### 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 .