Wave functions for positrons and electrons

[friend]

Is the wave function for the positron the complex conjugate of the wave function for the electron? I've tried to google this, but I can't seem to get a definite answer from a reliable source. It seems that antimatter is derived in quantum field theory which does not concentrate on wave functions. And there the charge is conjugated, which amounts to taking the complex conjugate. But then I'm told that you can look at the positron as if it were an electron moving backwards in time. The negative sign on the time variable can be shifted to a negative sign on the complex i, which gives us a complex conjugate. But a wave function can describe motion through time and space, so I suppose you'd have to assume that the positron was moving with the same momentum as the electron. Then is the wave function of the positron the complex conjugate of the electron? Thanks.

 

[king vitamin]

One problem is that, by "wave function," you seem to be referring to a position-space wave function, and such objects do not exist in relativistic quantum field theory. So the answer this question,

Is the wave function for the positron the complex conjugate of the wave function for the electron?

is no simply because neither have wave functions. One could try to take the "one-particle approximation," which is basically the limit Dirac was working in with his famous 1928 paper introducing his equation, in which case one can define approximate wave functions for both electrons and positrons (this theory breaks down in various limits, but works at low energy and short enough times). In this theory, there is no reason the electron and positrons wave functions should ever be complex conjugates of each other.

At the level of quantum field theory, charge conjugation is more complicated than just complex conjugation of the field operators. (In fact, one should be careful in even saying that, because it makes charge conjugation sound like it is an anti-unitary operation, when it is actually unitary.)

 

[friend]

So the answer this question, is no simply because neither have wave functions. One could try to take the "one-particle approximation," which is basically the limit Dirac was working in with his famous 1928 paper introducing his equation, in which case one can define approximate wave functions for both electrons and positrons (this theory breaks down in various limits, but works at low energy and short enough times). In this theory, there is no reason the electron and positrons wave functions should ever be complex conjugates of each other.

I can certainly understand that in general this is not the case because the momentum of each can be in arbitrary directions. And in the presence of any potential the momentum of the electron will become different from the momentum of a positron, so no there as well. However, what about if each is a free particle traveling between the same two points in the same time duration, if we simply substitute the free electron for the free positron, are they complex conjugates? I think this is asking whether the charge of the electron appears in the wave function. I don't remember that being the case. Thanks again.

 

[king vitamin]

However, what about if each is a free particle traveling between the same two points in the same time duration, if we simply substitute the free electron for the free positron, are they complex conjugates?

If you work in the one-particle approximation (with the usual caveats about its applicability), the wave functions of the free electron and free positron may be chosen to be equal to each other. (What I mean by "may be chosen" is that you can multiply the wave functions by phases without changing anything, but by choosing the same phase convention for both, the wave functions can be made equal.)

I think this is asking whether the charge of the electron appears in the wave function.

Only if the charge appears in the Hamiltonian. Otherwise, the Dirac equation for both wave functions is the same, and you get the same solutions for both.

 

[friend]

I have a bit riding on this, and I would be remiss if I did not turn over every stone.

It seems the electron positron being conjugates of each other is only relevant at the level of QFT, where a distinction needs to be made as to what particle may decay into an electron plus stuff or a positron plus stuff. So, could the wave functions of positrons and electrons be conjugates of each other only at the differential level? As I understand it, wave functions of quantum mechanics are elevated to operators in QFT. And QFT specifies the number of particles at each point. So it seems we have wave functions being used differentially at each point to specify how various kinds of particles emerge from a point. Does this question even make sense?

 

[king vitamin]

Unless you have some precise definitions of what you mean by "wave functions at the... differential level" or "wave functions being used differentially," I have no way to answer you.

I recommend that you buy a QFT textbook, I think that your confusion is very fundamental and will not be remedied by asking questions on an online forum. Characterizing QFT as elevating wave functions to operators or as specifying the number of particles at a point are very off-base (the latter notion is anathema to the very heart of QFT).