Geometric Representation Theory (Lecture 25)
Posted by John Baez
This time in the Geometric Representation Theory seminar, we showed how to categorify the commutation relations between annihilation and creation operators for the harmonic oscillator:
$a a^* = a^* a + 1$
obtaining an isomorphism of spans of groupoids:
$A A^* \cong A^* A + 1$
This reduces a basic ingredient of quantum field theory to pure combinatorics, not involving the continuum in any form.
Even better, we did an inclass experiment demonstrating these commutation relations!

Lecture 25 (Jan. 22)  John Baez on
groupoidifying the harmonic oscillator. The annihilation
and creation operators as spans of groupoids. The groupoidified
commutation relations:
$A A^* \cong A* A + 1$
Demonstration: an actual experiment proving these commutation
relations! Weak pullbacks for composing spans of groupoids.
Examples of weak pullbacks. Using weak pullbacks to compute $A A^*$
and $A^* A$.

Streaming
video in QuickTime format; the URL is
http://mainstream.ucr.edu/baez_01_22_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare

Streaming
video in QuickTime format; the URL is
Re: Geometric Representation Theory (Lecture 25)
In the lecture notes it says:
The answer to the second question has been postponed to next week, I assume?
Would you mind giving a hint for someone too lazy to go through this and figure it out?
Did you think about groupoidifying the exponentiated creation and annihilation operators $\exp( A )$ and $\exp(A^*)$?
Given that you can say what a sum of spans is, can you say what a series $\mathrm{Id} + A + \frac{1}{2} A \circ A + \frac{1}{6} A \circ A \circ A + \cdots$ of spans is?