Eigenvalue degeneracy in real physical systems

[ErikZorkin]

I think you mean continuous variables.

? POVMs are problematic when coupled with continuous variables? I thought in exactly the opposite way. Nevertheless, I don't care about continuous variables. The key problem of this story is: you can't computationally define the operator's spectrum, let alone eigenstates/projections. That's it. There is certainly a way to avoid such a treatment.

 

[bhobba]

I don't care about continuous variables. The key problem of this story is: you can't computationally define the operator's spectrum, let alone eigenstates/projections.

Then there is no issue.

QM has nothing to do with the practicalities of computing eigenvalues any more than classical mechanics has anything to do with the practicalities of numerically solving a differential equation. It leas to interesting things like the butterfly - but has nothing to do with the theories validity.

Thanks
Bill

 

[ErikZorkin]

Then there is no issue.

I am pretty sure! I thought I almost found an answer (POVMs), but then I got a bit concerned about the difficuleties as pointed out by rubi. Forgive me my ignorance, but do you apply spectral theorem when dealing the POVM--based QM?

 

[bhobba]

I am pretty sure! I thought I almost found an answer (POVMs), but then I got a bit concerned about the difficuleties as pointed out by rubi. Forgive me my ignorance, but do you apply spectral theorem when dealing the POVM--based QM?

No. Please, please read the link I gave. The spectral theorem only applies to what's called resolutions of the identity which are disjoint POVM's.

Thanks
Bill

 

[atyy]

Nice example! But, again, it's not always possible to split the eigenvalues, as far as I understand.

I think as long as we have experiment as a guide, it is ok. The theorem you cite assumes finite dimensions and fails for infinite dimensions. For finite dimensions, the degeneracy cannot be greater than the dimensionality. Let the dimensionality be N. Then we guess the number of eigenvalues m to range between 1 .. N. For each guess, we compute the predictions. Although we will never know with certainty which m is correct, we just take the provisional answer to be the one that matches observations most closely.

This will be fine, because in practice, even for non-degenerate Hamiltonians, our ability to do a brute-force diagonalization is already insufficient. For typical Hamiltonians in condensed matter, we quickly run out of electrons in the universe that can do the computation, eg. http://fqxi.org/data/essay-contest-files/Swingle_fqxi2012.pdf.

 

[ErikZorkin]

QT as a mathematical theory is consistent

What's QT? Quantum mechanics or QFT? First of all, the very formal foundation of QM is not done yet. Second, you can't prove its consistency. Because your axiomatic system would necessarily include arithmetic whose consistency can't be proven in arithmetic alone.

 

[bhobba]

First of all, the very formal foundation of QM is not done yet.

Von Neumann sorted that out ages ago. Dirac's elegant formulation is now rigorous since Rigged Hilbert Spaces have been worked out.

Please, please read a good book on QM such as Ballentine.

Thanks
Bill

 

[atyy]

First of all, the very formal foundation of QM is not done yet.

Formal foundations of QM are done, eg. http://arxiv.org/abs/1110.6815 (p9), which can be generalized to continuous variables, partial example http://arxiv.org/abs/0706.3526 (Eq 3, 4).

 

[bhobba]

Because your axiomatic system would necessarily include arithmetic whose consistency can't be proven in arithmetic alone.

Godels theorem is irrelevant in this context. Its as consistent as any other physical theory ie as consistent as geometry, arithmetic etc etc.

Thanks
Bill

 

[ErikZorkin]

Von Neumann sorted that out ages ago. Dirac's elegant formulation is now rigorous since Rigged Hilbert Spaces have been worked out.

Please, please read a good book on QM such as Ballentine.

Thanks
Bill

Well, the axioms of QM are far from what's called a formal mathematical axiom. cf. axioms of ZFC

 

[ErikZorkin]

Godells theorem is irrelevant in this context. Its as consistent as any other physical theory ie as consistent as geometry, arithmetic etc etc.

Thanks
Bill

Consistency of arithmetic can't even be proven. I believe you mean not the consistency, as mathematicians define it but rather informally since it allows for accurate predicitons.

 

[atyy]

@ErikZorkin, I googled around and found http://arxiv.org/abs/quant-ph/0111063 A reformulation of Hilbert's tenth problem through Quantum Mechanics, which you may find amusing.

 

[ErikZorkin]

Formal foundations of QM are done, eg. http://arxiv.org/abs/1110.6815 (p9), which can be generalized to continuous variables, partial example http://arxiv.org/abs/0706.3526 (Eq 3, 4).

Postulates are not formal axioms

 

[bhobba]

Well, the axioms of QM are far from what's called a formal mathematical axiom. cf. axioms of ZFC

If that what you want then here is the book:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - it what mathematicians call non trivial - meaning its HARD. But it does what you want.

Thanks
Bill

 

[atyy]

Postulates are not formal axioms

Sure, but no one doubts they can be formulated in ZFC.

 

[atyy]

Postulates are not formal axioms

For that matter, the article you cite in the OP is not a formal proof.

 

[ErikZorkin]

Then why are there still so many attempts at axiomatization of QM?

 

[ErikZorkin]

For that matter, the article you cite in the OP is not a formal proof.

No, sir, it is!

 

[ErikZorkin]

For this matter, such works as this are closer to what's called formal axiomatic foundation.

But I fear that this discussion went in a wrong direction a bit.

 

[atyy]

Then why are there still so many attempts at axiomatization of QM?

There are two major lines of foundational research nowadays.

The first takes the Copenhagen interpretation (and does not attempt to solve the measurement problem), and the axioms as given for example in the articles by Paris and Busch as correct. The research is to find a more "intuitive" derivation of the axioms, eg. http://arxiv.org/abs/1011.6451 or http://arxiv.org/abs/1303.1538 or http://arxiv.org/abs/1403.4621.

The second tries to solve the measurement problem of Copenhagen, eg. Bohmian mechanics, Many-Worlds, Consistent Histories etc

 

[bhobba]

Consistency of arithmetic can't even be proven. I believe you mean not the consistency, as mathematicians define it but rather informally since it allows for accurate predicitons.

Precisely what has that got to with QM not being fully worked out yet? All physical theories are like that even when expressed in highly abstract mathematics such as Symplectic geometry. Why are you shifting context?

Thanks
Bill

 

[atyy]

No, sir, it is!

Of course it is not. It is ordinary mathematical proof.

 

[ErikZorkin]

Of course it is not. It is ordinary mathematical proof.

Well, since it concerns computable analysis, the proof is constructive and as such can be formalized automatically. That's called proof normalization. And there is software out there that does the job. It's the calssical proof of the spectral theorem that can't be ever formalized.

 

[atyy]

For this matter, such works as this are closer to what's called formal axiomatic foundation.

But I fear that this discussion went in a wrong direction a bit.

Yes, the 3 links I gave in post #110 and that bhobba gave in #104 are in this spirit.

However, that people still work on deriving the axioms does not mean the axioms are not already at the same level of rigour as the article in the OP. It just means that people are looking for more "intuitive" ways to derive the axioms.

 

[ErikZorkin]

Why are you shifting context?

Well, if it was I, who shifted the focus, I apologize.

 

[atyy]

Well, since it concerns computable analysis, the proof is constructive and as such can be formalized automatically. That's called proof normalization. And there is software out there that does the job. It's the calssical proof of the spectral theorem that can't be ever formalized.

Sure that's the same as I would say for the axioms of QM.

 

[ErikZorkin]

Sure that's the same as I would say for the axioms of QM.

For God's sake of course NO!

 

[atyy]

For God's sake of course NO!

Why not?

 

[bhobba]

For this matter, such works as this are closer to what's called formal axiomatic foundation..

I think the book I suggested does just that. Its based on the formal logic approach of Von Neumann who was hardly ignorant of such things. It starts out from logic in a formal sense.

Thanks
Bill

 

[Paul Colby]

May be way off base here, but is what you're asking equivalent to "are there physical systems for which a complete set of mutually commuting observables is known not to exist?"