In introducing the concept of 'virtual photon',Halzen Martin writes (ch#1,P#7) "An ekectron emits a photon (the quantum of electromagnetic field) and as a result,recoils in order to conserve momentum.it is clearly impossible to conserve energy as well,so the emitted photon is definitely not a real photon"... Why energy and momentum conservation cannot be simultaneously satisfied?Is momentum conservation is a bit prefferred over energy conservation?
Familiar versions of uncertainty principle does not deal with conservation principles.The homogeneity of space and homogeneity in time do not seem to disturb one another.Then where is the crunch? Neither energy and momentum are conjugate variables...
It is relatively simple if you view the situation in the right reference frame: Consider an electron totally stationary. It suddenly emits a photon, recoiling in the opposite direction to conserve momentum. However, the system now suddenly has kinetic energy, and it had nothing before. The rest mass energy is unchanged, because the electron is still an electron and the photon has no rest mass. So the total energy of the system has gone from \(m_e\) to \(sqrt{m_e ^2 + \mathbf{p_e}^2}+\mathbf{p}_\gamma\), which is clearly more (I am using natural units here so c=1). This is bad. As for why momentum is conserved and not energy, well in the above scenario if you conserved energy the photon couldn't exist so you'd trivially conserve momentum. The only way the process can actually proceed is if energy conservation is violated, but since we can still conserve momentum then at least this much happens. We can always view such a process from a frame like this, so this applies to all virtual particle interaction vertices.
