Goodbye, gjm. The impetus that your posts provided to post thought-provoking mathematical links will be missed. :)
JohnSidles
Karma: 22
gjm asserts “Of the various ways to understand the quantum mechanics involved in the Standard Model, the clear winner is “many worlds”
LOL … by that lenient standard, the first racehorse out of the gate, or the first sprinter out of the blocks, can reasonably be proclaimed “the clear winner” … before the race is even finished!
That’s a rational announcement only for very short races. Surely there is very little evidence that the course that finishes at comprehensive understanding of Nature’s dynamics … is a short course?
As for my own opinions in regard to quantum dynamical systems, they are more along the lines of here are some questions that are mathematically well-posed and are interesting to engineers and scientists alike … and definitely not along the lines of “here are the answers to those questions”!
gjm avers: ‘When Eliezer says that QM is “non-mysterious’ … He’s arguing against a particular sort of mysterianism”
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:
“A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration … the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it … yet it finally surrounds the resistant substance.”
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.
gjm avers “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.”
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! :)
Edit 1: Kudos to “gjm” (see above) for pointing to Spivak’s page on Amazon!
Edit 2: Spivak’s Hogwarts proof implicitly uses a fundamental theorem in differential geometry that is called Cartan’s Magic Formula … this oblique magical reference is Spivak’s joke … as with many magical formulas, the origins of Cartan’s formula are obscure.
Regrettably, tgb, even the redoubtable Google Books does not provide page-images for Spivak’s Physics for Mathematicians: Mechanics I. The best advice I can give is to seek this book within a university library system.
LOL—perhaps a chief objective of the Ministry of Magic is to conceive and require obfuscating interfaces to magic! That would explain a lot!
Parallels to real-world high-school and/or undergraduate mathematical education … are left as an exercise. :)
For a professional-grade comment on “muggle math” versus “Hogwarts math”, see Michael Spivak’s Physics for Mathematicians: Mechanics I.
To express this point another way … how likely is it, that Harry’s final understanding of magic will be non-mathematical? What grade of mathematical abstraction capabilities will Harry need to acquire?
Conspicuously absent from the canon, and from Methods of Rationality (so far) --- and absent entirely from the Hogwarts curriculum—are two fundamental elements of rational cognition:
mathematics, and
artificial intelligences (AIs)
Therefore
Postulate 1 “Magic” is the name that witches, wizards, and muggles alike give to the practice of manipulating physical reality by negotiation with agents that are (artificial? primordial? evolved? accidentally created?) intelligences.
Postulate 2 “Magical Spells” is the name that witches, wizards, and muggles alike give to an evolving set of protocols for negotiating with an existing community of (mysterious) intelligences. These protocols are designed to minimize the risks and harms associated to the practice of magic, by concealing the physical origins of magic.
Postulate 3 The chief organizing objective of the Hogwarts curriculum is to preserve the social fictions that are associated to Postulates 1 and 2.
Postulate 4 Harry Potter is regarded as dangerous because he seeks to evade the restrictions associated to Postulates 1, 2, and 3, by inquiring into the true nature of magic and its actions.
Literary Remark Harry Potter would do well to reflect upon the words and fate of Captain Ahab:
“All visible objects, man, are but as pasteboard masks. But in each event — in the living act, the undoubted deed — there, some unknown but still reasoning thing puts forth the mouldings of its features from behind the unreasoning mask. If man will strike, strike through the mask! How can the prisoner reach outside except by thrusting through the wall? To me, the white whale is that wall, shoved near to me. Sometimes I think there’s naught beyond. But ’tis enough. He tasks me; he heaps me; I see in him outrageous strength, with an inscrutable malice sinewing it. That inscrutable thing is chiefly what I hate; and be the white whale agent, or be the white whale principal, I will wreak that hate upon him. Talk not to me of blasphemy, man; I’d strike the sun if it insulted me. For could the sun do that, then could I do the other; since there is ever a sort of fair play herein, jealousy presiding over all creations. Ohg abg zl znfgre, zna, vf rira gung snve cynl. Jub’f bire zr? Gehgu ungu ab pbasvarf.”
Conclusion Harry Potter’s quest to restore Hermione Granger may be leading him and the Hogwarts crew to a similarly disastrous fate as Ahab and the Pequod crew.
An elaboration of the above argument now appears on Shtetl Optimized, essentially as a meditation on the question: What strictly mathematical proposition would comprise rationally convincing evidence that the key linear-quantum postulates of “One Ghost in the Quantum Turing Machine* amount to “an unredeemed claim [that has] become a roadblock rather than an inspiration” (to borrow an apt phrase from Jaffe and Quinn’s arXiv:math/9307227).
Readers of Not Even Wrong seeking further (strictly mathematical) mathematical illumination in regard to these issues may wish to consult Arnold Neumaier and Dennis Westra’s textbook-in-progress Classical and Quantum Mechanics via Lie Algebras (arXiv:0810.1019, 2011), whose Introduction states:
“The book should serve as an appetizer, inviting the reader to go more deeply into these fascinating, interdisciplinary fields of science. … [We] focus attention on the simplicity and beauty of theoretical physics, which is often hidden in a jungle of techniques for estimating or calculating quantities of interest.”
That the Neumaier/Westra textbook is an unfinished work-in-progress constitutes proof prima facie that the final tractatus upon these much-discussed logico-physico-philosophicus issues has yet to be written! :)
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.
“These conversations [are] very stressful to all involved … there are language and even philosophical barriers to be overcome.”
The entanglement(s) of hot-noisy-evolved biological cognition with abstract ideals of cognition that Eliezer Yudkowsky vividly describes in Harry Potter and the Methods of Rationality, and the quantum entanglement(s) of dynamical flow with the physical processes of cognition that Scott Aaronson vividly describes in Ghost in the Quantum Turing Machine, both find further mathematical/social/philosophical echoes in Joshua Landsberg’s Tensors: Geometry and Applications (2012), specifically in Landsberg’s thought-provoking introductory section Section 0.3: Clash of Cultures (this introduction is available as PDF on-line).
E.g., the above discussions above relating to “map versus object” distinctions can be summarized by:
Aaronson’s Law of Ontic Mixing “We can’t always draw as sharp a line as we’d like between map and territory”.
as contrasted with the opposing assertion
Landsberg’s No-Mixing Principle “Don’t use coordinates unless someone holds a pickle to your head”
As Landsberg remarks
“These conversations [are] very stressful to all involved … there are language and even philosophical barriers to be overcome.”
The Yudkowsky/Aaronson philosophical divide is vividly mirrored in the various divides that Landsberg describes between geometers and algebraists, and mathematicians and engineers.
Question Has it happened before, that philosophical conundrums have arisen in the course of STEM investigation, then been largely or even entirely resolved by further STEM progress?
Answer Yes of course (beginning for example with Isaac Newton’s obvious-yet-wrong notion that “absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external”).
Conclusion It may be that, in coming decades, the philosophical debate(s) between Yudkowsky and Aaronson will be largely or even entirely resolved by mathematical discourse following the roadmap laid down by Landsberg’s outstanding text.
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:
Theorem The flow of any Hamiltonian vector field consists of canonical transformations
Proof (Hogwarts version) …
Proof (Muggles version) …
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.
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!
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):
Kohn’s Provocative Statement In general the many-electron wavefunction
) for a system of electrons is not a legitimate scientific concept, when , where .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)!
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?
Gjm asks “Along what vector field V are you taking the Lie derivative?
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:
Question For hot noisy quantum dynamical systems (like brains), what is the lowest-dimension varietal state-space for which Bob’s simulation data-record cannot be verifiably distinguished from Alice’s simulation data-record? In particular, do polynomially-many varietal dimensions suffice for Bob’s record to be indistinguishable from Alice’s?
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!
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!
The dynamicist Vladimir Arnold had a wonderful saying:
“Every mathematician knows that it is impossible to understand any elementary course in thermodynamics.”
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 equationthus 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!
Thank you gjm. To the best of my understanding, (1) all markup glitches are fixed; (2) all links are live; and (3) an added paragraph (fourth-from-last) now explicitly links dynamic-J methods to Scott’s notion of “freebits”.
A preprint would be terrific too.
A tough(?) question and a tougher(?) question: When self-modifying AI’s are citizens of Terry Tao’s Island of the Blue-Eyed People/AIs, can the AIs trust one another to keep the customs of the Island? On this same AI-island, when the AI’s play the Newcomb’s Paradox Game, according to the rules of balanced advantage, can the PredictorAIs outwit the ChooserAIs, and still satisfy the island’s ProctorAIs?
Questions in this class are tough (as they seem to me), and it is good to see that they are being creatively formalized.