## "Symplectic, Quaternionic, Fermionic", John Baez, 2014

#### Reviewed February 4, 2024

Citation: Baez, John. "Symplectic, Quaternionic, Fermionic", This Week's Finds (2014)

Tags: Philosophical, Pedagogical

This amazing blog post clarifies the different definitions of the symplectic group. The main tagline is that bosons are real and fermions are quaternionic. The idea for why fermions are quaternionic is that the usual space of operators on a spin 1/2 particle is a two-dimensional Hilbert space. When you add time-reversal into the mix you get another square root of -1 since time reversal squares to -1 on fermions. This is your j^2=-1 to make things quaternionic.

My favorite thing about this post is that Baez also writes out a discussion he had online with a friend Toby Bartels. Toby Bartels was unsure about the subject, had lots of questions, and was trying to build his intuition. To make the conversation go smoother, Baez spits a piece of pedagogical gold: "Don't calculate, guess!". This is a fantastic quote, which Baez uses twice in the post. To quote Baez: "I'm just testing your instincts".

The line "Don't calculate, guess!" is a great one, which I'll be sure to use going forward.

Another good article to read after this one: The Tenfold Way. There's a blog post , as well as a paper

> Baez, John C. "The tenfold way." arXiv preprint arXiv:2011.14234 (2020).

Another related paper is "The Quantization of Linear Dynamical Systems I: (Mostly!) Finite Systems", which can be found online here: https://personal.lse.ac.uk/robert49/teaching/partiii/pdf/QuantOfLinearFinite16Novem2021.pdf. This one has many nice quotes in it as well, including this one: “First quantization is a mystery, but second quantization is a functor.” — attributed to Edward Nelson.