Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Tuesday, June 12, 2007

Have a Nice Day

I apologise for moving off topic today, but I found this Smile Test very interesting. Apparently, most people are not as good at telling fake smiles as they think they are. The theory is that people are easily fooled because it is socially convenient not to know what people are thinking. Despite my expectations of doing badly, I actually did really well on this test (17/20). Now I realise that it's easier not to care what people think when one's ability to detect fakeness makes it impractical to take such things into account.

Ars Mathematica finally reports a retract of the claimed disproof of The Hypothesis. David Ben-Zvi has kindly provided notes on the recent Chicago conference, where Goncharov was talking about Motives, path integrals and trivalent graphs. Sounds intriguing. OK, I printed out the notes. Wow. OMG. Goncharov claims to have identified the category of mixed motives (a.k.a. the holy grail for ordinary real/complex geometry) in terms of path integrals for projective varieties. For instance, when the variety corresponds to modular subgroups indexed by $\hbar = \sqrt{N}^{-1}$, as in TGD or $N$-fold covers of moduli spaces, one gets Langlands from the cohomology. He concludes with a statement that Feynman integrals (with observables) are valued in motivic cohomology. Yeah, duh, the physicists know that. We just don't know how we're ever going to learn that much mathematics.

Ah! That means the S duality we need for the Riemann Hypothesis relies on the whole range of quantised $\hbar$, and is therefore necessarily omega-categorical. That was expected, because the surreal zeta arguments extend through the ordinals. It is fantastically exciting to have some confirmation of this link between $\hbar$ values and S duality. I wonder how string theory will deal with a variable $\hbar$. Oh, I see.

3 Comments:

Blogger Matti Pitkanen said...

Sad that the language used is incomprehensible to physicist like me. Did the dynamical hbar=1/sqrt(N) appear explicitly in the article? In TGD one has hbar=1/N.

I wonder to what degree the emergence of path integral representation is dictated by the belief that path integrals are fundamental mathematics because physicists invented them: their mathematical existence is highly questionable after all.

Personally I would be happy if these approaches would replace path integrals by something resembling what I believe to be nearer to what results in TGD. Functional integral in infinite-D symmetric space consisting deformations a minimal generalized Feynman diagram consisting of lightlike 3-surfaces representing incoming and outgoing particles with ends meeting at vertices. This should be exactly calculable and algebraic by the vanishing of radiative corrections.

June 12, 2007 6:41 PM  
Blogger Kea said...

Did the dynamical hbar=1/sqrt(N) appear explicitly in the article? In TGD one has hbar=1/N.

Hi Matti. Well, he claims to have proved some theorem in the N-->oo limit for his mathematical Feynman integrals, where hbar appears in the action. So one could take the TGD view that hbar=1/N and then his leading term turns out to go like \sqrt(hbar).

June 13, 2007 11:27 AM  
Blogger Matti Pitkanen said...

Hi Kea,

first of all sorry for hbar=1/N in TGD. I had just arrived from travel and was very sleepy which perhaps explains why I wrote this kind of nonsense.

*hbar defined Lie-algebraically via commutators is integer for hbar_0=1 and by convention I call this integeer n_a resp. n_b in M^4 resp. CP_2 degrees of freedom.

*The ratio n_a/n_b defines the hbar as it appears in Schrödinger equation. It has rational values in general. Ruler and compass rationals and -integers in particular are favored by their algebraic simplicity.

N--> infty limit for gauge theories with say SU(N) gauge group can be thought of as hbar-->N limit and I have considered this interpretation. At one particular very natural limit coupling strength

alpha =g^2/4*pi*hbar

goes to zero like 1/N.

One can consider also more general limits with m/n behaviour and for m/n<=1 one might ask whether limits for Farey numbers could be especially interesting.

June 13, 2007 4:56 PM  

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