Thus, for comparison with ordinary phenomena at
low energy we need to find a reasonable prescription for obtaining
a finite quantity of the noise kernel in the limit of ordinary
(point-defined) quantum field theory. Regularization schemes used
in obtaining a finite expression for the stress-energy tensor have
been applied to the noise kernel2
.
This includes the simple normal ordering [194, 295] and smeared field
operator [243] methods applied to
the Minkowski and Casimir spaces, zeta-function [87, 189, 53] for spacetimes with an
Euclidean section, applied to the Casimir effect [69] and the
Einstein Universe [242
], or the covariant
point-separation methods applied to the Minkowski [243
], hot flat space and
the Schwarzschild spacetime [245
]. There are
differences and deliberations on whether it is meaningful to seek a
point-wise expression for the noise kernel, and if so what is the
correct way to proceed - e.g., regularization by a subtraction
scheme or by integrating over a test-field. Intuitively the smear
field method [243] may better
preserve the integrity of the noise kernel as it provides a
sampling of the two point function rather than using a subtraction
scheme which alters its innate properties by forcing a nonlocal
quantity into a local one. More investigation is needed to clarify
these points, which bear on important issues like the validity of
semiclassical gravity. We shall set a more modest goal here, to
derive a general expression for the noise kernel for quantum fields
in an arbitrary curved spacetime in terms of Green functions and
leave the discussion of point-wise limit to a later date. For this
purpose the covariant point-separation method which highlights the
bi-tensor features, when used not as a regularization scheme, is
perhaps closest to the spirit of stochastic gravity.
The task of finding a general expression of the
noise-kernel for quantum fields in curved spacetimes was carried
out by Phillips and Hu in two papers using the “modified” point
separation scheme [282, 1, 284
]. Their first
paper [244
] begins with a
discussion of the procedures for dealing with the quantum stress
tensor bi-operator at two separated points, and ends with a general
expression of the noise kernel defined at separated points
expressed as products of covariant derivatives up to the fourth
order of the quantum field’s Green function. (The stress tensor
involves up to two covariant derivatives.) This result holds for
without recourse to renormalization of the Green
function, showing that
is
always finite for
(and off the light cone for
massless theories). In particular, for a massless conformally
coupled free scalar field on a four dimensional manifold, they
computed the trace of the noise kernel at both points and found
this double trace vanishes identically. This implies that there is
no stochastic correction to the trace anomaly for massless
conformal fields, in agreement with results arrived at
in [43
, 58
, 208
] (see also
Section 3). In their second paper [245
] a Gaussian
approximation for the Green function (which is what limits the
accuracy of the results) is used to derive finite expressions for
two specific classes of spacetimes, ultrastatic spacetimes, such as
the hot flat space, and the conformally- ultrastatic spacetimes,
such as the Schwarzschild spacetime. Again, the validity of these
results may depend on how we view the relevance and meaning of
regularization. We will only report the result of their first paper
here.