Photon propagator
The photon propagator is defined as $$\braket{0|T{A^\mu(x)A^\nu(y)}|0} = i\int\frac{d^4p}{(2\pi)^4}e^{ip(x-y)}\Pi^{\mu\nu}(p),$$ where the right side is the definition in momentum space.
To calculate the propagator from a lagrangian we would
- Derive the equations of motion
- Determine the greens function
For the simple lagrangian $\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} -A^\mu J_\mu$:
- Equation of motion: $\partial_\mu F^{\mu\nu} = J^\nu$, which is in momentum space $(-p^2 g_{\mu\nu}+p_\mu p_\nu)A_\mu=J_\nu$
- We want to find the momentum space greens function $\Pi^{\mu\nu}$ so that $A_\mu = \Pi_{\mu\nu}J^\nu$ However we cant invert the kinetic term since $\det(-p^2g_{\mu\nu}+p_\mu p_\nu)=0$. This follows from * ge invariance*, since for a given $J_\mu$ the $A_\mu$ is not uniquely determined.
We therefore have to fix the gauge like $\partial_\mu A^\mu=0$, to calculate the propagator. We will choose a more general gauge.
Definition: $R_\xi$ -gauge
$R_\xi$ gauges (also covariant gauge) are a generalization of the Lorenz gauge. Instead of a condition for the gauge field we add a gauge breaking term to the Lagrangian.
$\xi = 0$ is equivalent to Lorenz gauge ($\partial_\mu A^\mu =0$ mus be fulfilled for finite lagrangian).
$\xi = 1$ is Feynman-’t Hooft gauge
$\xi \to \infty$ is Unitary gauge (does not work for QED but weak interactions)
Final results of calculations in $R_\xi$ gauge should be independent of $\xi$
In $R_\xi$ gauge:
- Equation of motion: $$\left[-p^2 g_{\mu\nu} + (1-\frac{1}{\xi})p_\mu p_\nu\right]A^\nu = J_\mu$$
- The inverse of the operator $\Pi_{\mu\nu}$ for $A_\mu = \Pi_{\mu\nu}J^\nu$ is $$\Pi_{\mu\nu} = -\frac{g_{\mu\nu} - (1-\xi) \frac{p_\mu p_\nu}{p^2}}{p^2}$$ To add time ordering, we add $+i\epsilon$ to get
Definition: Photon propagator in $R_\xi$ gauge
The (time ordered $+i\epsilon$) photon propagator in $R_\xi$ gauge is given by $$i\Pi_{\mu\nu} = \frac{-i}{p^2 + i\epsilon} \left[g^{\mu\nu} - (1-\xi)\frac{p^\mu p^\nu}{p^2}\right]$$