In a question answered on this site concerning how group theory worked its way into quantum mechanics it was proposed that this was done by Weyl while investigating the many electron problem. Yes, but how then did a problem concerning many electrons by use of groups and symmetry permutations end up with formulations of Lie groups predicting the existence of the very particles that were assumed to exist from the many electron problem. This fanciful idea again worked its way into the Standard Model to predict quarks using the "eightfold" way. And to mystify this even more the essence of the unification of the particles is actually seen when the very symmetry used to predict the particles is broken. Very puzzling. Any historical maps of how symmetry flows through the Standard Model is welcome.

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    $\begingroup$ Group theory is just a mathematical tool, which once proved useful in one area started to be applied in others. Why is it puzzling? A big help was Yang and Mills introducing non-Abelian gauge theories in 1950-s, they explicitly involved Lie groups. Wikipedia's article on the Eightfold Way has a history section which describes how that idea came up. $\endgroup$ – Conifold Apr 21 '17 at 21:31
  • $\begingroup$ Your link "Yang and Mills introducing non-Abelian gauge theories in the 1950-s" answered my question! $\endgroup$ – Sedumjoy Apr 22 '17 at 17:23
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    $\begingroup$ In point of fact the connection to gauge theories is only half the story--the least interesting half. Global Lie group symmetries entered physics through the rotation group in the 1819th century; SU(3) in nuclear physics ("Elliott model") in the 1950s; and in particle physics ("Eightfold way") in 1960. They trained an entire generation of particle physicists in Lie algebras beyond SU(2) and enabled them to gauge those in the late 1960s. $\endgroup$ – Cosmas Zachos Mar 31 '18 at 21:40