Brian Cox and the Pauli Exclusion Principle

[Morberticus]

Hi,

I know this is old news at this stage, but I was watching his public lecture on quantum mechanics, and he says the energy levels of all the electrons in the universe shift to adjust when he adds energy to electrons in a diamond.

I understand that he should have used the phrase quantum state rather than energy level, and I understand that the shift is tiny, and effectively imperceptible, but even this does not sit well with me. Entanglement in quantum mechanics has always been about correlation, rather than causation. I don't see how manipulating the quantum states of electrons here could have an instantaneous effect on the quantum states of distant electrons without violating relativity.

 

[Bill_K]

That's the same question that was just asked and answered in this thread.

 

[Jilang]

I am thinking that he meant that to some imperceptible degree they are all correlated. As for spooky action at a distance, that just about sums up Quantum Mechanics.

 

[bhobba]

I answered that one in the thread Bill-K mentioned.

If you want to pursue it further Sean Carroll went into it in detail:
http://www.preposterousuniverse.com/blog/2012/02/23/everything-is-connected/

I have already posted my view. Nothing happens - the energy levels simply slot into the always infinite number of places available - no biggie. But opinions vary.

Thanks
Bill

 

[Ken G]

The problem is that Cox's comment could only apply if he was trying to be so precise that even remotely tiny effects should be viewed as relevant, but if one is being that precise, then there is already no such thing as "an electron in a diamond", for the simple reason that all electrons are indistinguishable particles, so just don't have personal identities. So the disconnect in the sentence by Cox comes from a fight between two different pictures that are mutually exclusive-- the Pauli exclusion principle, and the idea that electrons are individuals. This all stems from an incorrect way to state the principle, that "two electrons cannot be in the same state." States are how we predict the outcomes of experiments that involve electrons, but electrons are not distinguishable, so we should not pretend it makes precise sense to say "electron A is in state X and electron B is in state Y, where X and Y cannot be the same." If that sentence made perfect sense, there could not be a Pauli exclusion principle, because the correct way to say that principle is that the joint wave function of two electrons must acquire a minus sign when you swap the electron coordinates, which in turn implies that the joint wave function cannot involve two identical single-particle states. But notice it is still a joint wave function we are talking about, or we get no PEP!

Of course, Cox is speaking to non-physicists, so there is great latitude in how we try to communicate the special features of quantum mechanics, without going into the details of joint wave functions. Hence if we are really being precise, we should not hold Cox to a standard of precision in the first place! But I do agree with the OP that the main "sin" in Cox's remark is that it swaps in causation where it should really only cite sources of correlation. A joint wave function is a source of correlation, but attributing causation to it is a much stickier issue and really depends on arbitrary interpretations.

 

[Jilang]

I think your idea of indistinguishable is slightly wrong. If they are in a different place the are mostly distinguishable!

That said, the spatial overlap of the wave function, no matter how small, has to to be taken into account. It may be a blatant over-dramatisation, but hey that's show biz and Brian is doing a fabulous job in the U.K getting the youngsters interested in science; he's on prime-time TV a lot. All the best to him.

 

[Ken G]

I think your idea of indistinguishable is slightly wrong. If they are in a different place the are mostly distinguishable!

They are distinguishable in practice, sure, but Brian Cox is talking about imperceptible formal differences. If we are being perfectly precise, then we can never say two electrons "are in different places", because there is always some (incredibly tiny) probability we have lost track of which electron we are talking about, due to their indistinguishability. This manifests in ways that are related to the idea that entangled systems can present (very tiny) correlations even over vast distances. Entanglements can dwindle rapidly, and get to a point where we can ignore them and treat the electrons as though they were distinguishable, but it is never formally correct to do that. So it's the usual distinction between what quantum mechanics says as a formal theory, and the ways we really use it in practice (which involve idealizing the formal theory).

That said, the spatial overlap of the wave function, no matter how small, has to to be taken into account. It may be a blatant over-dramatisation, but hey that's show biz and Brian is doing a fabulous job in the U.K getting the youngsters interested in science; he's on prime-time TV a lot. All the best to him.

Exactly, the wave function is formally a global thing, ever since the Big Bang. So all electrons are entangled, in some formal sense, which also means that none of them are distinguishable or individual. I think that is what Brian Cox is referring to, though if he is taking that stance, he should avoid language that he is doing anything to a particular electron.

So he should not have said "Just gently warming it up, and put a bit of energy into it, so I’m shifting the electrons around. Some of the electrons are jumping into different energy levels." That language is not consistent with the rest of the point he is making, because it acts as though we have particular electrons within the diamond that we are doing something to. But what we are actually doing, if you take the perspective of quantum mechanics, is watching the wave function of the universe evolve in time, and any change in the outcome of observations on the diamond that involve electron behavior is entangled with the rest of the universe. If you say it that way, it's just a restatement of "spooky action at a distance," which is not controversial. It only sounds even weirder than that, and incorrectly so, if you say that doing something to some particular set of electrons is having an effect on some other particular set of electrons. It is the joint wave function that is involved, not particular electrons, or else you don't have a PEP.

 

[vanhees71]

I think, the following blog puts this nonsense right:

http://blogs.scienceforums.net/swansont/archives/11081

which is quoted by Carrol (see bhobba's posting) in his also very clear statement, but I think one should clearly say that something is wrong when it is wrong. To popularize science is a very difficult issue and thus to some extent I can excuse if something is not presented accurately, but I don't think that popularization serves the purpose of science if it mystifies issues. Science is the opposite of mystifying things, and the public who is paying for a lot of very expensive science projects like the LHC has the right to be informed correctly about what comes out of this endeavor and not being mystified. If they want to have esoterics they can get it everywhere, but if somebody watches a feature about science on TV, he or she expects to be informed about science and not esoterics. Such nonsense claims do more harm than good for science and it's funding with tax payer's money!

I don't know Brian Cox (I'm located in Germany), but such esoterics is quite common in popularizing science for the public. Once I've seen a TV feature in the German TV about the LHC, which only addressed bogus stuff like the creation of black holes that destroy the Earth etc. instead of explaining the really exciting science which is really going on there and in other labs to figure out the fundamental building blocks of our universe. Science popularization doesn't need esoterics to make an exciting narrative about the scientific endeavor. It's fascinating enough by itself to make an exciting story in TV, public lectures, popular science books, etc. There are really very good popular-science things out there, e.g., what I've seen when I was in the US in features like NOVA was excellent. There's also a German-French TV channel (ARTE) available here in Germany that produces very good science features, including very good documentaries about particle physics, cosmology, dark matter, dark energy, Bose-Einstein condensates, etc. So it is in fact possible to popularize these complicated issues without getting them totally wrong!

 

[Maui]

This blog post says

The second is that the Pauli Exclusion Principle doesn’t work this way. It applies to a single system in which you have all these identical electrons, and they can’t be in the same exact state. This is because of their QM behavior if you were to exchange them — something has to be different about the two electrons. In a crystal, the energies are slightly different as a result, and you get a band of energies. But this does not extend beyond the system, be it crystal or even individual atoms — the electrons belong to different systems, which are not co-located. Exchanging electrons meaning exchanging systems as well. That’s what’s different.

How do we know that 'separate' systems are not entangled at the quantum level(ever since the Big Bang). Or is he taking a semi-classical look in expressing his opinion suggesting electrons are little balls?

I probably shouldn't be saying this, but in general, the quantum physicists are a rather confused lot(due to the nature of the quantum world) so it's expected that noise will be generated when popularizing stuff they themselves don't understand. I am more worried for the ones who believe they truly understand it all well enough(now i am going into hiding).

 

[Morberticus]

How do we know that 'separate' systems are not entangled at the quantum level(ever since the Big Bang). Or is he taking a semi-classical look in expressing his opinion suggesting electrons are little balls?

I probably shouldn't be saying this, but in general, the quantum physicists are a rather confused lot(due to the nature of the quantum world) so it's expected that noise will be generated when popularizing stuff they themselves don't understand. I am more worried for the ones who believe they truly understand it all well enough(now i am going into hiding).

Separate systems would have a weak but permanent entanglement. This entanglement is "assumed" in standard quantum mechanics but it can be derived from the algebra of quantum field theory.

I think Ken G's post makes a lot of sense.

 

[Morberticus]

Also, quantum fields separated by spacelike intervals evolve independently (they don't commute), so there's now way interacting with the fields on Earth could affect the fields at the edge of the universe.

 

[Jilang]

I think, the following blog puts this nonsense right:

http://blogs.scienceforums.net/swansont/archives/11081

Sorry vanhees, whoever wrote this load of rubbish has no understanding of QM.

 

[bhobba]

How do we know that 'separate' systems are not entangled at the quantum level(ever since the Big Bang). Or is he taking a semi-classical look in expressing his opinion suggesting electrons are little balls?

Mate this is complicated stuff - people can and do make 'errors' all the time - I certainly do.

Regarding this issue I think you need to go right back to basics and look at the Pauli exclusion principle. In fact it doesn't say electrons can't be in the same state. It says when electrons are interchanged then the wavefunction changes sign. This means for composite systems if they are in the same state the wavefuiction cancels ie is zero - which is not possible:
http://www.physics.ohio-state.edu/~eric/teaching_files/writing.course/sample3.shortdraft.pdf

This applies to any electrons anywhere.

But now for some caveats. If they are bound in different atoms the particles they are bound with are also part of the composite systems state. Then we have other particles they are entangled with. This gives a lot more freedom for those electrons wavefunctions to not cancel on exchange of electrons.

Thanks
Bill

 

[bhobba]

Sorry vanhees, whoever wrote this load of rubbish has no understanding of QM.

Errrrr. Cant see how you arrive at that.

It looks spot on to me eg:
'Well, no. The issue isn’t the Pauli Exlusion Principle itself — that’s sound science. It’s what he’s done with it. The first, obvious problem is that relativity tells us that the communication can’t be instantaneous. The second is that the Pauli Exclusion Principle doesn’t work this way. It applies to a single system in which you have all these identical electrons, and they can’t be in the same exact state. This is because of their QM behavior if you were to exchange them — something has to be different about the two electrons. In a crystal, the energies are slightly different as a result, and you get a band of energies. But this does not extend beyond the system, be it crystal or even individual atoms — the electrons belong to different systems, which are not co-located. Exchanging electrons meaning exchanging systems as well. That’s what’s different.'

That's exactly the point I made. You have to go right back to what the Pauli exclusion principle says. Its a statement about electron exchange in composite systems. For two electrons its easy to see the electrons can't be in the same state because its state, in order to obey the exclusion principle is UaUb - UbUa. If Ua and Ub are the same then the wavefunction cancels so they can't be in the same state. For electrons in different atoms those atoms and electrons form the composite systems and there is a lot more freedom in preventing wavefunction cancellation on exchange ie the system state is a lot more complicated than UaUb - UbUa.

Thanks
Bill

 

[Ken G]

The problem there seems to come down to the fact that Cox appears to equate a "state of identically equal energy" with "the identically same state", and I'm not sure why he does that. Had he just said that all electrons are indistinguishable, so there's always a tiny chance that if I think I'm doing something to an electron in a diamond, I'm actually doing it to an electron halfway across the universe, he would have been on firmer footing. Like entanglement, electron exchange is not constrained by the rules of relativity. But like the original poster said, the way causation works always ends up still being constrained by relativity, even though the correlations mediated by a joint wave function, and its implied entanglements and exchanges, are not. So it just seems like Cox is not distinguishing the kinds of things that should be limited by the speed of light, with those that need not be. Still, he is hardly the first to do that-- people who popularize entanglement often make little effort to make those kinds of distinctions, so doing the same with the PEP is not that shocking. I don't really like failing to make that distinction in either the entanglement or PEP contexts, but we should at least be fair about how we hand out our passes!

 

[Jilang]

Errrrr. Cant see how you arrive at that.

Two ways:
The part about the constraint to the speed of light of communication seemed totally irrelevant to the subject matter and the fact that entanglement wasn't mentioned once.

Yes there was real science there, but short on QM.

 

[bhobba]

The part about the constraint to the speed of light of communication seemed totally irrelevant to the subject matter and the fact that entanglement wasn't mentioned once.

Its not irrelevant.

Obviously relativity prevents, as Brian stated, far away electrons instantaneously changing. That situation requires QFT - not QM - which is a whole new ball game.

And yes Brian did not mention entanglement, which a close analysis shows is actually very important to understanding what's going on.

The behavior of two electrons not entangled with anything, bound (which is really a form of entanglement) etc is contained by the fact the wave-function changes sign under electron exchange - that's why they can't be the same state. But when entangled its not that simple - not by a long shot eg the state depends on what its entangled with - so what state can't it be the same as?

Thanks
Bill

 

[Jilang]

I think we might be talking about different things. I was referring to the blog in post #12 by swansont.

With regard to the entanglement isn't every electron entangled with everything other one? If so wouldn't you expect a change in the Hamiltonian of the system to affect every one somewhat?

 

[bhobba]

I think we might be talking about different things. I was referring to the blog in post #12 by swansont.

I suspect we are.

With regard to the entanglement isn't every electron entangled with everything other one? If so wouldn't you expect a change in the Hamiltonian of the system to affect every one somewhat?

Of course not.

For example, even though for simplicity its not analysed that way, but rather as simply being in a potential well, electrons in an atom are entangled with the nucleus. Its easy to see this because, if you move the nucleus then the electrons go along with it ie the electrons state depends on the state of the nucleus which is the definition of entanglement. This means the electrons bound in one atom are distinguishable from the electrons in another atom. Only the electrons in that atom are affected by changes in that atoms nucleus. This breaks the fundamental indistinguishably for the exclusion principle to hold.

Thanks
Bill

 

[Bill_K]

With regard to the entanglement isn't every electron entangled with everything other one? If so wouldn't you expect a change in the Hamiltonian of the system to affect every one somewhat?

No. It's called the Cluster Decomposition Principle.

It is one of the fundamental principles of physics (indeed, of all science) that experiments that are sufficiently separated in space have unrelated results. If this principle were not valid, then we could never make any predictions about any experiment without knowing everything about the universe.

 

[Jilang]

Thanks Bills. This is a very interesting topic!
I have learned that in his foundation of QFT Weinberg derived micro causality and locality of the Hamiltonian from his cluster decomposition principle. This is a phenomenological constraint to the S-Matrix which requires that distant experiments give uncorrelated results. However Professor Zeh would contend that this principle cannot form a fundamental element of quantum theory since observable correlations may exist between distant systems and the concept of an S-Matrix can only be applied to sufficiently isolated microscopic systems. Macroscopic systems never cease to interact uncontrollably with their environment.
http://www.rzuser.uni-heidelberg.de/~as3/nonlocality.html
Is this now accepted as the source of decoherence or is it still to be agreed on?

 

[bhobba]

Is this now accepted as the source of decoherence or is it still to be agreed on?

I don't know what you mean by this.

Decoherence really has nothing to do with this issue except in a indirect way, that, since its a form of entanglement, you have the issue of how it is even possible to be in the same state.

Thanks
Bill

 

[vanhees71]

I forgot to write that indeed, according to the very foundations of local relativistic quantum field theory, which is, today, the most precise description about the behavior of matter (at least of the so far discovered part of the matter, i.e., the one that's composed of quarks, leptons, and gauge bosons of the strong and electro-weak interactions as well as the Higgs field) contain the "linked cluster principle" as an input. As Bill_K has written, this is very nicely explained in Weinberg's Quantum Theory of Fields, and it underlines that within local relativistic QFT there is NO spooky action at a distance. Of course, as any QT also local relativistic QFT fulfills the superposition principle and also the description of entanglement. Entanglement, however, describes correlations of quantum systems that can be far from our classical experience, but it's still correlations and no spooky interactions at a distance. If you heat up a diamond, you agitate the electrons somewhat (although the electrons take practically no part on the specific heat due to the Fermi statistics, but that's not the point here) and change the occupation of their states somewhat, but that doesn't instantly affect electrons far away on another galaxy only because all electrons are in principle entangled due to the Pauli principle (i.e., fermionic Fock space as spanned by the antisymmetrized single-particle product states).

The science blogger I quoted gave also a very convincing example with the Cs atoms defining our standard of time measurements (i.e., the unit second in the SI). The energy levels of the electrons in the Cs atom are not affected by what's happening to the electrons at some distant place or even nearby!

 

[Jilang]

I don't know what you mean by this.
Bill

I am referring to the entanglement of a microscopic system with the macroscopic environment that decoheres the entanglement microscopic system; the entanglement spreading out into the wider environment and making its effects less noticeable.

 

[DrewD]

I can't find his site right now, but a while back Prof. Cox posted here and linked to a page for a class that he (or a colleague) teaches for Undergrads. On the page they discuss two finite wells separated by small distance. Particles in these two wells clearly interact and the Pauli Exclusion Principle applies. The argument on the pages goes along the lines of "if these two wells were separated farther there would still be a slight effect of the energy levels from the other well. If the universe started causally connected and can be described by a universal wave function, then there is a slight (usually immeasurable) change whenever an electron changes energy levels."

This argument convinced me that his statement is not completely false. There are too many "ifs" for the statement to be really a safe scientific statement, but I was convinced that he could be correct. I don't know much about relativistic QM, so I could be wrong and perhaps there is no way that what he said is correct, but either way, I think it is inappropriate to speak like that to people that don't understand physics. Even if he is correct, the effect will have to be so small that it really isn't interesting beyond a purely theoretical discussion (that the audience certainly would not grasp). I think it is safe to say that Prof. Cox should not have said what he said even if, in some manner, it is correct.

 

[WannabeNewton]

Well there is a huge difference between the finite well model and an atomic model. Fully specifying a unique eigenstate of a particle in a finite well can be done using only energy eigenstates of the usual piece-wise Hamiltonian (free outside, constant potential inside along with the finite well boundary conditions) combined with a component of spin-a better example would probably be the quantum harmonic oscillator. But for an atomic model like Hydrogen one needs simultaneous eigenstates of commuting observables such as the Coulomb interaction Hamiltonian, the total orbital angular momentum, a component of the orbital angular momentum, and a component of spin so just focusing on two different energies doesn't tell you the full picture about the states. Furthermore if we're considering two independent atomic systems as subsystems of a two-component system then the substates will be independent of one another and their tensor products will form the joint state vector of the two-component system to which Pauli's exclusion principle applies.

 

[Ken G]

I think one problem is that it isn't clear if Cox is referring to tiny entanglements, or tiny overlaps in the single-particle state functions. Either of those things are relevant to the PEP, but in quite different ways. If it is overlap he is talking about, then the need for the joint wave function to be antisymmetric induces a shift in the energy because of the charge interactions. But those would not be instantaneous interactions, so he must not be talking about that. He must therefore be talking about entanglements, but as the OPer pointed out, those attributes should be described as constraints on correlations compared after the fact, not as "influences" that appear instantaneously.

More generally, interpreting what is happening in terms of what is happening to one electron having an effect on the rest is always going to lead to severe misconceptions, because the entire idea that you have something happening to "one electron" relies on the Hartree-Fock approximation that we can treat the joint wave function as a Slater determinant of single-electron wavefunctions. That approximation certainly breaks down in the situation Cox describes, so it sounds like his motivation for speaking in terms of instantaneous shifts in energy stems from an essentially incorrect treatment of the global wavefunction. We should probably point out that we really don't even know how such global wavefunction effects work in practice, because their effects over scales like half the universe have never been tested, and by now we should be skeptical of any physics predictions extrapolated into regimes way beyond any we have ever actually looked at!

ETA: In other words, it still bothers me that people are saying Cox is wrong because distant electrons don't interact, because that language suggests that "distant electrons" are individual things. If we are to be completely precise and include even the most incredibly tiny and unmeasurable issues that are predicted to exist by quantum mechanics, then we simply cannot talk about "different electrons", period, there just isn't any such thing. There are different observers, and different apparatuses, because they are distinguishable. This means the different coordinates we attach to experimental outcomes that we connect with electrons refer to these different observers and apparatuses-- not different electrons! So we should not even say that moving a nucleus moves an electron, we should say it moves the coordinates of electrons that we will use for predicting experiments involving electrons, but we never get to know which electrons we are experimenting on. I realize this is a very nitpicky objection, but so is the whole issue of a global wavefunction. My point is that the two come together-- any situation where we choose to talk about a global wavefunction of all the electrons in the universe is also a situation where we should not refer to individual electrons at all, we never know which electron we are measuring, we only know which apparatus we are using for the measurement.

 

[Bill_K]

Yes. What Ken G said. :thumbs: Except for the last paragraph.

 

[Ken G]

Sorry for the ETA!

 

[DrewD]

I think one problem is that it isn't clear if Cox is referring to tiny entanglements, or tiny overlaps in the single-particle state functions. Either of those things are relevant to the PEP, but in quite different ways. If it is overlap he is talking about, then the need for the joint wave function to be antisymmetric induces a shift in the energy because of the charge interactions. But those would not be instantaneous interactions, so he must not be talking about that. He must therefore be talking about entanglements, but as the OPer pointed out, those attributes should be described as constraints on correlations compared after the fact, not as "influences" that appear instantaneously.

FOUND IT! I haven't read through his (or somebody pretending to be him) explanation in a while, but my recollection is that he was talking about overlaps of the wavefunction. There is a link in post #15.

Wannabe I'm confused by this claim

Furthermore if we're considering two independent atomic systems as subsystems of a two-component system then the substates will be independent of one another and their tensor products will form the joint state vector of the two-component system to which Pauli's exclusion principle applies.

Certainly two atoms that are very close to each other can be in an inseparable state? Am I misunderstanding what you said here? Either way, I don't agree with Cox's interpretation.

 

[WannabeNewton]

Certainly two atoms that are very close to each other can be in an inseparable state?

Yes certainly, I should have said superpositions thereof. Regardless, the antisymmetry still applies to the total state.

 

[Ken G]

Actually, in my opinion, my last paragraph was the most rigorously correct thing I've said in the whole thread! If someone feels that is not true, I would love to know why. For example, I would like to know of a situation where we have to talk about the global wave function of the electrons involved, yet we can talk about what individual electrons are doing, rather than talk about what individual apparatuses are measuring electrons to be doing. The important distinction is that the apparatuses are distinguishable as independent or individual things, but the electrons are not, they just show up according to their total wave function and nothing more can be said. Is that not one of the most important things we know about electrons, the whole reason we have a PEP?

I think with Fermions, this point can actually get a bit muddled because we have this common idea that "two electrons cannot be in the same state" as the explanation of the PEP. But if you look at bosons, in a Bose-Einstein condensate, the situation is actually more clear. There, you might think we'd have nothing useful to say, because two bosons are allowed in the same state, so if being allowed, or not being allowed, to be in the same state was all that was going on here, then bosons would not do anything interesting. But they do, they exhibit weird behavior that can only be explained if the way you count their states requires that we not count them as if they were individual entities that each had their own state, that just gets the answer wrong. So the same is true of fermions, we cannot treat them as individual particles each with their own state, it's just less obvious we can't do that because the PEP makes us think, incorrectly, that we are getting away with doing just that.

 

[atyy]

I think one problem is that it isn't clear if Cox is referring to tiny entanglements, or tiny overlaps in the single-particle state functions. Either of those things are relevant to the PEP, but in quite different ways. If it is overlap he is talking about, then the need for the joint wave function to be antisymmetric induces a shift in the energy because of the charge interactions. But those would not be instantaneous interactions, so he must not be talking about that. He must therefore be talking about entanglements, but as the OPer pointed out, those attributes should be described as constraints on correlations compared after the fact, not as "influences" that appear instantaneously.

Why doesn't the wave function anti-symmetrization occur instantly? Naively, I'm thinking the wave function must always be antisymmetrized, so if one electron shifts, the entire wave function must immediately shift so that it remains antisymmetrized.

 

[Jano L.]

Actually, in my opinion, my last paragraph was the most rigorously correct thing I've said in the whole thread! If someone feels that is not true, I would love to know why. For example, I would like to know of a situation where we have to talk about the global wave function of the electrons involved, yet we can talk about what individual electrons are doing, rather than talk about what individual apparatuses are measuring electrons to be doing.

Theory of electronic states of atoms. One can talk about individual electrons. There is definite natural number $N$of them in each atom of any common element. They are different particles since they all contribute to mass and charge of the electronic cover of the atom and one can assume that they have definite positions without much difficulty (although apparently, this was not necessary to get many useful results from the model). There are no measuring apparatuses measuring their positions (or any other property of theirs) involved. Schroedinger's equation is just a mathematical model and $\psi(\mathbf r_1, \mathbf r_2, ...,\mathbf r_N)$ is an associated mathematical device useful for describing $N$ electrons in atoms and molecules. In the domain of atoms, it is not used as a device for describing what apparatuses will measure on the electrons. Rather the use of $\psi$ is to calculate expected average values of electronic quantities or probabilities of their configuration (momenta), irrespective of any apparatuses.

...So the same is true of fermions, we cannot treat them as individual particles each with their own state, it's just less obvious we can't do that because the PEP makes us think, incorrectly, that we are getting away with doing just that.

I do not see any reason to think that electrons cannot be treated as individual particles. Could you explain why you think that?

 

[vanhees71]

Of course the many-electron wave functions are always superpositions of antisymmetrized products of one-body wave functions. The Hilbert space is just that, i.e., any N-body wave function fulfills
$$\psi(t,\vec{x}_{P(1)},\sigma_{P(1)};\vec{x}_{P(2)},\sigma_{P(2)};\ldots;\vec{x}_{P(N)},\sigma_{P(N)}) = \sigma(P) \psi(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2;\ldots;\vec{x}_N,\sigma_N)$$
for all $P \in S_N$. The wave function is always totally antisymmetric under permutation of electrons. It must be taken antisymmetrized at the initial time and then quantum-theoretical dynamics keeps it in the antisymmetrized state, because the Hamiltonian must commute with all permutation operators. Otherwise electrons weren't indistinguishable from each other.

This means in the dynamics is nothing which must antisymmetrize the wave function at any time step, but that's fulfilled automatically.

Further, in the relativistic realm, we only use local QFTs today, and they are fulfilling the linked-cluster principle, clearly contradicting Cox's statements!