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July 28, 2025

The Duflo Isomorphism and the Harmonic Oscillator Hamiltonian

Posted by John Baez

Quick question. Classically the harmonic oscillator Hamiltonian is often written 12(p 2+q 2)\frac{1}{2}(p^2 + q^2), while quantum mechanically it gets some extra ‘ground state energy’ making the Hamiltonian

H=12(p 2+q 2+1) H = \frac{1}{2}(p^2 + q^2 + 1)

I’m wondering if there’s any way to see the extra +12+ \frac{1}{2} here as arising from the Duflo isomorphism. I’m stuck because this would seem to require thinking of HH as lying in the center of the universal enveloping algebra of some Lie algebra, and while it is in the center of the universal enveloping algebra of the Heisenberg algebra, that Lie algebra is nilpotent, so it seems the Duflo isomorphism doesn’t give any corrections.

Continue reading “The Duflo Isomorphism and the Harmonic Oscillator Hamiltonian” …
Posted at 1:48 PM UTC | Permalink | Followups (5)

July 25, 2025

The Clowder Project

Posted by John Baez

guest post by Emily de Oliveira Santos

I’d like to share here a personal project which might be of interest to the readers of this blog: the Clowder Project.

Clowder is a wiki and reference work for category theory built using the same general infrastructure and tag system of the Stacks Project, Gerby. The intention is for it to eventually become for category theory what the Stacks Project is for algebraic geometry.

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Posted at 11:18 AM UTC | Permalink | Followups (3)

July 24, 2025

2-Rig Conjectures Proved?

Posted by John Baez

Kevin Coulembier has come out with a paper claiming to prove some conjectures that Todd Trimble, Joe Moeller and I made in 2-Rig extensions and the splitting principle:

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Posted at 2:58 PM UTC | Permalink | Followups (5)

Lawvere’s Work on Arms Control

Posted by John Baez

Did you know that Lawvere did classified work on arms control in the 1960s, back when he was writing his thesis? Did you know that the French government offered him a job in military intelligence?

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Posted at 12:48 PM UTC | Permalink | Followups (5)

July 7, 2025

How to Count n-Ary Trees

Posted by John Baez

How do you count rooted planar nn-ary trees with some number of leaves? For n=2n = 2 this puzzle leads to the Catalan numbers. These are so fascinating that the combinatorist Richard Stanley wrote a whole book about them. But what about n>2n \gt 2?

I’ll sketch one way to solve this puzzle using generating functions. This will give me an excuse to talk a bit about something called ‘Lagrange inversion’.

Continue reading “How to Count n-Ary Trees” …
Posted at 8:24 AM UTC | Permalink | Followups (7)