Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 6.1

The S-matrix and time-ordered products — Schwartz M.D. z-lib.org

Physics — QFT · feynman-propagator-position-space

Problem

Calculate the Feynman propagator in position space. To get the pole structure correct, you may find it helpful to use Schwinger parameters (see Appendix B). Take the m→0m \to 0 limit of your result to find ⟨0∣T{ϕ0(x1)ϕ0(x2)}∣0⟩=−14π21(x1−x2)2−iε.(6.35)\langle 0|T\{\phi_0(x_1)\phi_0(x_2)\}|0\rangle = -\frac{1}{4\pi^2}\frac{1}{(x_1-x_2)^2 - i\varepsilon}. \tag{6.35}
Intuitive explanation

The Feynman propagator connects momentum and position space through a Fourier transform, but the integral has poles that require careful handling—this is where Schwinger parameters shine. By introducing an exponential regulator e−αm2e^{-\alpha m^2} in momentum space, you convert a tricky momentum integral into a Gaussian, then extract the position-space result by integrating over the regulator parameter. The iεi\varepsilon prescription emerges naturally from how the poles are bypassed, ensuring causality in the final answer.

Intuitive explanation

The Feynman propagator connects momentum and position space through a Fourier transform, but the integral has poles that require careful handling—this is where Schwinger parameters shine. By introducing an exponential regulator e−αm2e^{-\alpha m^2} in momentum space, you convert a tricky momentum integral into a Gaussian, then extract the position-space result by integrating over the regulator parameter. The iεi\varepsilon prescription emerges naturally from how the poles are bypassed, ensuring causality in the final answer.

Browse the problem library