The Einstein–Langevin equation (15) may also be derived by a method based on functional techniques
[258
]. Here we will summarize these techniques starting with semiclassical gravity.
In semiclassical gravity functional methods were used to study the backreaction of quantum fields in
cosmological models [109, 147, 153]. 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, contrast,
e.g., [137, 193, 194, 251] with the above references and [3, 4, 148, 154]. See also comments in Section 8 on
the black-hole backreaction problem comparing the approach by York et al. [380, 381
, 382
] to that of
Sinha, Raval and Hu [332
].
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 [14, 56, 82, 87, 227, 323, 343] closed-time-path (CTP) or ‘in-in’ effective action.
These techniques were adapted to the gravitational context [54, 72
, 94, 222, 223
, 296] 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 [58, 69
, 70
, 73
, 134
, 239, 247, 256, 258
, 267, 343
] to the influence-functional formalism of Feynman
and Vernon [108
], 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 [58
, 134, 191, 206
]. This approach to
semiclassical gravity is motivated by the observation [181] that in some open quantum systems
classicalization and decoherence [46, 131, 221, 352, 389, 390, 391, 392, 393] 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 [50, 51, 143, 182, 195, 231, 283, 374]. 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 [187
]. 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 [95
, 96
, 97
, 98
, 297, 298, 331], in which
such “high-momentum” gravitational modes are assumed to have already been “integrated
out.”
http://www.livingreviews.org/lrr-2008-3 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |