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I followed the link, that's what he means by 'cool stuff that's missing': > In short: > The key to Noether’s theorem is the requirement that we can freely rei

by sesm 2y ago

I followed the link, that's what he means by 'cool stuff that's missing': > In short: > The key to Noether’s theorem is the requirement that we can freely reinterpret observables as symmetry generators, and vice versa — in a way that’s consistent with the action of symmetry generators on both observables and symmetry generators. > In classical mechanics this is achieved by a hybrid structure: a Poisson algebra, whose elements are both observables and symmetry generators. > In an algebraic approach to quantum theory, this requirement singles out complex quantum mechanics. i =√−1 turns observables into symmetry generators, and vice versa. How would one explain this to the audience of Quanta Magazine?

graycat 2y ago
How? First have a clear statement of the context. Second, define and explain all the obscure terminology and mathematical notation -- at least give good references. Gee, in one discussion of Lagrangians, saw "configuration space" and "constraints". Okay, studied topological spaces, vector spaces, inner product spaces, measure spaces, Banach spaces, Hilbert spaces, probability spaces, but never saw a definition or explanation of a "configuration space". "Constraints"? Kuhn-Tucker optimization theory has a lot on constraints, for one of the issues there was a question, and I solved and published it. Thought I had some background in "constraints", but the Lagrangian explanations didn't make clear what they meant by "constraints". A "symmetry generator"? This is the first I ever heard of any such thing although ugrad math honors paper was on group representations. Then there was "phase space": What does that have to do with "phase" in light waves and sound waves? Sounds like one word with two different meanings? E.g., "Poisson algebra": Okay, there is the Poisson stochastic process, e.g., seems to get assumed in "half-life" calculations, and with more in Erhan \c Cinlar, {\it Introduction to Stochastic Processes,\/} ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ, 1975.\ \ and there is abstract algebra, e.g., groups, rings, fields, ... But a "Poisson algebra"?

by sesm 2y ago

graycat 2y ago

How? First have a clear statement of the context. Second, define and explain all the obscure terminology and mathematical notation -- at least give good references. Gee, in one discussion of Lagrangians, saw "configuration space" and "constraints". Okay, studied topological spaces, vector spaces, inner product spaces, measure spaces, Banach spaces, Hilbert spaces, probability spaces, but never saw a definition or explanation of a "configuration space". "Constraints"? Kuhn-Tucker optimization theory has a lot on constraints, for one of the issues there was a question, and I solved and published it. Thought I had some background in "constraints", but the Lagrangian explanations didn't make clear what they meant by "constraints". A "symmetry generator"? This is the first I ever heard of any such thing although ugrad math honors paper was on group representations. Then there was "phase space": What does that have to do with "phase" in light waves and sound waves? Sounds like one word with two different meanings? E.g., "Poisson algebra": Okay, there is the Poisson stochastic process, e.g., seems to get assumed in "half-life" calculations, and with more in Erhan \c Cinlar, {\it Introduction to Stochastic Processes,\/} ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ, 1975.\ \ and there is abstract algebra, e.g., groups, rings, fields, ... But a "Poisson algebra"?