Deep thoughts in shallow lagoons

I just got back from a conference in Reykjavik, Iceland (!), on “Foundational Questions in Physics and Cosmology.” Photos and trip report coming soon. For now, please content yourself with the following remarks, which I delivered to the assembled pontificators after a day of small-group conversation in a geothermally-heated lagoon.

I’ve been entrusted to speak for our group, consisting of myself, Greg Chaitin, Max Tegmark, Paul Benioff, Caslav Brukner, and Graham Collins.

Our group reached no firm conclusions about anything whatsoever.

Part of the problem was that one of our members — Max Tegmark — was absent most of the time. He was preoccupied with more important matters, like posing for the TV cameras.

So, we tried to do the best we could in Max’s absence. One question we talked about a lot was whether the laws of physics are continuous or discrete at a fundamental level. Or to put it another way: since, as we learned from Max, we’re literally living in a mathematical object, does that object contain a copy of the reals?

One of us — me — argued that this is actually an ill-posed question. For it’s entirely consistent with current knowledge that our universe is discrete at the level of observables — including energy, length, volume, and so on — but continuous at the level of quantum amplitudes. As an analogy, consider a classical coin that’s heads with probability p and tails with probability 1-p. To describe p, you need a continuous parameter — and yet when you observe the coin, you get just a single bit of information. Is this mysterious? I have trouble seeing why it should be.

We also talked a lot about the related question of how much information is “really” in a quantum state. If we consider a single qubit — α|0〉 + β|1〉 — does it contain one bit of classical information, since that’s how many you get from measuring the qubit; two bits, because of the phenomenon of superdense coding; or infinitely many bits, since that’s how many it takes to specify the qubit?

You can probably guess my answer to this question. You may have heard of the “Shut Up and Calculate Interpretation of Quantum Mechanics,” which was popularized by Feynman. I don’t actually adhere to that interpretation: I like to discuss things that neither I nor anyone else has any idea about, which is precisely why I came to this wonderful conference in Iceland. I do, however, adhere to the closely-related “What Kind of Answer Were You Looking For?” Interpretation.

So for example: if you ask me how much information is in a quantum state, I can show you that if you meant A then the answer is B, whereas if you meant C the answer is D, etc. But suppose you then say “yes, but how much information is really there?” Well, imagine a plumber who fixes your toilet, and explains to you that if the toilet gets clogged you do this; if you want to decrease its water usage you do that, etc. And suppose you then ask: “Yes, but what is the true nature of toilet-ness?” Wouldn’t the plumber be justified in responding: “Look, buddy, you’re paying me by the hour. What is it you want me to do?”

A more subtle question is the following: if we consider an entangled quantum state |ψ〉 of n qubits, does the amount of information in |ψ〉 grow exponentially with n, or does it grow linearly or quadratically with n? We know that to specify the state even approximately you need an exponential amount of information — that was the point Paul Davies made earlier, when he argued (fallaciously, in my opinion) that an entangled state of 400 qubits already violates the holographic bound on the maximum number of bits in the observable universe. But what if we only want to predict the outcomes of those measurements that could be performed within the lifetime of the universe? Or what if we only want to predict the outcomes of most measurements drawn from some probability distribution? In these cases, recent results due to myself and others show that the amount of information is much less than one would naïvely expect. In particular, the number of bits grows linearly rather than exponentially with the number of qubits n.

We also talked about hidden-variable theories like Bohmian mechanics. The problem is, given that these theories are specifically constructed to be empirically indistinguishable from standard quantum mechanics, how could we ever tell if they’re true or false? I pointed out that this question is not quite as hopeless as it seems — and in particular, that the issue we discussed earlier of discreteness versus continuity actually has a direct bearing on it.

What is Bohmian mechanics? It’s a theory of the positions of particles in three-dimensional space. Furthermore, the key selling point of the theory is that the positions evolve deterministically: once you’ve fixed the positions at any instant of time, in a way consistent with Born’s probability rule, the particles will then move deterministically in such a way that they continue to obey Born’s rule at all later times. But if — as we’re told by quantum theories of gravity — the right Hilbert space to describe our universe is finite-dimensional, one can prove that no theory of this sort can possibly work. The reason is that, if you have a system in the state |A〉 and it’s mapped to (where |A〉, |B〉, and |C〉 are all elements of the hidden-variable basis), then the hidden variable (which starts in state |A〉) is forced to make a random jump to either |B〉 or |C〉: you’ve created entropy where there wasn’t any before. The way Bohm gets around this problem is by assuming the wavefunctions are continuous. But in a finite-dimensional Hilbert space, every wavefunction is discontinuous!

We also talked a good deal about the many-worlds interpretation of quantum mechanics — in particular, what exactly it means for the parallel worlds to “exist” — but since there’s some other branch of the wavefunction where I told you all about that, there’s no need to do so in this one.

Oh, yeah: we also talked about eternal inflation, and more specifically the following question: should the “many worlds” of inflationary cosmology be seen as just a special case of the “many worlds” of the Everett interpretation? More concretely, should the quantum state you ascribe to your surroundings be a probabilistic mixture of all the inflationary “bubbles” that you could possibly find yourself in?

Other topics included Bell inequalities, the definition of randomness, and probably others I’ve forgotten about.

Finally, I wanted to take the liberty of mentioning a truly radical idea, which arose in a dinner conversation with Avi Loeb and Fotini Markopoulou. This idea is so far-out and heretical that I hesitate to bring it up even at this conference. Should I go ahead?

Moderator: Sure!

Well, OK then. The idea was that, when we’re theorizing about the nature of the universe, we might hypothetically want some way of, you know, “testing” whether our theories are right or not. Indeed, maybe we could even go so far as to “reject” the theories that don’t succeed in explaining stuff. As I said, though, this is really just a speculative idea; much further work would be needed to flesh it out.

34 Responses to “Deep thoughts in shallow lagoons”

  1. Domenic Says:

    “but since there’s some other branch of the wavefunction where I told you all about that, there’s no need to do so in this one.” I love it :D.

  2. wolfgang Says:

    “To describe p, you need a continuous parameter”
    really? what if one takes the frequentist position serious; in this case probabilities could be fractions n/m and as we know
    there are only as many fractions as there are natural numbers.

  3. Jamie McCarthy Says:

    “literally living in a mathematical object”

    Just checking: literally, really? As in literally?

    I don’t know the math to understand much of modern particle physics and I don’t want to be That Guy with the wacky theory that makes no sense. Forgive me for even bringing this up.

    But it’s always seemed plausible to me that the universe we observe and inhabit is an emergent property of the integers. Not in the sense of we’re part of a computer program being run by God. More like, the number sequence “2,3,5,7,11…” exists because it is impossible for integers not to exist, and our universe is just the number sequence formed by a more-complicated rule.

    (Not sure how Bell’s theorem impacts Wolfram’s idea of cellular automata being behind it all.)

    Anyway… when you say “living in a mathematical object” do you literally mean every object is just a solution to a formula, or were you just poetically saying we’re living in an object that obeys mathematical rules?

  4. Moshe Says:

    It’s the tautological principle, we must live in a universe that allows deep sounding tautologies to exist.

  5. Scott Says:

    Jamie: You’ll have to ask Max Tegmark; he’s the one who said it. Personally, I have no idea what it even means to say we’re “living in a mathematical object,” and was quoting Max — what’s the word? — ironically.

  6. Jonathan Vos Post Says:

    Examples of integer-encoded mathematical structures in Max Tegmark’s “The Mathematical Universe”.

    100, 105, 11120000, 113100120, 11220000110, 11220001110, 1132000012120201

    LINKS

    Max Tegmark, The Mathematical Universe, 5 Apr 2007, Table 1, p.3.
    EXAMPLE

    “Any mathematical structure can be encoded as a finite string of integers…”

    a(1) = 100 which encodes the empty set;

    a(2) = 105 which encodes the set of 5 elements;

    a(3) = 11120000 which encodes the trivial group C_1;

    a(4) = 113100120 which encodes the polygon P_3;

    a(5) = 11220000110 which encodes the group C_2;

    a(6) = 11220001110 which encodes Boolean algebra;

    a(7) = 1132000012120201 which encodes the group C_3.

    Abstract: “I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel’s sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.”

    follow my hotlink to the full page at OEIS, whiuch hotlinks in turn to Tegmark’s PDF,

  7. geez Says:

    Anyway… when you say “living in a mathematical object” do you literally mean every object is just a solution to a formula?
    A random graph is a mathematical object…

  8. alexC Says:

    there’s an article in physics today (May2004, Vol. 57 Issue 5, p10-11) where N. David Merman claims to have been a victim of the ‘Mathew effect’ – the tendency for ideas to be credited to famous personalities who used them, but did not originate them – in the case of the ‘shut up and calculate’ phrase. although he did some of his own digging, Merman casts it as a conjecture rather than a theorem (due to the ‘amnesia effect’) and asked for readers to disprove it by suppling an attribution to Feynman predating his first published use of the phrase. in a follow up article he indicated that of the 40 emails he had received in reply, no one had been able to do so.

    Scott, careful as ever with his phrasing, said ‘popularised by Feynman’, which was indeed the case. nonetheless it brought this article to mind. i enjoyed it because, among other things, Merman muses about the likelihood of Feynman originating such a phrase. e.g.:

    “Can you imagine the young Feynman ever having had a similar experience that seared ‘shut up and calculate’ into his tender consciousness? No, of course you can’t! Nobody could ever have had the slightest reason to direct the best human calculator that ever was to shut up and calculate.”

  9. Chris Says:

    What about the argument that it is the lack of a clear conceptual basis for quantum mechanics that prevents us from constructing theories for quantum gravity, quantum measurement and so on. Coming up with alternative models has a place from that perspective, even if they are empirically indistinguishable.

    And i didn’t get why the hidden variable for state |A> is forced to make a random jump, especially where the evolution to (|B>+|C>)/sqrt{2} is continuous.

  10. bob eldritch Says:

    “What is Bohmian mechanics? It’s a theory of the positions of particles in three-dimensional space. Furthermore, the key selling point of the theory is that the positions evolve deterministically: once you’ve fixed the positions at any instant of time, in a way consistent with Born’s probability rule, the particles will then move deterministically in such a way that they continue to obey Born’s rule at all later times. But if — as we’re told by quantum theories of gravity — the right Hilbert space to describe our universe is finite-dimensional, one can prove that no theory of this sort can possibly work.”

    But that’s assuming, of course, the scientific validity of the quantum theories of gravity, and none of which have been shown to be true by experimental tests. While wikipedia lists some 14 different kinds of quantum gravity theory that have been developed over the past 30 years or so.

    Couldn’t that begin to make one think there just may be something fundamentally wrong with the idea that you can combine the standard model of quantum theory with a gravity theory into a consistent and experimentally verifiable theory of everything, in any case?

    So, who knows? Bohm’s mechanics, which is, after all the only quantum interpretation that does provide a detailed a visualisable account of quantum objests in motion as waves and particles, could still turn out to be the correct account of quantum reality.

  11. Scott Says:

    Chris: Yes, there are many cases in history where a different philosophical perspective led to a scientific advance. To my mind, the best argument for Bohmian mechanics is that it led Bell to the Bell inequalities, and the best argument for the many-worlds interpretation is that it led Deutsch to quantum computing. But of course, this gives us a very different criterion for judging interpretations: not by whether they’re “true,” but by whether they’re useful stories.

    In my example, I was assuming that |A〉, |B〉, and |C〉 were all elements of the hidden-variable basis (i.e., whichever basis we’ve selected as “real” in our hidden-variable theory — in Bohmian mechanics this would be position). Thus, the point is that in finite-dimensional hidden-variable theories, you can evolve from an element of the hidden-variable basis to a superposition over two such elements.

  12. bob eldritch Says:

    Well OK Scott, you can, I suppose – since it’s your blog – ignore my point that despite three decades of attempts of appently thousand’s of theoretical physicist’s working on theories of quantum gravity (Lee Smolin says there have bee thousands working on String theories alone), none have been experimentally confirmed.

    But you could just consider that both quantum mechanics and the experimental evidence indicates that effects would not vary over any distnce between quantum objects entangled in composite states. And so one could think that the wave property of quantum objects could extend indefinetely as described in Bohmian mechanics.

  13. ScentOfViolets Says:

    Going to the point of whether ‘we live in a mathematical objectct’, let me ask a perhaps not terribly useful question: is it most people’s experience that they can directly apprehend the number two, or is it an emergent property? Iow, assuming there is some sense to Tegmark’s assertion (though not necessarily any truth), what sort of qualia are available to people who live in mathematical constructs? Could they directly percieve Artinian rings, for example, or projective modules? Or something humbler, a lowely natural? Or does the question even make sense?

  14. Scott Says:

    Bob: I do have some life outside this blog, and if I don’t reply to a comment immediately it doesn’t necessarily mean I’m ignoring it (though it might).

    What I believe is that there exists a quantum theory of gravity, and that it will almost certainly incorporate some form of the holographic entropy bound (≤1.4×1069 bits per square meter of surface area). But as for the specific quantum gravity proposals we have today, my views, as I’ve said before, are open to the highest bidder.

  15. James S Graber Says:

    “Shut up and calculate” is drastically wrong. The proper approach is “shut up and experiment”.

  16. Anonymous Says:

    “Shut up and calculate” is drastically wrong. The proper approach is “shut up and experiment”.

    If I am not mistaken, the technical term for “experimentation without calculation” is “goofing off”.

  17. James S Graber Says:

    Hi Anonymous.
    You do have a point. So let’s just turn off the LHC and calculate the results. Same thing with designing quantum computers.
    Best, Jim

  18. Scott Says:

    Settle down, guys… :-)

  19. James S Graber Says:

    Scott,
    Sorry if I was a little too snippy, but I want to restate my original point, reemphasizing the importance of experimentation. If the value of MWI was that it led Deutsch to quantum computation, and the value of the Bohm interpretation was that it lead Bell to his theorem, then the real value of Deutsch Shor et al is not that it breaks RSA, but that it is leading the world to experimentally investigate the cat death line aka the quantum classical boundary. For way too many years there has been speculation where there should have been experimentation.
    Best, Jim

  20. Anonymous (the same) Says:

    For way too many years there has been speculation where there should have been experimentation.

    Listen to what you’re saying. You’re not simply advocating that experiments be done along with theoretical calculation; you appear to be arguing that certain theoretical work of the past should not have been done, and that experiments should have been performed instead. So, I have three questions for you:

    1. How is it even possible to perform experiments in the absence of theoretical research?
    2. Assuming it is possible, what exactly would be learned from conducting such experiments, and why would anyone care?
    3. Exactly which theoretical work is so useless that humanity would have been better off if the researchers who performed it had been spending their time on experiments? (Get specific–no hiding behind general buzzwords here. Name names and publications.)

  21. James Graber Says:

    Hello again Anonymous (the same),
    I acknowledge that no serious experimentation can be done without some theory and some calculation. Really, I am not proposing to exterminate calculators or theoreticians.
    (I hope I’m wrong, but I think I detect some trolling here.)

    By juxtaposing “shut up and experiment” (SUE) with “shut up and calculate” (SUC), my context is clearly interpretations of QM. I am planning to attend the Perimeter Institute conference on Many Worlds at 50 to argue in some sense against the MWI, precisely on the grounds that the question of the QC boundary (or the existence of the collapse) has been treated as an interpretational issue rather than an experimental one. If you differ, please explain your point of view and I will try to understand it. I would also like to hear from others on this point. Thank you. Best. Jim Graber

  22. Greg Kuperberg Says:

    As an analogy, consider a classical coin that’s heads with probability p and tails with probability 1-p.

    I applaud this style of answer! A lot of the “paradox” of quantum physics is just smoke and mirrors that comes from not properly considering the classical analogue.

    I do, however, adhere to the closely-related “What Kind of Answer Were You Looking For?” Interpretation.

    Yes. The conclusive answer to, “how much information is there in a qubit?” is, “up to one qubit; the von Neumann entropy is at most log 2.” The real question is the relative utility of classical and quantum entropy, which of course depends on what you want to accomplish. (Well, you can give a range of relative utility. If a classical bit is worth one cent, then a qubit is worth anywhere from one cent to infinitely many cents.)

    If I am not mistaken, the technical term for “experimentation without calculation” is “goofing off”.

    That or worse. Experiment without theory is just as empty as theory without experiment. Perhaps my favorite illustration of that is the scene with the duck and the “witch” in Holy Grail.

  23. Greg Kuperberg Says:

    Also a comment about this:

    One question we talked about a lot was whether the laws of physics are continuous or discrete at a fundamental level.

    Just as with bits vs qubits, here too the answer may depend on the specific sense of the question. If the question is, “Is the universe microscopically raster?”, then my answer is, “Why should it be?” But if the question is, “Is the universe microscopically finite?”, then I would say, “I suppose so, but it’s not clear in what way.”

    Of course some things really are essentially raster, e.g., salt crystals, multicellular organisms, and raster computer displays ( :-) ). But the known laws of physics already reveal that things can be finite in new and surprising ways, without being raster at all; for example, quantum spin. So who is to say what the world really looks like at the Planck scale.

  24. KWRegan Says:

    Picking up from Scott’s last paragraph, here’s something that’s testable (at least in an “NP” sense) and IMHO relevant:

    Go to FourmiLab “HotBits” or some other source of “truly random” bits mentioned here. Get a string x of (say) 50,000 bits. The question is, can one find a string s of length (say) 500 bits such that U(s) = x? Here U is some fixed “natural and efficient” universal program (e.g. John Tromp’s low-footprint combinators) used as a basis for Kolmogorov complexity.

    A “yes” answer in even just one case would be a shot in the arm for determinism. At least one “universe-is-a-computer” theorist, Juergen Schmidhuber, predicts something like that here (see also here), with the computation U(s) = x itself being feasible. Mind you, as he says, this is verifiable but (likely) not feasibly falsifiable. Did anything like this come up in the “other topic” of the “definition of randomness”?

  25. James S Graber Says:

    “One question we talked about a lot was whether the laws of physics are continuous or discrete at a fundamental level.”

    I want to again try to reduce this question to an at least in principle experimental one.
    My understanding is that most if not all proposals with a minimum length have trouble with Lorentz invariance and so have observable consequences, possibly very small or only apparent at excessively high energies. I believe Scott has pointed out that there are significant mathematical differences between finite dimensional and infinite dimensional Hilbert spaces. Do these differences also lead to observable consequences? And is there a necessary connection between discrete physical laws and finite dimensional Hilbert spaces?

  26. Greg Kuperberg Says:

    The differences between finite-dimensional and infinite-dimensional Hilbert spaces have much more to do with the issues of consistency and Occam’s razor than with direct experimental predictions. We can all agree that predictions and experiments are a crucial part of any theory of physics. However, if a theory is inconsistent, or if it fails Occam’s razor, then that undermines any predictions that it may make.

    Both types of Hilbert spaces can lead to concerns about Occam’s razor, in different ways. An infinite-dimensional Hilbert space is so “big” that, depending on the way that it is used in a physical theory, it may lead to mathematical divergences or spurious free parameters. The most worrisome kind of Hilbert space in this respect is an inseparable one, by definition a Hilbert space that does not have a countable basis. On the other hand, the dimension of a finite-dimensional Hilbert space can be an arbitrary parameter. If you think that the Hilbert space of the universe is exactly 2^11783-dimensional, you could of course ask why it is that number and not 2^11782.

    A related point is that, even though experiments are always crucial in physics, new experiments may not be. It may happen that experiments that you have already done admit only one good explanation. That was exactly the case with Einstein’s theory of special relativity, for example. The existing Michelson-Morley experiments just didn’t have any other good explanation.

    Of course if experiments are cheap, then it’s not an issue, you can repeat them. But at the moment, direct tests of quantum gravity are anything but cheap; they are inaccessible at any price. On the other hand, if it turns out that short-range forces are indirect manifestations of quantum gravity, then we can in principle use a great wealth of past experiments and confirmed knowledge as valid tests of postdictions. Strings theorists say that short-range forces do in fact look like artifacts of quantum gravity. Thus they eventually hope to postdict known facts such as the mass of an electron.

  27. Anonymous (the same) Says:

    James S. Graber:

    By juxtaposing “shut up and experiment” (SUE) with “shut up and calculate” (SUC), my context is clearly interpretations of QM. I am planning to attend the Perimeter Institute conference on Many Worlds at 50 to argue in some sense against the MWI, precisely on the grounds that the question of the QC boundary (or the existence of the collapse) has been treated as an interpretational issue rather than an experimental one.

    I may have misunderstood. Because you made the contrast between “shut up and calculate” and “shut up and experiment”, it seemed to me that you were criticizing the “calculate” part of SUC, rather than the “shut up” part. In other words, you appeared to be arguing that (certain) theoretical studies of the foundations of QM were wrongheaded, because they did not properly relate to experimentally accessible phenomena. In response to this, I was trying to point out that such studies are necessary in order to lay the groundwork for future experiments.

    But the way I understand you now, you are agreeing with SUC proponents, in the sense of rejecting the idea that “interpretations” need to be superimposed upon the physics of QM, and holding that they should instead be part of the physics itself if they are to be meaningful. I certainly don’t have anything to say against this position.

    (And no, I am not a troll, as I hope is now clear.)

  28. Jonathan Vos Post Says:

    “If you think that the Hilbert space of the universe is exactly 2^11783-dimensional, you could of course ask why it is that number and not 2^11782.”

    Because 11783 is prime. You know, like 137.

  29. James S Graber Says:

    Greg Kupferberg,
    Thanks for addressing the Hilbert space dimension issue, That’s more or less what I was afraid was true. I agree that mathematical consistency of theories is extremely important. I also agree experimentation at the Planck scale is out of reach, but experimentation at the quantum computing scale is not. My point is that the results of these quantum computing experiments have foundational as well as practical significance, more so than most folks seem to give them credit for, as far as I can tell.
    Anonymous (the same),
    Your second paragraph is pretty much right on. I am trying to argue generally for the resolution of quantum questions by empirical means. (Experimentally confirmed theories are fine with me.) Specifically, I am arguing that the location of the quantum-classical boundary has been misclassified as an interpretational issue, not subject to experimental determination. This is in part a vocabulary issue, but I maintain that a useful boundary can be defined which is not only observable, but also contributes to the mathematical consistency of the theory. One way I argue for this conclusion is to emphasize the similarity of the quantum/classical boundary to the working-quantum-computer/ not-working-quantum-computer boundary. Almost everyone would agree this latter boundary is to be determined in the laboratory.
    Best, Jim

  30. anonymous Says:

    Greg Kuperberg said: “A lot of the “paradox” of quantum physics is just smoke and mirrors that comes from not properly considering the classical analogue.”

    Greg, I also liked Scott’s point about the coin toss. There is one point to keep in mind, though, before deciding that the usual attribution of “weirdness” to quantum phenomena is just smoke and mirrors. In the case of the coin, we know that the outcome is the result of “hidden variables” so to speak. We know that the movement of the coin is being affected by the momentum of many, many collisions with gas molecules, etc., and we know that these processes are deterministic, but too complicated to be used to predict how the coin will land. We also know empirically that the end result is entirely random, and we say there’s 50/50 odds for heads/tails. In the case of the quantum analogue, we know how to use the wave function to tell us what the “odds” are, and we know that, like the coin situation, we can’t predict the outcome with certainty. But the analogy with the coin breaks down because there’s nothing “behind” the outcome determining it, as in the case of the coin with it’s deterministic collisions etc… there’s nothing, no hidden variables (according to majority opinion anyway). I find this to be strange, personally.

  31. Greg Kuperberg Says:

    In the case of the coin, we know that the outcome is the result of “hidden variables” so to speak.

    Any time that you restrict yourself to a commuting set of observables, you can always proclaim that the outcome is the result of hidden variables. Indeed, any time all of your observables happen to commute, the hidden variables are palpably real and not really hidden. For instance, if you have a qubit, and for whatever reason you only know how to measure the X operator, you can say, yawn, that’s just a deterministic variable. Or on another day, if you are only allowed to measure Z, you can say, yawn, that’s just a deterministic variable. But of course X and Z don’t commute, so if later you can measure either one, you discover that your hidden determinism never really existed.

    So it goes with the gas molecules or whatever that determine a coin toss. Yes, the outcome of a coin flip can be placed in a much larger commuting set of sort-of hidden variables. But that set is in turn contained in another algebra of observables that don’t commute. Physical determinism is only a simulacrum of quantum non-determinism. Any kind of classical randomness, such as a coin flip, is ultimately just a restricted case of quantum randomness.

  32. John Sidles Says:

    Ashtekar and Schilling have summarized the geometric view of quantum mechanics as follows:

    [In geometric quantum mechanics] the linear structure which is at the forefront in text-book treatments of quantum mechanics is, primarily, only a technical convenience and the essential ingredients–—the manifold of states, the symplectic structure and the Riemannian metric—–do not share this linearity.

    We engineers embrace the conjugate point of view, namely, that linear quantum mechanics is “the truth”—however implausible that may seem!

    From this engineering point of view, geometric quantum mechanics is viewed as a highly ingenious approach to achieving high-fidelity model order reduction.

    Regardless of whether one believes that linear quantum mechanics is “true” or whether one believes Kählerian geometric quantum mechanics is “true”, the emerging bottom line seems to be that practical quantum simulation effectively has the same algorithmic difficulty as (say) simulating Navier-Stokes fluid dynamics (CFD).

    Proving this rigorously is, of course, similarly difficult for fluid dynamics as it is for quantum mechanics. :)

    But for those who wonder “what institutions will replace America’s now-vanished giant industrial research laboratories?” the clear trend seems to be that simulation and observation are supplanting theory and experiment … with no lessening of mathematical, scientific, or economic vigor.

    The rules have changed, for sure, but the fun has not!

    References:

    summary of Nambu quantum mechanicsreview by Ashtekar and SchillingNavier-Stokes Clay problemreview of CFD achievements and challenges

  33. John Sidles Says:

    Hmmm … apologies for the broken links in the above references (the preview worked fine, but WordPress subsequently removed some crucial formatting).

    Hopefully the following references will be OK:

    • summary of Nambu quantum mechanics, • review of geometric quantum mechanics by Ashtekar and Schilling, • The Navier-Stokes Millenium Prize Problem, • review of computational fluid dynamics achievements and challenges.

    A pleasant aspect of quantum system engineering is that we get to regard the above research as being about a single geometric topic, namely, the efficient coding of dynamics and the attendant trade-offs between information and entropy.

  34. anonymous Says:

    Greg Kuperberg — Thanks for your response to my comment about the coin. It makes good sense.