An article from Nature "Undecidability of the spectral gap" (arXiv:1502.04573 [quant-ph]) shows that finding the spectral gap based on a complete quantum level description of a material is undecidable (in Turing sense). No matter how completely we can analytically describe a material on the microscopic level, we can't predict its macroscopic behavior. The problem has been shown to be uncomputable as no algorithm can determine the spectral gap. Even if there is in a way to make a prediction, we can't determine what prediction is, as for a given a program, there is no method to determine if it halts.
Does this result eliminate once and for all the possibility of a theory of everything based on fundamental physics? Is Quantum physics undecidable? Is this an an epistemic result proving that undecidability places a limit on our knowledge of the world?
There isn't a limit to our knowledge of the nature. There is a limit to our knowledge of the supernatural world. If our study of the nature presents a limit it means our tools of research are inadequate or we are making errors in use of physico-mathematical models. On the other hand Quantum Mechanics is based on the Indeterminacy Principle that postulates an intrinsic indetermination into physical processes of the microscopic world. The problem of the spectral gap is also and above all a logical consequence of that view. No wonder.
No-it simply means that a particular mathematical approach has issues. However the spectral gap need not be defined in that way.
Undecidability is a property that makes sense within a fixed mathematical framework; physics represents a large number of equivalent, approximate, mathematical descriptions of natural phenomena. That's why undecidability doesn't make sense in physics-the mathematical framework isn't uniquely defined and is subject to improvement, as more information becomes available through experiments.
Once more: the uncertainty principle has to do with the relation between the variances of the distribution of certain pairs of physical quantities in quantum mechanics; the spectrum (and its properties, one of which is the gap) of an operator has to do with the average of the distribution. These are independent quantities.
I think you understood nothing about the Indeterminacy Principle.What you describe is only one aspect of the Indetermination Principle. That principle allows to one electron that doesn't have sufficient energy to surmount anyway a greater barrier of opposite potential energy. This is another aspect of indetermination.
The fact that a question was already discused in other place doesn't exclude it can be discused again independently of that. Perhaps you want to suggest all what there was to say it was said there and now there is nothing to say. But the thing isn't how you think.
Undecidability of a theory means that there are problems the theory cannot solve. A theory of everything, on the other hand, is a mathematical construct that describes the whole of reality. That the theory of everything might not be able to solve all problems is to be expected. So we could have a TOE, even in the presence of undecidability.
Given that ``the whole of reality'' is by no means well defined, a theory of it is, likewise, not well defined. So whether any mathematical framework for it might be incomplete, while inevitable, isn't relevant, because it's bound to be superseded.
Gödel's theorem didn't stop Turing and von Neumann from working out what questions it made sense to ask when designing computers.
Indetermination means just there are real problems of the nature that the science cannot solve and cannot know. But then indetermination allows things that any scientific logic would exclude. It is bizarre that in postmodern physics there are at the same time opposite concepts like indetermination, undecidability, theory of all. And all these concepts would be based on the same fundamental principles.
Well using words without paying attention to what they actually mean does lead to meaningless sentences. There's no such notion as ``indetermination'' nor ``postmodern physics''. The uncertainty principles of quantum mechanics are well-defined statements that don't have anything to do with whether certain statements are undecidable, nor with any, ``theory of all'' that's just a meaningless expression, since ``all'' isn't defined.
It is truly bizarre to claim words as "indetermination" and "postmodern physics" have no meaning. The "Uncertainty Principle" is called also "Indeterminacy Principle" and they are based on te same mathematical model. The word "uncertainty" leads to think indetermination in nature is due to the ineliminable systematic error introduced by the inadequacy of measure instruments that perturb physical phenomenon while the word "indeterminacy" indicates an intrinsic indetermination of nature independently of measure instruments. In any case both meanings lead to a limit for the knowledge of the nature. "Postmodern physics" then is part of postmodern culture or do you think physics isn't culture?
I have clarified the physical meaning of terms "uncertainty", "indeterminacy", "indetermination" that goes beyond semantics of different languages. The fact that classical trajectory is "just wrongful thinking" is your viewpoint. I can garantuee to you that in a cathode-ray tube , in the absence of intermediate electric potentials, the trajectory of electrons that go from cathode to anode is perfectly linear. The wave model of massive particles has to be considered in the right manner in order to avoid errors. Anyway thanks for your help to my English because cetainly you know English better than myself.
I have proved diffraction isn't an exclusive property of waves but also a beam of massive particles can produce diffraction (I can indicate to you the paper if you are intersted in). Besides the correct working of an oscilloscope isn't to obtain diffraction and therefore the effective apertures have to be great with respect to sizes of electrons. In that case there isn' t diffraction and the trajectory of electrons is perfectly rectilinear. I think then the effect that you see on the oscilloscope isn't due to diffraction but rather to bad focalization.
I wrote in concise manner what I think about the undecidability of the spectral gap. I am glad to inform you that soon I will write a paper on that question and I hope it will be welcome for you.
As a few have pointed out here, this is actually a few questions in one, and its constituent sub-questions are not necessarily related.
First, yours is just an example of the phenomenon of emergence, and irrespective of how you analyse things, unpredictable emergence will always arise. Second, as Marcelo Gleiser and others have argued, there does not necessarily exist such a thing as a TOE - the assumption that there is is unwarranted and may stem from subconscious cognitive bias. Third, there are unavoidably, on sound theoretical grounds, many undecidables in math & physics. There are even examples of math itself breaking down entirely (the Cantor anomaly, for instance.)
All of which do not necessarily preclude a TOE - the TOE would just have to include the fact that sometimes there exist undecidable or unanswerable things. A few people have come up with ways to work out nature's fundamental constants, for instance, and the undecidables did not stand in the way of their theories (other impediments did exist however, see e.g. Burkhard Heim's et al. theories.).
No, but one may change the research direction for the theory of everything.
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