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:influence action , or for short, defined by effective action for the gravitational field, , as
Expression (22) 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 (22). For this we need to work with the -dimensional actions corresponding to in Equation (22) and in Equation (9). For example, the parameters , , , and of Equation (9) are the bare parameters , , , and , and in , instead of the square of the Weyl tensor in Equation (9), one must use , which by the Gauss–Bonnet theorem leads to the same equations of motion as the action (9) 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 (24) in a form suitable for dimensional regularization. Since both and contain second-order derivatives of the metric, one should also add some boundary terms [206, 361]. 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 (8) can now be derived. Using the definition of the stress-energy tensor and the definition of the influence functional, Equations (22) and (23), we see that
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