Before answering the question more or less directly, I'd like to point out that this is a good question that provides an object lesson and opens a foray into the topics of singular integral equations, analytic continuation and dispersion relations. Here are some references of these more advanced topics: Muskhelishvili, Singular Integral Equations; Courant & Hilbert, Methods of Mathematical Physics, Vol I, Ch 3; Dispersion Theory in High Energy Physics, Queen & Violini; Eden et.al., The Analytic S-matrix. There is also a condensed discussion of `invariant functions' in Schweber, An Intro to Relativistic QFT Ch13d.
The quick answer is that, for m2∈R, there's no "shortcut." One must choose a path around the singularities in the denominator. The appropriate choice is governed by the boundary conditions of the problem at hand. The +iϵ "trick" (it's not a "trick") simply encodes the boundary conditions relevant for causal propagation of particles and antiparticles in field theory.
We briefly study the analytic form of G(x−y;m) to demonstrate some of these features.
Note, first, that for real values of p2, the singularity in the denominator of the integrand signals the presence of (a) branch point(s). In fact, [Huang, Quantum Field Theory: From Operators to Path Integrals, p29] the Feynman propagator for the scalar field (your equation) may be explicitly evaluated:
G(x−y;m)=limϵ→01(2π)4∫d4pe−ip⋅(x−y)p2−m2+iϵ={−14πδ(s)+m8π√sH(1)1(m√s) if s≥0−im4π2√−sK1(m√−s)if s<0.
where s=(x−y)2.
The first-order Hankel function of the first kind H(1)1 has a logarithmic branch point at x=0; so does the modified Bessel function of the second kind, K1. (Look at the small x behavior of these functions to see this.)
A branch point indicates that the Cauchy-Riemann conditions have broken down at x=0 (or z=x+iy=0). And the fact that these singularities are logarithmic is an indication that we have an endpoint singularity [eg. Eden et. al., Ch 2.1]. (To see this, consider m=0, then the integrand, p−2, has a zero at the lower limit of integration in dp2.)
Coming back to the question of boundary conditions, there is a good discussion in Sakurai, Advanced Quantum Mechanics, Ch4.4 [NB: "East Coast" metric]. You can see that for large values of s>0 from the above expression that we have an outgoing wave from the asymptotic form of the Hankel function.
Connecting it back to the original references I cited above, the +iϵ form is a version of the Plemelj formula [Muskhelishvili]. And the expression for the propagator is a type of Cauchy integral [Musk.; Eden et.al.]. And this notions lead quickly to the topics I mentioned above -- certainly a rich landscape for research.
This post imported from StackExchange Physics at 2014-07-13 04:38 (UCT), posted by SE-user MarkWayne