User talk:YohanN7

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Math reference desk[edit]

Hi, I quickly skimmed your post to the math reference desk; I can't reply there .. not allowed, but thought I'd give a quick reply here directly. In short -- some of the qft functionals can be made mathematically rigorous, but if requires a math background far exceeding that available to a typical physics PhD (and teaching it would add on another 5 years to the students time-in-class). Yes, the WP articles on QFT are mostly abysmal, and it will take huge amounts of work to fix them, and show when/where more rigorous approaches are possible. The latest research, though is quite interesting: here's one from the grand-master Alain Connes himself: from which I quote: "This paper gives a complete selfcontained proof of our result announced in hep-th/9909126 showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $\Hc$ which is commutative as an algebra....". Going back in time... one reason that supersymmetry initially got so exciting is that many variants were finite and required no renormaliztion. There was also a huge amount of interest getting a better grip on differential eqns in general, e.g. exactly solvable models, one poster-child of which was the Yang-Baxter equation .. which for example, describes the quantum deformations of the classical and affine Lie algebras (among other things). Another approach is the operator product expansion which avoids the problem of time-ordering of the quantum field operators by focusing solely on the operators for measureable quantities. So there's a huge amount of material here, its just so big its hard to grasp, and yes the WP articles in this area all need major improvements. (talk) 14:22, 19 September 2016 (UTC)

Re-reading your post -- maybe I can provide a much much simpler answer. First, there is no prescription for taking some diff eq describing some symplectic geometry (the diff eq of classical mechanics) and replacing the p's and q's by operators to get a valid first-quantized operator eqn. One can almost get there, the first steps are studied under the name of prequantization or geometric quantization, e.g. the Moyal bracket. One can perform a second quantization of lagrangians, though, and this is covered in almost any book on QFT. The mathematical methods thread their way through the theory of the Fredholm alternative, and lead into e.g. the trace-class operators and the study of topological vector spaces. The compact topological vector spaces are the same thing as universal enveloping algebras, thanks to the Gelfand–Naimark theorem (see also Tannaka-Krein duality). The quantum deformation of these leads to the quantum groups, for example, the Weyl algebra is the quantum deformation of the symplectic form --again anything with the word "symplectic" in it is really about classical mechanics. There are some marvelous books on classical mechanics that cover this: a very old but-still-modernish one is Abraham and Marsden from the 1970's. Another example are the clifford algebras which are deformations of the ordinary quadratic form aka metric tensor. Note that clifford algebras describe spin, which is why spin manifolds are studied. Lets see. Then the rotation group describes rotations in 3D space. Its covered by the spin group, which is covered by the string group which is covered by the 5-brane group in the Postnikov tower. So there is a huge effort to convert those squonky QFT theories, and to take any arbitrary lagrangian or hamiltonian, or poisson bracket and quantize it. Note that most possion brackets end up turning into Hopf algebras, which is why there's so much focus on that. The other key insight is that supersymmetry is the local, gauged version of the exterior algebra. Since the exterior algebra is central for solving differential equations, (by means of jet bundles, its not surprising that the locally gauged version of it is even more interesting.... (talk) 14:54, 19 September 2016 (UTC)
Thanks for your detailed reply, and I apologize for my late response (I have been taking a little break from Wikipedia). It will take me some time (probably plenty) to digest what you have written here. Some of the concepts I just barely know the definition (or statement) of. Thanks again! YohanN7 (talk) 07:59, 29 September 2016 (UTC)

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You might be interested in my remarks in Talk:Minkowski space. — Preceding unsigned comment added by (talk) 10:12, 21 October 2016 (UTC)

A further edit has now appeared. — Preceding unsigned comment added by (talk) 10:25, 21 October 2016 (UTC)
Thanks. I have that page on my watchlist. YohanN7 (talk) 11:22, 21 October 2016 (UTC)