# Supersection Rules

I was just reading that because of superselection rules we cannot superpose two particles with different electric charges. But when I look in my particles physics books it seems there are decay processes that do this all the time: consider $\pi^+ → μ^+ +\nu_\mu$, for example. Can anyone tell me what's going on here?

Thanks a lot!

stevendaryl
Staff Emeritus
I was just reading that because of superselection rules we cannot superpose two particles with different electric charges. But when I look in my particles physics books it seems there are decay processes that do this all the time: consider $\pi^+ → μ^+ +\nu_\mu$, for example. Can anyone tell me what's going on here?

Thanks a lot!

I thought that the rule was that there cannot be a superposition $|\Psi\rangle = \alpha |A\rangle + \beta |B\rangle$ with the total charge in $|A\rangle$ unequal to the total charge in $|B\rangle$. A decay such as the one in your example doesn't violate that.

But the antimuon has positive charge, and the neutrino no charge, no?

Bill_K
You're misunderstanding the meaning of the word "superposition". A state with a muon AND a neutrino is not a superposition. A superposition would be a state that is maybe a muon OR maybe a neutrino.

As stevendaryl illustrated, the state has probability |α|2 of being |A> and probability |β|2 of being |B>. This can't happen if |A> and |B> have different total charge.

Thanks. So in the decay above -- which I've written down just as you see it in the textbooks -- represents a mixed state, not a coherent superposition?

And what about when we mix quarks as in the CKM matrix -- is that a mixed state or a superposition also? Thanks a lot!

kith
So in the decay above -- which I've written down just as you see it in the textbooks -- represents a mixed state, not a coherent superposition?
Schematically, if you have a pion state $|\pi^+\rangle$ and let it evolve in time, you get something like $α(t)|\pi^+\rangle + β(t)|μ^+\rangle⊗|\nu_\mu\rangle + γ(t)|e^+\rangle⊗|\nu_e\rangle + ...$ If you perform a measurement on this superposition state, your final state will be one of the terms. Each of them is a definite state, not a mixture. For example, the term you were discussing is a state where you have exactly one anti-muon and one myon neutrino.

And what about when we mix quarks as in the CKM matrix -- is that a mixed state or a superposition also?
If you apply a matrix to a state vector you get a new state vector. You can't get statistical mixtures this way. In order to get them you need a so-called super operator: an operator which takes a state operator (density matrix) to another state operator. Such super operators become necessary only in open quantum systems.

Thanks Kith -- especially for your explanation of what's going on when we represent the pion decay in that way. Final question: when we represent neutrino mixing as sums of neutrinos of different flavors, are we making a superposition precluded by superselection rules then?

Thanks again!

kith