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4 The Einstein-Langevin Equation: Functional Approach

The Einstein–Langevin equation (15View Equation) may also be derived by a method based on functional techniques  [258Jump 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 [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. [380Jump To The Next Citation Point, 381Jump To The Next Citation Point, 382Jump To The Next Citation Point] to that of Sinha, Raval and Hu [332Jump 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 [14, 56, 82, 87, 227, 323, 343Jump To The Next Citation Point] closed-time-path (CTP) or ‘in-in’ effective action. These techniques were adapted to the gravitational context [54, 72Jump To The Next Citation Point, 94, 222, 223Jump To The Next Citation Point, 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 [58Jump To The Next Citation Point, 69Jump To The Next Citation Point, 70Jump To The Next Citation Point, 73Jump To The Next Citation Point, 134Jump To The Next Citation Point, 239, 247, 256, 258Jump To The Next Citation Point, 267, 343Jump To The Next Citation Point] to the influence-functional formalism of Feynman and Vernon [108Jump 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 [58Jump To The Next Citation Point, 134, 191, 206Jump To The Next Citation Point]. 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 [187Jump 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 [95Jump To The Next Citation Point, 96Jump To The Next Citation Point, 97Jump To The Next Citation Point, 98Jump To The Next Citation Point, 297, 298, 331], 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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