Thursday, August 16, 2007

Alain Connes: uniqueness of Standard Model

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Previous articles about Connes' non-commutative arts in particle physics:
Alain Connes: a TOE
Alain Connes et al.: predictions for masses
In a newer paper (shorter version here), Ali H. Chamseddine and Alain Connes work with the same model as before (so that all previous analyses and criticism still apply). What is new is that they argue that in their non-commutative setup, the Standard Model group with its spectrum (except for an unknown number of families) is a unique solution to a list of conditions. That sounds great except that we must ask: What are these conditions?

First, we must understand the objects. They assume that the following objects must be a part of the construction:
  • algebra: at the end, the physically relevant algebra $A_F$ will be isomorphic to $C \oplus H \oplus Matrices_C (3\times 3)$. This subalgebra of a larger, auxiliary algebra $A$ is essentially the Standard Model group except that it is kind of projective - the determinants don't have to be one
  • Hilbert space $H$: this is the space of all components of the fermionic fields in your theory
  • an anti-linear isometry $J$ on $H$. It satisfies $J^2 = \pm 1$
  • a $Z_2$ grading $\gamma$ on $H$ that (anti)commutes with $J$: $J\gamma=\pm\gamma J$
  • a Hermitean Dirac operator $D$ acting on $H$ that satisfies $JD=\pm DJ$.

All three $\pm$ signs above are independent and a priori arbitrary; in the ultimate Standard Model case, the only minus sign is in the $\gamma J = -J \gamma$ rule. The only other conditions we must satisfy to uniquely get the right Standard Model with the right spectrum are:

  • the greater algebra $A$ must be over real numbers and it must satisfy:
    $\forall a,b\in A$, ... $[a, J b^\dagger J^{-1}] = 0$
  • the twisted double commutators of the Dirac operator with $A_F$ must vanish:
    $\forall a,b\in A_F$, ... $[[D,a],J b^\dagger J^{-1}]=0$

That's it. How should you interpret all players in this jungle of assumptions?




First, all the involutions have to do with the type of spinors or with the pseudoreal representations of $SU(2)$. The dimension of their whole spacetime is $10$, just like in heterotic string theory on Calabi-Yaus. However, they can only calculate it modulo 8: it is a K-theoretical dimension that only cares about the reality and chirality of the spinors, a tiny portion of the wisdom of string theory. I think that all their comments about quaternions are inserted to make the stuff more sexy but at the end, it is nothing else than known facts about reality and pseudoreality of various representations in the Standard Model that all of us know.

The first condition that $a,b$ from the greater algebra essentially "commute", up to the involution $J$ and the daggers, may actually be almost trivial to satisfy because $J$ effectively exchanges the order of multiplication. But I still don't understand why it's true.

The main role of $\gamma$, the grading, is to exclude the case in which $A$ is a single copy of $Matrices_{R/C/H} (k \times k)$. Because suddenly they require that $\gamma$ and $J$ must actually anti-commute, it follows that $A$ must be a direct sum. After some extra work, some of which looks like if they were adding new conditions, they end up with the large algebra to be

$A = Matrices_H (2 \times 2) \oplus Matrices_C (4 \times 4)$
This would be analogous to a "Connes GUT" gauge group $USp(4) \times SU(4)$. Recall that the latter is locally isomorphic to $SO(5) \times SO(6)$. Finally, this Connes GUT gauge group $A$ (that is probably never unbroken) is reduced down to $A_F$, the connesized Standard Model gauge group by the condition for the Dirac operator. Even though the "Connes GUT" group looks very different from normal GUT groups, they argue that its consequences are analogous to those of $SO(10)$ GUTs.

I still don't quite understand why the commutators and double commutators (even with the $J$'s and daggers inserted) are zero for such a non-Abelian algebra. And I have no idea what would go wrong with their assumptions if they, for example, doubled the size of all matrices in $A$. Even if the grading singled out the four-dimensional spaces, it is still unsurprising that one can say the number "4" in a form of a difficult conundrum. But that doesn't mean that each such a conundrum is relevant for physics, does it?

The assumptions contain a lot of structure here. Did they actually get something that wasn't put in? One of the more non-trivial features of the Standard Model is that the number of colors is equal to three. They definitely do seem to argue that this number is uniquely predicted by the assumptions above. How could it be? What would be wrong about $U(1) \times SU(2) \times SU(5)$, among other choices?

Well, on page 10, they argue that the dimensions of the algebras must be bounded from above, and assuming that 12=20 (more convincingly: 10 + 2 = 10 x 2), they also show that they maximize the dimension. I don't understand these inequalities at all but I suspect that they arise because the $SU(3)$ QCD factor of $A_F$ must be inside $SU(4)$ in the algebra $A$. This reduces the question about the number of colors to a question about the numbers in the greater, so far unphysical Connes GUT group encoded in $A$. I fail to see why their constructions and conditions for $A$ are physically natural or meaningful in any way. For example, why the condition is any better than just saying that $A$ should be whatever it is in their picture.

If you have found some more non-trivial interpretation of these constructions, let me know. Right now, I think that they obtain as many results as the number of assumptions they insert. Except that they don't write the assumptions about the gauge group and representations on two lines as we usually do but spread them over 13 pages instead. ;-)

Also, the previous criticism applies: gravity is only unified with other forces at the level of words because they are controlled by independent constants and different components of a physically non-unified framework. And the predictions are inconsistent with the renormalization group - they either contradict the existence of running or they at least don't say what is the scheme and scale in which the predictions should be valid.

So as far as I can say, the work remains a physically vacuous artistic construction of a cool mathematician and his colleague.

And that's the memo.

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