Using the Klein–Gordon equation (2) and expression (4
) for the stress-energy tensor and the
corresponding operator, we can write
Substituting Equation (38) into the
-dimensional version of the Einstein–Langevin Equation (15
),
taking into account that
satisfies the semiclassical Einstein equation (8
), and substituting
expression (39
), we can write the Einstein–Langevin equation in dimensional regularization as
and the fact that the first term on the right-hand side of this identity is real, whereas the second one is
purely imaginary. Once we perform the renormalization procedure in Equation (41), setting
will
yield the physical Einstein–Langevin equation. Due to the presence of the kernel
, this
equation will usually be non-local in the metric perturbation. In Section 6 we will carry out
an explicit evaluation of the physical Einstein–Langevin equation, which will illustrate the
procedure.
When the expectation values in the Einstein–Langevin equation are taken in a vacuum state , such as,
an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the
Wightman and Feynman functions, defined as
From Equations (43), we see that the kernels
and
are the real and imaginary
parts, respectively, of the bitensor
. From expression (5
) we see that the stress-energy operator
can be written as a sum of terms of the form
, where
and
are
differential operators. It then follows that we can express the bitensor
in terms of the
Wightman function as
Similarly, the kernel can be written in terms of the Feynman function as
Finally, the causality of the Einstein–Langevin equation (41) can be explicitly seen as follows. The
non-local terms in that equation are due to the kernel
, which is defined in Equation (29
) as the
sum of
and
. Now, when the points
and
are spacelike separated,
and
commute and, thus,
, which is real. Hence, from the
above expressions, we have that
, and thus
. This fact is
expected since, from the causality of the expectation value of the stress-energy operator [359
], we know that
the non-local dependence on the metric perturbation in the Einstein–Langevin equation, see Equation (15
),
must be causal. See [208
] for an alternative proof of the causal nature of the Einstein–Langevin
equation.
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