Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Sunday, November 25, 2007

M Theory Lesson 129

Jacques Distler bemoans the pitiable standards of the physics blogosphere in his post on Lisi's paper. Apparently, triality for the generations is a no starter. If one sticks to ordinary classical representation theory, no doubt this is correct.

But in M Theory one does not do this. The term triality is clearly a ternary analogue of the ubiquitous stringy term duality. Now let's look at a bit more lattice theory.

Recall (see Ebeling) that the subgroup $\Gamma (3)$ of $SL(2, \mathbb{Z})$ has modular forms $\theta_0$ and $\theta_1$ of the form, for $q = e^{2 \pi i z}$,

$\theta_0 = 1 + 6(q + q^3 + q^4 + 2 q^7 + q^9 + \cdots)$
$\theta_1 = 3 q^{\frac{1}{3}} (1 + q + 2q^2 + 2q^4 + \cdots)$

The theta function for the $E8$ lattice takes the form

$\Theta = \theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3}$

and this function appears three times in the celebrated j-invariant

$j = \frac{1728 (\theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3})^{3}}{- 4^{3} (\theta_{1}^{4} - \theta_{0}^{3} \theta_{1})^{3}}$

The group $\Gamma (3)$ appears naturally when studying ternary codes. The quotient of $SL(2, \mathbb{Z})$ by this group gives the group $PSL(2, F_{3})$, which is otherwise known as the group $A_4$, studied by Ernest Ma in his derivation of the tribimaximal mixing matrix.

5 Comments:

Blogger CarlBrannen said...

Good for you, Kea, the reason the string theorists are hopelessly lost is that they will only judge correct a solution to the theory of everything that is derived according to the rules that they believe in.

The standard model is completely compatible with all known experiments. So any method that derives the standard model from a simple set of assumptions is compatible with experiment. If it violates Coleman Mandula or makes Einstein roll over in his grave it doesn't matter. All that matters is that it gets the standard model.

It is human nature to look at those experiments and speculate on what they mean in terms of how nature works. But none of that matters to new physics. All that matters is that the experiments be explained with a small number of assumptions. We are not restricted to the conclusions that our elders jumped to.

November 25, 2007 2:44 PM  
Blogger L. Riofrio said...

Carl is correct that some people judge new theories only as they relate to old ones. Modifications to Relativity must be expressed in standard tensors with upper and lower indices, for example.

I would also add that all theories should not be judged as "theories of everything," which may be an impossbly high standard. If a Theory of Everything were discovered, there would be no more need for theorists.

Someone is accused of "not having a theory" when all these standards are not met. Believing that the Sun will set constitutes a theory. For these and other reasons new theories are often not given a fair hearing.

November 25, 2007 5:34 PM  
Blogger Matti Pitkanen said...

My viewer did not show all formulas of Distler but I think that his G is standard model gauge group. What Distler finds that neither of the non-compact forms of E_8 allows imbedding of G such that one obtains three generations in the fundamental representation so that the model is dead.

November 25, 2007 6:43 PM  
Blogger Kea said...

I must be very slow. I only recently realised what my referees really meant when they rejected my (better) papers. They kept telling me that I had to make careful contact with the physics, or I was just doing mathematics. Since these people work on brane worlds, or strings, or loops, this had me scratching my head trying to figure out how all their papers made contact with physics. It finally dawned on me, after 20 years of thinking I was just an idiot, that their criteria included a clause that anything discussed carefully in terms of established physics, such as GR, was good enough to be called physics. I failed to see how one could possibly understand quantum gravity by sticking so steadfastly to what was already known, so I persisted in my pig-headedness and became known as a lost cause.

November 26, 2007 9:35 AM  
Blogger L. Riofrio said...

I sympathise with you, Kea. Since we promote the successful ideas, it is the "moderators" with their strings, branes, etc. who are lost causes.

November 26, 2007 4:38 PM  

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