Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, September 19, 2007

M Theory Lesson 105

An argument $(n_1 , n_2 , n_3 , \cdots , n_k )$ of an MZV is a list of elements $n_i \in \mathbb{N}$. Observe that this is just a 2-ordinal in the sense of Batanin, as shown by the 2-level tree. The total number of leaves $n_1 + n_2 + \cdots + n_k$ is the weight of the MZV and the number $k$ is the depth. These are important parameters for characterising MZVs.

Aside: Carl Brannen tells us that in his next few blog posts he will run through the snuark derivation of the Koide mass formula for leptons, and also a formula that describes masses for baryon and meson resonances, numbering in the several hundred. That sounds kind of interesting to me. Long live blogging.

Check out this conference title!

Also in the news: it looks like the Martians have finally resorted to biowarfare to kill off the human plague on Earth.

3 Comments:

Blogger L. Riofrio said...

I look forward to Carl's post and yours too. LOL, the meteorite situation sounds like opening chapters in "War of the Worlds."

September 20, 2007 10:53 AM  
Blogger Doug said...

Hi Kea,

The Asteroid Belt, not Mars, may be the culprit in Peru?

Killer asteroid fingered
Astronomical forensics pins down dinosaur killer. 5 September 2007

http://www.nature.com/news/infocus/dinosaurs.html

or

September 05, 2007
Asteroid Smashup May Have Wiped Out the Dinosaurs
Simulations point to an asteroid collision that sealed the dinosaurs' fate before their reign was half over ... By JR Minkel
"Using computer simulations, researchers reconstructed the trajectories of several thousand asteroids between Mars and Jupiter that are clustered near a 25-mile-wide rock called 298 Baptistina."

http://www.sciam.com/article.cfm?articleID=D75F54EC-E7F2-99DF-39A13F68A7FF1409&chanID=sa007

September 21, 2007 12:31 AM  
Blogger Matti Pitkanen said...

Hi Kea,

I continue my out-of-topicing.

I ended up with a further extension of the 8-D imbedding space from physical motivations: one can say that one half of generalized imbedding space was still missing. This is discussed at my blog. As a by-product a result emerged which might be interesting from the point of view of M-theory in the 11-D sense.


The discrete subgroups of SU(2) with fixed quantization axes possess a well defined multiplication with product defined as the group generated by forming all possible products of group elements as elements of SU(2). This product is commutative and all elements are idempotent and thus analogous to projectors. Trivial group G_1, two-element group G_2 consisting of reflection and identity, the cyclic groups Z_p, p prime, and tedrahedral, octahedral, and icosahedral groups are the generators of this algebra.


By commutativity one can regard this algebra as an 11-dimensional module having natural numbers as coefficients ("rig"). The trivial group G_1, two-element group G_2 generated by reflection, and tedrahedral, octahedral, and icosahedral groups define 5 generating elements for this algebra. The products of groups other than trivial group define 10 units for this algebra so that there are 11 units altogether. The cyclic groups Z_p generate a structure analogous to natural numbers acting as analog of coefficients of this structure.

Clearly, one has effectively 11-dimensional commutative algebra in 1-1 correspondence with the 11-dimensional "half-lattice" N^11 (N denotes natural numbers) which is p-adically very natural if one allows also "supernaturals".
Leaving away reflections, one obtains N^7.

The projector property suggests a deep connection with von Neumann algebras and Jones inclusions. An interesting question concerns the possible Jones inclusions assignable to the subgroups containing infinitely manner elements. The inclusion of both singular coverings and singular factor spaces H/G_axG_b (only these were included earlier) extends the "half lattice" to Z^11.

The intriguing question is whether dimensions 11, 7 and their difference 4 might relate somehow to the mathematical structures of M-theory with 7 compactified dimensions. Momentum lattice comes in mind first. What if anything it could mean that 11-D momenta label the sectors of the generalized imbedding space (phases of matter with different Planck constants)?

September 21, 2007 8:03 PM  

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