Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Friday, September 05, 2008

Banks on Holography

Carl, Louise ... and everybody else ... you simply must see T. Banks' outstanding talk at the current PI Multiverse conference. He outlines a relatively mainstream, but completely original, analysis of holography using basic quantum mechanics and general relativity.

An FRW type cosmology is obtained from a dense black hole fluid which satisfies causality (causal diamond) constraints arising from finite dimensional holographic pixel Hamiltonians on a lattice, which in turn are analysed in terms of noncommutative function algebras inspired by matrix models and M theory. Their main model considers a physical region of a dilute black hole gas, arising from a large fluctuation of the maximal entropy stable fluid.

He is inclined (but reluctant in the end) to give up inflation altogether, because the model generates homogeneity and isotropy without it. Banks also points out that many field theory prejudices regarding the nature of an emergent geometry from fluctuating classical spacetimes simply cannot be correct from this point of view, which advocates a large $N$ (matrix size) limit construction from towers of causal diamonds of increasing cosmic time, for each observer.

Of course Banks prefers Dark Energy to a varying speed of light, and this leads to some criticism of the landscape on the grounds that it cannot reproduce universes of this kind, with suitable values for $\Lambda$. On the other hand, he recognises the need for a richer mathematical discussion of the quantum operators (his Hamiltonians are only supposed to be a simple solution to the constraints).


Blogger kneemo said...

According to the talk, pixels in the model are assigned to basis elements of a finite dimensional non-abelian function algebra. Moreover, it was mentioned that such basis elements are one on each pixel and zero everywhere else. The fuzzy sphere analogy was also mentioned.

Such a construction immediately reminded me of projective space, where say for CP^N, there exists N patches corresponding to the N distinct primitive idempotents (up to automorphism) that are elements of the basis for the C* algebra of NxN complex matrices. Such a C* algebra, in noncommutative geometry, is interpreted as a function algebra over three points. The Jordan algebra of NxN Hermitian matrices can then be interpreted as the algebra of 'smooth' functions over these three points.

Given such a pixel/primitive idempotent correspondence, pixels would be assigned to a subset of the full N^2 basis of the C* algebra, namely the N independent primitive idempotents with 1 on the kth diagonal place and zero everywhere else. U(N) transformations would then transform the pixels, as U(N) is the isometry group of CP^N, which results in a change of basis for the function algebra.

September 06, 2008 5:09 PM  
Blogger kneemo said...


Such a C* algebra, in noncommutative geometry, is intepreted as a function algebra over N points. The Jordan algebra of NxN Hermitian matrices becomes the algebra of 'smooth' functions over these N points.

September 08, 2008 7:31 AM  
Blogger L. Riofrio said...

Thanks, finally got around to reading the pdf. He would bre more inclined to give up inflation if he knew there are alternatives. Neither the landscape nor QFT predict a suitable value of lambda, which should give them a clue.

September 09, 2008 10:49 AM  
Blogger CarlBrannen said...

I didn't get much out of it, possibly because I'm having too much fun with category theory at the same time.

September 09, 2008 6:19 PM  
Blogger Kea said...

Hi kneemo, Louise and Carl. Great, Carl! (evil laugh) Another convert, at last!

September 09, 2008 7:33 PM  

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