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

- the validity of semiclassical gravity [237], e.g., whether the fluctuations to mean ratio is a correct criterion [10, 11, 113, 115, 198, 303];
- whether the fluctuations in the vacuum energy density, which drive some models of inflationary cosmology, violate the positive-energy condition;
- the physical effects of black-hole horizon fluctuations and Hawking radiation backreaction – to begin with, are the fluctuations finite or infinite?
- general relativity as a low-energy effective theory in the geometro-hydrodynamic limit towards a kinetic-theory approach to quantum gravity [185, 187, 188].

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.

5.1 Point separation

5.1.1 n-tensors and end-point expansions

5.2 Stress-energy bitensor operator and noise kernel

5.2.1 Finiteness of the noise kernel

5.2.2 Explicit form of the noise kernel

5.2.3 Trace of the noise kernel

5.1.1 n-tensors and end-point expansions

5.2 Stress-energy bitensor operator and noise kernel

5.2.1 Finiteness of the noise kernel

5.2.2 Explicit form of the noise kernel

5.2.3 Trace of the noise kernel

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