- #1

- 79

- 11

I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand.

When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like

$$

\begin{align}

H_D S_F(x-y) &= (i \hbar \gamma^\mu \partial_\mu - mc) \int \tilde{S}_F(p) \cdot e^{i p (x - y) / \hbar} \, \frac{d^4p}{(2 \pi)^4} \notag \\

&= \int (\gamma^\mu p_\mu - mc) \tilde{S}_F(p) e^{i p (x - y) / \hbar} \, \frac{d^4p}{(2 \pi)^4} \notag

\end{align}

$$

After that, the Dirac-Delta is used in Integral-representation and by comparing coefficiants, a momentum-space Propagator can be written as ##\tilde{S}_F(p) = \frac{\gamma^\mu p_\mu + mc}{p^2 - m^2 c^2}##. What I don't get is how in line two of the above equation the Hamiltonian can be dragged into the integral, especially in it's momentum-space form, which I also can't find a derivation for. I have attempted it in the following way

$$

\begin{align}

\tilde{H}_D &= \int (i \hbar \gamma^\mu \partial_\mu - mc) e^{-i p x / \hbar} \, d^4x \notag \\

&= \int (i \hbar \gamma^\mu (-i \frac{p_\mu}{\hbar}) - mc) e^{-i p x / \hbar} \, d^4x \notag \\

&= \int (\gamma^\mu p_\mu - mc) e^{i p x / \hbar} \, d^4x \notag

\end{align}

$$

Obviously, this already displays what has been used above, however, within an integral, so I cannot just assume ##\tilde{H}_D = (\gamma^\mu p_\mu - mc)##.

So, can someone please explain to me how this is valid? I can't seem to understand it myself and I cannot find anything on the Internet either that would make sense out of this.

Thank you in advance and have a great day everyone :)