Dr No (...Need for a zoo of new particles?)

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our more geometric and conceptual interpretation of the standard model gives a clear indication that particle physics is not so much a long list of elementary particles as the unveiling of the fine geometric structure of space-time. 

Noncommutative geometry

Alain Connes 1994

Poincaré and Einstein showed how to infer the correct flat spacetime geometry (known as Minkowski space geometry) underlying special relativity from experimental evidence associated to Maxwell’s theory of electromagnetism. In fact, the Maxwell equations are intrinsically relativistic. This flat geometry is then extended to the curved Lorentzian manifolds of general relativity. From the particle physics viewpoint, the Lagrangian of electromagnetism is ... just a very small part of the full Standard Model Lagrangian... Thus, it is natural to wonder whether the transition from the Lagrangian of electrodynamics to the Standard Model can be understood as a further refinement of the geometry of spacetime, rather than the introduction of a zoo of new particles.  

The spectral formalism of noncommutative geometry ... makes it possible to consider spaces which are more general than ordinary manifolds. This gives us more freedom to obtain a suitable geometric setting that accounts for the additional terms in the Lagrangian. The idea to obtain the spectral geometry (A, H, D) is that: 

• The algebra A is dictated by the comparison of its group of inner automorphisms with the internal symmetries. • The Hilbert space H is the Hilbert space of Euclidean fermions. • The line element ds is the propagator for Euclidean fermions. 

Using this idea together with the spectral action principle ... allowed to determine a very specific finite noncommutative geometry F such that pure gravity (in the form of the spectral action) on the product M×F, with the metric given by the inner fluctuations of the product metric, delivers the Standard Model coupled to gravity. Thus in essence what happens is that the scrutiny of spacetime at very small scales (of the order of 10^-16 cm) reveals a fine structure which replaces an ordinary point in the continuum by {a} finite geometry F. In the {chronological first} approach this finite geometry was taken from the phenomenology i.e. put by hand to obtain the Standard Model Lagrangian using the spectral action. The algebra A_F, the Hilbert space H_F and the operator D_F for the finite geometry F were all taken from the experimental data. The algebra comes from the gauge group, the Hilbert space has as a basis the list of elementary fermions and the operator is the Yukawa coupling matrix. 

... instead of taking the finite geometry F from experiment, one should in fact be able to derive it from first principles. The main intrinsic reason for crossing by a finite geometry F has to do with the value of the dimension of spacetime modulo 8. We would like this KO-dimension to be 2 modulo 8 (or equivalently 10) to define the Fermionic action, since this eliminates the doubling of fermions in the Euclidean framework. In other words the need for crossing by F is to shift the KO-dimension from 4 to 2 (modulo 8). This suggested to classify the simplest possibilities for the finite geometry F of KO-dimension 6 (modulo 8) with the hope that the finite geometry F corresponding to the Standard Model would be one of the simplest and most natural ones. This was finally done ... in our joint work with A. Chamseddine ([8])... 

On the fine structure of spacetime

Alain Connes

... we classify the irreducibe geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem... It gives, almost uniquely, the Standard Model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input. The geometrical model is valid at the unification scale, and has relations connecting the gauge couplings to each other and to the Higgs coupling... We thus manage to have the advantages of both SO(10) and Kaluza-Klein unification, without paying the price of plethora of Higgs fields or the infinite tower of states.

Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model

Ali H. Chamseddine, Alain Connes

(Submitted on 25 Jun 2007 (v1), last revised 21 Nov 2007 (this version, v3))

Image capture from the short video : CERN The Standard Model of Particle Physics

 

A comment and some contextualisation

The title of this post sticks to the rule for my August-holiday series and it is a wink to Saying No makes physicists what they are a recent post by Lubos Motl. Here are some extracts I think relevant for what is at stake in my blog (one may appreciate the not so usual carefulness of Lubos in his statements):

Physics simply cannot try to incorporate every idea that is out there.

...an overwhelming majority of theoretical particle physicists don't try to construct Connes-like models of gauge theories coupled to fermions formulated as "theories on noncommutative geometries" because they have been shown something from these ideas and they didn't see any evidence or nontrivial tantalizing hints in these ideas. Or they simply didn't make sense... 

They wrote off this research program not because they are "arrogant" but because according to their evaluation of the content of this program building upon their expertise, they either concluded that it was downright wrong or they just didn't see enough value to join that research. Physicists and scientists in general must be allowed to make this conclusion. The right to say No is a defining part of the culture of the scientific epoch. It was a key development that allowed the scientists to stop saying Yes to all the church officials and the group think of the public and investigate ideas impartially and carefully, with Yes and No being given approximately equal chances to become the final answer of any research...

To pick mathematical papers and declare them relevant in physics randomly or according to purely mathematical criteria cannot lead anywhere, at least not systematically. At the end, physicists are usually ahead of mathematicians when it comes to the discovery of basic concepts or patterns that are ultimately parts of both fields (mirror symmetry is a beloved modern example). Mathematicians are usually followers in the grandest scheme of things. Sometimes, mathematicians are ahead.