M Theory Lesson 197
Speaking of platonic groups in neutrino physics, Lieven Le Bruyn beautifully clarifies the story in a post on Galois. As he points out, these three groups, the tetrahedral, octahedral and icosahedral,
in turn correspond to the three exceptional Lie algebras $E_6$, $E_7$, $E_8$ via the McKay correspondence (wrt. their 2-fold covers).Yesterday we came across $\Gamma (3)$ in connection with the neutrino mixing tetrahedron. Recall that the generating function for $\Gamma (3)$ is $j^{\frac{1}{3}}$, where the dimension of $E_8$ appears in the second term of the expansion. But these connections to the exceptional Lie groups have much more to do with lattices and operads than with strings or toes, as Lieven promises to explain soon. M Theory is the theory that explains the structure of stringy geometry, not the theory that confirms so called stringy physics.
posted by Kea | 12:22 PM
8 Comments:
Kea:
Can you add "Recent Comments" widget for your side-bar? Goto this link:
http://blogger-templates.blogspot.com/2007/03/recent-comments.html
It will make your blog a lot easier to navigate. You might have to have Blogger upgrade your blog to the new version.
Chimpanzee, when I get a higher density of on topic comments, I will add the widget. Thanks for the link anyway.
I am sure that more people will move to your M-theory as stringy physics loses favour.
Thank you for the link to the article about Galois's last letter before his fatal duel. He must have led a very exciting life, making breakthroughs in mathematics and fighting duels. Dueling was a very permanent way to settle a dispute, unlike the uncivilized, interminable, tiresome squabbles which now take the place of duels.
The discussion of groups is interesting. I didn't know that geometric solids correspond to Lie algebras. Does category theory have any bearing on group theory in physics, e.g. symmetry groups representing basic aspects of fundamental interactions and particles?
E.g., the Standard Model group structure of particle physics, U(1)*SU(2)*SU(3) is equivalent to the S(U(3)*U(2)) subgroup of SU(5), and E(8) contains many elements, including S(U(3)*U(2)) subgroups, so SU(5) and E(8) have been considered candidate theories of everything on mathematical grounds.
Do you think that these platonic symmetry searching methods are the wrong way to proceed in physics? Woit writes in http://arxiv.org/PS_cache/hep-th/pdf/0206/0206135v1.pdf that there the Standard Model problems are not tied to making the symmetry groups appear from some grand theory like a rabbit from a hat, but are concerned with explaining things like why the weak isospin SU(2) force is limited to action on just left-handed particles, why the masses of the standard model particles - including neutrinos - have the values they do, whether some kind of Higgs theory for mass and electroweak symmetry breaking is really solid science or whether it is like epicycles (there are quite a landscape of different versions of the Higgs theory with different numbers of Higgs bosons, so by ad hoc selection of the best fit and the most convenient mass, it's a quite adjustable theory and not extremely falsifiable), and how quantum gravity can be represented within the symmetry group structure of the Standard Model at low energies (where any presumed grand symmetry like SU(5) or E(8) will be broken down into subgroups by various symmetry breaking mechanisms).
What worries me is that, because gravity isn't included within the Standard Model, there is definitely at least one vital omission from the Standard Model. Because gravity is a long-range, inverse-square force at low energy (like electromagnetism), gravity will presumably involve a term similar to part of the electroweak SU(2)*U(1) symmetry group structure, not to the more complex SU(3) group. So maybe the SU(2)*U(1) group structure isn't complete because it is missing gravity, which would change this structure, possibly simplifying things like the Higgs mechanism and electroweak symmetry breaking. If that's the case, then it's premature to search for a grand symmetry group which contains SU(3)*SU(2)*U(1) (or some isomorphism). You need to empirically put quantum gravity into the Standard Model, by altering the Standard Model, before you can tell what you are really looking for.
Otherwise, what you are doing is what Einstein spend the 1940s doing, i.e., seaching for a unification based on input that fails to include the full story. Einstein tried to unify all forces twenty before the weak and strong interactions were properly understood from experimental data, so he was too far ahead of his time to have sufficient understanding of the universe experimentally to be able to model it correctly theoretically. E.g., parity violation was only discovered after Einstein died. Einstein's complete dismissal of quantum fields was extremely prejudiced and mistaken, but it's pretty obvious that he was way off beam not just for his theoretical prejudices, but for trying to build a theory without having sufficient experimental input about the universe. In Einstein's time there was no evidence of quarks, no colour force, no electroweak unification, and instead of working on trying to understand the large number of particles being discovered, he preferred to stick to classical field theory unification attempts. To the (large) extend that mainstream ideas like string theory tend to bypass experimental data from particle physics entirely, such theories seem to suffer the same fate as Einstein's efforts at unification. To start with, they ignore most of the real problems in fundamental physics (particle masses, symmetry breaking mechanisms, etc.), they assume that existing speculations about unification and quantum gravity are headed in the correct direction, then they speculatively unify those guesses without making any falsifiable predictions. That's what Einstein was doing. To those people this approach seemed like a very good idea at the time, or at least it seemed to be the best choice available at the time. However, a theory that isn't falsifiable experimentally may still be discarded for theoretical reasons when a better theory comes along.
Interesting fact about the mesons. The simplest meson spectrums are those of the upsilon (which is made of a b and anti-b), and the J/psi (made of a c and anti-c).
In each of these two cases, there are exactly six very well defined resonances. Their masses are not spread by very much. They certainly do not look like "radial excitations"...
anonymous, you are right to point out that GUT approaches have failed to make headway in understanding quantum gravity. This is not surprising, because symmetry is a mathematical convenience, not a physical principle in itself. The application of Platonic symmetries to neutrino mixing is usually couched in terms of some SUSY model, belonging to a string framework which shares the big problem of GUTS, namely, having nothing whatsoever to do with empirical reality.
Here we prefer to think of the Standard Model (along with the gauge symmetries and any extra Lie or discrete symmetries) as arising from a more fundamental algebraic language that describes measurement processes, including quantum gravitational ones. Carl works on Schwinger type algebras using Clifford algebra. Matti Pitkanen has his well developed TGD picture. I work on deriving such structures from categorical axioms.
Dear Kea,
I share your belief that category theory will be important part of the story but having learned my Gödel I do not believe that any axiomatics is enough for physics.
I have already earlier made concrete proposals in this direction. The attempts to understand various aspects in the relationship between subjective time (sequence of quantum jumps) and geometric time in terms of zero energy ontology, p-adic length scale hypothesis, and hierarchy of Planck constants, led to a little step forward in this respect.
Zero energy states are assigned to causal diamonds formed by pairs of future and past directed lightcones.
Very special kind of inclusions of causal diamonds inside larger causal diamonds needed to explain basic facts about conscious experience of time, and interactions between disjoint causal diamonds might allow natural description in terms of category theory. See my blog.
Another -not obviously related - step of progress came as I developed TGD based view about CMB fluctuations and its anomalies.
What I realized was that a symplectic analog of conformal field theory with n-point functions satisfying fusion rules and constructible in terms of canonical invariants emerges very natural description of the fluctuations. The reason is that symplectic transformations of CP_2 and of sphere of last scattering act as natural symmetries of the situation in TGD.
What is remarkable that the n-point functions vanish as two arguments co-incide rather than being singular (as does also the correlation function for temperature fluctuations). Since these n-point functions are also basic building blocks of interaction vertices in TGD, divergences are absent in TGD. This is of course a basic prediction of TGD and symplectic QFT would realize it concretely.
A fascinating possibility is that cosmic quantum coherence implied by gigantic values of gravitational Planck constants might mean that the strange correlations of fluctuation spectrum with galactic and perhaps even solar geometry might be a result of quantum correlations generated by cosmic time-like entanglement in dark degrees of freedom. May be the measurements or cosmic state function reductions before them would have created these weird correlations. This would realize Wheeler's observer participancy in cosmic scales. If dark matter density fluctuations determine the structures formed by visible matter we might act like Gods just by measuring microwave fluctuation spectrum;-)!
Of course, every physicist in his right winged mind would say that these correlations are pure accident or result of erratic elimination of foreground. For conservative view peppered with "intelligence below 20" type comments, see the blog of Lubos.
As already became clear, symplectic QFT fits excellently also the needs of quantum TGD since super-canonical invariances (symplectic (contact-) transformations of light-cone boundaryxCP_2) having structure of conformal algebra with lightlike radial coordinate replacing complex variable are isometries of the "world of classical worlds".
The classification of symplectic QFT:s would mean entire industry for many years if people would be ready to take seriously anything proposed by a person outside acedemic hegenomony. See my blog containing also a link to an article.
A fascinating possibility is that cosmic quantum coherence implied by gigantic values of gravitational Planck constants might mean that the strange correlations of fluctuation spectrum with galactic and perhaps even solar geometry might be a result of quantum correlations generated by cosmic time-like entanglement in dark degrees of freedom.
Yes, this has been my view for some time, although I tend to use stringy terminology to describe it.
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