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 (8). 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 (25), 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 (22), we can interpret these derivatives as expectation values of operators. The relevant
second order derivatives are
where
with
defined in Equation (13);
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 (4). 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 (27), are real and also
is free of ultraviolet
divergences in the limit as
.
From Equations (25) and (25), 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 (26) and (30) 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 [38, 58
, 73
, 132, 206
, 308, 379]: It consists of deriving a new “stochastic”
effective action from the observation that the effect of the imaginary part of the influence action (30) 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 (14) with
given by Equation (12), 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 (32) 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 (15); notice that only the real part of
contributes to the
expectation value (26). To be precise, we first get 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.