In this section we explore further the properties of the noise kernel and the stress-energy bitensor. Similar to
what was done for the stress-energy tensor, it is desirable to relate the noise kernel defined at separated
points to the Green’s function of a quantum field. We pointed out earlier [187] that field quantities defined
at two separated points may possess important information, which could be the starting point for probes
into possible extended structures of spacetime. Of more practical concern is how one can define a finite
quantity at one point or in some small region around it from the noise kernel defined at two separated
points. When we refer to, say, the fluctuations of energy density in ordinary (point-wise) quantum field
theory, we are in actuality asking such a question. This is essential for addressing fundamental issues like
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. It is well-known that several regularization methods can work equally well for the removal of ultraviolet divergences in the stress-energy tensor of quantum fields in curved spacetime. Their mutual relations are known and discrepancies explained. This formal structure of regularization schemes for quantum fields in curved spacetime should remain intact when applied to the regularization of the noise kernel in general curved spacetimes; it is the meaning and relevance of regularization of the noise kernel, which is of more concern (see comments below). Specific considerations will, of course, enter for each method. But for the methods employed so far (such as zeta-function, point separation, dimensional and smeared-field) applied to simple cases (Casimir, Einstein, thermal fields) there is no new inconsistency or discrepancy.
Regularization schemes used in obtaining a finite expression for the stress-energy tensor have been
applied to the noise kernel. This includes the simple normal-ordering [237, 378] and smeared-field
operator [303] methods applied to the Minkowski and Casimir spaces, zeta-function [68, 106, 232] for
spacetimes with an Euclidean section, applied to the Casimir effect [86] and the Einstein universe [302
], or
the covariant point-separation methods applied to the Minkowski [303
], hot flat space and Schwarzschild
spacetime [305
]. 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 [303] 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’s functions and leave the discussion of point-wise limit to a
later date. For this purpose the covariant point-separation method that highlights the bitensor
features, when used not as a regularization scheme, is perhaps closest to the spirit of stochastic
gravity.
The task of finding a general expression for 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 [1, 358, 360
].
Their first paper [304
] 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 for the noise kernel, defined at
separated points expressed as products of covariant derivatives up to the fourth order of the quantum field’s
Green’s function. (The stress tensor involves up to two covariant derivatives.) This result holds for
without recourse to renormalization of the Green’s 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 [58
, 73
, 258
] (see also Section 3). In their second paper [305
] a
Gaussian approximation for the Green’s 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.
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