Rancor commonly arises when STEM discussions in general, and discussions of quantum mechanics in particular, focus upon personal beliefs and/or personal aesthetic sensibilities, as contrasted with verifiable mathematical arguments and/or experimental evidence and/or practical applications.
In this regard, a pertinent quotation is the self-proclaimed “personal belief” that Scott asserts on page 46:
“One obvious way to enforce a macro/micro distinction would be via a dynamical collapse theory. … I personally cannot believe that Nature would solve the problem of the ‘transition between microfacts and macrofacts’ in such a seemingly ad hoc way, a way that does so much violence to the clean rules of linear quantum mechanics.”
Scott’s personal belief calls to mind Nature’s solution to the problem of gravitation; a solution that (historically) has been alternatively regarded as both “clean” or “unclean”. His quantum beliefs map onto general relativity as follows:
General relativity is “unclean” “We can be confident that Nature will not do violence to the clean rules of linear Euclidean geometry; the notion is so repugnant that the ideas of general relativity CANNOT be correct.”
as contrasted with
General relativity is “clean” “Matter tells space how to curve; space tells matter how to move; this principle is so natural and elegant that general relativity MUST be correct!”
Of course, nowadays we are mathematically comfortable with the latter point-of-view, in which Hamiltonian dynamical flows V are naturally associated to non-vanishing Lie derivatives L of metric structures g, that is mathcal{L}_{V}gne0.
This same mathematical toolset allow us to frame the ongoing debate between Scott and his colleagues in mathematical terms, by focusing our attention not upon the metric structure g, but similarly upon the complex structure J.
In this regard a striking feature of Scott’s essay is that it provides precisely one numbered equation (perhaps this a deliberate echo of Stephen Hawking’s A Brief History of Time, which also has precisely one equation?). Fortunately, this lack is admirably remedied by the discussion in Section 8.2 “Holomorphic Objects” of Andrei Moroianu’s textbook Lectures on Kahler Geometry. See in particular the proof arguments that are associated to Moroianu’s Lemma 8.7, which conveniently is freely available as Lemma 2.7 of an early draft of the textbook, that is available on the arxiv server as arXiv:math/0402223v1. Moroianu’s draft textbook is short and good, and his completed textbook is longer and better!
Scott’s aesthetic personal beliefs naturally join with Moroianu’s mathematical toolset to yield a crucial question: Should/will 21st Century STEM researchers embrace with enthusiasm, or reject with disdain, dynamical theories in which mathcal{L}_{V}Jne0?
Scott’s essay is entirely correct to remind us that this crucial question is (in our present state-of-knowledge) not susceptible to any definitively verifiable arguments from mathematics, physical science, or philosophy (although plenty of arguments from plausibility have been set forth). But on the other hand, students of STEM history will appreciate that the community of engineers has rendered a unanimous verdict: mathcal{L}_{V}Jne0 in essentially all modern large-scale quantum simulation codes (matrix product-state calculations provide a prominent example).
So to the extent that biological systems (including brains) are accurately and efficiently simulable by these emerging dynamic-J methods, then Scott’s definition of quantum dynamical systems may have only marginal relevance to the practical understanding of brain dynamics (and it is plausible AFAICT that this proposition is entirely consonant with Scott’s notion of “freebits”).
Here too there is ample precedent in history: early 19th Century textbooks like Nathaniel Bowditch’s renowned New American Practical Navigator (1807) succinctly presented the key mathematical elements of non-Euclidean geometry (many decades in advance of Gauss, Riemann, and Einstein).
Will 21st Century adventurers learn to navigate nonlinear quantum state-spaces with the same exhilaration that adventurers of earlier centuries learned to navigate first the Earth’s nonlinear oceanography, and later the nonlinear geometry of near-earth space-time (via GPS satellites, for example)?
Conclusion Scott’s essay is right to remind us: We don’t know whether Nature’s complex structure is comparably dynamic to Nature’s metric structure, and finding out will be a great adventure! Fortunately (for young people especially) textbooks like Moroianu’s provide a well-posed roadmap for helping mathematicians, scientists, engineers—and philosophers too—in setting forth upon this great adventure. Good!
Rancor commonly arises when STEM discussions in general, and discussions of quantum mechanics in particular, focus upon personal beliefs and/or personal aesthetic sensibilities, as contrasted with verifiable mathematical arguments and/or experimental evidence and/or practical applications.
In this regard, a pertinent quotation is the self-proclaimed “personal belief” that Scott asserts on page 46:
Scott’s personal belief calls to mind Nature’s solution to the problem of gravitation; a solution that (historically) has been alternatively regarded as both “clean” or “unclean”. His quantum beliefs map onto general relativity as follows:
as contrasted with
Of course, nowadays we are mathematically comfortable with the latter point-of-view, in which Hamiltonian dynamical flows V are naturally associated to non-vanishing Lie derivatives L of metric structures g, that is mathcal{L}_{V}gne0.
This same mathematical toolset allow us to frame the ongoing debate between Scott and his colleagues in mathematical terms, by focusing our attention not upon the metric structure g, but similarly upon the complex structure J.
In this regard a striking feature of Scott’s essay is that it provides precisely one numbered equation (perhaps this a deliberate echo of Stephen Hawking’s A Brief History of Time, which also has precisely one equation?). Fortunately, this lack is admirably remedied by the discussion in Section 8.2 “Holomorphic Objects” of Andrei Moroianu’s textbook Lectures on Kahler Geometry. See in particular the proof arguments that are associated to Moroianu’s Lemma 8.7, which conveniently is freely available as Lemma 2.7 of an early draft of the textbook, that is available on the arxiv server as arXiv:math/0402223v1. Moroianu’s draft textbook is short and good, and his completed textbook is longer and better!
Scott’s aesthetic personal beliefs naturally join with Moroianu’s mathematical toolset to yield a crucial question: Should/will 21st Century STEM researchers embrace with enthusiasm, or reject with disdain, dynamical theories in which mathcal{L}_{V}Jne0?
Scott’s essay is entirely correct to remind us that this crucial question is (in our present state-of-knowledge) not susceptible to any definitively verifiable arguments from mathematics, physical science, or philosophy (although plenty of arguments from plausibility have been set forth). But on the other hand, students of STEM history will appreciate that the community of engineers has rendered a unanimous verdict: mathcal{L}_{V}Jne0 in essentially all modern large-scale quantum simulation codes (matrix product-state calculations provide a prominent example).
So to the extent that biological systems (including brains) are accurately and efficiently simulable by these emerging dynamic-J methods, then Scott’s definition of quantum dynamical systems may have only marginal relevance to the practical understanding of brain dynamics (and it is plausible AFAICT that this proposition is entirely consonant with Scott’s notion of “freebits”).
Here too there is ample precedent in history: early 19th Century textbooks like Nathaniel Bowditch’s renowned New American Practical Navigator (1807) succinctly presented the key mathematical elements of non-Euclidean geometry (many decades in advance of Gauss, Riemann, and Einstein).
Will 21st Century adventurers learn to navigate nonlinear quantum state-spaces with the same exhilaration that adventurers of earlier centuries learned to navigate first the Earth’s nonlinear oceanography, and later the nonlinear geometry of near-earth space-time (via GPS satellites, for example)?
Conclusion Scott’s essay is right to remind us: We don’t know whether Nature’s complex structure is comparably dynamic to Nature’s metric structure, and finding out will be a great adventure! Fortunately (for young people especially) textbooks like Moroianu’s provide a well-posed roadmap for helping mathematicians, scientists, engineers—and philosophers too—in setting forth upon this great adventure. Good!