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:
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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