# Question about schwinger dewitt expansion

1. Jul 19, 2013

### dmp1

I have been trying to derive the schwinger dewitt expansion described in Avramidi's book. The Green's function solution to the Laplace Beltrami equation is, in n dimensions:

$G[x,\acute{x}]=$$\frac{1}{(4 \pi \tau)^{n/2}}$$\sqrt{\Delta}$$\;$$\mathcal{P}$$\;$$e^{-\sigma/(2 \:\tau)}$$\;$$\;$$\Omega$

where

• σ is the geodesic length squared from $\acute{x}$ to $x$
Δ is the Van Fleck Morette determinant
Ω is the transfer function
$\mathcal{P}$ is the parallel displacement operator
Here is my question: is the following interpretation correct?

$\mathcal{P}$ satisfies the relation:

$v^i \mathcal{D}_i \:\mathcal{P}=0$

where $v^i$ is the vector along the geodesic

$\mathcal{P}$ will transport a scalar from $\acute{x}$ to $x$. If the connection $\Gamma$ which defines $\mathcal{D}_i$ were the Levi-Civitas connection, scalars would be transported unchanged and $\mathcal{P}=1$. However as is stated in Adler Bazin and Schiffer, one doesn't need to use the Levi-Civitas connection; as long as $\Gamma$ transforms under change of coordinates in the correct way it is allowable as a connection. For general $\Gamma$ therefore
$\:\mathcal{P}≠1$ and this is the reason Avramidi includes the factor in his formula
Is my understanding correct?