

6.1 Perturbations around
Minkowski spacetime
The Minkowski metric
, in a manifold
which is topologically
, and the usual
Minkowski vacuum, denoted as
, are the class
of simplest solutions to the semiclassical Einstein
equation (7), the so-called
trivial solutions of semiclassical gravity [91
]. They constitute
the ground state of semiclassical gravity. In fact, we can always
choose a renormalization scheme in which the renormalized
expectation value
. Thus, Minkowski spacetime
and the vacuum state
are a solution
to the semiclassical Einstein equation with renormalized
cosmological constant
. The fact that the vacuum
expectation value of the renormalized stress-energy operator in
Minkowski spacetime should vanish was originally proposed by
Wald [283], and it may be
understood as a renormalization convention [100, 113]. Note that
other possible solutions of semiclassical gravity with zero vacuum
expectation value of the stress-energy tensor are the exact
gravitational plane waves, since they are known to be vacuum
solutions of Einstein equations which induce neither particle
creation nor vacuum polarization [107, 73, 104].
As we have already mentioned the vacuum
is an eigenstate of the total four-momentum operator
in Minkowski spacetime, but not an eigenstate of
. Hence, even in the Minkowski background, there are
quantum fluctuations in the stress-energy tensor and, as a result,
the noise kernel does not vanish. This fact leads to consider the
stochastic corrections to this class of trivial solutions of
semiclassical gravity. Since, in this case, the Wightman and
Feynman functions (37), their values in the
two-point coincidence limit, and the products of derivatives of two
of such functions appearing in expressions (38) and (39) are known in
dimensional regularization, we can compute the Einstein-Langevin
equation using the methods outlined in Sections 3
and 4.
To perform explicit calculations it is convenient
to work in a global inertial coordinate system
and in the associated basis, in which the components
of the flat metric are simply
. In Minkowski spacetime, the components of the
classical stress-energy tensor (3) reduce to
where
, and the formal expression for the
components of the corresponding “operator” in dimensional
regularization, see Equation (4), is
where
is the differential operator (5), with
,
, and
. The field
is the field operator in the Heisenberg
representation in an
-dimensional Minkowski spacetime, which
satisfies the Klein-Gordon equation (2). We use here a
stress-energy tensor which differs from the canonical one, which
corresponds to
; both tensors, however, define the same
total momentum.
The Wightman and Feynman functions (37) for
are well known:
with
where
and
. Note
that the derivatives of these functions satisfy
and
, and similarly for the
Feynman propagator
.
To write down the semiclassical Einstein
equation (7) in
dimensions for this case, we need to compute the vacuum expectation
value of the stress-energy operator components (79). Since, from (80), we have that
, which is a constant (independent of
), we have simply
where the integrals in dimensional regularization have been
computed in the standard way [209
], and where
is Euler’s gamma function. The semiclassical
Einstein equation (7) in
dimensions before renormalization reduces now to
This equation, thus, simply sets the value of the bare coupling
constant
. Note, from Equation (82), that in order to
have
, the renormalized and
regularized stress-energy tensor “operator” for a scalar field in
Minkowski spacetime, see Equation (6), has to be defined as
which corresponds to a renormalization of the cosmological constant
where
with
being Euler’s constant. In the case of a massless
scalar field,
, one simply has
. Introducing this renormalized coupling constant
into Equation (83), we can take the
limit
. We find that, for
to satisfy the semiclassical Einstein
equation, we must take
.
We can now write down the Einstein-Langevin
equations for the components
of the stochastic metric
perturbation in dimensional regularization. In our case, using
and the explicit expression
of Equation (34), we obtain
The indices in
are raised with the Minkowski metric,
and
; here a superindex
denotes the components of a tensor linearized around
the flat metric. Note that in
dimensions the two-point
correlation functions for the field
is written as
Explicit expressions for
and
are given by
with the differential operators
and
.

