Fred E.J. Linton wrote:
Peter Selinger offered the thought that, considering
... the category of finite dimensional complex vector spaces vs. the category of finite dimensional Hilbert spaces. They are equivalent ...
Hmmm ... you mean just *any* linear transformation is allowed between two Hilbert spaces?
In applications to quantum mechanics people really want to work with both unitary and self-adjoint operators, and often others as well. So they work with the category of finite-dimensional Hilbert spaces and *all* linear maps between these. As a mere category this is equivalent to the category of finite-dimensional vector spaces - so to understand the "Hilbertness" of Hilbert spaces, they introduce a dagger structure as well. (The infinite-dimensional case would introduce extra wrinkles, like unbounded self-adjoint operators. It's possible that only after we treat this case correctly can we declare that we know what's going on. Perhaps trying to treat both unitary and self-adjoint operators as morphisms in the same category is simply a bad idea. There are a lot of options worth exploring.) If so, I'm not so sure my Hilbert spaces are the same as yours :-) .
Indeed! If you treat Hilbert spaces as "sets with structure", the obvious morphisms are isometries - inner-product-preserving linear operators. But in quantum theory, Hilbert spaces are being used for something quite different. And so there's a struggle going on to understand this. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]