Feynmann diagram - positron representation

[prajor]

Hello, this might be a basic question. In feynman diagram's we represent the positron as traveling backwards in time. Is that correct ? How do we interpret this ?

 

[Bill_K]

prajor, Nothing travels backward in time, including positrons. Quantum field theory is carefully designed to preserve causality, which means that signals cannot propagate faster than light or into the past. The interpretation of antiparticles has been modified several times as we came to understand them better.

Originally it was thought that relativistic quantum mechanics would be described by a wavefunction φ(x, t) analogous to the Schrödinger wavefunction. As usual, what you do is Fourier transform φ and speak of plane waves, φ ~ exp(i(k·x - ωt)) But among these now are solutions with ω < 0. However they do not violate causality. On the contrary, including them is necessary to preserve causality.

Since the usual quantum interpretation is p = hk, E = hω, the first thought was that these states represented negative energy solutions. But the difficulty with this was quickly realized: any interaction would cause positive energy solutions to decay into negative energy ones, and consequently normal states including the vacuum state would be unstable.

So Dirac proposed his 'hole theory', in which the negative energy levels are fully occupied. The Pauli principle would prevent the transition of any more particles to those levels. In this theory the positron was seen as an unoccupied negative energy level. But hole theory is wrong. In the first place it only works for fermions. In the second place the fully occupied sea would result in the vacuum having infinite energy and infinite charge, with real physical consequences which are not observed.

Feynman's idea of positrons moving backwards in time was next, and while this, along with his diagram expansion, made his calculations much easier, it is wrong also.

Here's what we now understand. Relativistic quantum theory is not described by a wavefunction analogous to the Schrödinger wavefunction. The quantity φ(x, t) is an operator. Its negative frequency components do not represent negative energy particles, nor do they represent particles traveling backwards in time. There simply are no negative energy states. The Hilbert space does not contain them. What it contains instead are positive energy states for the antiparticles. The positive frequency terms in φ are particle creation operators, and the mysterious negative frequency terms are antiparticle annihilation operators. Feynman's idea was right in that the amplitudes for electron emission and positron absorption are related. But in such a case the positron does not travel to the past, it comes from it.

 

[prajor]

Bill, thanks a lot for the elaborate answer. That really helps.

BTW, may I know if there is a comprehensive source (book / site) covering other areas, leading to the current understanding as you explained ?

 

[Dickfore]

This is the paper that discussed it:

http://link.aps.org/doi/10.1103/PhysRev.76.749