Fermionic Fock space superselected?

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Discussion Overview

The discussion revolves around the concept of superselection in fermionic Fock space, particularly focusing on the implications of half-integer spin and the effects of spatial rotations on fermionic states. Participants explore whether fermionic particle states can be coherently superposed and the implications of such superpositions in the context of quantum field theory (QFT).

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that applying a full 2π rotation to a fermionic Fock state alters its relative phase, suggesting that coherent superpositions of different particle numbers may not be physically realizable.
  • Others point out that the existence of a QFT interaction capable of creating such superpositions from non-superposed states raises questions about the hermiticity of the corresponding Hamiltonian.
  • A participant challenges the idea that a superposition of even and odd particle numbers can be created, citing the need for hermiticity and the implications of the spin-statistics theorem.
  • Another participant introduces a thought experiment involving a box containing a single electron, suggesting that this scenario could lead to a superposition of states, but acknowledges that the argument may not hold under scrutiny.
  • Some participants note that the discussion of superselection may not be commonly addressed in the context of constructing fermionic multi-particle state spaces, indicating a gap in the literature.

Areas of Agreement / Disagreement

There is no consensus on whether fermionic particle states can coherently superpose. While some participants agree on the common acceptance of superselection into odd and even sectors, others raise questions about the implications and feasibility of such superpositions in practical QFT.

Contextual Notes

Participants express uncertainty regarding the implications of the spin-statistics theorem and the role of relativistic invariance in the discussion of superselection. The limitations of the thought experiments presented are acknowledged, particularly in relation to measurement and state collapse.

Who May Find This Useful

This discussion may be of interest to those studying quantum field theory, superselection rules, and the foundations of quantum mechanics, particularly in relation to fermionic systems.

Jazzdude
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If you construct a fermionic Fock state

|psi> = a|psi1> + b|psi2>

where |psi1> is a 1-particle state and |psi2> is a 2-particle state, and you apply a full 2pi spatial rotation to it, then each particle contributes a factor of -1 to the amplitude because of the half-integer spin SO(3) representation it comes with.

R(2pi) |psi> = -a|psi1> + (-1)(-1)b|psi2> = -a|psi1> + b|psi2>

Now this state is not equivalent to the one we started with because the relative phase has changed. But because the 2pi rotation is an exact symmetry for the physical state the usual argument is that this coherent superposition should not be possible. That's the argument used in fermion-boson (univalent) superselection. The usual solution is to say that the different spin states are in different superselection sectors, and there are no physically constructible observables that can determine the relative phase, so that we can as well mix them incoherently.

So is it commonly accepted that the fermionic particle space separates into two superselection sectors of odd and even particle numbers? If not, why not?

In case I've made a silly mistake in my argument please forgive me. I didn't get a lot of sleep recently.

Thanks,
Jazz
 
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Jazzdude said:
So is it commonly accepted that the fermionic particle space separates into two superselection sectors of odd and even particle numbers?
Yes, it's common.

But I like to think of it in a different, perhaps more fundamental way. Is there a QFT interaction which can create such a superposition from an initial state which is not in such a superposition? If you have some experience with practical QFT, you can find out that interaction Hamiltonian leading to such a production would not be represented by a hermitian operator, while we need hermiticity for the sake of unitarity (probability conservation).
 
Demystifier said:
Yes, it's common.

But I like to think of it in a different, perhaps more fundamental way. Is there a QFT interaction which can create such a superposition from an initial state which is not in such a superposition? If you have some experience with practical QFT, you can find out that interaction Hamiltonian leading to such a production would not be represented by a hermitian operator, while we need hermiticity for the sake of unitarity (probability conservation).
I fail to see this. E.g. [itex]1+\psi(x)+\psi^+(x)[/itex] would create a superposition of even and odd particle number states out of a state with only even or odd particle number. However, this operator, while being hermitian, is not invariant under a 360 degree rotation.
 
DrDu said:
I fail to see this. E.g. [itex]1+\psi(x)+\psi^+(x)[/itex] would create a superposition of even and odd particle number states out of a state with only even or odd particle number. However, this operator, while being hermitian, is not invariant under a 360 degree rotation.
Hm, but psi must have a spinor index (due to the spin-statistics theorem), while a Lagrangian should not have one. So you must contract it with something having another spinor index, which, I think, should contain another fermion field. But then you cannot create a single fermion, but only a pair of them.

But then again, the spin-statistics theorem, as well as the claim that Lagrangian should not have a spinor index, rest on the assumption of relativistic invariance. On the other hand, in recent years it is quite popular to study field theories which break relativistic invariance. So after all, maybe you right. Maybe superpositions of even and odd numbers of fermions are not impossible.
 
Last edited:
Demystifier said:
Hm, but psi must have a spinor index (due to the spin-statistics theorem), while a Lagrangian should not have one.

That the lagrangian should not be spinorial is a good argument. However, the spin statistics theorem is not important here, as Jazzdudes argument will hold for any particle of half integer spin.

Consider the following situation: Take a long box containing exactly one electron at low temperature so that only the symmetric ground state will be populated. Now insert a wall in the middle of the box. Each box will then contain a superposition of 0 and 1 electron if such a superposition were possible at all. Now rotate one of the boxes by 360 degrees and recombine the two halves: Tataa, the electron has disappeared!
 
DrDu said:
Consider the following situation: Take a long box containing exactly one electron at low temperature so that only the symmetric ground state will be populated. Now insert a wall in the middle of the box.
So far so good.

DrDu said:
Each box will then contain a superposition of 0 and 1 electron ...
That's wrong. Your trick with boxes is not different from a beam-splitter experiment with photons. In essence, there is no such thing as "state in the box". There is only a total state, which in this case has a form
|1>=|1L>+|1R>
which is a superposition of particle in the Left box and particle in the Right box. If you perform a measurement which determines in which box the particle is, the state will "collapse" to either |1L> or |1R>. But the vacuum state |0> plays no role here.
 
Demystifier said:
There is only a total state, which in this case has a form
|1>=|1L>+|1R>
which is a superposition of particle in the Left box and particle in the Right box. If you perform a measurement which determines in which box the particle is, the state will "collapse" to either |1L> or |1R>. But the vacuum state |0> plays no role here.

After writing the previous post I had gnarling doubts that this thought experiment is really due to my ingenuity and in fact I saw that it is discussed in the article by Wightman cited in a previous post and also I realized that the electron would not disappear. Rather, the rotation should transform a symmetric state into an antisymmetric state what has been observed experimentally.
The discussion of Wightman I find somewhat dissatisfying:
"As far as the relevance of these experiments to the
univalence superselection rule is concerned, the standard response is that the
experiments are beautiful, but they are not the ones involved in the univalence
superselection rule where you must rotate the entire isolated system. "
 
Thanks for the clarification! I know about univalent superselection, but I was just surprised that I've never seen any mentioning of superselection when constructing the fermionic multi particle state space. But I'm more a foundations of QT guy than a field theorist. So thanks for confirming my thoughts!

Cheers,

Jazz
 

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