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
baez@math.removethis.ucr.andthis.edu
