Re: quantum logic

[Pedro Resende <pmr@math.ist.utl.pt>]

Hi John,

One of the ideas behind the theory of quantales is that the category of
"sheaves" on a given quantale should be a topos in *some* generalized
sense, whose subobject classifier would be a quantale (which is then
related to the multiplicative fragment of a noncommutative linear
logic). There are a few papers by various authors addressing sheaves on
quantales, however none getting near a satisfactory definition of
"quantum topos", but I don't think the field is exhausted. In
particular Chris Mulvey wrote a paper with his student Nawaz (you can
download it from Chris' web page), but restricted to idempotent
right-sided quantales, which form a rather limited class. Nevertheless
that paper gives you a category of sheaves which actually is a topos in
the classical sense, but equipped with additional structure that
provides the "quantum" part. I know that currently he has been working
with another student on an extension of this to a more general
situation encompassing all involutive quantales (= involutive monoids
in the monoidal category of sup-lattices), and last time I heard about
it the results looked promising.

The significance of this wrt Hilbert spaces is that once you consider
involutive quantales of the form "Max(A)" (ie, those consisting of all
the closed linear subspaces of a unital C*-algebra A), there is a
notion of "irreducible representation" of Max(A) that classifies up to
unitary equivalence the irreducible representations of A, and, to a
certain extent still in need of further clarification (very preliminary
material is in a paper of mine which is due to appear in the J. Algebra
and is downloadable from my web page - still a couple of typos and
minor bugs in the on-line version, I'm afraid), the category of
representations of A is approximated by the corresponding category of
quantale modules over Max(A). (Each representation of A on a Hilbert
space H induces in a natural way an action of Max(A) on the lattice of
closed linear subspaces of H.) By all of this I mean that ultimately
the category of sheaves on Max(A) should provide a logical handle on
the category of representations of A, and it seems reasonable to expect
that what you are saying about the category of Hilbert spaces and
bounded linear maps may relate to this general scheme.

Best,

Pedro.



On Sunday, October 12, 2003, at 01:57 AM, John Baez wrote:

> Dear Categorists -
>
> Do any of you know particularly insightful treatments of
> quantum logic via category theory?  I'm more or less familiar
> with quantum logic as the theory of the complete orthocomplemented
> lattice of closed subspaces of a given Hilbert space.  But now I'm
> interested in developing quantum logic starting as much as possible
> from general properties of and structures on the category of
> Hilbert spaces and bounded linear maps - for example, the fact
> that it's an abelian category, and becomes a *-category and symmetric
> monoidal category in a nice way (with Hilbert tensor product as the
> monoidal structure).  And I'm interested in things like how the
> 2-dimensional Hilbert space acts a bit like a subobject classifier.
>
> I don't mind sticking with finite-dimensional Hilbert spaces for now
> to avoid certain subtleties.
>
> On a related note: I've repeatedly heard people say something
> like "the multiplicative fragment of linear logic is the internal
> logic of (closed symmetric?) monoidal categories", but I've never heard
> a precise result along these lines.  Has anyone worked out a
> sufficiently
> general concept of "the internal logic of a category" or "the
> internal logic of a certain 2-category of categories" so that one
> could take something like a monoidal category, or a symmetric monoidal
> category, or a closed symmetric monoidal category - or maybe the
> 2-category of all such - and extract by some systematic method the
> corresponding "internal logic"?  I'm vaguely imagining some class
> of generalizations of the Mitchell-Benabou language of a topos, or
> something like that - but I'm really interested in the nonCartesian
> case.
>
> The reason I ask this is that it would be nice if you could
> throw the (closed, symmetric, monoidal, *, etcetera...) category
> of Hilbert spaces into some big machine and have "quantum logic"
> pop out - and then throw in other similar categories, and have other
> kinds of logic pop out.
>
> Best,
> jb
>
>
>
>