3.5 The role of the rotational symmetry

We emphasize that the presence of the rotational symmetry with respect to the Killing vector ∂ϕ, which generates rotations in the 2-plane orthogonal to the entangling surface Σ, plays an important role in our construction. Indeed, without such a symmetry it would be impossible to interpret Tr ρα for an arbitrary α as a partition function in some gravitational background. In general, two points are important for this interpretation:

i) that the spacetime possesses, at least locally near the entangling surface, a rotational symmetry so that, after the identification ϕ → ϕ + 2πα, we get a well-defined spacetime E α, with no more than just a conical singularity; this holds automatically if the surface in question is a Killing horizon;

ii) and that the field operator is invariant under the “rotations”, ϕ → ϕ + w; this is automatic if the field operator is a covariant operator.

In particular, point ii) allows us to use the Sommerfeld formula (more precisely its generalization to a curved spacetime) in order to define the Green’s function or the heat kernel on the space E α. As is shown in [184Jump To The Next Citation Point] (see also discussion in Section 2.13) in the case of the non-Lorentz invariant field operators in flat Minkowski spacetime, the lack of the symmetry ii) makes the whole “conical space” approach rather obscure. On the other hand, in the absence of rotational symmetry i) there may appear terms in the entropy that are “missing” in the naively applied conical space approach: the extrinsic curvature contributions [204Jump To The Next Citation Point] or even some curvature terms [134Jump To The Next Citation Point].

In what follows we consider the entanglement entropy of the Killing horizons and deal with the covariant operators so that we do not have to worry about i) or ii).


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