Feynman propagator

Feynman propagator

[′fīn·mən ′präp·ə‚gād·ər]
(quantum mechanics)
A factor (ρ + m)/ (ρ2 - m 2 + i ε) in a transition amplitude corresponding to a line that connects two vertices in a Feynman diagram, and that represents a virtual particle.
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where [[SIGMA].sub.F](x-y) is the so-called Feynman propagator, obeying
where [[SIGMA].sub.F](x - y) is the Feynman propagator, obeying
However, as had been indicated earlier, this imaginary number cannot be arbitrarily altered as it is connected with the phase factor iS/[??] in the Feynman propagator. If it is necessary to include a Wick rotation, the power of should be changed to v + 1 in Naber's equation (N9) instead of just v.
We shall show in the next Section that from this photon propagator by space-time statistics we can get a propagator with the [k.sub.E] replaced by the energy-momemtum four-vector k which is similar to the Feynman propagator (with a mass-energy parameter [[lambda].sub.1] > 0).
Having displayed Wyler's expression of the fine structure constant [[alpha].sub.EM] in terms of the ratio of dimensionless measures, we shall present a Fiber Bundle (a sphere bundle fibration over a complex homogeneous domain) derivation of the Wyler expression based on the bundle [S.sup.4] [right arrow] E [right arrow] [D.sub.5], and explain below why the propagation (via the determinant of the Feynman propagator) of the electron through the interior of the domain [D.sub.5] is what accounts for the "obscure" factor V [([D.sub.5]).sup.1/4] in Wyler's formula for [[alpha].sub.EM].
The Feynman propagator of a massive scalar particle (inverse of the Klein-Gordon operator) [([D.sub.[mu]][D.sup.[mu]] - [m.sup.2]).sup.-1] corresponds to the kernel in the Feynman path integral that in turn is associated with the Bergman kernel [K.sub.n](z, z') of the complex homogenous domain [D.sub.n], proportional to the Bergman constant [k.sub.n] [equivalent to] 1/V([D.sub.n]), i.
where we have introduced a momentum scale [mu] to match units in the Feynman propagator expression, and the Bergman Kernel [K.sub.n]([??], [bar.[??]]') of [D.sub.n] whose dimensionless entries are [??] = ([z.sub.1], [z.sub.2], ..., [z.sub.n]), [??]' = ([z'.sub.1], [z'.sub.2], ..., [z'.sub.n]) is given as