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5.1 Point separation

The point separation scheme introduced in the 1960s by DeWitt [92] was brought to more popular use in the 1970s in the context of quantum field theory in curved spacetimes [83Jump To The Next Citation Point, 84Jump To The Next Citation Point, 93] 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 x into a pair of closely separated points x and x′. The problematic terms involving field products such as ˆ 2 φ (x ) become ˆ ˆ ′ φ(x)φ(x ), 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’s 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’s function G (x,y) is an example of a bi-scalar; it transforms as a scalar at both points x and y. We can also define a bitensor Ta1...anb′1...b′m(x,y): Upon a coordinate transformation, this transforms as a rank n tensor at x and a rank m tensor at y. We will extend this up to a quad-tensor Ta ...a b′...b′ c′′...c′′d′′′...d′′′ 1 n1 1 n2 1 n3 1 n4, which has support at four points x,y,x′ andy ′, transforming as rank n1,n2, n3 and n4 tensors at each of the four points. This also sets the notation we will use: unprimed indices refer to the tangent space constructed above x, single primed indices to y, double primed to x′ and triple primed to y′. For each point, there is the covariant derivative ∇a at that point. Covariant derivatives at different points commute, and the covariant derivative at, say, point x′ does not act on a bitensor defined at, say, x and y:

Tab′;c;d′ = Tab′;d′;c and Tab′;c′′ = 0. (48 )
To simplify notation, we will henceforth 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 [G (x,y)] ≡ G (x,x ) is used. This extends to n-tensors as

[ ] Ta1...an1b′1...b′n2c′1′...c′n′3d′′1′ ...d′n′′4 = Ta1...an1b1...bn2c1...cn3d1...dn4, (49 )
i.e., this becomes a rank (n1 + n2 + n3 + n4) tensor at x. The multi-variable chain rule relates covariant derivatives acting at different points, when we are interested in the coincident limit:
[ ] [ ] [ ] Ta1...amb′1...b′n ;c = Ta1...amb′1...b′n;c + Ta1...amb′1...b′n;c′ . (50 )
This result is referred to as Synge’s theorem in this context; we follow Fulling’s discussion [121Jump To The Next Citation Point].

The bitensor of parallel transport gab′ is defined such that when it acts on a vector vb′ at y, it parallel transports the vector along the geodesics connecting x and y. This allows us to add vectors and tensors defined at different points. We cannot directly add a vector va at x and vector wa ′ at y. But by using ′ gab, we can construct the sum ′ va + gab wb′. We will also need the obvious property [ ] gab′ = gab.

The main bi-scalar we need is the world function σ(x,y ). This is defined as a half of the square of the geodesic distance between the points x and y. It satisfies the equation

σ = 1σ;pσ . (51 ) 2 ;p
Often in the literature, a covariant derivative is implied when the world function appears with indices, a ;a σ ≡ σ, i.e., taking the covariant derivative at x, while a′ σ means the covariant derivative at y. This is done since the vector a − σ is the tangent vector to the geodesic with length equal to the distance between x and y. As σa 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 S (x,y) is of the form
S(x,y ) = A(0) + σpA (p1) + σpσqA (p2q)+ σpσq σrA(p3q)r + σpσqσrσsA (p4q)rs + ..., (52 )
where, following our convention, the expansion tensors (n) A a1...an with unprimed indices have support at x (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 --- pa = σa ∕√ 2σ. Then the series expansion can be written
(0) 1 (1) (2) 3 (3) 2 (4) S (x,y) = A + σ2A + σA + σ2 A + σ A + .... (53 )
The expansion scalars are related, via A (n) = 2n∕2A(pn)...p pp1 ...ppn 1 n, to the expansion tensors.

The last object we need is the VanVleck–Morette determinant D (x,y), defined as D (x,y) ≡ − det(− σ;ab′). The related bi-scalar

( ) 12 Δ1 ∕2 = ∘D-(x,y-)-- (54 ) g(x)g(y)
satisfies the equation
Δ1∕2 (4 − σ;pp) − 2Δ1∕2;pσ;p = 0 (55 )
with the boundary condition [ ] Δ1 ∕2 = 1.

Further details on these objects and discussions of the definitions and properties are contained in [83, 84] and [90, 301Jump To The Next Citation Point]. There it is shown how the defining equations for σ and Δ1 ∕2 are used to determine the coincident limit expression for the various covariant derivatives of the world function ([σ;a], [σ;ab], etc.) and how the defining differential equation for Δ1∕2 can be used to determine the series expansion of 1∕2 Δ. We show how the expansion tensors (n) A a1...an are determined in terms of the coincident limits of covariant derivatives of the bi-scalar S(x,y). ([301] 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.)

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