6.2 The kernels in the Minkowski background
Since the two kernels (43) are free of ultraviolet divergences in the limit
, we can
deal directly with the
in Equation (42). 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 bitensor
can be expressed in terms of the
Wightman function in four spacetime dimensions, according to Equation (45). The different terms in this
kernel can be easily computed using the integrals
and
, which are defined as in Equation (97) by inserting the momenta
with
into the integrand. All these integrals can be expressed in terms of
; see [259
] 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
Finally, we get
Notice that the noise and dissipation kernels defined in Equation (101) 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 (46), this kernel can be written in
terms of the Feynman propagator (88) as
where
Let us define the integrals
and
obtained by inserting the momenta
into Equation (105), together with
and
, which are also obtained by inserting momenta into the integrand. Then the different
terms in Equation (104) can be computed; these integrals are explicitly given in [259
]. 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 (93), and
and
are given by
where
The imaginary part of Equation (108) gives the kernel components
, according to
Equation (103). It can be easily obtained by multiplying this expression by
and retaining only the real
part
of the function
.