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4 The Einstein–Langevin Equation: Functional Approach

The Einstein–Langevin equation (14View Equation) may also be derived by a method based on functional techniques [208Jump To The Next Citation Point]. 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 [12390129]. 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 [20311515815912434]. See also comments in Sec. 5.6 of [169Jump To The Next Citation Point] on the black hole backreaction problem comparing the approach by York et al. [297Jump To The Next Citation Point298Jump To The Next Citation Point299Jump To The Next Citation Point] versus that of Sinha, Raval, and Hu [264Jump To The Next Citation Point].

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 [2571118466272Jump To The Next Citation Point4270]. These techniques were adapted to the gravitational context [7618140182Jump To The Next Citation Point23657Jump To The Next Citation Point] 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 [272Jump To The Next Citation Point44Jump To The Next Citation Point58Jump To The Next Citation Point201112Jump To The Next Citation Point54Jump To The Next Citation Point55Jump To The Next Citation Point216196208Jump To The Next Citation Point206] to the influence functional formalism of Feynman and Vernon [89Jump To The Next Citation Point], 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 [44Jump To The Next Citation Point167Jump To The Next Citation Point156112]. This approach to semiclassical gravity is motivated by the observation [148] that in some open quantum systems classicalization and decoherence [30330430530618033279307109] 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 [1881192281493637160293]. 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 [154Jump To The Next Citation Point]. 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 [78Jump To The Next Citation Point77Jump To The Next Citation Point79Jump To The Next Citation Point80Jump To The Next Citation Point263237238], in which such “high-momentum” gravitational modes are assumed to have already been “integrated out.”

 4.1 Influence action for semiclassical gravity
 4.2 Influence action for stochastic gravity
 4.3 Explicit form of the Einstein–Langevin equation
  4.3.1 The kernels for the vacuum state

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