Re: quantum information and foundation

[Urs Schreiber <urs.schreiber@googlemail.com>]

On Mon, Dec 28, 2009 at 6:54 PM, Joyal, André <joyal.andre@uqam.ca> wrote:

> 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.


I would like to expand on this remark, and point out an application of
(higher) category theory that might deserve more attention from
mathematicians.


First a remark concerning the detachment of string theory from experiment:

much of theoretical physics, not just string theory, is far remote
from experiment, but -- in principle -- for a good reason: if
experiment shows that a certain incarnation of mathematical structure
X is relevant for the description of the physical world, then for
understanding it well we ought to study also all other incarnations of
structure X, even if they are not (yet) known to be relevant for the
description of the world themselves.

As a simple example: not all solutions of Einstein's equations
describe anything in the real world. But we want theoretical
physicists to understand as many as possible of them: while some
particular cosmological model (say one with closed timelike geodesics)
may look utterly irrelevant for the description of the real world
(given the present state of experimental knowledge!), it is the
understanding of the collection of all such models and their
interrelation that helps with understanding the particular one that
does describe the real world.

This idea, that we may study a theory in terms of the collection of
its models, should resonate with category theorists.

>From that perspective string theory strongly deserves to be studied by
theoretical physicsists, even in the absence of experimental evidence:
the string perturbation series is a conceptually compelling variation
of Feynman's celebrated sum over correlators of a 1d QFT. Every
theoretical physicist worth his or her money should feel an itch to
explore the analogous sums over correlators of 2d QFTs. And that's
what (perturbative) string theory is.

http://ncatlab.org/nlab/show/string+theory

And indeed, the above idea that for understanding one model it helps
to understand all its variations, is at work here, too: studying the
string perturbation series has led to a better understanding of
Feynman's perturbation series, since a few years quite spectacularly
resulting in a previously undreamed of understanding of the higher
loop Feynman terms in supergravity theories.

The fact that the discovery of many other suggestive aspects of the
string perturbation series made a whole community become so excited
about it that they threw some care and scientific discipline in the
wind is a problem, but one of the sociology of science, not a fault of
the topic.

The reason why I feel saying all this is worthwhile on a mailing list
devoted to category theory, is that a closer look shows that the
mathematical structures involved in string theory are not only an
impressive source of examples of applications of higher category
theory, but in some cases even their archetypical motivational
examples.

The cobordism hypothesis/theorem

http://ncatlab.org/nlab/show/cobordism+hypothesis

is arguably comparatively pivotal for higher category theory as, say,
the Yoneda lemma is for ordinary category theory. (I really think it
is.) With that in mind, it should not be forgotten that both its roots
in the ideas of Witten, Atiyah and Segal, as well as its present
rather impressive applications in the work of
Freed-Hopkins-Lurie-Teleman

  http://arxiv.org/abs/0905.0731

are situated in the conceptual framework that was opened by the step
from the Feynman perturbation series to string theory:

as John Baez mentioned in a previous message, cobordism
representations are being speculated to encode quantized general
relativity, but that speculation should not make us forget that what
made theoretical physicists eventually pass from the study of quantum
field theories defined on Minkowski space or similar, to "full"
quantum field theories defined on all possible cobordisms was the idea
that the Feynman perturbation series ought to have a generalization
from a sum over graphs to a sum over cobordisms of higher dimension:
conformal field theory used to be studied on R^2 for years until
string theory opened the perspective that a CFT ought to be defined on
general surfaces. Today the classification of such full 2dCFT -- the
representation theory of 2-dimensional conformal cobordism categories
-- is an impressive result in the theory of modular tensor categories.

 http://golem.ph.utexas.edu/string/archives/000813.html

Indeed, it seems to me that the most substantial conceptual progress
on the grand perspective exhibited by the passage to the string
perturbation series has recently come not out of the physics
departments (which seem to be curiously stuck with throwing
insufficient formal tools at their grand targets), but out of the math
departments, those math departsments where higher category theory has
an influence in one way or other.

In order to proliferate this observation, with AMS publishing we are
currently preparing a book volume that is devoted to exhibiting
aspects of the full story behind this claim.

http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory

The text at that link may provide more details on the point that I am
trying to make here.

I can summarize this point maybe as follows: pure mathematicians and
especially category theorists and higher category theorists should not
be tricked by complaints such as voiced in Smolin's book into thinking
that it is ill-advized to have a closer look at  the mathematical
structures to be found in string theory, well hidden under physicist's
nonsense as they may be.

On the contrary: much of what makes the present practice of string
theory so tiresome is that the lively activity of the 1980s of
mathematically inclined researchers looking into the mathematical
structures of the theory has largely vanished, at least in the physics
departments. The theory is much more interesting than the average talk
of its current practicioners. And much deeper.

One of the foremost powers of category theory is its ability to
unravel hidden structures and make them become mathematically active.
String theory is a vast reservoir of crucial (higher) categorical
structures that is, while recently beginning to be investigated as
such, largely like a huge bag of disjoint LEGO pieces which physicist
dream of putting together to a grand edifice, but which is waiting for
the higher category theorist to actually assemble it.

Best,
Urs


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