From my professor lecture:
The Photon field is defined as: $$A_{\mu}(x)= \sum_{s=1,2}\int \frac{d^3 \vec{p}}{{(2 \pi)^{3}\sqrt{2 E_{p}}}}\left(\epsilon^{s}_{\mu}(p) a^s_{\hspace{0.05cm}\vec{p}}e^{-ip^{\nu}x_{\nu}}+\epsilon^{*s}_{\mu}(p)a^{\dagger s}_{\hspace{0.05cm}\vec{p}}e^{ip^{\nu}x_{\nu}}\right)$$ with the commutation relations: $$[a^s_{\hspace{0.05cm}\vec{p}},a^{\dagger r}_{\hspace{0.05cm}\vec{k}}]=(2 \pi)^{3}\delta^{sr}\delta{\left(\vec{p}-\vec{k}\right)}.$$ The Feynman rules for external lines can be obtained by acting the field $A_{\mu}$ on the initial and final state particles:
Incoming photon ~~~~• :$\hspace{2.5 cm}$ $A_{\mu}|\gamma(k,r)\rangle=\epsilon^{r}_{\mu}(k)$
Outgoing photon •~~~~ :$\hspace{2.5 cm}$ $\langle\gamma(k,r)|A_{\mu}=\epsilon^{*r}_{\mu}(k)$
Where we used the commutation relations, the expression of the field and the relation: $$|\gamma(k,r)\rangle = \sqrt{2E_{k}} a^{\dagger r}_{\hspace{0.05cm}\vec{k}}|0\rangle$$
Then the lecture follows with the same thing for a fermionic and a scalar field, so far so good, but there are three things that I don't understand:
- From what I see we use just "half" of the expression of the field:
$$A_{\mu}(x)|\gamma(k,r)\rangle= \sum_{s=1,2}\int \frac{d^3 \vec{p}\sqrt{2E_{k}}}{{(2 \pi)^{3}\sqrt{2 E_{p}}}}\epsilon^{s}_{\mu}(p)e^{-ip^{\nu}x_{\nu}}a^s_{\hspace{0.05cm}\vec{p}} a^{\dagger r}_{\hspace{0.05cm}\vec{k}}|0\rangle$$
But not the additional term with $a^{\dagger s}_{\hspace{0.05cm}\vec{p}} a^{\dagger r}_{\hspace{0.05cm}\vec{k}}$.
And the same for the outgoing photon for the term with both annihilation operators.
I think this is because the fist term ($a^{\dagger s}_{\hspace{0.05cm}\vec{p}}$) of the field creates a photon while the second ($a^{s}_{\hspace{0.05cm}\vec{p}}$) annihilates a photon, but then the notation $A_{\mu}|\gamma(k,r)\rangle$ shouldn't be wrong? And the same for the premise "The Feynman rules for external lines can be obtained by acting the field $A_{\mu}$ on the initial and final state particles" ?
- Redoing the same calculation I actually obtain and additional prefactor:
$$A_{\mu}|\gamma(k,r)\rangle=\epsilon^{r}_{\mu}(k)e^{-ik^{\nu}x_{\nu}}$$
$\hspace{0.75cm}$ And $e^{ik^{\mu}x_{\mu}}$ for the outgoing particle. Why these prefactors are neglected?
- When we consider the S-matrix element we write: $\langle i|S|f \rangle$ (with i the initial stante and f the final state).
Now, since the incoming photon ~~~~• is an initial state why we use the ket $|\gamma\rangle$ and not the bra $\langle\gamma|$? , that would lead to: ~~~~•$=\langle\gamma(k,r)|A_{\mu}=\epsilon^{*r}_{\mu}(k)$ (and viceversa for the outgoing one).