From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2477 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: quantum logic Date: Sat, 18 Oct 2003 16:57:00 -0400 (EDT) Message-ID: References: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018687 4338 80.91.229.2 (29 Apr 2009 15:24:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:47 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Mon Oct 20 14:51:18 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 20 Oct 2003 14:51:18 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1ABeBR-0006nf-00 for categories-list@mta.ca; Mon, 20 Oct 2003 14:51:09 -0300 In-Reply-To: <200310122208.h9CM8sf26075@math-cl-n01.ucr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 28 Original-Lines: 82 Xref: news.gmane.org gmane.science.mathematics.categories:2477 Archived-At: 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 = \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.) > > > > >