Non-commutative geometry and physics
Matthew MacIntyre at the National University of Senegal
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.
John C. Baez
>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.
Alejandro Rivero
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.