Connes: Physics to Number Theory

In summary, this paper discusses the identification of the Prime Vacuum as an initial condition for string theory. It suggests that this vacuum state is identified with a self-dual state under phase-space duality. This suggests a more general mathematical principle in operation.
  • #1
marcus
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today Alain Connes (with co-author Matilde Marcolli)
posted
http://arxiv.org/abs/hep-th/0411114

Physics to Number Theory via Non-Commutative Geometry, Part II

Part I got a big play on SPR, we should know something about this.
Maybe only a little. But something.

Part One of "Physics to Number Theory" was
http://arxiv.org/math.NT/0404128

I can't do more than flag these two papers. If this thread goes anywhere
it will have to be by other people having comments to make about non-commutative geometry applied to physics.
 
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  • #2
a must read

marcus said:
today Alain Connes (with co-author Matilde Marcolli)
posted
http://arxiv.org/abs/hep-th/0411114

Physics to Number Theory via Non-Commutative Geometry, Part II

I've had a preliminary look at this paper: EVERYONE should
read this. Universality for renormalisation appears
to be properly worked out.

Also: on page 8 " ... relevant physical quantities, including the
coupling constants, share this implicit dependebnce on the scale..."

Fantastic!
Kea
 
  • #3
Kea said:
I've had a preliminary look at this paper: EVERYONE should
read this. Universality for renormalisation appears
to be properly worked out.

Also: on page 8 " ... relevant physical quantities, including the
coupling constants, share this implicit dependebnce on the scale..."

Fantastic!
Kea

dont wait for help or companionship Kea,
comment some more

on this board it is not considered bad to post two or three in a row
as you think of more to say. I hope very much you will comment some more---and other people may help too
 
  • #4
Connes and Marcolli have defined a new object they call Q-lattices. I have only glanced oved it, I wonder if it is a generalization of Eratosthenes's sieve. Remember that this sieve is basically to scale an integer lattice and to map it over itself. After considering all the possible scalings, the sites not receiving any element are the prime numbers.
 
  • #5
Ok:...

It will take me a long time to understand this paper...but
a few more immediate thoughts
 
  • #6
...

The Riemann-Hilbert problem is important in soliton theory -
ie. the inverse scattering method - which was generalised
to a quantum inverse scattering method. It was the study of
the Sine-Gordon equation in this context (by Kulish and
Re****ikhin) that led to the discovery of quantum groups
by physicists. And of course, quantum group Hopf algebras
have a great deal to do with knots (and Category Theory,
which is what I'm trying to convert you all to)...so it's
very nice to see this tie in by good mathematicians.

The Hopf algebras they discuss have been studied by the
causal set people (Markopoulou and others). More lattices.
This is because lattices are basic to the structure of a
Heyting algebra - that is, the intuitionistic logic of a topos.

A good understanding of RepG (which I'm not claiming to have)
relies on thinking of it in terms of abelian group objects
in the functor category 'Set to the G', which is a topos.

Now, from the perspective of String theory one might ask:
How does one understand multidimensional extensions
of this rich algebraic structure underlying the standard
model? My guess (actually, I'm not really guessing) is that
this paper demonstrates quite clearly that higher category
theory is absolutely essential. Moreover, this provides the
link to LQG.

Cheers
Kea
:smile:
 
  • #7
arivero said:
Connes and Marcolli have defined a new object they call Q-lattices. I have only glanced oved it, I wonder if it is a generalization of Eratosthenes's sieve. Remember that this sieve is basically to scale an integer lattice and to map it over itself. After considering all the possible scalings, the sites not receiving any element are the prime numbers.

The Riemann-Hilbert problem was solved some years ago:

http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm
o:)

The webpage was left 'unfinished' for historical reasons, but you bet your bottom dollar 2005 would be a very significant year to post it's completion!

LoOk ClosEly, do not take note of its 'apparent' clumsyness :biggrin:
 
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  • #8
Wave's_Hand_Particle said:
The Riemann-Hilbert problem was solved some years ago:
Just a precision: the Riemann-Hilbert problem, so called because it comes from the list of Hilbert's problems, is different from the conjectures on Riemann's Zeta function.
 
  • #9
look at this

hep-th/0411118
 
  • #10
Kea said:
hep-th/0411118


Identifying the Prime Vacuum INITIAL condition, within string theory has been just an hobby for some(self included :rolleyes: ).

Quote from paper linked:In this context it
is demonstrated that the trivial state, with V (q) = E = 0, is identified with the
self–dual state under phase–space duality. These observations suggest a more general mathematical principle in operation. In physical systems that exhibit a duality structure, the self–dual states under the given duality transformations correspond to critical points

The non-dynamic (static) Transformations that string-DUAL-theory are trying to incorperate will obviously lead to the asking of this: Will string theorists eventually admit defeat in their 'Vacuum Solutions' of phase evolution for generalized string worldlines, by the creation of a model that is on a par with the Wave-Particle interpretation, a model infact that goes 'BOTH-WAYS'..the correct way, and the in-correct way!

Critical points can mean many things.
 
  • #12
arivero said:
http://philo.at/pipermail/phil-logic/2001-September/000033.html
http://www.arxiv.org/abs/math.QA/0211199
http://www.math.fsu.edu/~marcolli/
(check "notes on Q-lattices, etc there)
http://www.matrix.ua.ac.be/wp-print.php?p=42


Really interesting that your first link reveals Osher, I take it that this is the same person as here:http://www.superstringtheory.com/forum/dualboard/messages11/770.html

Which brings 'me' back to here:http://www.superstringtheory.com/forum/dualboard/messages9/136.html

I recall Osher with some great respect, I only wish that I could find out if he/she is ok..I Hope so.

Great links, I do not need to express the fact that my original website, with some very interesting images and text, was demolished by persons unknown, ok no big deal, but I am working towards a '05' deadline, in honour of Einstein and his miracle year :biggrin:
 
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1. What is the connection between physics and number theory in Connes' work?

In his groundbreaking work, Alain Connes has shown that there is a deep connection between physics and number theory. Specifically, he has demonstrated that the principles of non-commutative geometry, which are used in theoretical physics, can also be applied to number theory. This means that concepts from physics, such as symmetry and space-time, can be used to study and understand numbers.

2. How has Connes' work impacted the fields of physics and number theory?

Connes' work has had a significant impact on both physics and number theory. His theory of non-commutative geometry has opened up new avenues for research in both fields, allowing for a deeper understanding of the underlying structures and patterns. It has also led to the development of new techniques and approaches that have advanced our understanding of the universe and the nature of numbers.

3. What are some practical applications of Connes' work?

While Connes' work is primarily theoretical, it has potential practical applications in areas such as cryptography and quantum computing. By using principles from non-commutative geometry to study numbers and their properties, it may be possible to develop more secure encryption methods and more efficient algorithms for quantum computers.

4. What are some key concepts in Connes' theory of non-commutative geometry?

Some key concepts in Connes' theory include the use of operator algebras, which describe the properties of non-commuting operators, and spectral triples, which are used to connect geometry and algebra. He also uses the concept of a "space of states" to describe the behavior of physical systems and their underlying structures.

5. What are some potential future developments in Connes' work?

Connes' work is still an active and evolving field of research, with many potential future developments. One area of interest is the application of non-commutative geometry to other branches of mathematics, such as algebraic geometry and topology. Additionally, there is ongoing research into the implications of Connes' theory for quantum field theory and the unification of quantum mechanics and general relativity.

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