It is convenient to introduce the two new kernels
where 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 [73
], where the coupling constant
was fixed to be zero. See also [208
] 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 [259
]. The results agree with the general form found by Horowitz [169
, 170
] using an axiomatic
approach and coincide with that given in [110
]. 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 [347].
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