As in the Lorentz invariant case to compute the entanglement entropy associated with a surface we
choose
spatial coordinates
, where
is the
coordinate orthogonal to the surface
and
are the coordinates on the surface
.
Then, after going to Euclidean time
, we switch to the polar coordinates,
,
. In the Lorentz invariant case the conical space, which is needed for calculation of the
entanglement entropy, is obtained by making the angular coordinate
periodic with period
by
applying the Sommerfeld formula (22
) to the heat kernel. If Lorentz invariance is broken, as it
is for the operator (45
), there are certain difficulties in applying the method of the conical
singularity when one computes the entanglement entropy. The difficulties come from the fact that
the wave operator
, if written in terms of the polar coordinates
and
, becomes an
explicit function of the angular coordinate
. As a result of this, the operator
is not
invariant under shifts of
to arbitrary
. Only shifts with
, where
is an integer are allowed. Thus, in this case one cannot apply the Sommerfeld formula since
it explicitly uses the symmetry of the differential operator under shifts of angle
. On the
other hand, a conical space with angle deficit
is exactly what we need to compute
for the reduced density matrix. In [184
], by using some scaling arguments it was shown
that the trace of the heat kernel
on a conical space with
periodicity, is
In the rest of the review we shall mostly focus on the study of Lorentz invariant theories, with field
operator quadratic in derivatives, of the Laplace type, .
http://www.livingreviews.org/lrr-2011-8 |
Living Rev. Relativity 14, (2011), 8
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