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.

]]>[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

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.

]]>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.

]]>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 ]]>

Because 11783 is prime. You know, like 137.

]]>*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.)

]]>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.

]]>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?