Abstract
Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.
Highlights
Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute
We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts
Peierls succeeded in deriving the covariant formulation of a relativistic quantum field theory that he had in mind
Summary
We briefly discuss aspects of a general quantum field theory with a real scalar field. Given a spacetime (M, g) with manifold M and metric g, for any region R ⊂ M we can associate a unital -algebra of observables called AR. We will on think of the algebras AR as being generated by a Hermitian scalar field φ(x) and the expectation values from a given vacuum state |0. In order to construct this Hilbert space, we do not need the full spacetime algebra AM. We would expect that operators on a Cauchy surface Σ would be enough to generate the full Hilbert space. Surprisingly according to the Reeh-Schleider theorem [31, 32], in quantum field theory much less is needed; any open set R is sufficient to generate a dense subspace of H
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