quantum information and foundation

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Bob, Thorsten and Dusko,

I thank you for expressing frankly your position.

Thorsten wrote:

>I have to admit that I am quite ignorant about many of the areas 
>mentioned in the previous email. On the other hand developments in 
>Computer Science I know about don't seem to feature. Maybe they are 
>too mundane for Mathematicians.

Dusko wrote:

>is it just my impression, or are category theorists a little more
>sceptical about the value of applications than most mathematical
>communities? they seem to seek a recognition that categories are useful
>across mathematics, but then hesitate to recognize the depth and value of
>the applications in the other areas. --- can it be that we suffer from a
>superiority complex of some sort?

Bob wrote:

>In this context there is the highly unfortunate fact that there are
>certain quite prominent people in the category theory community who think
>that any deviation from treating category as a branch of pure mathematics
>and pure mathematics only is a bad thing!

You seems to suggest that this is a debate between pure and applied category theorists.
I disagree to the extend that "quantum foundation" and "quantum information" 
are very speculative subjects. The "Foundational Question Institute",

http://www.fqxi.org/

which is known to support speculative research projects exclusively,
is funding a project on Quantum Foundation by Bob Coecke: 

http://www.fqxi.org/grants/large/awardees/view/__details/2008/coecke

It has funded a project called "Topos Quantum Theory" by Christopher Isham

http://www.fqxi.org/grants/large/awardees/view/__details/2006/isham

It is funding a project "Categorifying Fundamental Physics" by John Baez: 

http://www.fqxi.org/grants/large/awardees/view/__details/2008/baez

Physics is in bad shape today according to Lee Smolin:

http://www.amazon.ca/Trouble-Physics-String-Theory-Science/dp/061891868X/

His main critic is that string theory has lost contact with experiments. 
It has become an academically driven discipline.
Maybe we should stop calling it physics. 
Of course, it can be interesting mathematically. 
In mathematics, the word "quantum" is often used as a prefix to express some vague
connection to quantum physics, like non-commutative algebras and Feynman diagrams.
By itself it is no proof that the named notion is fitting something in the natural world.
There are quantum groups, quantum algebras, quantum Grassmanians,
quantum planes, quantum bundles, quantum Schubert cells, quantum cohomology theories,
quantum fields, quantum Yan-Baxter operators, etc.
The theory of quantum groups is mathematically very interesting
but it has no applications that I know to real quantum physics:

http://en.wikipedia.org/wiki/Quantum_group

I have a Phd student working on quantum quasi-shuffle algebras
and he needs not to know about quantum physics because it is irrelevant.

The notion of dagger compact closed category is interesting
and purely mathematical, like the notion of quantum group.


Quantum information science is also quite speculative:

http://en.wikipedia.org/wiki/Quantum_Information_Science
http://en.wikipedia.org/wiki/Topological_quantum_computer

Again, there is nothing wrong with highly speculative research.
So the present debate is not about real applications of category theory.

Bob wrote:

>What the quantum information `hype' has done is injected some new blood in
>foundations of quantum mechanics research, an area which for several
>reasons had been suffocated by by the end of the previous century, despite
>the universal discomfort of the physics community with quantum mechanics.
>(a typical slogan which reflects this is: ``don't ask questions just
>compute'')  One would expect that this surge of quantum foundations, which
>meanwhile has led to many novel ideas, approaches, and radically different
>manners to think about physics in general, will ultimately lead to new
>mathematics. Moreover, the natural guise of many of these new ideas is
>within category theory, a message that some including myself have been
>trying to pass on within the foundations of physics communitee, with
>moderate success.

Feynam introduced his diagram as a method for computing the solutions
of QED field equations. It is essentially a technique for enumerating 
the terms arising in perturbation theory.  The method was extended to all physical fields
and Penrose understood the connection between the diagrams and tensor calculus. 
The geometry of tensor calculus is just an abstraction of this connection.
Feynman diagrams are very useful in physics and mathematics.
But the mystery of quantum physics lies elsewere: the extraction of a 
probability distribution from the complex values of a wave function.
I dont think that a categorical formalism based on Feynman diagrams 
is very different from what the physicists are currently doing.
This maybe why your formalism is having a moderate success among the physicists.
Of course, a good formalism can stimulate new developements.
But it should not be presented as radically new if it is not.
Too much hype may backfire. It is not good  for the reputation of category theory.


Happy New Year to all!

Best,
André






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