Using the Klein-Gordon equation (2), and
expressions (3
) for the stress-energy
tensor and the corresponding operator, we can write
Substituting Equation (31) into the
-dimensional version of the Einstein-Langevin
Equation (14
), taking into account
that
satisfies the semiclassical Einstein
equation (7
), and substituting
expression (32
), 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 pure
imaginary. Once we perform the renormalization procedure in
Equation (34), setting
will yield the physical Einstein-Langevin equation.
Due to the presence of the kernel
, this
equation will be usually 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, for instance, 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 (36), we see that the
kernels
and
are the
real and imaginary parts, respectively, of the bi-tensor
. From the expression (4
) we see that the
stress-energy operator
can be written as a sum of
terms of the form
, where
and
are some differential operators. It then follows
that we can express the bi-tensor
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 (34) can be explicitly
seen as follows. The non-local terms in that equation are due to
the kernel
which is defined in Equation (22
) 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 [283
], we know that the
non-local dependence on the metric perturbation in the
Einstein-Langevin equation, see Equation (14
), must be causal.
See [169
] for an alternative
proof of the causal nature of the Einstein-Langevin equation.