Talk:Propagator

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[edit] Suspect formulas

Formulas in this article are suspect in many ways.

(1) Position-space KG propagator, m>0 case, does not reduce to m=0 case for m \to 0. (For small x, J1(x) ~ x, so it becomes constant)

(2) There's propagation outside the light cone in both cases.

(3) See http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/31/02/ and http://www.physicsforums.com/showthread.php?t=161235 for other definitions of position-space KG propagator.

(4) There is a well known formula relating propagators for Dirac and Klein-Gordon equations: S_F(x-x') = (i \gamma^{\mu} \partial_{\mu} - m) \Delta_F(x-x') and it does not appear to hold for Dirac propagator in this article (whichever KG propagator is used) because differentiation of J1 will necessarily produce either J0, J2 or both.

(5) What happens to fermions if m \to 0? Position-space Dirac propagator seems to vanish!

<off topic> I've checked several books on QFT; not one of them gives the expression for a position-space Dirac propagator. Not P&S, not Weinberg, not even Zee. Unbelievable.</off topic> --Itinerant1 08:55, 3 August 2007 (UTC)

I'm not certain the last comment is really off topic. Not being able to verify information in the article is of some concern.

Regarding concern 1, a constant is indeed a solution to the Klein-gordon equation, it doesn't have the necesary singularity at x=x' however. (For a propagator or Green function we want the differential equation to equal a delta function not 0.) You should be able to combine these two solutions to get the boundary conditions you want.

When m is not equal to 0, my guess is that we should use the Bessel function of the second type Y_1(m|x-x'|) which does have the proper limit. (Or to be more accurate, use a linear combination of the two which matches the desired boundary conditions.) Choosing boundary conditions is roughly (or at least partially) equivalent to chosing a Feynman propagator, a retarded Green Function, or an advanced Green function.

Lee Loveridge Ph.D. —Preceding unsigned comment added by 157.185.96.158 (talk) 17:38, 3 October 2007 (UTC)