Why Pauli's Exclusion Principle?

[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.