Quantum Quandaries-Baez' new paper

[marcus]

Quantum Quandaries---Baez' new paper

quant-ph/0404040

just out

gets down on the key QG issues
sample:

"...propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime. Furthermore, I claim that we can only see this from a category-theoretic perspective — in particular, one that de-emphasizes the primary role of the category of sets and functions.

Part of the difficulty of combining general relativity and quantum theory is that they use different..."

 

[marcus]

a moment of recognition
it is not enough for humans to arrive at a
quantum theory of gravity
we have also to learn why it has
been so hard to get there

what can be learned from the 75 years
of trying to merge quantum theory
with general relativity and them acting
like oil and water

 

[marcus]

selfAdjoint had an earlier link to this paper
I was just reading back in the "Rovelli's program" thread
where John Baez had dropped in and sA mentioned he
was reading this one.
It could be important. The idea of a *-category
Who invented the idea of a *-category?
Baez does not attribute it or cite a reference
It is possible that it is new (I mean in the mathematical sense
of not having been already studied and all the easy theorems picked off).

It is a good idea. one can see that it has possibilities.
I wonder if Baez invented it himself.
But perhaps not. It may be very common and wellknown to people
who are more familiar than I am with category theory.

It is interesting that both Hilbert spaces and cobordisms form
categories which are *-categories. We should discuss this

 

[marcus]

...

It is interesting that both Hilbert spaces and cobordisms form
categories which are *-categories. We should discuss this

reading further, the definition of *-category is on page 10

-----quote-----

What is a *-category? It is a category C equipped with a map sending each morphism f:X → Y
to a morphism f*: Y → X, satisfying
1_X* = 1_X

(fg)* = g*f*,
and
f** = f.

To make Hilb into a *-category we define T* for any bounded linear operator T → H′ to be the adjoint operator T* ′ → H, given by
$$\langle T^* \psi,\phi \rangle = \langle \psi, T\phi \rangle$$
We see by this formula that the inner product on both H and H′ are required to define the adjoint of T.
In fact, we can completely recover the inner product on every Hilbert space from the *-category structure of Hilb...

 

[marcus]

It's nice how he recovers the inner product from the * relation

its like wearing comfortable clothes instead of a suit
if you define Hilb naively then since (,) is structure the maps have
to preserve the innerproduct so only unitaries get to be morphisms
what a drag, very restrictive

but he says look physicists use selfadjoints and other kind of maps besides
unitary, so let's include as morphisms all the bounded (complex-) linear
operators
so the category is basically just the complex vector spaces
there is a low-level topological issue that they should have a topopology
that is compatible with some innerproduct or other and the morphisms should
be continuous in that topology (here already a few little theorems about
hilbertizable topol. linear spaces, clarifying the meaning of hilbertizable, but still nothing hard or surprising)

So we have this category of a kind of linear-space with cts linear mappings as morphisms----about as vanilla as one can get. And he says ahah I have a little trick to show you, notice that the complex numbers C are an object in this category!

And for every space X and for every x in X
there is a natural map T_x: C -> X
injecting C into X, such that it takes the complex number one to the vector x.

Now if this category is a *-category then each morphism
T_x has a buddy-morphism T_x*
satisfying the three axioms above

an easy calculation shows
that
(x,y) = T_x* T_y
where the rhs is just composition of mappings
or to be precise the lhs number is what you get by applying
the rhs map to the number one.

that gives an inner product on the space X and in fact gives the
original innerproduct that the * relation defined on the category came from.

so if you have a *relation you can forget that it came from the objects
being originally hilbertspaces, and then you can recover all the innerproducts
of all the hilbertspaces from the *relation.

the idea of *-category has a quality of gentle insistence, it may get in thru the door by sheer niceness

and then we have to see that a category of Cobordisms which is basic to General Relativity is also a *-category even tho the morphisms are not
even mappings (they are physicomental umbilicals like feynmann diagrams that connect up one situation to another by a kind of trunk or storyline, well you say it however you like)

 

[marcus]

by sometime today (given the ways of the web)
baez has probably put the term
"star-functor"
into mathematics-language

a topological quantum field theory, TQFT,
is among other things a *-functor
from nCob to Hilb

 

[Janitor]

"baez has probably put the term..."

As well as being famous for using "tensor" as a transitive verb!

 

[marcus]

As well as being famous for using "tensor" as a transitive verb!

seriously though, I couldn't find the idea of a star-category
in any of the category-theory stuff I looked in
had you heard or read of the term?

he actually says that *-category talk makes ordinary category
theorists nervous. I don't understand their alleged reluctance
I'm curious about your take on this, please tell me straight (no irony)
so I can be sure I understand

 

[matt grime]

Using star category is just a nicely suggestive way of talking about the opposite category in this familiar setting (to physicists at least). Given any category C, there is the opposite category C^{op} whose objects are those of C and with Hom_{C^op}(X,Y) identified with Hom_{C}(Y,X) by reversing the arrows. The simplest case is Vect the category of vector spaces where op corresponds to taking transposes. It so happens that for Hilb we need not just transpose but conjugate transpose, often denoted by * or $$\dagger$$ (dagger in case the tex doesn't work). Star doesn't make us nervous, it's just that stars or daggers don't fit into the nice slogans about preserving structures that we often make.

 

[eforgy]

an easy calculation shows
that
(x,y) = T_x* T_y
where the rhs is just composition of mappings
or to be precise the lhs number is what you get by applying
the rhs map to the number one.

that gives an inner product on the space X and in fact gives the
original innerproduct that the * relation defined on the category came from.

so if you have a *relation you can forget that it came from the objects
being originally hilbertspaces, and then you can recover all the innerproducts
of all the hilbertspaces from the *relation.

This is a neat trick and Urs used something like this to define a metric in our paper (something I had failed to do on my own for 6 years!). I think I'll point out a few things though. First, if you have another map $T:\mathcal{H}\to\mathcal{H}$, then

$$T T_\phi = T_{T\phi}.$$

From this it follows trivially that

$$\langle T \psi,\phi\rangle = (T_{T \psi})^* T_\phi = (T T_\psi)^* T_\phi = T^*_\psi T^* T_\phi = T^*_\psi T_{T^* \phi} = \langle\psi, T^* \phi\rangle$$

as you expect. Now, if you had another invertible map $g:\mathcal{H}\to\mathcal{H}$ that was self-adjoint with respect to this metric, we can define a g-modified metric

$$\langle \psi,\phi\rangle_g = T^*_\psi g T_\phi$$

Then we've got

$$\langle T\psi, \phi\rangle_g = (T_{T\psi})^* g T_\phi = T^*_\psi T^* g T_\phi = T^*_\psi g g^{-1} T^* g T_\phi = \langle \psi, g^{-1} T^* g \phi\rangle_g$$

so we could let

$$T^{*_g} = g^{-1} T^* g$$

denote the adjoint with respect to the g-modifed metric in terms of the adjoint with respect to the original metric.

This could be used, for example, to turn a Riemannian metric into a Lorenztian one. It could also be used to deform a flat metric into a curved one. Let your imagination wander :)

Eric

 

[marcus]

I found what I think may be the seed of the *-category idea
on page 22 of this paper by Baez and Dolan:
http://arxiv.org/q-alg/9503002
"Higher-dimensional algebra and topological quantum field theory"
This appeared in 1995 in Journal of Mathematical Physics.

a *-relation on the morphisms of a category is described, but
the term *-category isn't used, instead they use a
circumlocution and don't really give that type of category a name.
but they are clearly getting at the idea.