B In QFT, do Fermion fields belonging to distinguishable particles commute or anti commute?

[Paul Colby]

In the standard model fermion field components collect quite a few labels. The basic fermion field has 4 components that obey anti commutation relations. If one has two types of fermions, say electrons and muons. Do these commute or anti commute? Same question for other labels like gauge group labels?

 

[gentzen]

they don‘t anti commute

 

[PeterDonis]

If one has two types of fermions, say electrons and muons. Do these commute or anti commute?

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]

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.

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]

What I find confusing is exchanging a positron of at position x1 with the electron at position x2, the states are distinct

Because the electric charges are opposite, yes.

these field operators obey anti commutation relations.

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.

Electrons and muons, OTOH, aren't CPT conjugates and no such proof goes through.

 

[Demystifier]

they don‘t anti commute

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

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.

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?

 

[Paul Colby]

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.

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]

nulls in scattering cross sections

Can you be more specific?

 

[Paul Colby]

Well, take electron-proton forward scattering for example. Wouldn’t the cross section be forced to zero at small angles?

 

[Demystifier]

Well, take electron-proton forward scattering for example. Wouldn’t the cross section be forced to zero at small angles?

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]

Yeah, I see your point. I was thinking a 180 degree rotation swaps final states and should change the sign of the scattering amplitude. This would force a zero for forward scattering. To be honest, I don’t know.

 

[Demystifier]

Another argument. In path integral quantization, fermionic fields are described by Grassmann variables, and they anticommute even for fields of different particle species. There is no way that such anticommuting numbers are compatible with canonical operator quantization, unless the field operators also satisfy anticommutation relations for different particle species.

 

[gentzen]

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.

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.

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?

Yes, I still don‘t see any need for exchanging distinguishable stuff, even in this specific case.

 

[pines-demon]

In non-field theory, there is no Pauli exclusion principle for wave-functions related to two fermions of a different kind. So I would expect their associated field operators to commute.

 

[Paul Colby]

I don‘t understand why exchanging distinguishable stuff should make sense.

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.

 

[Demystifier]

A similar discussion is here:
https://physics.stackexchange.com/q...-different-species-and-anti-commutation-rules

 

[gentzen]

this question may reduce to a mater of convention.

A similar discussion is here:
https://physics.stackexchange.com/q...-different-species-and-anti-commutation-rules

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]

Electron is certainly distinguishable from positron, right? And yet I think nobody doubts that electron and positron field operators anticommute.

 

[gentzen]

Electron is certainly distinguishable from positron, right?

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]

@gentzen it seems that you are right. Weinberg, The Quantum Theory of Fields vol 1, pages 171-172, says that it is the convention that fermionic fields of different species anticommute.