Questions about Identical Particles

[hgandh]

From Weinberg's Quantum Theory of Fields Vol. 1, Chapter 4. Under interchange of two identical species of particles we have:
\begin{equation}
\Phi_...p,\sigma,n...p',\sigma',n...= \pm \Phi_...p',\sigma',n...p,\sigma,n...
\end{equation}
Plus sign for bosons and minus for fermions. As far as I understand, this is just a statement on labeling identical particles. The labeling of particle 1 with $p,\sigma$ and particle 2 with $p',\sigma'$ is the same physical state as vice versa because we cannot distinguish the two particles. Now this is where I get confused.

"What about interchanges of particles belonging to different species? If we like, we can avoid this question by simply agreeing from the beginning to label the state-vector by listing all photon momenta and helicities first, then all electron momenta and spin z-components, and so on through the table of elementary particle types. Alternatively, we can allow the particle labels to appear in any order and define the state-vectors with particle labels in an arbitrary order as equal to to the state-vector with particle labels in some standard order times phase factors, whose dependence on the interchange of particles of different species can be anything we like. In order to deal with symmetries like isospin invariance that relate particles of different species, it is convenient to adopt a convention that generalizes (1): the state-vector will be taken to be symmetric under interchange of any bosons with each other, or any bosons with any fermions, and antisymmetric with respect to interchange of any fermions with each other, in all cases, whether the particles are of the same species or not."

I am not sure if I understand this correctly but this is my interpretation: In the first case, we can treat all particles of different species as distinguishable and therefore fix them in some order. In the second, some particles of different species are indistinguishable and we define the state-vector in accordance with some standard state-vector. Lastly, we treat all bosons as indistinguishable from each other as well as all fermions. If somebody could help explain this quote it would be much appreciated.

 

[Gene Naden]

some particles of different species are indistinguishable

I don't think this is right. I think particles of different species are always distinguishable.

I think the quote is saying that you can write the state vector as if fermions of different species were indistinguishable, and as if bosons of different species were indistinguishable. Why he wants to do this is not clear to me. Maybe that is explained later.

 

[PeterDonis]

I think particles of different species are always distinguishable.

The issue is what counts as "different species".

For example, the quote given in the OP mentions isospin invariance. The electron and neutrino form a weak isospin doublet. So are the electron and neutrino "different species"? Suppose we have a state which is rotated in weak isospin space so it is part electron and part neutrino. How do we treat it when it comes to the Pauli exclusion principle?

This is the sort of issue Weinberg is addressing.

 

[DrDu]

I find Weinberg sometimes quite cryptic. For some reason, he avoids the concept of superselection, which I consider helpful in this context.

 

[Gene Naden]

I find it hard to imagine such a mixed state, part electron and part neutrino. Different charge, very different mass.

 

[stevendaryl]

I'm not sure, but it seems that he might be considering all fermions to be indistinguishable, and the fact that one is an electron and the other is a neutrino is a matter of what state the fermion is in. So the antisymmetry under particle exchange just means (in the case of one electron and one neutrino) that making the change from "particle A is in the electron state, and particle B is in the neutrino state" to "particle B is in the electron state and particle A is in the neutrino state" causes a change of sign. My feeling is that it can't possibly make any difference whether or not you allow such operations.

I don't think that the proposal violates superselection, though. Superselection says there can be no superposition of states with different total charges, but thinking of neutrinos and electrons as different states of the same particle doesn't violate that rule.

 

[stevendaryl]

I'm not sure, but it seems that he might be considering all fermions to be indistinguishable, and the fact that one is an electron and the other is a neutrino is a matter of what state the fermion is in. So the antisymmetry under particle exchange just means (in the case of one electron and one neutrino) that making the change from "particle A is in the electron state, and particle B is in the neutrino state" to "particle B is in the electron state and particle A is in the neutrino state" causes a change of sign. My feeling is that it can't possibly make any difference whether or not you allow such operations.

I don't think that the proposal violates superselection, though. Superselection says there can be no superposition of states with different total charges, but thinking of neutrinos and electrons as different states of the same particle doesn't violate that rule.

I looked it up, and that's not exactly what superselection means. Instead, it means (in the case of total charge) that if $|\psi \rangle$ and $|\phi\rangle$ are states with different total charges, then for any observable $A$, $\langle \phi|A|\psi \rangle = 0$.

In any case, viewing neutrinos and electrons as different states of the same particle doesn't violate this, it just implies that the operator that turns one into another is not an observable.

 

[PeterDonis]

find it hard to imagine such a mixed state, part electron and part neutrino. Different charge, very different mass.

Sure, lots of things in quantum field theory are difficult to imagine. That doesn't mean they aren't there. The Standard Model of particle physics makes it perfectly clear that such states are possible. We don't observe them because we don't know how to build a detector that would detect them; we only know how to build detectors that detect properties like mass and charge. (This is a "B" level thread so it's really out of scope here to go into detail about why this is; it would need to be at least an "I" level and quite possibly an "A" level discussion.)

Mathematically, the electron-neutrino doublet works exactly like the spin space of a spin-1/2 particle: "electron" and "neutrino" correspond to basis states like, say, "up" and "down" spin about the z-axis. "Mixed electron/neutrino" states correspond to spin states like "up" and "down" about the x-axis or y-axis.

Superselection says there can be no superposition of states with different total charges, but thinking of neutrinos and electrons as different states of the same particle doesn't violate that rule.

That's correct. A "mixed electron/neutrino" state such as I described would not be an eigenstate of the charge operator, so it would not have a well-defined charge; but it would have an expectation value of charge (which in the simple mixed state I described would be $- \frac{1}{2}$).