Undecidability and unpredictability: not limitations, but triumphs of science by Markus P Mueller

Thank you Marcus, for asking an important and interesting question. I try to explain what I mean by non-locality in this matrix dynamics, and why it does not imply superluminal signalling. In this dynamics at the Planck scale, there is no space-time. There is only a new notion of time - the Connes time. All processes take place in a Hilbert space, where there is no conventional notion of distance [space-time emerges subsequently, from this Hilbert space, after spontaneous localisation]. So, whereas Alice and Bob are two space-like separated observers from the viewpoint of a conventional Minkowski spacetime, who are making their respective measurements, the picture of the same set-up is very different in matrix dynamics. From the viewpont of this new dynamics, a correlated pair of say electron and positron in an entangled state are represented by operators evolving with time, but this evolution does not imply that the electron and positron are moving away from each other. We must not think of them as spatially separated. Also, one talks of simultaneity in Connes time, which plays the role of an absolute [reversible] time. When Alice makes a measurement on the electron, it simultaneously changes the state of the positron [simultaneous in Connes time]. But no travel or signalling is involved.

I explain this in some detail in this paper:

https://arxiv.org/abs/1903.05402

starting at the bottom of p. 26. Basically, there are two different ways of lookimg at an EPR event. One is the space-time-less matrix dynamics way [non-local but no signalling], and the conventional way..involves signalling. Quantum non-locality appears to violate relativity if we accept that QM needs space-time. But qm does not need spacetime - in fact spacetime is external to qm and must be removed so as to find a better description of qm. The matrix dynamics achieves that - because there is an absolute time, but no light-cones. Lorentz invariance is emergent.

I will be very happy to discuss this point further with you. Do let me know what you think.

Thanks,

Tejinder

I really enjoyed this essay. Relating the incompleteness theorem to Euclid's axiom is a great illustration of the point which really puts the issue of incompleteness in a new perspective. And I agree that much of our trouble with the interpretation of quantum mechanics comes from asking the wrong questions and attempting to force the theory into an over-specific ontological structure.

I wondered about your phrasing of the 'unanswerable questions' in quantum mechanics - 'What is, at some given moment, the actual configuration of the world?' Relativistically the concept of 'the state of the world at some given moment' isn't well-defined, so it would seem that it follows directly from relativity that this question is unanswerable, and therefore quantum mechanics wouldn't be adding anything very new here. Or did you mean 'at some given moment' to refer to 'on some spacelike hyperplane of simultaneity'?

I also think there's an important difference between the case of quantum mechanics and the case of 'the same time.' In the case of relativity, Einstein did not simply assert that it so happens that questions about 'the same time' have no answer - he argued convincingly that these questions are meaningless (in our universe and in a large class of universes like ours). Whereas quantum mechanics doesn't seem to show us that 'What is the actual configuration of the world (on some suitable spacelike hyperplane)'? isn't a well-posed question - rather it's just a contingent fact that in our actual world this question has no answer (if it is indeed a fact!). So the claim that this question is unanswerable seems less logically compelling then the claim that 'same time' questions are unanswerable (though of course that doesn't mean it isn't true!). I wondered if you agree, or if you think there's a stronger claim to be made to the effect that questions about the actual configuration of the world aren't even meaningful?

The idea of actual, physical randomness is an odd one. If it isn't meant to be due to a complex of unrelated extrinsic causes, to be extrinsically uncaused, and intrinsically foundationless, would be the best explanation for nothing happening at all.

Dear Markus Mueller,

I just tried to elaborate on what I directly indicated with be careful when calculating as if. Please find possible implications concerning QM yourself.

Incidentally, I live for many decades in Magdeburg.

Best, Eckard

Dear Markus,

Brilliant essay, I liked the structure of arguments, the ideas, and the general gist of it. The sensation is that of regaining a freedom considered lost. It may be enough that always exists in a mathematical sense a possible structure that fits all the data, like in Wheeler's version of the twenty questions game. It may also worth trying to fit a solution that is unitary, i.e. unbroken by projections, this is one of the things that interest me (such a solution can't be fixed just by any initial conditions at a given time, it depends on future experimental settings). Best way is to keep open all possibilities. Thanks for the essay, it was a pleasure to read it!

Cheers,

Cristi

Dear Markus,

I very much enjoyed your thoughtful, masterfully written essay. It was so refreshing with its message of hope in comparison to all the usual discussions that want to turn back the clock of quantum mechanics to something more akin to classical physics or, metaphorically, Hilbert's program.

Years ago (22 or so!), I wrote a job application which I've just looked up. It started with these words, "The world we live in is well-described by quantum mechanics. What should we make of that? In a way, the answer to this question was once less positive than it is today. For although quantum theory is a tool of unprecedented accuracy ... the intellectual lesson we have come to

derive from it has been one ... of limitations. The best place to see this attitude is in a standard presentation of the Heisenberg uncertainty relations. It is almost as if the world were holding something back that we really had every right to possess: The task of physics, or so it was believed, is simply to sober up to this and make the best of it. ... In contrast to this ... the last ten years have seen the start of a significantly more positive, almost intoxicating, attitude about the basic role of quantum mechanics. This is evidenced no more clearly than [with quantum information and computing]. The point of departure in these disciplines is not to ask what limits quantum mechanics places upon us, but instead what novel, productive things we can do in the quantum world that we could not have done otherwise. In what ways can we say that the quantum world is fantastically better than the classical world?" Your paper brought back to me the romance of those lost days, but you did it so much better!

I had never previously thought about Goedel's incompleteness theorem in the positive way that you do, even though some other writers should have led me close to it. When I read your words on that point, I immediately thought, that's got to be right! "It is not a fundamental limit to what we can know, but a precious piece of knowledge about a non-property of the structure that we have discovered"--Beautiful!

Incidentally, in this paper of mine,

https://arxiv.org/pdf/1601.04360.pdf

I transcribed an entry from one of John Wheeler's notebooks that blew me away when I first ran across it. I don't think it's exactly what you have in mind, but here's the little story I wrote when introducing it: "Despite the dubious connection to anything firmly a part of QBism, I report Wheeler's idea because it seems to me that it conveys some imaginative sense of how the notion of 'birth' described here carries a very different flavor from the 'intrinsic randomness' that [Adan Cabello] and others seem to be talking about. ... Imagine along with Wheeler that the universe can somehow be identified with a formal mathematical system, with the universe's life somehow captured by all the decidable propositions within the system. Wheeler's 'crazy' idea seems to be this. Every time an act of observer-participancy occurs (every time a quantum measurement occurs), one of the undecidable propositions consistent with the system is upgraded to the status of a new axiom with truth value either TRUE or FALSE. In this way, the life of the universe as a whole takes on a deeply new character with the outcome of each quantum measurement. The 'intrinsic randomness' dictated by quantum theory is not so much like the flicker of a firefly in the fabric of night, but a rearrangement of the whole meaning of the universe."

You caused other thoughts in me as well. In the paper linked to above, I emphasized in a small piece of it that QBism shares *some* of its elements with a structural realism. But most philosophers of science I've told this to have been (predictably) dismissive. It's hard to say what stands in their way, except possibly that if an idea is associated with QBism, it's got to be bad! Upon reading you, however, I got a vision on how I might break the impasse: Make up a new name, a new distinction! Thus, from here out I will dub QBism's distinctive flavor "normative structural realism." But I will discuss this with you offline sometime.

In any case, I write all of this to let you know, in my eyes your essay has everything that should be expected of a winner of this contest. I learned a number of things from you, but mostly your essay caused me to think over and over about its contents all week. It hasn't left my mind, and that's a mark of distinction in an old doddering mind like mine!

All the best,

Chris Fuchs

Hi Markus,

"To say what a pattern is, you have to choose a compression algorithm, or a universal machine (which is analogous to a choice of language). For finite data, the notion of compressibility will depend on this choice. Any ultimate definition of a real pattern will have to deal with this issue in some sense..."

I can see how this might be a practical problem in choosing what predicates/basic concepts you might use as a basis for the construction of different sorts of compression algorithms for resolving simplified but predictive patterns from complex data sets ... I'm just not sure that an ultimate definition of a 'real pattern' can be derived from that algorithmic perspective.

Could it be the case that an ontological (natural language) definition of a 'real pattern' derived from one's own observational experience is precisely what you would need as a basic concept for the construction of algorithms that might then more or less model that observational reality? But if you already define a 'real pattern' as something that's derived from a background of raw data vs noise, then your ontology is already Quinean at best, and at worst an externalist realism of mere pattern appearances weakly emerging from the external data/noise background. Thus a definition of a 'real pattern' is derived from the 'data' (patterns)...

From a phenomenal (empirical) perspective, we don't observe raw data and noise but rather we observe the patterns that we call 'raw data' and 'noise' ... pattern recognition as the basic form of phenomenal experience is in this sense a priori, and the concepts of data and its inverse in noise, are both derived from that basis. What else is structural realism than intuiting structure within real patterns, where there is nothing beyond or behind the real patterns themselves?

And is a definition of 'reality' then nothing more or less than the degree of correlation observed between different algorithmic pattern recognition systems? Non-correlated systems would be orthogonal systems incapable of the communication of any recognisable pattern, whereas 'we' obviously already exist in a world of observable and very concrete pattern correlations. At least, that's how I understand a perspectival (intersubjective), observer dependent reality.

Cheers,

Malcolm

Dear Professor Fuchs,

There is no experiment that contradicts quantum mechanics to date. However, quantum mechanics has been tested and verified only upto TeV energy scales or so. Thus when we try to make theories of quantum gravity valid at the Planck scale, we can make them by assuming that quantum theory holds at the Planck scale. Or we can make them by assuming that quantum theory is violated at the Planck scale, but recovered at lower energies. Then, if the predictions of the two approaches are different, experimentalists can try to find out which approach, if either one, is correct.

I hope we can agree on this much.

It could well be that dynamics at the Planck scale is deterministic; yet the emergent low energy dynamics, being QM, is indeterministic:

Nature does not play dice at the Planck scale

But of course this deterministic Planck scale dynamics is not a return to Newtonian days: it is a non-local, non-unitary matrix-valued Lagrangian dynamics. There is no space-time here: space-time, along with quantum mechanics, are emergent.

Thanks and best regards,

Tejinder

Dear Professor Mueller,

thanks a lot for these great insights and new viewpoint!

While reading, I was tempted to compare your viewpoint to objective programming. If your structure S defined in Fig. 1 is an abstract class, we cannot derive specific instances from it and thus, without an instance at hand, we cannot give answers to certain questions about S.

I hope to be able to apply your ideas to my special case of under-constrained problems in cosmology as well.

All the best for the contest and your future research!

Jenny Wagner