Vacuum expectation value of time-ordered functions in QED

Question

I have been asked to take the Fourier transform of the two-point EM current function $$G_{\mu \nu}(x) = \langle \Omega \lvert T\{J_{\mu}(x) J_{\nu}(0) \} \rvert \Omega \rangle,$$ and to prove something specific about said result.

However, I'm having trouble with the following. Should I calculate $$\langle \Omega \lvert T\{J_{\mu}(x) J_{\nu}(0) \} \rvert \Omega \rangle =\frac{\int \mathcal{D}\psi \mathcal{D}\bar{\psi} J_{\mu}(x) J_{\nu}(0) e^{iS[\psi, \bar{\psi}]}}{\int \mathcal{D}\psi \mathcal{D}\bar{\psi} e^{iS[\psi, \bar{\psi}]}},$$ or is there a better approach? I'm aware that the vacuum expectation value could also be computed by taking the functional derivative of the generating functional, but said derivatives are to be taken with respect to the sources, and I'm not sure if in that case I should take the functional derivative with respect to the fields $A_{\mu}(x)$ and $A_{\nu}(0)$.

Any help would be much appreciated.

Answer 1

There's a difference between a source of the quantum field in the path integral formalism, and a current operator, despite of them being denoted by the same letter $J_{\mu}$. In this answer, $J_{\mu}$ is the current operator.

So we want to calculate $\left< J_{\mu} (x) J_{\nu} (y) \right>$. First thing to do is to plug the definition of the E/M current operator. This, of course, depends on which charged particles are present in your theory. In case of spinors, it is given by $$ J_{\mu} (x) = i e \, \bar{\psi} (x) \, \gamma^{\mu} \, \psi (x). $$

Now you have to calculate $$ \left< J_{\mu} (x) J_{\nu} (y) \right> = \int D\psi D\bar{\psi} e^{i S[\bar{\psi}, \psi]} J_{\mu} (x) J_{\nu} (y), $$

where I allowed myself to omit the normalization factor in the denominator.

Plug your expression for $J_{\mu} (x)$ and use Wick's theorem! You will end up with three terms.

The first two terms will contain the trace of products of fermion propagators. By a fermion propagator I mean $$ \left< \bar{\psi} ^a (x) \psi_b (y) \right> = S^a_{\;b}(x-y). $$

The third term will contain $S(0)^2$. This term is singular, and it is to be artificially removed by redefining the quantum operator $J_{\mu}$ as normal-ordered.

I hope this helps.