Double cover of SO(3) is SU(2)! Hurrah!
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Replying to @MathPrinceps
And also, the usual 2d complex rep of SU(2) is a 1d quaternionic rep! All self-dual complex irreps of groups are either complexifications of real reps or the underlying complex reps of quaternionic reps.
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Replying to @johncarlosbaez
I wish more people would read Penrose and Rindler. Assiduously. What a marvelous thing it is.
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Replying to @MathPrinceps
I should have read Penrose and Rindler more assiduously; for some reason I didn't have it at hand when I was focused on general relativity.
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Replying to @johncarlosbaez
I confess to considerable frustration with contemporary expositions of general relativity. The best of these have many excellences, but none of them starts at the beginning.
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Replying to @MathPrinceps @johncarlosbaez
What is spacetime, and why should it be (modeled by) a manifold? Why should it have the various geometrical structures ascribed to it by general relativity? Why should these be interrelated and constrained in the way the theory posits? These questions have interesting answers.
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Replying to @MathPrinceps @johncarlosbaez
It has always frustrated me that these answers, though illuminating and elegant and mostly more than 45 years old, still find no place in contemporary expositions of the theory, which for that reason remains more obscure than necessary. (C.f., e.g.: http://bit.ly/2wcF5jp )
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Replying to @MathPrinceps @johncarlosbaez
Perhaps the most frustrating aspect of this situation is that these answers, largely anticipated by Hermann Weyl, cohere so beautifully -- just as do the constituent geometries that Hilbert recognized in Euclid -- to account for the naively postulated geometry of spacetime.
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Replying to @MathPrinceps @johncarlosbaez
It's profoundly sad that classical projective geometry is now largely a lost art; for it is not only fundamental to understanding the mathematical structures of quantum mechanics, but also to understanding those of general relativity. (Dirac, too, bemoaned this loss.)
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Here, for example, is a marvelous geometrical gem, now long forgotten, which ought to be far better known: http://bit.ly/2vRhk0W
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Replying to @MathPrinceps @johncarlosbaez
Aha! I have failed to keep up properly, it seems. Now, at last, perhaps this book fills the gap that has so long irritated me: http://bit.ly/2PjacCE I have yet to study it, but it certainly looks promising. Thank god!
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Replying to @MathPrinceps
For projective geometry and QM, I found Varadarajan's book nice. I was really into the foundations of QM back when I was a grad student. https://www.amazon.com/Geometry-Quantum-Theory-V-S-Varadarajan/dp/0387493859 …
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