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.