

4.2 Influence action for
stochastic gravity
In the spirit of the previous derivation of the Einstein-Langevin
equation, we now seek a dynamical equation for a linear
perturbation
to the semiclassical metric
, solution of Equation (7). Strictly speaking,
if we use dimensional regularization we must consider the
-dimensional version of that equation. From the
results just described, if such an equation were simply a
linearized semiclassical Einstein equation, it could be obtained
from an expansion of the effective action
. In particular, since, from Equation (18), we have that
the expansion of
to
linear order in
can be obtained from an expansion of
the influence action
up to second order in
.
To perform the expansion of the influence action,
we have to compute the first and second order functional
derivatives of
and then set
. If we do so using the path integral
representation (15), we can interpret
these derivatives as expectation values of operators. The relevant
second order derivatives are
where
with
defined in Equation (12);
denotes the commutator and
the anti-commutator. Here we use a Weyl ordering
prescription for the operators. The symbol
denotes the following ordered operations: First,
time order the field operators
and then apply the
derivative operators which appear in each term of the product
, where
is the
functional (3). This
“time ordering” arises because we have path
integrals containing products of derivatives of the field, which
can be expressed as derivatives of the path integrals which do not
contain such derivatives. Notice, from their definitions, that all
the kernels which appear in expressions (20) are real and also
is free of ultraviolet divergences in the limit
.
From Equation (18) and (20), since
and
, we can write the expansion for the influence action
around a background metric
in terms of the previous kernels. Taking into
account that these kernels satisfy the symmetry relations
and introducing the new kernel
the expansion of
can be finally written as
where we have used the notation
From Equations (23) and (19) it is clear that the
imaginary part of the influence action does not contribute to the
perturbed semiclassical Einstein equation (the expectation value of
the stress-energy tensor is real), however, as it depends on the
noise kernel, it contains information on the fluctuations of the
operator
.
We are now in a position to carry out the
derivation of the semiclassical Einstein-Langevin equation. The
procedure is well known [43
, 167
, 58
, 110, 26, 296, 246]: It consists
of deriving a new “stochastic” effective action from the
observation that the effect of the imaginary part of the influence
action (23) on the corresponding
influence functional is equivalent to the averaged effect of the
stochastic source
coupled linearly to the perturbations
. This observation follows from the identity first
invoked by Feynman and Vernon for such purpose:
where
is the probability distribution functional of a
Gaussian stochastic tensor
characterized by the
correlators (13) with
given by Equation (11), and where the path
integration measure is assumed to be a scalar under diffeomorphisms
of
. The above identity follows from the
identification of the right-hand side of Equation (25) with the
characteristic functional for the stochastic field
. The probability distribution functional for
is explicitly given by
We may now introduce the stochastic effective action as
where the “stochastic” influence action is defined as
Note that, in fact, the influence functional can now be written as
a statistical average over
:
The stochastic equation of motion for
reads
which is the Einstein-Langevin equation (14); notice that only the
real part of
contributes to the expectation
value (19). To be precise, we
get first the regularized
-dimensional equations with
the bare parameters, with the tensor
replaced by
, where
Of course, when
these tensors are related,
. After that we renormalize and take the limit
to obtain the Einstein-Langevin equations in the
physical spacetime.

