2.8 Heat kernel and the Sommerfeld formula
Consider for concreteness a quantum bosonic field described by a field operator
so that the
partition function is
. Then, the effective action defined as
where parameter
is a UV cutoff, is expressed in terms of the trace of the heat kernel
. The latter is defined as a solution to the heat equation
In order to calculate the effective action
we use the heat kernel method. In the context of manifolds
with conical singularities this method was developed in great detail in [69
, 101
]. In the Lorentz invariant
case the invariance under the abelian symmetry
plays an important role. The heat kernel
(where we omit the coordinates other than the angle
) on regular flat space then depends
on the difference
. This function is
periodic with respect to
. The heat
kernel
on a space with a conical singularity is supposed to be
periodic.
It is constructed from the
periodic quantity by applying the Sommerfeld formula [207]
That this quantity still satisfies the heat kernel equation is a consequence of the invariance under the
abelian isometry
. The contour
consists of two vertical lines, going from
to
and from
to
and intersecting the real axis between the poles of the
:
,
and
,
, respectively. For
the integrand in Eq. (22) is
a
-periodic function and the contributions of these two vertical lines cancel each other.
Thus, for a small angle deficit the contribution of the integral in Eq. (22) is proportional to
.