In semiclassical gravity functional methods were
used to study the backreaction of quantum fields in cosmological
models [123, 90, 129]. The primary
advantage of the effective action approach is, in addition to the
well-known fact that it is easy to introduce perturbation schemes
like loop expansion and nPI formalisms, that it yields a fully self-consistent solution. For a
general discussion in the semiclassical context of these two
approaches, equation of motion versus effective action, see, e.g.,
the work of Hu and Parker (1978) versus Hartle and Hu (1979)
in [203, 115, 158, 159, 124, 3, 4]. See also comments in
Sec. 5.6 of [169] on the black hole
backreaction problem comparing the approach by York et
al. [297
, 298
, 299
] versus that of
Sinha, Raval, and Hu [264
].
The well known in-out effective action method
treated in textbooks, however, led to equations of motion which
were not real because they were tailored to compute transition
elements of quantum operators rather than expectation values. The
correct technique to use for the backreaction problem is the
Schwinger-Keldysh closed-time-path (CTP) or ‘in-in’ effective
action [257, 11, 184, 66, 272, 41, 70]. These
techniques were adapted to the gravitational context [76, 181, 39, 182
, 236, 57
] and applied to
different problems in cosmology. One could deduce the semiclassical
Einstein equation from the CTP effective action for the
gravitational field (at tree level) with quantum matter fields.
Furthermore, in this case the CTP functional
formalism turns out to be related [272, 43
, 58
, 201, 112
, 54
, 55
, 216, 196, 208
, 206] to the
influence functional formalism of Feynman and Vernon [89
], since the full
quantum system may be understood as consisting of a distinguished
subsystem (the “system” of interest) interacting with the remaining
degrees of freedom (the environment). Integrating out the
environment variables in a CTP path integral yields the influence
functional, from which one can define an effective action for the
dynamics of the system [43
, 167
, 156, 112]. This approach
to semiclassical gravity is motivated by the observation [149] that in some open
quantum systems classicalization and decoherence [303, 304, 305, 306, 180, 33, 279, 307, 109] on the system
may be brought about by interaction with an environment, the
environment being in this case the matter fields and some
“high-momentum” gravitational modes [188, 119, 228, 150, 36, 37, 160, 293]. Unfortunately, since
the form of a complete quantum theory of gravity interacting with
matter is unknown, we do not know what these “high-momentum”
gravitational modes are. Such a fundamental quantum theory might
not even be a field theory, in which case the metric and scalar
fields would not be fundamental objects [154
]. Thus, in this
case, we cannot attempt to evaluate the influence action of Feynman
and Vernon starting from the fundamental quantum theory and
performing the path integrations in the environment variables.
Instead, we introduce the influence action for an effective quantum
field theory of gravity and matter [79
, 78
, 77
, 80
, 263, 237, 238], in which such
“high-momentum” gravitational modes are assumed to have already
been “integrated out.”