### 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 [441675811026296246]: 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.