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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