John Baez

Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, Princeton University

October 3, 2007

Spans in Quantum Theory

Many features of quantum theory — quantum teleportation, violations of Bell's inequality, the no-cloning theorem and so on — become less puzzling when we realize that quantum processes more closely resemble pieces of spacetime than functions between sets. In the language of category theory, the reason is that Set is a "cartesian" category, while the category of finite-dimensional Hilbert spaces, like a category of cobordisms describing pieces of spacetime, is "dagger compact". Here we discuss a possible explanation for this curious fact. We recall the concept of a "span", and show how categories of spans are a generalization of Heisenberg's matrix mechanics. We explain how the category of Hilbert spaces and linear operators resembles a category of spans, and how cobordisms can also be seen as spans. Finally, we sketch a proof that whenever C is a cartesian category with pullbacks, the category of spans in C is dagger compact.

You can see the transparencies for this talk in PDF and Postscript.

For more on this subject try these introductory papers:

Also try these somewhat more technical ones:

Text © 2007 John Baez
Image at top © Aaron Lauda