5.2 Stress-energy bitensor 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 (4), 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
two tensors at
and scalars at
.
To carry out point separation on Equation (56), 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 values, 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-dimensional 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 (12), 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 (60), 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 the problem 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 the free fields that 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 (66) 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 (66) has been cancelled by those present in the pair of Green’s functions
in the second term, in agreement with the results of Section 3.
5.2.2 Explicit form of the noise kernel
We will keep 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 (57), 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’s function
and the Dyson (anti-time ordered) Green’s function
:
This can be connected with the zeta-function approach to this problem [302] as follows: Recall that 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 [77, 102]. When the trace of the
stress tensor
is evaluated for a field configuration that satisfies 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 [360]
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 (74) yields
For the massless conformal case, this reduces to
which holds for any function
. For
as the Green’s function, it satisfies the field equation (2):
We will assume only that the Green’s 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 [305
] is completely general since it is assumed that the Green’s function is only satisfying the field
equations in its first variable; an alternative proof of this result was given in [258]. This condition holds not
just for the classical field case, but also for the regularized quantum case, where one does not expect the
Green’s 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 [58
, 73
, 206
]
derived from the Feynman–Vernon influence-functional formalism [107, 108] for some particular
cases.