At the moment, I’m taking a quick break from reading some rather dense mathematical prose, and I spent yesterday plugging away at a draft of my research group’s next technical publication. This led me to reflect on a lesson that I think a lot of science education leaves out: Even in a technical article, you have to have a story to carry the progression through. “These are all the boffo weird roadside attractions we found while proving the theorems in our last paper” is honest, but not adequate.
Our research project is the reconstruction of the mathematical formalism of quantum theory from physical principles. We tease apart the theory, identify what is robustly strange about it — for many more quantum phenomena can be emulated with classical stochasticity than are often appreciated — and try to build a new representation that brings the most remarkable features of the physics to the forefront. In special relativity, we have Einstein’s postulates, and the dramatic tension between them: Inertial observers can come to agree upon the laws of physics, but they cannot agree upon a standard of rest. In thermodynamics, we have the Four Laws, which come with their own dramatic tension, in that energy is conserved while entropy is nondecreasing. Both of these theories are expressed in terms of what agents can and cannot do, yet they are more than “mere” engineering, because they apply to all agents. Or, to say it another way, it is to the benefit of any agent to pick up the theory and use it as a guide.
What, then, is the analogue for quantum theory? If the textbook presentation of quantum physics is like the formulae for the Lorentz transform, with all those square roots and whatnot, or the Maxwell relations in thermo, with all those intermingling partial derivatives that we invent hacks about determinants to remember, what is quantum theory’s version of Einstein’s postulates or the Four Laws?
That’s the grandiose version, anyway. The reason I got invited to speak at an American Mathematical Society meeting is that the geometric structures that arise in this work are vexingly fascinating. You want about Galois fields and Hilbert’s 12th problem? We’ve got ’em! How about sphere packing and unexpected octonions? We’ve got those, too! And the structure that leads down the latter path turns out, on top of that, to yield a new way of thinking about Mermin’s 3-qubit Bell inequality. It is all lovely, and it is all strange.
The SIC problem gives us the opportunity to travel all throughout mathematics, because, while the definition looks pretty small, the question is bigger on the inside.
