Defining entanglement without tensor factoring: A Euclidean hourglass prescription

We consider entanglement across a planar boundary in flat space. Entanglement entropy is usually thought of as the von Neumann entropy of a reduced density matrix, but it can also be thought of as half the von Neumann entropy of a product of reduced density matrices on the left and right. The latter form allows a natural regulator in which two cones are smoothed into a Euclidean hourglass geometry. Since there is no need to tensor factor the Hilbert space, the regulated entropy is manifestly gauge invariant and has a manifest state-counting interpretation. We explore this prescription for scalar fields, where the entropy is insensitive to a nonminimal coupling, and for Maxwell fields, which have the same entropy as d ??? 2 scalars.