• Octonions and the Standard Model (Part 13) Posted by John Baez When Lee and Yang suggested that the laws of physics might not be invariant under spatial reflection - that there’s a fundamental difference between left and right - Pauli was skeptical. • In a letter to Victor Weisskopf in January 1957, he wrote: “Ich glaube aber nicht, daß der Herrgott ein schwacher Linkshänder ist.” (I do not believe that the Lord is a weak left-hander.) But just two days after Pauli wrote this letter, Chien-Shiung Wu’s experiment confirmed that Lee and Yang were correct. • There’s an inherent asymmetry in nature. • We can trace this back to how the ’left-handed’ fermions and antifermions live in a different representation of the Standard Model gauge group than the right-handed ones. • And when we try to build grand unified theories that take this into account, we run into the fact that while we can fit the Standard Model gauge group into in various ways, not all these ways produce the required asymmetry. • There’s a way where it fits into , which is too symmetrical to work… and alas, this one has a nice octonionic description!
Article Summaries:
- John Baez’s latest post explores how the octonionic structure of the Standard Model’s gauge group can be understood through the representation theory of Spin groups. He cites Will Sawin’s theorem that Spin(9) contains exactly two conjugacy classes of subgroups isomorphic to Spin(8): one embedded inside the familiar Spin(8) subgroup and a second, “diagonal” subgroup that is not. Baez shows how these arise from natural 2‑to‑1 homomorphisms and distinguishes them by restricting the Weyl spinor representation. The left‑handed subgroup reproduces the observed fermion spectrum, while the diagonal subgroup-though mathematically distinct-fails to match physical data yet appears in octonionic grand‑unified models.
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