Feynman propagator
Definition: Feynman propagator
The Feynman propagator is the propagator of a free scalar theory $$D_F(x_1, x_2) = \braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0},$$ which can be calculated to $$D_F(x_1, x_2) = \int \frac{d^4k}{(2\pi)^4}\frac{i}{k^2-m^2+i\epsilon}e^{ik(x_1-x_2)}.$$
Note:
- The propagator is lorentz invariant
- It contains the contributions from both the advanced and the retarded propagators from Old fashioned perturbation theory
- Since we integrate over all possible $d^4k$ the four-momentum contributions can be off-shell $k_0 \neq \sqrt{\vec{k}^2+m^2}$
- The pole description with $i\epsilon$ corresponds to the time ordering with $\Theta(t)$ functions
Derivation of the Feynman propagator
Starting from $$D_F(x_1, x_2) = \braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0},$$ we can write the time ordering as $$\braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0} = \braket{0|\phi_0(x_1)\phi_0(x_2)|0}\Theta(t_1-t_2) + \braket{0|\phi_0(x_2)\phi_0(x_1)|0}\Theta(t_2-t_1)$$ Inserting the field operators and factorizing part of the exponential function with an integral subsitution leads to $$\braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0} = \int \frac{d^3}{(2\pi)^3}\frac{1}{2w_k} e^{-i\vec{k}(\vec{x}1 - \vec{x}2)}(e^{iw_k\tau}\Theta(-\tau)+e^{-iw_k\tau}\Theta(\tau)),$$ with $\tau = t_1 - t_2$. The two terms with different time ordering correspond to the advanced and retarted propagators from Old fashioned perturbation theory Using $$e^{iw_k\tau}\Theta(-\tau)+e^{-iw_k\tau}\Theta(\tau) = \lim{\epsilon \to 0} \frac{-2w_k}{2\pi i} \int^{\infty}{-\infty}\frac{dw}{w^2 - w^2_k + i\epsilon}e^{iw\tau},$$ we obtain the given results when pulling together the $d^3k$ and $dw$ integrals.
The last relation can be seen when executing the integral using the Cauchy’s integral formula