The influence functional corresponding to the action (1) describing a scalar field in a spacetime (coupled to a metric field) may be introduced as a functional of two copies of the metric, denoted by and , which coincide at some final time . Let us assume that, in the quantum effective theory, the state of the full system (the scalar and the metric fields) in the Schrödinger picture at the initial time can be described by a density operator which can be written as the tensor product of two operators on the Hilbert spaces of the metric and of the scalar field. Let be the matrix element of the density operator describing the initial state of the scalar field. The Feynman–Vernon influence functional is defined as the following path integral over the two copies of the scalar field:

Alternatively, the above double path integral can be rewritten as a CTP integral, namely, as a single path integral in a complex time contour with two different time branches, one going forward in time from to , and the other going backward in time from to (in practice one usually takes ). From this influence functional, the influence action , or for short, defined by carries all the information about the environment (the matter fields) relevant to the system (the gravitational field). Then we can define the CTP effective action for the gravitational field, , as This is the effective action for the classical gravitational field in the CTP formalism. However, since the gravitational field is treated only at the tree level, this is also the effective classical action from which the classical equations of motion can be derived.Expression (15) contains ultraviolet divergences and must be regularized. We shall assume that dimensional regularization can be applied, that is, it makes sense to dimensionally continue all the quantities that appear in Equation (15). For this we need to work with the -dimensional actions corresponding to in Equation (15) and in Equation (8). For example, the parameters , , , and of Equation (8) are the bare parameters , , , and , and in , instead of the square of the Weyl tensor in Equation (8), one must use , which by the Gauss–Bonnet theorem leads to the same equations of motion as the action (8) when . The form of in dimensions is suggested by the Schwinger–DeWitt analysis of the ultraviolet divergences in the matter stress-energy tensor using dimensional regularization. One can then write the Feynman–Vernon effective action in Equation (17) in a form suitable for dimensional regularization. Since both and contain second order derivatives of the metric, one should also add some boundary terms [284, 167]. The effect of these terms is to cancel out the boundary terms which appear when taking variations of keeping the value of and fixed at and . Alternatively, in order to obtain the equations of motion for the metric in the semiclassical regime, we can work with the action terms without boundary terms and neglect all boundary terms when taking variations with respect to . From now on, all the functional derivatives with respect to the metric will be understood in this sense.

The semiclassical Einstein equation (7) can now be derived. Using the definition of the stress-energy tensor and the definition of the influence functional, Equations (15) and (16), we see that

where the expectation value is taken in the -dimensional spacetime generalization of the state described by . Therefore, differentiating in Equation (17) with respect to , and then setting , we get the semiclassical Einstein equation in dimensions. This equation is then renormalized by absorbing the divergences in the regularized into the bare parameters. Taking the limit we obtain the physical semiclassical Einstein equation (7).http://www.livingreviews.org/lrr-2004-3 |
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