Gerardus 't Hooft doubt about power of calculus..

[Yuri Danoyan]

Gerardus 't Hooft in his Nobel Lecture "A Confrontation with Infinity" did some important comment about applicability of differential equations:

 

"The mathematics of differential equations grew out of this

and nowadays it such a central element in theotetical physics that we often do not realize how important and how non-trivial these observations actually were.In modern theories of physics we send distances and time intervals to zero all the time also in multidimensional field theories,assuming that the philosopy of differential equations applies".p.360

 

"I have this critical note.As string theory makes heavy use of differential equations it is clear that some sort of continuity is counted on.We schould attempt to find an improved short-distance formulation of theories of this sort,if only to justify the use differential equations or even functional integral".p.370

 

http://nobelprize.org/nobel_prizes/physics/laureates/1999/thooft-lecture.pdf

 

Somebody know about string community reaction on this important notes?

[ajb]

It is a bit more than just differential equations. Really the question is of the applicability of differential geometry to fundamental physics. The objection to this is that "applying quantum mechanics" to the space-time manifold suggest that space-time should have some kind of discrete or fuzzy structure. Different approaches to this exist under the umbrella of "non-commutative (differential) geometry" (See for example [1,2]).

 

The ethos here is that smooth manifolds should arise as the "classical limit" of these non-commutative spaces. I am not sure if all approaches to NCG really have smooth manifolds as the classical limit. The category in work in do.

 

I work on a form of NCG, but the non-commutativity is about as mild as you can get. Some of the coordinates anticommute. These are the so called supermanifolds (See for example [3]). There represent some of the basic examples of NCG and come from allowing the algebra of smooth functions to no longer being commutative. This gives a "fuzzy picture" and we do not have a very good notion of a point. This is often the case with any NCG.

 

However, most of what we can do on smooth manifolds can be done on supermanifolds (and more general NCGs) if we formulate what we know about smooth manifolds in an algebraic way independent of the notion of a point.

 

Things like functions , vector fields and vector bundles generalise to supermanifolds straight forwardly. Thus, as differential equations can be thought of as vector fields on a manifold one would expect most of the theory of differential equations to pass to the noncommutative case. I have not really investigated this outside the category of supermanifolds. So, I don't fully know the status.

 

Now as for the string theory community I cannot really speak for them, but for sure they know of NCG and supermanifolds. How much will depend greatly on the individual and what they are trying to do.

 

References

[1]Connes Noncommutative geometry. Available at http://www.alainconnes.org/en/downloads.php

 

[2] Madore An introduction to noncommutative differential geometry and its physical applications Lond. Math. Soc. Lect. Note Ser. 257, 2000.

 

[3] Rogers Supermanifolds: Theory and applications, Hackensack, USA: World Scientific (2007) 251 p.

 

(as an aside, I have met John Madore once at a conference in Oxford and have met/spoken to Alice Rogers on a number of occasions. Alan Connes I once saw but never actually spoke to him. )

[Yuri Danoyan]

I am familiar with Alain Connes view from Scientific American

 

http://www.sciam.com/article.cfm?id=the-geometer-of-particle

 

"The picture that emerges from the Standard Model, then, is that of spacetime as a noncommutative space that can be viewed as consisting of two layers of a continuum, like the two sides of a piece of paper. The space between the two sides of the paper is an extra discrete (noncontinuous), noncommutative space. The discrete part creates the Higgs, whereas the continuum parts generate the gauge bosons, such as the W and Z particles, which mediate the weak force"

 

It seems to me non esthetic.

Edited August 19, 2008 by Yuri Danoyan

[ajb]

Indeed, I am not completely convinced by the ideas of Connes on the NCG origin of the standard model. I went to a talk he was giving and although it seemed a nice idea I just could not really accept it. Not sure why.

[Norman Albers]

Thank you for excellent discussion.