re: quantum logic

["John Baez" <baez@math.ucr.edu>]

Michael Barr wrote:

> I will let others answer about the connection between closed monoidal
> categories and MLL, but I just wanted to say that I am not sure what you
> mean by the category of Hilbert spaces. If you want the inner product
> preserved, then only isometric injections are permitted.  If you want just
> bounded linear maps then you are not making any real use of the inner
> product.

Right.  I wanted to leave things flexible so different readers could
interpret my question in different ways, but I also tried to hint
that I think it's crucial to work with the *-category Hilb whose objects
are Hilbert spaces, whose morphisms are bounded linear maps, and whose
*-structure sends the bounded linear map f: H -> H' to its Hilbert
space adjoint f*: H' -> H.  This *-structure can be used to define
concepts crucial for quantum mechanics, like "self-adjoint" and
"unitary" operators, as well as "isometric injections".  Isometric
injections are a nice way to study subobjects in Hilb, but they're
not good enough for doing full-fledged quantum mechanics, nor is
ignoring the inner product altogether.

Category theorists are often a bit uncomfortable with *-categories
because they prefer "adjoints" that are defined using other structure
rather than put in by brute force.  However, I'm convinced that we
can only understand how quantum field theory exploits the analogy
between differential topology and Hilbert space theory if we think
about *-categories.  For example, a topological quantum field theory
is a symmetric monoidal functor from some *-category of cobordisms
to the *-category Hilb - but the most physically realistic TQFTs are
the "unitary" ones, which preserve the *-structure.

I've talked about this *-stuff and the nascent concept of "n-categories
with duals" in my papers on 2-Hilbert spaces

http://math.ucr.edu/home/baez/2hilb.ps

and 2-tangles

http://math.ucr.edu/home/baez/hda4.ps

and now I want to say a bit about how it impinges on quantum
logic - but to avoid reinventing the wheel, I'd like to hear
anything vaguely relevant anyone knows about approaching quantum
logic with an eye on category theory.

(I know a bit about quantales, but maybe there's other stuff
I've never heard of.)