It is convenient to introduce the two new kernels

where is given by the restriction to of expression (102). The renormalized coupling constants , , and are easily computed as it was done in Equation (85). Substituting their expressions into Equation (104), we can take the limit , using the fact that, for , , we obtain the corresponding semiclassical Einstein–Langevin equation.For the massless case one needs the limit of Equation (104). In this case it is convenient to separate in Equation (86) as , where

and use that, from Equation (102), we have The coupling constants are then easily renormalized. We note that in the massless limit, the Newtonian gravitational constant is not renormalized and, in the conformal coupling case, , we have that . Note also that, by making in Equation (94), the noise and dissipation kernels can be written as where It is also convenient to introduce the new kernel This kernel is real and can be written as the sum of an even part and an odd part in the variables , where the odd part is the dissipation kernel . The Fourier transforms (109) and (110) can actually be computed and, thus, in this case we have explicit expressions for the kernels in position space; see, for instance, [179, 56, 137].Finally, the Einstein–Langevin equation for the physical stochastic perturbations can be written in both cases, for and for , as

where in terms of the renormalized constants and the new constants are and . The kernels and are given by Equations (105) when , and , when . In the massless case, we can use the arbitrariness of the mass scale to eliminate one of the parameters or . The components of the Gaussian stochastic source have zero mean value, and their two-point correlation functions are given by , where the noise kernel is given in Equation (95), which in the massless case reduces to Equation (108).It is interesting to consider the massless conformally coupled scalar field, i.e., the case , which is of particular interest because of its similarities with the electromagnetic field, and also because of its interest in cosmology: Massive fields become conformally invariant when their masses are negligible compared to the spacetime curvature. We have already mentioned that for a conformally coupled field, the stochastic source tensor must be traceless (up to first order in perturbation theory around semiclassical gravity), in the sense that the stochastic variable behaves deterministically as a vanishing scalar field. This can be directly checked by noticing, from Equations (95) and (108), that, when , one has , since and . The Einstein–Langevin equations for this particular case (and generalized to a spatially flat Robertson–Walker background) were first obtained in [58], where the coupling constant was fixed to be zero. See also [169] for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [269, 280, 132].

Note that the expectation value of the renormalized stress-energy tensor for a scalar field can be obtained by comparing Equation (111) with the Einstein–Langevin equation (14), its explicit expression is given in [209]. The results agree with the general form found by Horowitz [137, 138] using an axiomatic approach, and coincides with that given in [91]. The particular cases of conformal coupling, , and minimal coupling, , are also in agreement with the results for these cases given in [137, 138, 270, 57, 182], modulo local terms proportional to and due to different choices of the renormalization scheme. For the case of a massive minimally coupled scalar field, , our result is equivalent to that of [276].

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