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Question about schwinger dewitt expansion

  1. Jul 19, 2013 #1
    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:

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

    where

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

    [itex]\mathcal{P}[/itex] satisfies the relation:

    [itex]v^i \mathcal{D}_i \:\mathcal{P}=0[/itex]

    where [itex]v^i[/itex] is the vector along the geodesic

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