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schieghoven

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This question is quite relevant to some other posts at the end of the very long "very simple QFT questions" thread, but I've decided to start a new thread with a heading which is more indicative of what I wish to ask the group. As a question, it's a fairly concise, but the analysis is lengthy and I've tried to sum up what I've done, and I hope it's of interest.

The Feynman propagator is almost exclusively used and referred to in terms of its Fourier transform. For instance, for a scalar particle we write:

[tex] G(k) = \frac{1}{k^2 - m^2 + i\epsilon} [/tex]

However, I wish to evaluate the propagator in its position representation (i.e. as a function of spacetime (t,x)). Doing so may provide some useful insight. This means evaluating the Fourier transform[tex] G(x) = \frac{1}{(2 \pi)^4} \int d^4k \; \frac{e^{-ikx}}{k^2 - m^2 + i\epsilon} [/tex]

Evaluation of this integral was discussed on another recent thread of this forum ("Propagator - Hankel function - infinite? - Weinberg"), which I replied to. I've now more or less worked through the whole problem and my solution is developed below. My solution disagrees with the Wikipedia article 'Propagator', so I agree with Hans de Vries in the long 'very simple QFT' thread that the Wikipedia article needs correcting.

My position is that this integral is formally not convergent for any x. For instance, for t > 0 we do the k_0 integral first, completing the contour in the lower half plane and collecting the pole at [tex] k_0 = \sqrt{k^2 + m^2} [/tex]; this gives

[tex] G(x) = \frac{-i}{(2\pi)^3} \int \frac{d^3k}{2 E_k} e^{-ikx} [/tex]

where [tex] E_k = \sqrt{k^2 + m^2} [/tex]. This integral is clearly divergent - try to do it in polar coordinates, for instance: the denominator only goes as 1/k for large 3-momenta k, while the volume of integration goes as k^3.This indicates that the propagator cannot be a function, but rather a generalised function (distribution). Generalised functions are defined by their action on some suitable class of test functions, typically smooth functions with compact support. For instance, the delta function acts on test functions h according to

[tex] \delta^4(x) : h \rightarrow h(0) . [/tex]

Fourier analysis for distributions is a rigorous theory, though it is not always expressible in terms of convergent integrals, and this is what we see above. Carrying out the analysis, I find that the propagator acts on test functions h according to[tex] G : h \rightarrow \lim_{\epsilon \rightarrow 0} \int_{|t^2-x^2|>\epsilon^2} d^4x \; B(x) h(x) [/tex]

where[tex] B(x) = \frac{-im H_1(m\tau)}{8\pi \tau} \qquad (\tau = \sqrt{t^2 - x^2}) [/tex]

[tex] B(x) = \frac{m K_1(m\tau)}{4\pi^2 \tau} \qquad (\tau = \sqrt{x^2 - t^2}) [/tex]

for timelike and spacelike x, respectively. H_1 is a Hankel function of order 1, and K_1 is a modified Bessel function of order 1. (The spacelike result is the result mentioned on page 202 of Weinberg QTF1 and discussed in the other thread.) The function B is a good indicator of the spacetime dependence of the propagator - for instance, the exponential decay of K_1 with distance means that the propagator is mostly confined to the forward and backward timelike regions, consistent with causality. [tex] B(x) = \frac{m K_1(m\tau)}{4\pi^2 \tau} \qquad (\tau = \sqrt{x^2 - t^2}) [/tex]

The \epsilon -> 0 construction in the solution is necessary because the Bessel functions -iH_1 and K_1 in the integrand diverge at the light cone, one to -infinity and the other to +infinity, and both diverge too strongly to simply define the action of G on h in terms of an integral over all space time. However, nonzero epsilon excludes the light cone from the domain of integration, and a (delicate) proof shows that the limit exists as epsilon tends to zero. It's like trying to define the integral of 1/x over [-1,1]; the best we can do is define a Cauchy 'principal value' of the divergent integral.

Having defined the propagator G, it is then necessary to show that it satisfies the defining relation

[tex] (\partial^\mu \partial_\mu + m^2 ) G = \delta^4(x) [/tex]

This means proving the remarkable claim that [tex] \lim_{\epsilon \rightarrow 0} \int_{|t^2-x^2|>\epsilon^2} d^4x \;

B(x) (\partial^\mu \partial_\mu + m^2 )h(x) = h(0) [/tex]

for all test functions h. This (lengthy) calculation uses Green's theorem (integration by parts); surface integral terms from the boundaries (\tau = \epsilon near the light cone) can be collected to give the h(0) as required. We use the fact that B satisfies the massive wave equation at all x except on the lightcone. It's really tricky to get the whole thing watertight because the Bessel functions are diverging as you move toward the light cone. However, it's done now and I'm fully convinced of the validity of the claim, give or take minor errors. I'm looking for references to check the analysis and answer against, but I haven't found any that do the problem in full mathematical rigour. Feynman (Theory of Fundamental Forces, Benjamin 1961) mentions a similar result but includes a delta function, which I disagree with, and doesn't indicate that the epsilon -> 0 limit is necessary.B(x) (\partial^\mu \partial_\mu + m^2 )h(x) = h(0) [/tex]

Has anyone ever seen this problem worked through before? I'm puzzled why none of the textbooks discuss this problem from a formal standpoint. It's such an ubiquitous quantity and I think it's crucial to know what it means and under what conditions it can be used.

Thanks,

Dave