### 2.9 An explicit calculation

Consider an infinite -plane in -dimensional spacetime. The calculation of the entanglement
entropy for this plane can be done explicitly by means of the heat kernel method. In flat spacetime, if the
operator is the Laplace operator,
one
can use the Fourier transform in order to solve the heat equation. In spacetime dimensions one has
Putting and choosing in the polar coordinate system , that
we have that , where and is the angle between the
-vectors and . The radial momentum and angle , together with the other
angles form a spherical coordinate system in the space of momenta . Thus, one has for the
integration measure , where is the area of a unit
radius sphere in dimensions. Performing the integration in Eq. (23) in this coordinate system we find
For the trace one finds
where is the area of the surface . One uses the integral for
the derivation of Eq. (25). The integral over the contour in the Sommerfeld formula (22) is calculated
via residues ([69, 101])
Collecting everything together one finds that in flat Minkowski spacetime
where is the volume of spacetime and is the area of the surface .
Substituting Eq. (27) into Eq. (20) we obtain that the effective action contains two terms. The one
proportional to the volume reproduces the vacuum energy in the effective action. The second term
proportional to the area is responsible for the entropy. Applying formula (16) we obtain the
entanglement entropy
of an infinite plane in spacetime dimensions. Since any surface, locally, looks like a plane, and a
curved spacetime, locally, is approximated by Minkowski space, this result gives the leading contribution to
the entanglement entropy of any surface in flat or curved spacetime.