# Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $$\phi^r$$ (where $$r$$ could be a vector or spinor index, or even a multiindex if $$\phi$$ is a tensor field etc.) as $$\Delta^{rs}_F(x - x') = \langle 0| \mathcal{T}\{\phi^r(x), \phi^s(x')\}|0\rangle$$

Suppose the equations of motion for the field(s) $$\phi^r$$ can be written as $$L^{r s}\phi^s = 0$$ for some differential operators $$L^{rs}$$. Then supposedly the Feynman propagator is a Green's function for $$L^{rs}$$ (up to sign), i.e. $$L^{rs} \Delta^{s t}_F(x) = - \delta^4(x) \delta^{r t}$$ Q: Is this true, and if so, why?

It certainly is true for a Klein-Gordon field, but what about the general case? I can also see why we have $$L^{r s} \Delta^{s t}_F(x) = 0$$ for $$t \neq 0$$.

• Well, you could take this as the definition of the propagator (see e.g. this PSE post, where I use the notation $\mathscr D$ instead of $L$). More precisely, you should let $\mathcal T$ be defined such that this condition is true (which sometimes agrees with the naive definition of time-ordering, but sometimes acquires extra contact terms, cf. this PSE post). Feb 10 '20 at 1:20
• As a start, you could just consider the propagators as being Green functions of PDEs. You can do the same for the Dirac equation $(i\hbar\gamma^{\mu}\partial_{\mu} - m)\psi$ and define a fermion propagator. Feb 10 '20 at 5:09