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14 abr 1993, 1:47:3814/4/93

a

Some time ago at CERN I heard Alain Connes speaking about his attempts to

use non-commutative geometry to construct a "theory of everything". At

that time he was working on the electroweak theory. He seemed to be very

enthusiastic and well-informed about QFT. Can anyone report briefly on the

progress,if any, made by Connes and company in this direction. I know of

course that NC geometry is an active field, but I mean specifically

regarding the "theory of everything" claims. One has heard rumours.

18 abr 1993, 1:04:0018/4/93

a uunet!sci-physics-research

In article <1993Apr14....@nuscc.nus.sg> matm...@nuscc.nus.sg (Matthew MacIntyre at the National University of Senegal) writes:

>Some time ago at CERN I heard Alain Connes speaking about his attempts to

>use non-commutative geometry to construct a "theory of everything". At

>that time he was working on the electroweak theory. He seemed to be very

>enthusiastic and well-informed about QFT. Can anyone report briefly on the

>progress,if any, made by Connes and company in this direction.

>Some time ago at CERN I heard Alain Connes speaking about his attempts to

>use non-commutative geometry to construct a "theory of everything". At

>that time he was working on the electroweak theory. He seemed to be very

>enthusiastic and well-informed about QFT. Can anyone report briefly on the

>progress,if any, made by Connes and company in this direction.

I periodically describe this work, with less and less precision as time

goes on since I haven't been thinking about it. On me I have one of the

first things Connes wrote on this, "Essay on Physics and Noncommutative

Geometry. The key idea of this work is to use the notion of a "universal

connection" from noncommutative geometry to treat gauge theories on

spacetmies more general than the usual sort of manifold. In this essay

Connes considers several models, the most sophisticated of which

develops electroweak theory i the context of "a two-sheeted S^4," as

(Euclideanized) spacetime. Thus we take two copies of S^4, M, and M', better

thought of as S^4 x {0,1}, "every point x of M being at a distance of

~10^{-16} cm from some point x' of M'. Thus we are dealing with the

simplest possible Kaluza-Klein theory where the fibre is: two points!

OF course ordinary differential geometry does not do anything with a

two-point space but non-commutative differential geometry does. (Even

though the algebra A we shall be dealing with is commutative the

abandonment of local charts and replacement by operator theoretic data

gives much more freedom to manuouevre.) Thus the Higgs fields will

appear essentially from the quantized differential (f(x') - f(x))/L of a

function on X = M union M', where L is the distance between the two

sheets, and the disconnetedness of the fibre will be used to get

nontrivial bundles on X whose dimensions differ on the two copies of M."

(Namely, dimension 1 on M and 2 on M'.) Briefly, the Higgs field falls

out quite naturally as an aspect of the gauge fields in this context,

and need not be put in "by hand."

More recently he has dealt with the strong interaction as well but I

don't have these papers on me. His book in French has recently been

translated into English as "Noncommutative Geometry," and has tripled in

size due to additions in the process. This should be a good place to

learn about the stuff, as well as the review articles by Kastler (sorry,

no refs). Unfortunately the only preprint of the book we have here at

UCR was stolen before I got a good look at it.

18 abr 1993, 14:10:5418/4/93

a

In article 87...@galois.mit.edu, "John C. Baez" <jb...@bourbaki.mit.edu> writes:

> In article <1993Apr14....@nuscc.nus.sg> matm...@nuscc.nus.sg (Matthew MacIntyre at the National University of Senegal) writes:

> >Some time ago at CERN I heard Alain Connes speaking about his attempts to

> >use non-commutative geometry to construct a "theory of everything". At

> >that time he was working on the electroweak theory. He seemed to be very

> >enthusiastic and well-informed about QFT. Can anyone report briefly on the

> >progress,if any, made by Connes and company in this direction.

>

> More recently he has dealt with the strong interaction as well but I

> don't have these papers on me. His book in French has recently been

> translated into English as "Noncommutative Geometry," and has tripled in

> size due to additions in the process. This should be a good place to

> learn about the stuff, as well as the review articles by Kastler (sorry,

> no refs). Unfortunately the only preprint of the book we have here at

> UCR was stolen before I got a good look at it.

The reviews from Kastler are PrePrints from the Centre de Physique Teorique

at Marseille, france.

usually titled "state-of-art of the Connes-Lott version

of the standard model of... in non-commutative differential geometry".

or something so.

My personal opinion (Im a student) is that the Connes-Lott model

have a lot of a gadget (as Connes says in the french version of the book),

but it is evolving. The general theory is a very interesting (and powerfull)

one; it have conexions with the Grothendiek-...-... theory of foliated

spaces and pointless spaces, which have links with (Maclane-Moordijk)

sheaves and (Tanakka-...) group duals, which is related to

(Doplicher-Roberts) C*-things duals and (...) quantum grups, some of these

sammed things relating to (Araki-Haag-Kastler-...) Local

Quantum Field Theory, which is linked to (Wighman-Osterw..-...) Axiomatic QFT

which we all know relates to... (sorry this is going long, and Im not

very good on this, after all . So Stop)

(But trust me, all the things I have named exist, I have seen them :-)

Which is the editor of the english version of the book? I could

be interested on getting a DEFINITIVE version, not the always evolving

preprint.

Alejandro Rivero

Zaragoza University, Theoretical Physics Dep

Spain

PS: next meeting is May 30-June13, in French West Indies.

Im not going, as It is difficult to get the money for

a stage in the Caribbean sea... Nobody trust you are going

to study. Anyway, If you can convince to your boss, try it.

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