An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 2.3

The Klein-Gordon Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · klein-gordon-propagator-spacelike

Problem

Evaluate the function

\langle 0|\,\phi(x)\phi(y)\,|0\rangle = D(x-y) = \int \frac{d^3p}{(2\pi)^3}\,\frac{1}{2E_{\mathbf{p}}}\,e^{-ip\cdot(x-y)},

for

(x - y)

spacelike so that

(x-y)^2 = -r^2

, explicitly in terms of Bessel functions.