- #106
atyy
Science Advisor
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ErikZorkin said:Postulates are not formal axioms
For that matter, the article you cite in the OP is not a formal proof.
ErikZorkin said:Postulates are not formal axioms
atyy said:For that matter, the article you cite in the OP is not a formal proof.
ErikZorkin said:Then why are there still so many attempts at axiomatization of QM?
ErikZorkin said: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.
ErikZorkin said:No, sir, it is!
atyy said:Of course it is not. It is ordinary mathematical proof.
ErikZorkin said: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.
bhobba said:Why are you shifting context?
ErikZorkin said: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 said:Sure that's the same as I would say for the axioms of QM.
ErikZorkin said:For God's sake of course NO!
ErikZorkin said:For this matter, such works as this are closer to what's called formal axiomatic foundation..
Because those axiomatics that are used in common QM framework are not computable. But, again, there is no comprehensive formalization of QM.atyy said:Why not?
ErikZorkin said:Because those axiomatics that are used in common QM framework are not computable. But, again, there is no comprehensive formalization of QM.
ErikZorkin said:But, again, there is no comprehensive formalization of QM.
atyy said:If QM is not axiomatizable, how can you prove it is uncomputable?
ErikZorkin said:POVMs are in bijection with PVMs. That's not the actual issue, as soon as we don't directly use the spectral theorem to construct POVMs.
atyy said:It is of course true that no finite axiomatization can produce all true statements of number theory.
atyy said:If QM is not axiomatizable, how can you prove it is uncomputable?
bhobba said:Now your concern seems to be given a Hermitian operator actually determining the Ei. Of course that's a hard computational problem like many in physics but its not a big issue at the foundations of QM any more than the computability of differential equations in classical physics is
ErikZorkin said:With POVMs you seem to start with defining measurement operators that partition the unity somehow and then do straightforward procedures to compute probabilites and final states. Where do you actually use spectral decomposition here and who gives you the measurement operators?
bhobba said:
bhobba said:Because of that measurement theory on QN is now based on POVM's.
ErikZorkin said:This is an informal foundation. Let's just finish discussing axiomatization of QM. That's not the topic of the thread.
bhobba said:Ok.
Then what is your issue precisely? That computing eigenvalues and eigenvectors is a computationally difficult problem? So?
Thanks
Bill
ErikZorkin said:My question is, do you use spectral decomposition theorem to derive POVMs or if yes, where?
ErikZorkin said:A side question: can you formulate the Stern-Gerlach experiment in terms of POVMs instead of PVMs?
ErikZorkin said:It's not about "difficulty" (or what you would call complexity, if stated precisely). It's about the fact that, in general, eigenvectors/spaces and projections are UNCOMPUTABLE. There is no algorithm to compute them. If you reread the OP, you'll see why I am interested in computable versions of spectral theorem and why I am looking for more suitable explanation than Operator/Decomposition/Measurement/Projection.
bhobba said:Thinking about it a bit more are you asking if it could be simulated on a computer - I would say - no. Feynman commented on this and reached the same conclusion which he always found rather amazing.
ErikZorkin said:I started with the words: let's pretend we simulate the universe. Would you argue that?
ErikZorkin said:Say, in Stern-Gerlach experiment, if the beam splitting is beyond Planck scale even if the beam traveled though the whole observable Universe, you can't deduce whether spin is up or down. So you land at a non-verifiable statement, scientifically speaking. You'd need a more rigorous theory.
bhobba said:Beam splitting is beyond Planck scale? What you mean by that beats me.
Thanks
Bill
ErikZorkin said:\So, POVMs in case of Stern-Gerlach experiment would be the same as the projections