# Pair Production, electron and positron from isolated photon

1. Apr 10, 2014

### cooev769

I don't completely understand why an electron positron pair cannot be created from an isolated photon. I understand it must have something to do with 4 momentum conservation, but I really can't see a problem given the photon has enough energy for the mass to energy vice versa conversion.\

The only seemingly sound argument I could seem to convince myself with is that if 4 momentum is conserved and we say pf=pi. And we take pi of the photon in the photons rest frame we have (0,0,0,E/c), but the electrons can never move at the speed of light and hence the 4 momentum of any two of the electrons will have a negative momentum value (-x,0,0,E/c) and hence it is therefore impossible. But I don't believe this is correct. Any suggestions appreciated thanks!

2. Apr 10, 2014

### snorkack

Pretty simple and trivial argument: equality of reference frames. You can always pick a reference frame where the isolated photon is Doppler shifted to have too little energy for pair production. Since the event must happen in all reference frames or in none of them, and there are frames where it is impossible, therefore it is impossible in all frames.

Even simpler argument: an isolated photon has no time whatsoever and therefore no events whatsoever can happen to it. Splitting a photon into several photons of lower energy travelling in the same initial direction would not violate conservation of momentum (unlike production of particles with rest mass) but it is still impossible because of no time at the speed of light.

3. Apr 10, 2014

### cooev769

Cheers. What i'm really looking for though is a rigorous violation of the conservation of 4 momentum.

4. Apr 10, 2014

Staff Emeritus
You won't find a "rigorous violation", since it is conserved.

5. Apr 10, 2014

### cooev769

No it isn't as an isolated photon cannot produce and electron and an anti electron, therefore writing the 4 momentum before and after will produce a violation in conservation of 4 momentum, which I know occurs, I just don't know the proof and i've been trying for a while.

6. Apr 10, 2014

### ChrisVer

Can't photons of energy E>~1MeV lead to pair production?

7. Apr 10, 2014

### cooev769

Yep, but apparently in the presence of a nucleus or something, not an isolated photon which is why i'm trying to prove it mathematically.

8. Apr 10, 2014

Staff Emeritus
No, momentum is conserved.

9. Apr 10, 2014

### Bill_K

Take as your rest frame the center of mass frame of the electron-positron pair. Their total momentum in this frame is zero. Now what is the momentum of the photon? If momentum is conserved, the photon would also have to have momentum zero. But the momentum of a photon can't be zero, since it would have to be standing still.

10. Apr 10, 2014

### ofirg

Just to follow up on Bill_K thread, it shows that momentum and Energy can't both be conserved. In the center of mass frame of the electron-positron pair their momentum is zero but there energy has to be at least twice the mass of the electron. A "zero momentum" photon also has zero energy and therefore can't have a four vector which is equal to the four vector of the electron-positron pair.

I'm not sure this argument is true. I think that in general a massless scalar field can decay to two collinear scalar fields ( or more)

11. Apr 10, 2014

### ofirg

scalar particles not fields