

5.2 Stress-energy bi-tensor
operator and noise kernel
Even though we believe that the point-separated results are more
basic in the sense that it reflects a deeper structure of the
quantum theory of spacetime, we will nevertheless start with
quantities defined at one point, because they are what enter in
conventional quantum field theory. We will use point separation to
introduce the bi-quantities. The key issue here is thus the
distinction between point-defined (pt)
and point-separated (bi) quantities.
For a free classical scalar field
with the action
defined in
Equation (1), the classical
stress-energy tensor is
which is equivalent to the tensor of Equation (3), but written in a
slightly different form for convenience. When we make the
transition to quantum field theory, we promote the field
to a field operator
. The fundamental
problem of defining a quantum operator for the stress tensor is
immediately visible: The field operator appears quadratically.
Since
is an operator-valued distribution,
products at a single point are not well-defined. But if the product
is point separated,
, they are finite and well-defined.
Let us first seek a point-separated extension of
these classical quantities and then consider the quantum field
operators. Point separation is symmetrically extended to products
of covariant derivatives of the field according to
The bi-vector of parallel displacement
is
included so that we may have objects that are rank 2 tensors at
and scalars at
.
To carry out point separation on
Equation (49), we first define the
differential operator
from which we obtain the classical stress tensor as
That the classical tensor field no longer appears as a product of
scalar fields at a single point allows a smooth transition to the
quantum tensor field. From the viewpoint of the stress tensor, the
separation of points is an artificial construct, so when promoting
the classical field to a quantum one, neither point should be
favored. The product of field configurations is taken to be the
symmetrized operator product, denoted by curly brackets:
With this, the point separated stress-energy tensor operator is
defined as
While the classical stress tensor was defined at the coincidence
limit
, we cannot attach any physical meaning
to the quantum stress tensor at one point until the issue of
regularization is dealt with, which will happen in the next
section. For now, we will maintain point separation so as to have a
mathematically meaningful operator.
The expectation value of the point-separated
stress tensor can now be taken. This amounts to replacing the field
operators by their expectation value, which is given by the
Hadamard (or Schwinger) function
and the point-separated stress tensor is defined as
where, since
is a differential operator, it can be
taken “outside” the expectation value. The expectation value of the
point-separated quantum stress tensor for a free, massless (
) conformally coupled (
) scalar field
on a four dimension spacetime with scalar curvature
is
5.2.1
Finiteness of the
noise kernel
We now turn our attention to the noise kernel
introduced in Equation (11), which is the
symmetrized product of the (mean subtracted) stress tensor
operator:
Since
defined at one point can be ill-behaved
as it is generally divergent, one can question the soundness of
these quantities. But as will be shown later, the noise kernel is
finite for
. All field operator products present in
the first expectation value that could be divergent, are canceled
by similar products in the second term. We will replace each of the
stress tensor operators in the above expression for the noise
kernel by their point separated versions, effectively separating
the two points
into the four points
. This will allow us to express the noise kernel in
terms of a pair of differential operators acting on a combination
of four and two point functions. Wick’s theorem will allow the four
point functions to be re-expressed in terms of two point functions.
From this we see that all possible divergences for
will cancel. When the coincidence limit is taken,
divergences do occur. The above procedure will allow us to isolate
the divergences and to obtain a finite result.
Taking the point-separated quantities as more
basic, one should replace each of the stress tensor operators in
the above with the corresponding point separated version (53), with
acting at
and
and
acting at
and
. In this framework the noise kernel is defined as
where the four point function is
We assume that the pairs
and
are each within their respective Riemann normal
coordinate neighborhoods so as to avoid problems that possible
geodesic caustics might be present. When we later turn our
attention to computing the limit
, after
issues of regularization are addressed, we will want to assume that
all four points are within the same Riemann normal coordinate
neighborhood.
Wick’s theorem, for the case of free fields which
we are considering, gives the simple product four point function in
terms of a sum of products of Wightman functions (we use the
shorthand notation
):
Expanding out the anti-commutators in Equation (59) and applying Wick’s
theorem, the four point function becomes
We can now easily see that the noise kernel defined via this
function is indeed well defined for the limit
:
From this we can see that the noise kernel is also well defined for
; any divergence present in the first expectation
value of Equation (59) have been cancelled
by those present in the pair of Green functions in the second term,
in agreement with the results of Section 3.
5.2.2
Explicit form of
the noise kernel
We will let the points separated for a while so
we can keep track of which covariant derivative acts on which
arguments of which Wightman function. As an example (the complete
calculation is quite long), consider the result of the first set of
covariant derivative operators in the differential
operator (50), from both
and
, acting on
:
(Our notation is that
acts at
,
at
,
at
, and
at
.) Expanding out the differential operator above, we
can determine which derivatives act on which Wightman function:
If we now let
and
, the contribution to the noise kernel is (including
the factor of
present in the definition of the noise
kernel):
That this term can be written as the sum of a part involving
and one involving
is a general
property of the entire noise kernel. It thus takes the form
We will present the form of the functional
shortly. First we note, that for
and
time-like separated, the above split of the noise
kernel allows us to express it in terms of the Feynman (time
ordered) Green function
and the Dyson (anti-time
ordered) Green function
:
This can be connected with the zeta function approach to this
problem [242] as follows: Recall
when the quantum stress tensor fluctuations determined in the
Euclidean section is analytically continued back to Lorentzian
signature (
), the time ordered product results. On
the other hand, if the continuation is
, the
anti-time ordered product results. With this in mind, the noise
kernel is seen to be related to the quantum stress tensor
fluctuations derived via the effective action as
The complete form of the functional
is
with
5.2.3
Trace of the noise
kernel
One of the most interesting and surprising
results to come out of the investigations of the quantum stress
tensor undertaken in the 1970s was the discovery of the trace
anomaly [61, 84]. When the trace of the
stress tensor
is evaluated for a field
configuration that satisties the field equation (2), the trace is seen to
vanish for massless conformally coupled fields. When this analysis
is carried over to the renormalized expectation value of the
quantum stress tensor, the trace no longer vanishes.
Wald [284] showed that this was
due to the failure of the renormalized Hadamard function
to be symmetric in
and
, implying that it does not necessarily satisfy the
field equation (2) in the variable
. (The definition of
in the
context of point separation will come next.)
With this in mind, we can now determine the noise
associated with the trace. Taking the trace at both points
and
of the noise kernel functional (67) yields
For the massless conformal case, this reduces to
which holds for any function
. For
being the Green function, it satisfies the field
equation (2):
We will only assume that the Green function satisfies the field
equation in its first variable. Using the fact
(because the covariant derivatives act at a
different point than at which
is supported), it follows
that
With these results, the noise kernel trace becomes
which vanishes for the massless conformal case. We have thus shown,
based solely on the definition of the point separated noise kernel,
that there is no noise associated with the trace anomaly. This
result obtained in [245
] is completely
general since it is assumed that the Green function is only
satisfying the field equations in its first variable; an
alternative proof of this result was given in [208]. This
condition holds not just for the classical field case, but also for
the regularized quantum case, where one does not expect the Green
function to satisfy the field equation in both variables. One can
see this result from the simple observation used in Section
3:
Since the trace anomaly is known to be locally determined and
quantum state independent, whereas the noise present in the quantum
field is non-local, it is hard to find a noise associated with it.
This general result is in agreement with previous
findings [43
, 167
, 58
], derived from the
Feynman-Vernon influence functional formalism [89, 88] for some
particular cases.

