Why Pauli's Exclusion Principle?

[dustball]

The Fermi exclusion principle can be explained by saying that electrons are
spinorial objects, i.e. the wave function of an electron is multiplied by -1 when
the observer (or an electron itself) undergoes a full (2pi) rotation. Now take a
belt (or a paper ribbon) by its ends with 2 hands. When you interchange the
positions if the hands, you will get a kink on the belt. Therefore, if you have
a pair of electrons (one for each hand) and you interchange them, they undergo
a full rotation with respect to each other, and their collective wave function
must be multiplied by -1. Therefore the wave functions of the electrons must
be antisymmetric. This explanation is called "the Feynman belt trick," he came up
with it in the mid-80s. The standard explanation uses relativistic quantum field
theory and is much more technical. There is still quite a bit of a controversy
about the spin-statistics connection, and the papers are still written about it.
From a practical point of view the issue is well settled since the experimental
support for the exclusion principle is overwhelming. Historically the idea that
electrons are spinorial came from the observation of the splitting of the spectral
lines in the magnetic field, and each line splits into 2, so the electron must
be described by a 2-dimensional irreducible projective representation of the
rotation (or Lorentz) group. The representations of these groups are all known,
and the spinoriality of the electron is forced upon us as soon as we accept the
basic dogmas of quantum mechanics. It is amusing that for a long time people
thought that there is no classical analog to spin, but it turned out that there is.
If we quantize the classical system the phase space of which is a unit 2-d sphere,
we will get exactly a non-relativistic particle of spin 1/2 sitting at the origin.
Take a look at http://gregegan.customer.netspace.net.au/APPLETS/21/21.html
for a nice demonstrration of the Dirac belt trick that is closely related to
Feynman's. The projective representations pop up because the rotation and
Lorentz groups are not simply-connected, i.e. there are some loops in these groups
that can not be continuously collapsed to a point, so on the deepest mathematical
level the reason for Fermi exclusion principle is topological in nature.
This makes all the topologists feel warm and fuzzy.
By the way, Roger Penrose in his new book "The Road to Reality" gives
nice explanations of spinorial objects.

I hope it helps.

 

[ZapperZ]

ZapperZ

I am tired of this silly game where you consistently misrepresent whatever I write in the worst possible way you can find and jump to the most aggressive but inaccuratel conclusions. Even when I correct this you continue to repeat the same misrepresentations which you justify from selected partial readings of what I write. I presume this gives you some sort of perverted pleasure. I would have preferred a respectful dialogue, but I guess it's not to be found with you.

In the meantime, I suggest you get up to date on the origins of the spin-statistics connection and the Pauli rule if you intend to contribute to a thread about it.

You can correct this VERY easily by simply by showing what I requested - a many-body BCS equivalent, something that you CLAIM support your theory. This, you have continually refused to do.

Zz.

 

[mikeyork]

Now Mike York's solution has elements of interpretation incorporated, it's not really only 'plain QM' (for ex. the assumption of completeness is clearly part of the interpretation)

You are correct that I have replaced the symmetrization postulate by an alternative axiomatic approach -- one which requires a method for choosing unique state vectors (the reason for defining "completeness" of state descriptions) and which makes the simple recognition that particle permutation is not a physical transformation.

but it is too sketchy at this point to count as a causal deduction of Pauli's principle. Even if he would eventually provide one his solution should prove to be at least theoretically progressive, being capable to accommodate more facts than its alternatives but in a sizeable way.

Even if this doesn't pan out, it should be understood that my approach is far more intuitive -- even, I would argue, necessary, since you can't calculate QM interference effects reliably without uniqueness and the permutation axiom is simply a statement of the obvious -- than the ad hoc symmetrization postulate which just comes out of nowhere and should really be called the unsymmetrization postulate. On top of which, the symmetrization postulate is actually insufficient to do the job unless properly qualified by the condition that spin frames are related in a particular order-dependent way. This alone is good reason for preferring my approach.

 

[mikeyork]

Now take a belt (or a paper ribbon) by its ends with 2 hands. When you interchange the positions if the hands, you will get a kink on the belt. Therefore, if you have a pair of electrons (one for each hand) and you interchange them, they undergo a full rotation with respect to each other, and their collective wave function must be multiplied by -1. Therefore the wave functions of the electrons must be antisymmetric. This explanation is called "the Feynman belt trick," he came up with it in the mid-80s.

The "belt" explanation is simply a topological analogy to the proof I gave in my 1975 pre-print and Broyles in his 1976 publication. What you haven't explained however, is why the belt should be there at all. To continue the topological analogy, the explanation is that the belt represents an implicit order dependence that connects the two state descriptions (i.e. one in terms of the other). If you use order-independent state descriptions (for which wave functions can be chosen symmetric under exchange) there is no such connection and therefore, by analogy, no belt and no kink. However, there are limits to this analogy (as with all analogies) and it is no substitute for the rigorous proof.

 

[dustball]

The "belt" explanation is simply a topological analogy to the proof I gave in my 1975 pre-print and Broyles in his 1976 publication. What you haven't explained however, is why the belt should be there at all. To continue the topological analogy, the explanation is that the belt represents an implicit order dependence that connects the two state descriptions (i.e. one in terms of the other). If you use order-independent state descriptions (for which wave functions can be chosen symmetric under exchange) there is no such connection and therefore, by analogy, no belt and no kink. However, there are limits to this analogy (as with all analogies) and it is no substitute for the rigorous proof.

Sorry, I didn't know about your preprint or Broyles paper.
To me "the true reason," for the exclusion principle is the fact that the 2-dimensional
representation of the 3-dimensional rotation group is 2-valued projective.
The dimensionality of the representation is an experimental fact (the spectral lines
split in 2 in the magnetic field). Being spinorial, electrons have belts attached to them,
that's the whole point. In any case, the purpose of my posting was to describe the
intuitive ideas behind the exclusion principle, not to present a "rigorous" proof.
I very much doubt that the student who originally asked the question would benefit
from such a proof, I'm not even sure if (s)he could make any sense of our discussion,
most likely (s)he is not following it any more.

 

[mikeyork]

To me "the true reason," for the exclusion principle is the fact that the 2-dimensional representation of the 3-dimensional rotation group is 2-valued projective. The dimensionality of the representation is an experimental fact (the spectral lines split in 2 in the magnetic field). Being spinorial, electrons have belts attached to them, that's the whole point.

Then how do you account for the boson exclusion rules (the observable property that people intend by wavefunction symmetrization)? Or the fact that both fermion and boson exclusion rules can be summed up in a common generalized rule regarding allowed composite spin?

The two-valuedness is the reason that fermion wavefunctions change sign under 2pi rotations. It is not the reason for the generalized exclusion rule, although it has a role in explaining why the Pauli rule (a special case of the generalized rule) applies to fermions only.

 

[dustball]

Then how do you account for the boson exclusion rules (the observable property that people intend by wavefunction symmetrization)? Or the fact that both fermion and boson exclusion rules can be summed up in a common generalized rule regarding allowed composite spin?

The two-valuedness is the reason that fermion wavefunctions change sign under 2pi rotations. It is not the reason for the generalized exclusion rule, although it has a role in explaining why the Pauli rule (a special case of the generalized rule) applies to fermions only.

Well, I guess the odd-half-spin objects are spinorial, they correspond to the 2-valued
representations of the rotation group, the whole-spin objects are tensorial, they
correspond to the true representations of the rotation group,
and the identical objects are indistinguishable, the rest of it is just an icing on the cake.
To tell you the truth, I don't consider this subject particularly stimulating, it's too
ideological to my taste, sorry about it. Although I have to admit that I have never seen
an explanation with all the physical and mathematical assumptions stated explicitly.
Have you?

 

[mikeyork]

To tell you the truth, I don't consider this subject particularly stimulating, it's too
ideological to my taste, sorry about it. Although I have to admit that I have never seen
an explanation with all the physical and mathematical assumptions stated explicitly.
Have you?

Yes, I authored it:

http://xxx.lanl.gov/abs/quant-ph/0006101

(There is a question mark over the interpretation of "full anti-symmetrization" in the case of three or more particles and, if I have time, I will write up an amendment to clarify it; but the rest of the paper provides exactly what you say you have never seen, by identifying critical features necessary to QM, for reasons that have nothing to do with spin or even identical particles, and showing that they inexorably lead to the spin-statistics connection via a generalized exclusion rule.)

 

[dustball]

There are some other nice articles on the subject in arxiv.org. I especially liked
the ones by Robert Oeckl and by Bernd Kuckert. It looks like a lot of confusion is
created by forgetting that the individual particles don't exist, the n-particle
states live on the n-configuration space, not on the direct product of n coppies
of the space where one particle is located.

 

[mikeyork]

There are some other nice articles on the subject in arxiv.org. I especially liked
the ones by Robert Oeckl and by Bernd Kuckert.

Can you give the link? I searched on the names you mention to no avail.

 

[dustball]

http://www.arxiv.org/abs/hep-th/0008072
http://www.arxiv.org/abs/quant-ph/0208151
Type in spin statistics into the title window for a lot more, search both physics and math, enjoy.

 

[mikeyork]

... I was the one who pointed out several examples in condensed matter in which, not only is there no such thing as being "unresolved", but CLEAR antisymmetrization were used to describe those phenomena! (Mott insulator, BCS ground state, let's include He3 superfluidity in it too). And these ARE NOT just "two particle" systems, but a many-particle system where you claim that "conventional theory" and your theory should deviate. Yet, the conventional theory works VERY well here. On the other hand, there's nothing from your end to make the same level of success.

I have at last found some time to look more deeply into your assertions and I can report that you are incorrect to say that the phenomena you mention are described by "CLEAR antisymmetrization". On the contrary, the theory is actually permutation symmetric -- as I claim it should be. It only appears to be anti-symmetric because anti-commuting operators are assumed. But this ignores the fact (shown in my paper) that this assumption implies an anti-symmetric choice of spin quantization frames. When this is taken into account, it cancels the anti-symmetry from the anti-commutation, leaving the Hamiltonian symmetric overall.

The reason this is obscured is because it is implicitly assumed that one can choose a common spin quantization frame for both particles in each pair in a symmetric way. But as my paper shows, this cannot be done. Indeed the asymmetry which arises when choosing a common frame must be explcitily accounted for or, if the state vectors are to be unique, it results in the usual spurious anti-symmetry in the case of half-integer spin particles due to an implicitly anti-symmetric choice of spin quantization frames -- even when both frames have common axes.

The problem with the cases you cite appears to be simply that half-integer anti-symmetrization continues to be assumed, incorrectly, to be equivalent to the observed patterns regarding allowed and excluded states of particle pairs, whereas in reality this antisymmetrization is illusory since it is mutually cancelling with the reference frame antisymmetry it depends on (when this is not ignored).

 

[ZapperZ]

I have at last found some time to look more deeply into your assertions and I can report that you are incorrect to say that the phenomena you mention are described by "CLEAR antisymmetrization". On the contrary, the theory is actually permutation symmetric -- as I claim it should be. It only appears to be anti-symmetric because anti-commuting operators are assumed. But this ignores the fact (shown in my paper) that this assumption implies an anti-symmetric choice of spin quantization frames. When this is taken into account, it cancels the anti-symmetry from the anti-commutation, leaving the Hamiltonian symmetric overall.

The reason this is obscured is because it is implicitly assumed that one can choose a common spin quantization frame for both particles in each pair in a symmetric way. But as my paper shows, this cannot be done. Indeed the asymmetry which arises when choosing a common frame must be explcitily accounted for or, if the state vectors are to be unique, it results in the usual spurious anti-symmetry in the case of half-integer spin particles due to an implicitly anti-symmetric choice of spin quantization frames -- even when both frames have common axes.

The problem with the cases you cite appears to be simply that half-integer anti-symmetrization continues to be assumed, incorrectly, to be equivalent to the observed patterns regarding allowed and excluded states of particle pairs, whereas in reality this antisymmetrization is illusory since it is mutually cancelling with the reference frame antisymmetry it depends on (when this is not ignored).

Other than digging something that is rather old, can you, for example, show where, in a Mott insulator, that the ground state is actually consistent with what you just said? I mean, you have GOT to expect that when you keep referring to your "paper" that I would require something more substantial than simply accepting your claim wholesale. Would you care to derive the Mott insulator ground state here, or would you prefer to submit one to one of the Phys. Rev. journals?

Zz.

 

[mikeyork]

Other than digging something that is rather old, can you, for example, show where, in a Mott insulator, that the ground state is actually consistent with what you just said? I mean, you have GOT to expect that when you keep referring to your "paper" that I would require something more substantial than simply accepting your claim wholesale. Would you care to derive the Mott insulator ground state here, or would you prefer to submit one to one of the Phys. Rev. journals?

Zz.

I hope to re-submit a revised version of my paper giving a clearer explanation of the pairwise symmetry in the multi-fermion case to a refereed journal some time later this year. I'll let you know when.

In the meantime, the impossibility of choosing a common spin frame in a symmetric way that is erroneously ignored in the usual anti-commutation formulation (the apparently anti-symmetric Hamiltonian is actually symmetric) is fully explained in my 2000 conference paper that I referred you to before. That is all you actually need.

As regards the Mott insulator in particular, please indicate specifically where you think the explanation I have given is inadequate. For example, where does the ground state violate the otherwise well-verified generalized exclusion rule (in whichever frame of reference you think is appropriate) or why does the explanation of the real permutation symmetry underlying the illusory anti-commutation not apply? If you state clearly your objections to what I have written I will try to address it once more.

 

[ZapperZ]

As regards the Mott insulator in particular, please indicate specifically where you think the explanation I have given is inadequate. For example, where does the ground state violate the otherwise well-verified generalized exclusion rule (in whichever frame of reference you think is appropriate) or why does the explanation of the real permutation symmetry underlying the illusory anti-commutation not apply? If you state clearly your objections to what I have written I will try to address it once more.

Fine. In the band structure formulation, Mott insulators are half-filled states in which hopping from one site to another isn't supressed. So band structure Hamiltonian predicts that such material are good conductors. However, in reality, it isn't. For antiferromagnetic mott insulators, there is a huge supression of such hopping due to the requirement of antisymmetrization. You have half-filled states, but spin antisymmetrization surpresses such charge hopping. The material becomes an insulator, and it does. It is only upon doping that you get any form of charge conductivity.

So get me the ground state of a Mott insulator without applying any antisymmetrization rules for fermions.

Zz.

 

[mikeyork]

Fine. In the band structure formulation, Mott insulators are half-filled states in which hopping from one site to another isn't supressed. So band structure Hamiltonian predicts that such material are good conductors. However, in reality, it isn't. For antiferromagnetic mott insulators, there is a huge supression of such hopping due to the requirement of antisymmetrization. You have half-filled states, but spin antisymmetrization surpresses such charge hopping.

This supposed antisymmetrization is illusory. The requirement is actually permutation symmetry. The appearance of antisymmetrization comes about only if you ignore the additional antisymmetrization the anticommutation rule implies for the spin quantization frames -- due to a 2*pi rotation on one particle's frame when you interchange them. To remove this rotation, you have to reverse it, thereby getting an extra minus sign to cancel that due to the anticommutation.

The illusion arises because the field creation operators are not single valued. Uniqueness requires a physically complete description of the state they are creating, just as is required for unique state vectors. This requires the methodology described in my paper for eliminating possible 2*pi rotations from spin quantization frames. When this is included in the specification of single-valued creation operators you will see how anticommutation implies the 2*pi relative rotation on one particle.

The material becomes an insulator, and it does. It is only upon doping that you get any form of charge conductivity.

So get me the ground state of a Mott insulator without applying any antisymmetrization rules for fermions.

Zz.

Done.

 

[ZapperZ]

This supposed antisymmetrization is illusory. The requirement is actually permutation symmetry. The appearance of antisymmetrization comes about only if you ignore the additional antisymmetrization the anticommutation rule implies for the spin quantization frames -- due to a 2*pi rotation on one particle's frame when you interchange them. To remove this rotation, you have to reverse it, thereby getting an extra minus sign to cancel that due to the anticommutation.

The illusion arises because the field creation operators are not single valued. Uniqueness requires a physically complete description of the state they are creating, just as is required for unique state vectors. This requires the methodology described in my paper for eliminating possible 2*pi rotations from spin quantization frames. When this is included in the specification of single-valued creation operators you will see how anticommutation implies the 2*pi relative rotation on one particle.
Done.

No, you haven't. You haven't derived anything, much less the Mott ground state. And frankly, I would rather you wait touting your paper till you get published. You don't want get the Fleishsman and Pons disease, do you?

Zz.

 

[mikeyork]

No, you haven't. You haven't derived anything, much less the Mott ground state.

You tell me it follows from the supposed antisymmetrization. I have shown that it is actually symmetrization. Just follow the logic. It's all there. If you want to keep demanding more detail and more explanation because you haven't understood that which is already provided, then there will be no end to this process. At some point you have to come to terms with what I have given you or acknowledge that you simply haven't understood it.

 

[ZapperZ]

You tell me it follows from the supposed antisymmetrization. I have shown that it is actually symmetrization. Just follow the logic. It's all there. If you want to keep demanding more detail and more explanation because you haven't understood that which is already provided, then there will be no end to this process. At some point you have to come to terms with what I have given you or acknowledge that you simply haven't understood it.

No, it is because you said this:

On the contrary, the theory is actually permutation symmetric -- as I claim it should be. It only appears to be anti-symmetric because anti-commuting operators are assumed. But this ignores the fact (shown in my paper) that this assumption implies an anti-symmetric choice of spin quantization frames. When this is taken into account, it cancels the anti-symmetry from the anti-commutation, leaving the Hamiltonian symmetric overall.

You haven't produce this "symmetry overall" Hamiltonian that can reproduce the SAME identical experimental observation. THAT is what you claim and THAT is what I want you to show.

If all you can prove here is "oh, I can do this slight of hand and get back the SAME thing", then what are we trying to prove here? Is there any experimental observation that can distinguish or verify your starting point as being valid? Even you, I hope, know enough about how physics work that to simply say "oh, I have a different starting point and can rederive everything, but it doesn't show anything new that can be tested to verify it" doesn't really get you anyhere!

Give me the "symmetric" Hamiltonian for the Mott insulator and derive for me the Mott-Hubbard gap and ground state.

Zz.

P.S. You WAITED this long to follow up on this thread. Why can't you wait a few more months until AFTER your paper is published?

 

[mikeyork]

You haven't produce this "symmetry overall" Hamiltonian that can reproduce the SAME identical experimental observation. THAT is what you claim and THAT is what I want you to show.

That is exactly what I did do. If you are expecting me to rewrite my paper here then the answer is no. You can read the original.

If you follow that paper and the additional details regarding Hamiltonians and field operators I have provided here (remember that field operators, just like state vectors are functions of the state descriptions they create and have the same uniqueness requirements) you will have no trouble seeing that anticommuting operators are the result of an order-dependent method of choosing spin frames and that that order-dependence must be reversed under permutation. Alternatively, you could use commuting operators and order-independent spin frames. Either way the Hamiltonian is symmetric under permutation.

If there is some part of this you do not understand then be specific.

The simple and obvious fact is that permutation symmetry of the Hamiltonian follows from (1) uniqueness and (2) the physical insignificance of permutation.

If all you can prove here is "oh, I can do this slight of hand and get back the SAME thing", then what are we trying to prove here?

There is no sleight of hand. Everything is in plain sight. If you believe what I have written to be erroneous, then show me the error. I am not going to waste any more time on this until you do. Good luck!

Is there any experimental observation that can distinguish or verify your starting point as being valid? Even you, I hope, know enough about how physics work that to simply say "oh, I have a different starting point and can rederive everything, but it doesn't show anything new that can be tested to verify it" doesn't really get you anyhere!

I am not certain if there might be testable predictions. If there are they will require genuine multi-particle interactions not just pairwise interactions. Pairwise exclusion rules and the statistical behavior that results are fully covered already -- it just helps to have a complete theory if you want to explain them fully and not one that ignores two-valuedness and relies on coincidence to guess away the problem. The simple fact is that conventional theory DOES NOT explain the well-known pairwise results adequately because of this hidden reliance on coincidence.

Even you, I hope, know enough about how physics works to understand that the first complete proof of the spin-statistics theorem, and one that does not require field theory or local causality, is not just a "new starting point" that "doesn't show anything new".

But enough of this argumentative shifting of goalposts around in circles. If you want to start at the beginning again, then feel free but don't expect me to join you.

 

[ZapperZ]

That is exactly what I did do. If you are expecting me to rewrite my paper here then the answer is no. You can read the original.

If you follow that paper and the additional details regarding Hamiltonians and field operators I have provided here (remember that field operators, just like state vectors are functions of the state descriptions they create and have the same uniqueness requirements) you will have no trouble seeing that anticommuting operators are the result of an order-dependent method of choosing spin frames and that that order-dependence must be reversed under permutation. Alternatively, you could use commuting operators and order-independent spin frames. Either way the Hamiltonian is symmetric under permutation.

But that is what I am asking. Start with your "equivalent" symmetry whatever. Now derive the same result that one would get without resorting to going back to the antisymmetric case. You haven't shown this at all. All you can do is show that "look, we got back to what we already know, and we can go on from there". You have done nothing here that can distinguish that what you have done is correct (assuming it is valid and will be published).

But enough of this argumentative shifting of goalposts around in circles. If you want to start at the beginning again, then feel free but don't expect me to join you.

As I recall just a few hours ago, it was you who came back and specifically re-engaged me in this thread without offering anything new on the progress of your paper. Again, if you have waited THIS long to resuscitate this thread, couldn't you have waited just a little bit longer until after your paper is published? I would have paid a bit more attention to it when that happens.

Zz.

 

[mikeyork]

But that is what I am asking. Start with your "equivalent" symmetry whatever.

The facility with which you finds new ways to "misunderstand" continues to astound me. There is no "equivalent" symmetry. There is only a permutation symmetric Hamiltonian. Period. The equivalence is in different ways of writing it using either commuting or anti-commuting operators. No doubt you'll find a new way to misunderstand that.

Now derive the same result that one would get without resorting to going back to the antisymmetric case.

There is no antisymmetric case. Only a case that appears to be antisymmetric when an important detail required for uniqueness is omitted. Since this case gets the correct result (due to the usual happy coincidence regarding the non-uniqueness and the fortunate implicit and hidden choice of order-dependent spin frames), then any other equivalent way of writing the same symmetric Hamiltonian (once the symmetry is recognized by imposing uniqueness on the otherwise non-unique case) will get the same result.

You haven't shown this at all. All you can do is show that "look, we got back to what we already know, and we can go on from there". You have done nothing here that can distinguish that what you have done is correct (assuming it is valid and will be published).

Don't you get tired of telling me that black is white?

You keep harping on about publication as if it makes a difference to whether or not the paper is correct. I have news for you, the paper is the same before or after refereeing.

As I recall just a few hours ago, it was you who came back and specifically re-engaged me in this thread without offering anything new on the progress of your paper.

No. I simply returned to fill in a detail hanging over from last year that I thought some readers might be concerned about. Why you think this was about engaging you is something for you to sort out. If I thought you were the only person of interest in this forum, I would unsubscribe. Looking over this forum, it seems that you think everything is about you. Perhaps you hadn't noticed but other people are watching this thread (and others) too.

Again, if you have waited THIS long to resuscitate this thread, couldn't you have waited just a little bit longer until after your paper is published? I would have paid a bit more attention to it when that happens.

Whereas instead you prefer to pay a great deal of attention to your fantasies about it. You would have wasted less of your time if you'd paid attention to what I actually wrote or, if you think it is worthless until published in a refereed journal, then a more rational thing might have been to ignore it. I look forward to such rationality. It would sure beat the hell out of your usual stuff.

 

[ZapperZ]

Don't you get tired of telling me that black is white?

You keep harping on about publication as if it makes a difference to whether or not the paper is correct. I have news for you, the paper is the same before or after refereeing.

Great. Then why don't you come back here AFTER it has been published. You keep referring to your "paper" as IF it is a valid treatment. I have news for you - that assumption right now is purely yours. So don't go around telling me that *I* am the one thinking everything here is about me. You suffer from the same problem too IF this is true.

You start from the false premise (delusion?) that your paper is valid even when it hasn't been published. That is what I do not understand. Why you couldn't just wait until AFTER it was published, I have no idea. Yet, you keep referring to it as if it was the bible.

So when I can look forward to it appearing in... where, PRL?

BTW, if this place is "all about me", then I would have halted this discussion a long time ago and asked you to continue this in the IR forum per our Guidelines. But nooooo... silly me thought it would be appropriate to let you continue with this. It is obvious (for both you and I) that this is not going to go anywhere. So this is where it will end. If you have something new to add, such as a citation to your paper when it is finally published, then you are welcome to let me know and I'll reopen this thread. But not till then.

Zz.