Re: quantum logic

[Michael Barr <barr@barrs.org>]

After giving the matter some thought, I finally decided that the
category of Hilbert spaces should have as its morphisms norm-reducing
linear maps.  At the very least that will ensure that an isomorphism is
an isometry.  And yes the category of finite dimensional Hilbert spaces
is *-autonomous.  The internal hom of two is the space of all linear
maps and the inner product <f,g> = \sum f(u_i)g(u_i), taken over an
orthonormal basis of the domain.  This can be shown to be invariant
under orthogonal change of  basis.  The norm of the identity on an
n-dimensional space is sqrt(n).  The dual of a space is itself, of
course with the duality being adjunction (or transpose).  Then the
tensor product H # G = (H --o G^*)^*.

Here is another approach to the same structure.  Consider a pair
(V,\phi) where V is a finite dimensional space and \phi is an
isomorphism of V with its dual space.  You have to add positive
definiteness and symmetry, but that is no problem.  Maps again are norm
reducing.  Now we can define (V,\phi) # (W,\psi) = (V # W,\phi # \psi),
(V,\phi)^* = (V,\phi^{-1}), and (V,\phi) --o (W,\psi) = (V # W,\phi^{-1}
# \psi).  The resultant category is exactly the same as before.

BTW, it is easy to see that the transpose of a norm-reducing map is norm
reducing.


On Sun, 12 Oct 2003, John Baez wrote:

> 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.)
>
>
>
>
>