Why Does the First Order Term in QED Scattering Matrix Give No Contribution?

[QuantumDevil]

Can some explain me why first order term in perturbation expansion of scattering matrix gives no contribution for every possible IN and OUT states? It is said that this is connected with the fact that condition of energy-momentum conservation cannot be satisfied for real photons and electrons. For me it isn't so obvious :(

 

[blechman]

You mean, why can't a (free) electron just suddenly spit out a photon giving a process: $e\rightarrow e\gamma$? Or why can't a (free) electron just suddenly absorb a (free) photon to become a new electron: $e\gamma\rightarrow e$? Or why can't a (free) photon suddenly turn into a (free) electron-positron pair: $\gamma\rightarrow e^+e^-$?

Choose any of these processes. Now write down the conservation law for energy and momentum (so there are 4 equations in all). Also write down the on-shell conditions for the electron ($E_e^2-\vec{p}_e\cdot\vec{p}_e=m_e^2$) and the photon ($E_\gamma^2-\vec{p}_\gamma\cdot\vec{p}_\gamma=0$), where I have set c=1 for simplicity. Using these equations, solve for $E_e,\vec{p}_e,E_\gamma,\vec{p}_\gamma$.Can't do it, can you?!

 

[nrqed]

You mean, why can't a (free) electron just suddenly spit out a photon giving a process: $e\rightarrow e\gamma$? Or why can't a (free) electron just suddenly absorb a (free) photon to become a new electron: $e\gamma\rightarrow e$? Or why can't a (free) photon suddenly turn into a (free) electron-positron pair: $\gamma\rightarrow e^+e^-$?

Choose any of these processes. Now write down the conservation law for energy and momentum (so there are 4 equations in all). Also write down the on-shell conditions for the electron ($E_e^2-\vec{p}_e\cdot\vec{p}_e=m_e^2$) and the photon ($E_\gamma^2-\vec{p}_\gamma\cdot\vec{p}_\gamma=0$), where I have set c=1 for simplicity. Using these equations, solve for $E_e,\vec{p}_e,E_\gamma,\vec{p}_\gamma$.


Can't do it, can you?!

To the OP: A second way but equivalent is to use four-momentum conservation if you are at ease with this. Isolate the four-momentum of one of the photons and square both sides. After a little algebra it will be clear that the two sides can't be equal.

 

[QuantumDevil]

To the OP: A second way but equivalent is to use four-momentum conservation if you are at ease with this. Isolate the four-momentum of one of the photons and square both sides. After a little algebra it will be clear that the two sides can't be equal.

Indeed this leads to contradiction