### 5.1 Point separation

The point separation scheme introduced in the 1960s by DeWitt [74] was brought to more popular use in the 1970s in the context of quantum field theory in curved spacetimes [756768] as a means for obtaining a finite quantum stress tensor. Since the stress-energy tensor is built from the product of a pair of field operators evaluated at a single point, it is not well-defined. In this scheme, one introduces an artificial separation of the single point to a pair of closely separated points and . The problematic terms involving field products such as becomes , whose expectation value is well defined. If one is interested in the low energy behavior captured by the point-defined quantum field theory – as the effort in the 1970s was directed – one takes the coincidence limit. Once the divergences present are identified, they may be removed (regularization) or moved (by renormalizing the coupling constants), to produce a well-defined, finite stress tensor at a single point.

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.

#### 5.1.1 n-tensors and end-point expansions

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 :

To simplify notation, henceforth we will eliminate the semicolons after the first one for multiple covariant derivatives at multiple points.

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

i.e., this becomes a rank tensor at . The multi-variable chain rule relates covariant derivatives acting at different points, when we are interested in the coincident limit:
This result is referred to as Synge’s theorem in this context; we follow Fulling’s discussion [100].

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

Often in the literature, a covariant derivative is implied when the world function appears with indices, , i.e., taking the covariant derivative at , while means the covariant derivative at . This is done since the vector is the tangent vector to the geodesic with length equal to the distance between and . As records information about distance and direction for the two points, this makes it ideal for constructing a series expansion of a bi-scalar. The end point expansion of a bi-scalar is of the form
where, following our convention, the expansion tensors with unprimed indices have support at (hence the name end point expansion). Only the symmetric part of these tensors contribute to the expansion. For the purposes of multiplying series expansions it is convenient to separate the distance dependence from the direction dependence. This is done by introducing the unit vector . Then the series expansion can be written
The expansion scalars are related, via , to the expansion tensors.

The last object we need is the VanVleck–Morette determinant , defined as . The related bi-scalar

satisfies the equation
with the boundary condition .

Further details on these objects and discussions of the definitions and properties are contained in [6768] 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.)