It is convenient to introduce the two new kernels
For the massless case one needs the limit as of Equation (111). In this case it is convenient to separate in Equation (93) as , where
Finally, the Einstein–Langevin equation for the physical stochastic perturbations can be written in both cases, for and for , as
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 to 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 perturbations 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 (102) and (115) 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 , where the coupling constant was fixed to be zero. See also  for a discussion of this result and its connection to the problem of structure formation in the trace anomaly driven inflation [162, 339, 355].
Note that the expectation value of the renormalized stress-energy tensor for a scalar field can be obtained by comparing Equation (118) with the Einstein–Langevin equation (15); its explicit expression is given in . The results agree with the general form found by Horowitz [169, 170] using an axiomatic approach and coincide with that given in . The particular cases of conformal coupling, , and minimal coupling, , are also in agreement with the results for these cases given in [72, 169, 170, 223, 340], 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 .
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