A distinguished vacuum state for a quantum field in a curved spacetime: formalism, features, and cosmology

Abstract

We define a distinguished "ground state" or "vacuum" for a free scalar quantum field in a globally hyperbolic region of an arbitrarily curved spacetime. Our prescription is motivated by the recent construction [1, 2] of a quantum field theory on a background causal set using only knowledge of the retarded Green's function. We generalize that construction to continuum spacetimes and find that it yields a distinguished vacuum or ground state for a non-interacting, massive or massless scalar field. This state is defined for all compact regions and for many noncompact ones. In a static spacetime we find that our vacuum coincides with the usual ground state. We determine it also for a radiation-filled, spatially homogeneous and isotropic cosmos, and show that the super-horizon correlations are approximately the same as those of a thermal state. Finally, we illustrate the inherent non-locality of our prescription with the example of a spacetime which sandwiches a region with curvature in-between flat initial and final regions.

The fermionic signature operator in de Sitter spacetime

“…We denote the corresponding spinor bundle by SM. Its fibres S x M are endowed with an inner product ≺.|.≻ x of signature (2,2), which we refer to as the spin scalar product; for details see [3,33]). Clifford multiplication is described by a mapping γ which satisfies the anti-commutation relations,…”

“…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. More recently, a similar construction was proposed for scalar fields in [2,28,39], but only in space-times of finite lifetime. Yet, the ensuing state fails to obey to the Hadamard property as first observed in [13].…”

Sorkin-Johnston vacuum for a massive scalar field in the 2D causal diamond

“…As we have stated earlier, the SJ modes Eqn. (8) are also solutions of the KG equation. A natural starting point for constructing these modes is therefore to start with a complete set of solutions {s k } in the space S = ker( KG ) where KG ≡ − m 2 , and to find the action of i∆ on this set.…”

‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes

“…As stated in the introduction, the original definition of the Sorkin-Johnston states aimed at constructing distinguished states on any globally hyperbolic spacetime [3]. This was supposed to fill the gap left open by the absence of a vacuum state on nonstationary spacetimes, as well as serving as initial state for application in cosmological problems.…”