# Should the parallel propagator appear in the point-split stress-energy tensor?

The first step in Hadamard regularization of the stress-energy tensor of a free Dirac field is to write out the point-split expression

$$\langle T_{\mu \nu} \rangle \equiv \frac{1}{4} \lim_{x'\to x} \mathrm{Tr} \gamma_{(\mu} \left(\nabla_{\nu)}G^{(1)}(x, x') - \nabla_{\nu')} G^{(1)}(x, x') \right),$$ where $$G^{(1)}$$ is the Hadamard two-point function for the Dirac field, primed indices refer to the field point, $$\nabla_\mu$$ is the covariant derivative, the trace is over the suppressed spinor indices on $$G^{(1)}$$, and the limit is understood to be along the unique geodesic from $$x'$$ to $$x$$, which lie within a convex normal neighbourhood.

This is the form given in e.g. Christensen's original paper (PRD 17:4 1978), but it strikes me that it doesn't make a ton of sense since there are primed tensor and spinor indices on the RHS only. One might (?) more properly write

$$\langle T_{\mu \nu} \rangle \equiv \frac{1}{4} \lim_{x'\to x}\mathrm{Tr}\mathcal{J}^{-1} \gamma_{(\mu} \left(\nabla_{\nu)}G^{(1)}(x, x') - g^{\kappa'}_{\nu)} \nabla_{\kappa'} G^{(1)}(x, x') \right),$$

where $$g^{\kappa'}_{\nu}$$ and $$\mathcal{J}$$ are the parallel propagators of tensor and spinor indices.

Of course if the limit is taken exactly, these expressions are equivalent, since both parallel propagators become identities over the relevant indices. However, I would like to evaluate the limit only approximately, by evaluating the point-separated expression at successive finite separations and then extrapolating to zero geodesic distance.

Would the inclusion of the parallel propagators improve the convergence rate?