On separable states in relativistic quantum field theory

We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions A and B we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any R > 0 we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice t=const. all of whose points have a distance >R . We also show that the normal ordered energy density of these states can be made <= 10(31)m(4)/((mR))8 e(-1/4mR) (in Planck units). To achieve these results we use a rather explicit construction of test-functions f of positive type for which we can get sufficient control on lower bounds on f .

Automated summary

In this study, we investigate separable and physically reasonable states in relativistic quantum field theory. Specifically, we examine the minimum energy density required for the existence of separable states between spacelike separated regions. This research provides insights into the interplay between entanglement entropy and energy density, which is important in various fields including black hole thermodynamics. We focus on a linear scalar quantum field in a four-dimensional globally hyperbolic spacetime with a trivial topology. For general spacelike separated regions A and B, we prove the existence of a separable quasi-free Hadamard state. Moreover, we present a more precise construction for massive free scalar fields in Minkowski spacetime, ensuring the separability between any two regions in an inertial time slice with distance greater than R. The energy density of these states is also shown to be bounded.