Thus the first order of business is the construction of the stress tensor and then to derive the symmetric stress-energy tensor two point function, the noise kernel, in terms of the Wightman Green function. In this section we will use the traditional notation for index tensors in the point-separation context.
An object like the Green function is an example of a bi-scalar: It transforms as scalar at both
points
and
. We can also define a bi-tensor
: Upon a coordinate transformation, this transforms
as a rank
tensor at
and a rank
tensor at
. We will extend this up to a quad-tensor
which has support at four
points
, transforming as rank
tensors at each of the four points. This also sets
the notation we will use: unprimed indices referring to the tangent
space constructed above
, single primed indices to
, double primed to
and triple primed
to
. For each point, there is the covariant derivative
at that point. Covariant derivatives at different
points commute, and the covariant derivative at, say, point
does not act on a bi-tensor defined at, say,
and
:
Having objects defined at different points, the
coincident limit is defined as
evaluation “on the diagonal”, in the sense of the spacetime support
of the function or tensor, and the usual shorthand is used. This extends to
-tensors as
The bi-tensor of parallel
transport is defined such that when it acts on a
vector
at
, it parallel transports the
vector along the geodesics connecting
and
. This allows us to add vectors and tensors defined
at different points. We cannot directly add a vector
at
and vector
at
. But by using
, we can construct the sum
. We will also need the obvious property
.
The main bi-scalar we need is the world function . This is
defined as a half of the square of the geodesic distance between
the points
and
. It satisfies the equation
The last object we need is the VanVleck-Morette determinant , defined as
. The related bi-scalar
Further details on these objects and discussions
of the definitions and properties are contained in [67, 68] and [240]. There it is shown
how the defining equations for
and
are used to determine the coincident limit
expression for the various covariant derivatives of the world
function (
,
, etc.) and how
the defining differential equation for
can be used to
determine the series expansion of
. We show how the
expansion tensors
are determined in terms of
the coincident limits of covariant derivatives of the bi-scalar
. ([240] details how point
separation can be implemented on the computer to provide easy
access to a wider range of applications involving higher
derivatives of the curvature tensors.)