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. The limes $\epsilon\to 0$ is always taken implicitly.
- For $\epsilon=0$ this is just a Greensfunction for the Klein-Gordon equation, which makes sense, since we are just computing propagation for a classical field
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 gives $$\braket{0|\phi_0(x_1)\phi_0(x_2)|0}=\int \frac{d^3k_1}{(2\pi)^3} \frac{d^3k_2}{(2\pi)^3} \frac{1}{\sqrt{2w_{k_1}2w_{k_2}}}\braket{0|a_{k_1}a^\dagger_{k_2}|0}e^{i(k_2x_2-k_1x_1)},$$ with $k_0=w_k=\sqrt{m^2+\vec{k}^2}$, since only the $aa^\dagger$ term survives when multiplying out the field operators when acting on $\ket{0}$. Factorizing part of the exponential function with an integral subsitution leads to $$\braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0} = \int \frac{d^3k}{(2\pi)^3}\frac{1}{2w_k} e^{-i\vec{k}(\vec{x}_1 - \vec{x}_2)}(e^{iw_k\tau}\Theta(-\tau)+e^{-iw_k\tau}\Theta(\tau)),$$ 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. Putting this into the expression for the propagator we get $$D_F(x_1,x_2)=\braket{0|T{\phi_0(x_1)\phi_0(x_2)}|0}=\int \frac{d^4k}{(2\pi)^4}\frac{i}{k^2-m^2+i\epsilon}e^{ik(x_1-x_2)}$$