

6.2 The kernels in the
Minkowski background
Since the two kernels (36) are free of
ultraviolet divergences in the limit
, we can
deal directly with the
in Equation (35). The kernels
and
are actually the components
of the “physical” noise and dissipation kernels that will appear in
the Einstein-Langevin equations once the renormalization procedure
has been carried out. The bi-tensor
can be
expressed in terms of the Wightman function in four spacetime
dimensions, according to Equation (38). The different terms
in this kernel can be easily computed using the integrals
and
, which are defined as the previous one
by inserting the momenta
with
in the integrand. All these integrals can be
expressed in terms of
; see [209
] for the explicit
expressions. It is convenient to separate
into its even and odd parts with respect to the
variables
as
where
and
.
These two functions are explicitly given by
After some manipulations, we find
where
. The real and imaginary parts of the
last expression, which yield the noise and dissipation kernels, are
easily recognized as the terms containing
and
, respectively. To write them
explicitly, it is useful to introduce the new kernels
and we finally get
Notice that the noise and dissipation kernels defined in
Equation (94) are actually real
because, for the noise kernels, only the
terms of the
exponentials
contribute to the integrals, and, for
the dissipation kernels, the only contribution of such exponentials
comes from the
terms.
The evaluation of the kernel
is a more involved task. Since this kernel contains
divergences in the limit
, we use
dimensional regularization. Using Equation (39), this kernel can be
written in terms of the Feynman propagator (81) as
where
Let us define the integrals
and
obtained by inserting the momenta
into the previous integral, together with
and
which are also obtained by inserting
momenta in the integrand. Then, the different terms in
Equation (97) can be computed;
these integrals are explicitly given in [209
]. It is found that
, and the remaining integrals can be written in terms
of
and
. It is useful to introduce
the projector
orthogonal to
and the tensor
as
Then the action of the operator
is simply
written as
, where
is an arbitrary function of
.
Finally after a rather long but straightforward
calculation, and after expanding around
, we get,
where
has been defined in Equation (86), and
and
are given by
where
The imaginary part of Equation (101) gives the kernel
components
, according to Equation (96). It can be easily
obtained multiplying this expression by
and retaining only
the real part
of the function
.

