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 (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 [285, 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