In quantum field theory (QFT), fermion fields belonging to distinguishable particles, such as electrons and muons, do not anti-commute. The discussion highlights that anti-commutation relations apply primarily to indistinguishable fermions, while distinct fermions like electrons and muons are not subject to the same rules. The participants reference the standard model and CPT conjugation principles to support their claims, emphasizing that the anti-commutation of fermionic fields is a matter of convention, as noted in Weinberg's "The Quantum Theory of Fields".
PREREQUISITESParticle physicists, quantum field theorists, and students studying the standard model who seek to deepen their understanding of fermion behavior and anti-commutation relations.
Mu: the question doesn't even make sense, because you can't take a state with an electron and a muon in it and exchange the two and have a physically indistinguishable state, and "commute or anti commute" only makes sense when the exchange gives a physically indistinguishable state and you then ask whether that state is mathematically equal to the original state, or to minus the original state.Paul Colby said:If one has two types of fermions, say electrons and muons. Do these commute or anti commute?
Well, perhaps. What I find confusing is exchanging a positron of at position x1 with the electron at position x2, the states are distinct yet these field operators obey anti commutation relations.PeterDonis said:Mu: the question doesn't even make sense, because you can't take a state with an electron and a muon in it and exchange the two and have a physically indistinguishable state, and "commute or anti commute" only makes sense when the exchange gives a physically indistinguishable state and you then ask whether that state is mathematically equal to the original state, or to minus the original state.
Because the electric charges are opposite, yes.Paul Colby said:What I find confusing is exchanging a positron of at position x1 with the electron at position x2, the states are distinct
Because they're CPT conjugates of each other, so you can show that, if electrons anticommute with electrons and positrons anticommute with positrons, then electrons and positrons must also anticommute. But you can't just assume that electrons and positrons anticommute because they're indistinguishable fermions. They're not. You actually have to show that CPT conjugate fermion fields must obey the same anticommutation relations with each other as each field individually does with itself.Paul Colby said:these field operators obey anti commutation relations.
gentzen said:they don‘t anti commute
Do you have references supporting these claims? I am pretty much convinced that they anticommute and that exchanging them makes perfect sense, but I also don't know reference where it is stated explicitly.PeterDonis said:Mu: the question doesn't even make sense, because you can't take a state with an electron and a muon in it and exchange the two and have a physically indistinguishable state
After thinking about this for a bit, distinct fermions anti commuting would have pretty clear effects, like nulls in scattering cross sections and such. People would have noticed.Demystifier said:I am pretty much convinced that they anticommute and that exchanging them makes perfect sense, but I also don't know reference where it is stated explicitly.
Can you be more specific?Paul Colby said:nulls in scattering cross sections
It wouldn't, because ##b_{\rm electron}(k) \neq b_{\rm proton}(k)##, so the anticommutativity does not imply that their product is zero.Paul Colby said:Well, take electron-proton forward scattering for example. Wouldn’t the cross section be forced to zero at small angles?
In which context where exchanging them makes perfect sense do you think that they should anti commute? For measurement operators of spatially separated spacetime regions, exchanging them makes sense, but in this case they simply commute. And as states, I don‘t understand why exchanging distinguishable stuff should make sense.Demystifier said:Do you have references supporting these claims? I am pretty much convinced that they anticommute and that exchanging them makes perfect sense, but I also don't know reference where it is stated explicitly.
Yes, I still don‘t see any need for exchanging distinguishable stuff, even in this specific case.Demystifier said:For instance, consider two electron neutrinos emitted from the Sun. As they travel to Earth, they suffer neutrino mixing. Do you think you could describe it correctly without assuming that electron and muon neutrino field operators anticommute?
I’m contemplating writing a computer program to numerically solve an approximate truncated Hamilton matrix for hadron bound states. I’m concerned with sign ambiguities in the evaluation of matrix elements. There really shouldn’t be any. The nice thing about the standard model is the Hamiltonian is known. So far as SM Hamilton matrix elements go, this question may reduce to a mater of convention.gentzen said:I don‘t understand why exchanging distinguishable stuff should make sense.
Paul Colby said:this question may reduce to a mater of convention.
My conclusion from those discussions is also that if you insist to exchange distinguishable stuff, then you are free to choose any phase factor that you like. Maybe it is convenient to choose -1 for Fermions, because then you don't even need to figure out whether they are truly distinguishable or not in the concrete case.Demystifier said:A similar discussion is here:
https://physics.stackexchange.com/q...-different-species-and-anti-commutation-rules
No, it is not really a different field, hence it is not distinguishable. But this example shows why the -1 convention is used, because it is easy to make mistakes otherwise.Demystifier said:Electron is certainly distinguishable from positron, right?