2]).…”

Partial Differential Equations and Quantum States in Curved Spacetimes

“…Amongst numerous other nice properties, they ensure that quantum fluctuations of observables are bounded and allow for an extension of the algebra of fields to encompass Wick polynomials [32][33][34][35][36][37][38][39]. Over the years, the notion of Hadamard states has proved successful in a wide range of different settings, see, e.g., [40][41][42][43][44][45][46][47][48][49][50][51][52][53], to name a few.…”

The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions

“…Proposition 5.1) which play a pivotal rôle in the algebraic approach to linear quantum field theory, see e.g. [22,45] for textbooks, [8,9,17,43,48] for recent reviews, [18][19][20][21] for homotopical approaches and [23][24][25][30][31][32][33][34][35][36] for some applications. However, differently from [46,47], the Green operator will not have the usual support property due to the non-local behavior of the APS boundary condition.…”