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shminux comments on Quotes and Notes on Scott Aaronson’s "The Ghost in the Quantum Turing Machine" - Less Wrong Discussion
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Comments (82)
I honestly don't understand why you invoke Killing vectors to make your point. I am also not sure what this "complex structure J" means (is it some tensor?) in the QM context and what it would mean to take a Lie derivative of J with respect to some vector field.
The dynamicist Vladimir Arnold had a wonderful saying:
This saying is doubly true of quantum mechanics. For example, the undergraduate quantum physics notion of "multiply a quantum vector by
" is not so easy to convey without mentioning the number "
." Here's how the trick is accomplished. We regard Hilbert space as a real manifold 
that is equipped with a symplectic form 
and a metric 
. Given an (arbitrary) vector field 
on 
, we can construct an endomorphism 
by first "flatting" with 
and then "sharping" with 
, that is 
. The physicist's equation 
thus is naturally instantiated as the endomorphic condition 
.
The Point To a geometer, the Lie derivative of
has no very natural definition, but the Lie derivative of the endomorphism 
is both mathematically well-defined and (on non-flat quantum state-spaces) need not be zero. The resulting principle that "
is not necessarily constant" thus is entirely natural to geometers, yet well-nigh inconceivable to physics students!
Are you describing
It sounds as if your endomorphism J is supposed to play the role of i somehow, but how? What do you actually do with it, and why?
What is your manifold M actually supposed to be, and why? Is it just a formal feature of the theory, or is it meant to be spacetime, or some kind of phase space, or what?
JLM, the mathematically natural answer to your questions is:
• the quantum dynamical framework of (say) Abhay Ashtekar and Troy Schilling's Geometrical Formulation of Quantum Mechanics arXiv:gr-qc/9706069v1, and
• the quantum measurement framework of (say) Carlton Caves' on-line notes Completely positive maps, positive maps, and the Lindblad form, both pullback naturally onto
• the varietal frameworks of (say) Joseph Landsberg's Tensors: Geometry and Applications
Textbooks like Andrei Moroianu's Lectures on Kahler Geometry and Mikio Nakahara's Geometry, Topology and Physics are helpful in joining these pieces together, but definitely there is at present no single textbook (or article either) that grinds through all the details. It would have to be a long one.
For young researchers especially, the present literature gap is perhaps a good thing!
(Who's JLM?)
I don't think you actually answered any of my questions; was that deliberate? Anyway, it seems that (1) the general description in terms of Kähler manifolds is a somewhat nonstandard way of formulating "ordinary" quantum mechanics; (2) J does indeed play the role of i, kinda, since one way you can think about Kähler manifolds is that you start with a symplectic manifold and then give it a local complex structure; (3) yes, M is basically a phase space; (4) you see some great significance in the idea that some Lie derivative of J might be nonzero, but haven't so far explained (a) whether that is a possibility within standard QM or a generalization beyond standard QM, or (b) along what vector field V you're taking the Lie derivative (it looks -- though I don't understand this stuff well at all -- as if it's more natural to take the derivative of something else along J, rather than the derivative of J along something else), or (c) why you regard this as importance.
And I still don't see that there's any connection between this and Scott's stuff about free will. (That paragraph you added -- is it somehow suggesting that "dynamic-J methods" for simulation can somehow pull out information that according to Scott is in principle inaccessible? Or what?)
The natural answer is, along a Hamiltonian vector field. Now you have all the pieces needed to ask (and even answer!) a broad class of questions like the following:
Alice possesses a computer of exponentially large memory and clock speed, upon which she unravels the Hilbert-space trajectories that are associated to the overall structure)
, where 
is a Hilbert-space (considered as a manifold), 
is its metric, 
is its symplectic form, 
is the complex structure induced by )
, and )
are the (stochastic,smooth) Lindblad and Hamiltonian potentials that are associated to a physical system that Alice is simulating. Alice thereby computes a (stochastic) classical data-record as the output of her unraveling.
Bob pulls-back)
onto his lower-dimension varietal manifold (per Joseph Landsberg's recipes), upon which he unravels the pulled-back trajectories, thus obtaining (like Alice) a classical data-record as the output of his unraveling (but using far-fewer computational resources).
Then It is natural to consider questions like the following:
Is this a mathematically well-posed question? Definitely! Is it a scientifically open question? Yes! Does it have engineering consequences (and even medical consequences) that are practical and immediate? Absolutely!
What philosophical implications would a "yes" answer have for Scott's freebit thesis? Philosophical questions are of course tougher to answer than mathematical, scientific, or engineering questions, but one reasonable answer might be "The geometric foaminess of varietal state-spaces induces Knightian undertainty in quantum trajectory unravelings that is computationally indistinguishable from the Knightian uncertainty that, in Hilbert-space dynamical systems, can be ascribed to primordial freebits."
Are these questions interesting? Here it is neither feasible, nor necessary, nor desirable that everyone think alike!
I cannot tell whether your writing style indicates an inability to bridge an inferential gap or an attempt at status smash ("I'm so smart, look at all the math I know, relevant or not!"). I will assume that it's the former, but will disengage, anyway, given how unproductive this exchange has been so far. Next time, consider using the language appropriate for your audience, if you want to get your point across.
I believe you're being uncharitable. JS is a bit effervescent and waxes poetic in a few places, but doesn't say anything obviously wrong.
I would have assumed (perhaps wrongly) that you'd know how to take the Lie derivative of a (1, 1)-tensor field, and there's only a short Googling necessary to ascertain that complex structures are certain kinds of (1, 1)-tensor fields. The linked draft is pretty clear about what L_X(J) = 0 means, and that makes it clear what L_X(J) != 0 means -- X doesn't generate holomorphic flows.
In another comment, he shows how to construct a compatible (almost) complex structure J from a Riemannian structure g and a symplectic structure w. This is actually a special case of a theorem of Arnol'd, which states that fixing any two yields a compatible choice of the third. (I've always heard this called the "two out of three" theorem, but apparently some computer science thing has overtaken this name.) This shows that J is actually relevant to the dynamics of the underlying system -- just as relevant as the symplectic structure is.
From that, it's not too much of a stretch to make a metaphor between this situation and contrasting the study of non-conservative flows with the study of conservative flows. Seems reasonable enough to me!
Thank you for your gracious remarks, Paper-Machine. Please let me add, that few (or possibly none) of the math/physics themes of the preceding posts are original to me (that's why I give so many references!)
Students of quantum history will find pulled-back/non-linear metric and symplectic quantum dynamical flows discussed as far back as Paul Dirac's seminal Note on exchange phenomena in the Thomas atom (1930); a free-as-in-freedom review of the nonlinear quantum dynamical frameworks that came from Dirac's work (nowadays called the "Dirac-Frenkel-McLachlan variational principle") is Christian Lubich's recent On Variational Approximations In Quantum Molecular Dynamics (Math. Comp., 2004).
Shminux, perhaps your appetite for nonlinear quantum dynamical theories would be whetted by reading the most-cited article in the history of physics, which is Walter Kohn and Lu Jeu Sham's Self-Consistent Equations Including Exchange and Correlation Effects (1965, cited by 29670); a lively followup article is Walter Kohn's Electronic Structure of Matter, which can be read as a good-humored celebration of the practical merits of varietal pullbacks ... or as Walter Kohn calls them, variational Ansatzes, having a varietal product form.
There is a considerable overlap between Scott Aaronson's "freebit" hypothesis and the view of quantum mechanics that Walter Kohn's expresses in his Electronic Structure of Matter lecture (views whose origin Kohn ascribes to Van Vleck):
Scott's essay would (as it seems to me) be stronger if it referenced the views of Kohn (and Van Vleck too) ... especially given Walter Kohn's unique status as the most-cited quantum scientist in all of history!
Walter Kohn's vivid account of how his "magically" powerful quantum simulation formalism grew from strictly "muggle" roots---namely, the study of disordered intermetallic alloys---is plenty of fun too, and eerily foreshadows some of the hilarious scientific themes of Eliezer Yudkowsky's Harry Potter and the Methods of Rationality.
In view of these nonpareil theoretical, experimental, mathematical (and nowadays) engineering successes, sustained over many decades, it is implausible (as it seems to me) that the final word has been said in praise of nonlinear quantum dynamical flows! Happy reading Shminux (and everyone else too)!
Quantum aficionados in the mold of Eliezer Yudkowsky will have fun looking up "Noether's Theorem" in the index to Michael Spivak's well-regarded Physics for Mathematicians: Mechanics I, because near to it we notice an irresistible index entry "Muggles, 576", which turns out to be a link to:
Remark It is striking that Dirac's The Principles of Quantum Mechanics (1930), Feynman's Lectures on Physics (1965), Nielsen and Chuang's Quantum Computation and Quantum Information (2000)---and Scott Aaronson's essay The Ghost in the Turing Machine (2013) too---all frame their analysis exclusively in terms of (what Michael Spivak aptly calls) Muggle mathematic methods! :)
Observation Joshua Landsberg has written an essay Clash of Cultures (2012) that discusses the sustained tension between Michael Spivak's "Hogwarts math versus Muggle math". The tension has historical roots that extent at least as far back as Karl Gauss' celebrated apprehension regarding the "the clamor of the Boeotians" (aka Muggles).
Conclusion Michael Spivak's wry mathematical jokes and Eliezer Yudkowsky's wonderfully funny Harry Potter and the Methods of Rationality both help us to appreciate that outdated Muggle-mathematical idioms of standard textbooks and philosophical analysis are a substantial impediment to 21st Century learning and rational discourse of all varieties---including philosophical discourse.
The thing you link to is not anything by Joshua Landsberg, but another of your own comments.
That in turn does link to something by Landsberg that has a section headed "Clash of cultures" but it could not by any reasonable stretch be called an essay. It's only a few paragraphs long and about half of it is a quotation from Plato. (It also makes no explicit allusion to Spivak's Hogwarts-Muggles distinction, though I agree it's pointing at much the same divergence.)
LOL ... gjm, you must really dislike Lincoln's ultra-short Gettysburg Address!
More seriously, isn't the key question whether Landsberg's essay is correct to assert that "there are language and even philosophical barriers to be overcome", in communicating modern geometric insights to STEM researchers trained in older mathematical techniques?
Most seriously of all, gjm, please let me express the hope that the various references that you have pursued have helped to awaken an appreciation of the severe and regrettable mathematical limitations that are inherent in the essays of Less Wrong's Quantum Physics Sequence, including in particular Eliezer_Yudkowsky's essay Quantum Physics Revealed As Non-Mysterious.
The burgeoning 21st century literature of geometric dynamics helps us to appreciate that the the 20th century mathematical toolkit of Less Wrong's quantum essays perhaps will turn out to be not so much "less wrong" as "not even wrong," in the sense that Less Wrong's quantum essays are devoid of the geometric dynamical ideas that are flowering so vigorously in the contemporary STEM literature.
This is of course very good news for young researchers! :)
No, I think it's excellent (though I prefer the PowerPoint version), but it isn't an essay.
That sentence appears to me to embody some assumptions you're not in a position to make reliably. Notably: That I think, or thought until John Sidles kindly enlightened me, that Eliezer's QM essays are anything like a complete exposition of QM. As it happens, that wasn't my opinion; for that matter I doubt it is or was even Eliezer's.
In particular, when Eliezer says that QM is "non-mysterious" I don't think he means that everything about it is understood, that there are no further scientific puzzles to solve. He certainly doesn't mean it isn't possible to pick a mathematical framework for talking about QM and then contemplate generalizations. He's arguing against a particular sort of mysterianism one often hears in connection with QM -- the sort that says, roughly, "QM is counterintuitive, which means no one really understands it or can be expected to understand it, so the right attitude towards QM is one of quasi-mystical awe", which is the kind of thing that makes Chopraesque quantum woo get treated with less contempt than it deserves.
Even Newtonian mechanics is mysterious in the sense that there are unsolved problems associated with it. (For instance: What are all the periodic 3-body trajectories? What is the right way to think about the weird measure-zero situations -- involving collisions of more than two particles -- in which the usual rules of Newtonian dynamics constrain what happens next without, prima facie, fully determining it?) But no one talks about Newtonian mechanics in the silly way some people talk about quantum mechanics, and it's that sort of quantum silliness Eliezer is (at least, as I understand it) arguing against.
I think at least one of us has a serious misunderstanding of what's generally meant by the phrase "not even wrong". To me, it means "sufficiently vague or confused that it doesn't even yield the sort of testable predictions that would allow it to be refuted", which doesn't seem to me to be an accurate description of conventional 20th-century quantum mechanics. It might, indeed, turn out that conventional 20th-century QM is a severely incomplete description of reality, and that in some situations it gives badly wrong predictions, and that some such generalization as you favour will do better, much as relativity and QM improved on classical physics in ways that radically revised our picture of what the universe fundamentally is. But classical physics was not "not even wrong". It was an excellent body of ideas. It explained a lot, and it enabled a lot of correct predictions and useful inventions. It was wrong but clear and useful. It was the exact opposite of "not even wrong".
Finally and incidentally: What impression do you think your tone gives to your readers? What impression are you hoping for? I ask because I think the reality may not match your hopes.
That may or may not be the case, but there is zero doubt that this assertion provides rhetorical foundations for the essay * And the Winner is... Many-Worlds!*.
A valuable service of the mathematical literature relating to geometric mechanics is that it instills a prudent humility regarding assertions like "the Winner is... Many-Worlds!" A celebrated meditation of Alexander Grothendieck expresses this humility:
Surely in regard quantum mechanics, the water of our understanding is far from covering the rocks of our ignorance!
As for the tone of my posts, the intent is that people who enjoy references and quotations will take no offense, and people who do not enjoy them can simply pass by.
I don't believe I claimed that he did. When I expressed not understanding the relevance or even meaning of his notation, all I got in reply was more poetic waxing. By contrast, your clear and to the point explanation made sense to this lowly ex-physicist. So, I am not sure which part of my reply was uncharitable... Anyway, JS is now on my rather short list of people not worth engaging with in an online discussion.
The part where you accuse him of a "status smash" after he directly answered what J was (that is, the composition of those two maps) and why it was important (that is, because it reflects the symplectic structure). The only lack of productivity is this thread is yours.
These sorts of declarations always remind me of Sophist:
Huh, I guess I am not alone in being Sidles-averse, for the same reasons.
Shminux, perhaps some Less Wrong readers will enjoy the larger reflection of our differing perspectives that is provided by Arthur Jaffe and Frank Quinn’s ‘Theoretical mathematics’: Toward a cultural synthesis of mathematics and theoretical physics (Bull. AMS 1993, arXiv:math/9307227, 188 citations); an article that was notable for its biting criticism of Bill Thurston's geometrization program.
Thurston's gentle, thoughtful, and scrupulously polite response On proof and progress in mathematics (Bull. AMS 1994, arXiv:math/9307227, 389 citations) has emerged as a classic of the mathematical literature, and is recommended to modern students by many mathematical luminaries (Terry Tao's weblog sidebar has a permanent link to it, for example).
Conclusion It is no bad thing for students to be familiar with this literature, which plainly shows us that it is neither necessary, nor feasible, nor desirable for everyone to think alike!
That's a little out of context...I think the stranger literally means the union of concepts, not the union of opinions or points of view.
I see. Well, I very much appreciate your feedback, it's good to know how what I say comes across. I will ponder it further.
Shminux, it may be that you will find that your concerns are substantially addressed by Joshua Landsberg's Clash of Cultures essay (2012), which is cited above.
Shminux, there are plenty of writers---mostly far more skilled than me!---who have attempted to connect our physical understanding of dynamics to our mathematical understanding of dynamical flows. So please don't let my turgid expository style needlessly deter you from reading this literature!
In this regard, Michael Spivak's works are widely acclaimed; in particular his early gem Calculus on Manifolds: a Modern Approach to Classical Theorems of Advanced Calculus (1965) and his recent tome Physics for Mathematicians: Mechanics I (2010) (and in a comment on Shtetl Optimized I have suggested some short articles by David Ruelle and Vladimir Arnold that address these same themes).
Lamentably, there are (at present) no texts that deploy this modern mathematical language in service of explaining the physical ideas of (say) Nielsen and Chuang's Quantum Computation and Quantum Information (2000). Such a text would (as it seems to me) very considerably help to upgrade the overall quality of discussion of quantum questions.
On the other hand, surely it is no bad thing for students to read these various works---each of them terrifically enjoyable in their own way---while wondering: How do these ideas fit together?