Connes on Spectral Geometry of the Standard Model, IV
Posted by Urs Schreiber
With some background material in place # # # I’ll now try to indicate how we can get something like the standard model as the effective target space theory of our “superparticle”.
(I’ll stick to this superparticle imagery #, but you should remember that this is my prose around Connes et al.’s formulas.).
We need to find the right compactification geometry that will produce an effective standard model on the compactified space.
(I’ll no longer distinguish between “ordinary spaces and geometry” and “generalized spectral spaces and geometry”. Similarly, I’ll no longer distinguish between “metric” and “Dirac operator”, nor between “topological space” and “algebra”, and so on. We are now firmly in the quantum world, where spacetime is nothing but the world as seen by our quantum superparticle.)
The first proposal for such compactifications, which already came very close to the standard model, is decades old by now.
Alain Connes & John. Lott
Particle Models And Noncommutative Geometry
Nucl.Phys.Proc.Suppl.18B:29-47,1991
(spires).
See for instance the review
Daniel Kastler, Thomas Schucker
The Standard Model a la Connes-Lott
hep-th/9412185 .
As far as I can tell, at that time the Yang-Mills terms were stilll included by hand. The main point of the spectral approach was to realize that it could nicely explain the Higgs boson and its Yukawa coupling terms to the fermions. It did (and does) so by realizing the Higgs boson as an internal part of an ordinary minimally coupled connection 1-form - an internal gauge boson.
A few years later Connes apparently realized that also the gravitational and gauge kinetic Yang-Mills terms had an inherent operator-theoretic formulation, namely the spectral action principle.
This is indicated in
A. Connes
Gravity coupled with matter and foundation of non-commutative geometry
hep-th/9603053
and fully formulated in
Ali H. Chamseddine, Alain Connes
The Spectral Action Principle
hep-th/9606001,
both of which start by presenting the general spectral idea and then working out how to realize the standard model in detail.
I find that a very useful discussion of the details of the realization of the standard model in this approach is
C. P. Martin, Jose M. Gracia-Bondia, Joseph C. Varilly
The Standard Model as a noncommutative geometry: the low energy regime
hep-th/9605001 .
This is in particular designed to take the ordinary physicst by the hand and introduce him or her gently to the operator-theoretic spectral description by motivating these by the structure of the standard model.
So already over ten years ago people had a pretty good idea that and how the standard model action has an elegant descritption as a spectral action.
The only problem was: this description was wrong.
But only in the sense in which ideas in physics tend to be wrong - not entirely wrong but not quite right.
Namely, instead of containing the fermionic particle content of the standard model, these spectral models produced four copies of every fermion in the standard model. A slight overkill.
By a remarkable synchronicity, it seems that there was no progress on this aspect for about ten years, and now two preprints appear almost simultaneously, presenting the solution:
John W. Barrett
A Lorentzian version of the non-commutative geometry of the standard model of particle physics
hep-th/0608221.
Alain Connes
Noncommutative Geometry and the standard model with neutrino mixing
hep-th/0608226.
And the modification needed to get this solution is rather tiny, just a small change in the real structure $J$ of the spectral triple from ten years ago.
All right, I should finally say now what this compactification geometry is. Up to slight technical details that I will get wrong, it’s this:
First of all, the topology of our 4+6-dimensional compactified space is given by the algebra of functions
on it, where
is simply the algebra of smooth functions on a 4-dimensional spin manifold $X$ and
is the algebra “of functions on the internal space” $F$, where
is the algebra of complex numbers
is the algebra of quaternions and
that of $3\times 3$ matrices with entries complex numbers.
Sometimes people call this an almost commutative algebra. This is in order to distinguish it from the case where also the algebra $A_\mathrm{ext}$ of functions on the observable macroscopic space is non-commutative, which might be for instance true if you look at 4-dimensional space very closely, at extremely small scales.
Of course this choice of algebra is related to the gauge group of the standard model. In the algebraic approach, the natural group to study is the group
of automorphisms of the algebra. This splits into an inner and an outer part
By comparison with the case of ordinary geometries, we address the outer part as the group of diffeomorphisms of our target space. The inner part, however, is identified with the group of gauge transformations.
Next we need a Hilbert space on which this algebra $A$ is represented by compact operators. We take
where
is the Hilbert space of square integrable sections of a spinor bundle on our 4-manifold $X$ and
is a certain representation of the algebra $A_\mathrm{int}$ which is spanned by a basis which contains precisely the list of elementary fermions that appear in the standard model.
So this basis consists of
- the quarks $\array{ u_R & u_L \\ d_R & d_L }$ each of which comes in three colors;
- and the leptons $\array{ & \nu_L \\ e_R & e_L }$
and the antiparticles to each of these… and two more copies of all of this, giving the three generations of fermions.
All-in-all this is a list of
particles (three generations of particles and and antiparticles, each consisting of 4 quarks in three colors, a neutrino and a left- and a right-handed part of the electron) and our Hilbert space is simply the span of these 90 basis elements (everything here is over $\mathbb{C}$)
Next, we need a representation of $A$ on our Hilbert space $H$. The rep of $A_\mathrm{ext}$ on $H_\mathrm{ext}$ is the obvious one, obtained by multiplying spinors with complex numbers.
The rep of $A_\mathrm{int}$ on $H_\mathrm{int}$, encodes crucial information about the particle physics we want to obtain.
Writing down this rep in detail is not particularly enlightning. In his latest paper, however, Connes explains (in prop. 2.2) that we can understand this representation naturally as the direct sum of all inequivalent irreducible odd $A_\mathrm{int}$-bimodules.
Next, we need to specify the grading $\gamma$ and the real structure $J$ on our Hilbert space $H$.
This involves a choice of a couple of signs, and it is just this choice of signs which the recent progress consists of. With
and
we take the external part to be given by the standard choice
and
(hm, I think - I need to check that, corrections are appreciated).
Writing down the components of $\gamma_\mathrm{int}$ and $J_\mathrm{int}$ is again not really illuminating (though $J$ can essentially be thought of as charge conjugation), but, as with the rep of $A_\mathrm{int}$, Connes shows that there is natural way to understand both in terms of the isomorphism of the above mentioned bimodule with its contragradient one.
Finally we need of course to specify our Dirac operator
As described before, we take
to be the ordinary Dirac operator $D_0$ acting on sections of the given spinor bundle on our Riemannian 4-manifold $X$ plus a part $\delta D$ containing what for ordinary geometries is called the minimal coupling of fermions to gauge forces.
Since we think of $D$ as encoding a metric, we may think of $\delta D$ as “internal parts” of the metric, as in Kaluza-Klein theory.
One can derive the form of this part nicely in terms of inner automorphisms of the algebra $A$, and it turns out to be of the form
where $A = \sum_i a_i [D,a_i]$ for $a_i \in I$.
For ordinary geometries, these terms evidently look like the usual minimal coupling terms $\sum_i a_i [\gamma^\mu \nabla_\mu, a_i] = \gamma^\mu A_\mu$ to the gauge bosons.
The crucial point here is that we also get a contribution from the internal metric
and this gets interpreted as the Higgs boson.
The internal Dirac operator $D_\mathrm{int}$ can, since it acts on a finite dimensional space, simply be thought of as a large matrix, satisfying a couple of conditions. After computing the spectral action (but we need just the obvious term $\langle \psi , D \psi \rangle$ to see that) it will turn out that this matrix is essentially the matrix $Y$ of Yukawa couplings. So we read off all these couplings from our favorite particle accelerator and write them as entries into our internal Dirac operator.
That’s it. A spectral triple.
Maybe to summarize, here is what the main ingredients of the standard model correspond to in the spectral description:
- gauge group $\leftrightarrow$ inner automorphisms of algebra $A$
- diffeomorphism group $\leftrightarrow$ outer automorphisms of algebra $A$
- fermion species $\leftrightarrow$ basis elements of $H_\mathrm{int}$
- number of generations $\leftrightarrow$ choice of size of $H_\mathrm{int}$
- gauge bosons $\leftrightarrow$ external/internal part of the metric, as in Kaluza-Klein
- Higgs bosons $\leftrightarrow$ another external/internal part of the metric
- gauge couplings $\leftrightarrow$ encoded in external/internal part of the metric
- Yukawa couplings (fermion masses) $\leftrightarrow$ encoded in the internal metric
The next step is to analyze this spectral triple. To compute the dimensions of the space it describes and to determine the spectral action it induces.
history
A very complete listing, partly clickable, of the articles and eprints related to the first ten years of this model can be found in my webpage ncactors.html