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mitchell porter

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**A group-theorist's perspective on symmetry groups in physics**

Robert Arnott Wilson

*[Submitted on 29 Sep 2020 (v1), last revised 12 Nov 2020 (this version, v3)]*

R.A. Wilson (physics blog) worked on finite simple groups such as the famous "Monster". He writes:There are many Lie groups used in physics, including the Lorentz group of special relativity, the spin groups (relativistic and non-relativistic) and the gauge groups of quantum electrodynamics and the weak and strong nuclear forces. Various grand unified theories use larger Lie groups in different attempts to unify some of these groups into something more fundamental. There are also a number of finite symmetry groups that are related to the finite number of distinct elementary particle types. I offer a group-theorist's perspective on these groups, and suggest some ways in which a deeper use of group theory might in principle be useful. These suggestions include a number of options that seem not to be under active investigation at present. I leave open the question of whether they can be implemented in physical theories.

"One thing that immediately strikes a group-theorist... is that macroscopic physics is mostly described by real and/or orthogonal Lie groups, while quantum physics is mostly described by complex and/or unitary Lie groups... The two main questions to decide are (a) whether to try to transplant the macroscopic real/orthogonal groups to quantum physics, or the quantum complex/unitary groups to macroscopic physics, and (b) how to relate the finite groups to the Lie groups."

His speculations seem rather cavalier (e.g. hoping that "QCD can be implemented with the split real form... of the gauge group"), and it may be that much of the paper actually warrants a rebuttal. But perhaps there are valid ideas in there too. Also, the section on Clifford algebras may interest some readers here.