From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5442 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: quantum information and foundation Date: Tue, 29 Dec 2009 13:13:46 +0100 Message-ID: References: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1262182393 12007 80.91.229.12 (30 Dec 2009 14:13:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 30 Dec 2009 14:13:13 +0000 (UTC) To: =?ISO-8859-1?B?Sm95YWwsIEFuZHLp?= Original-X-From: categories@mta.ca Wed Dec 30 15:13:06 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NPzIN-0001fb-Nc for gsmc-categories@m.gmane.org; Wed, 30 Dec 2009 15:13:04 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NPypI-0001VQ-Al for categories-list@mta.ca; Wed, 30 Dec 2009 09:43:00 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5442 Archived-At: On Mon, Dec 28, 2009 at 6:54 PM, Joyal, Andr=E9 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]