Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Thursday, March 22, 2007

Time and Again

Connes has a good post on emergent Time at NCG. Recall that Jones' subfactors for a Galois theory for the $II_{1}$ case gave rise to the planar operads that we are interested in. Apparently Connes' thesis was on the reduction of type $III$ automorphisms to the $II_{1}$ ones. This naturally brings to mind our discussion with Matti Pitkanen on the appearance of braids in the boundary of his light-like 3-spaces. The convergence of ideas between these different NCG approaches is beginning to look promising.

For those in the UK, don't forget to hear Louise Riofrio talk about emergent time at the cosmology conference at Imperial.

5 Comments:

Anonymous a quantum diaries survivor said...

Kea,

you guys are unbelievable. The only thing I understand is that what you study is really tough but also so cool...

Cheers,
T.

March 22, 2007 10:38 AM  
Blogger Kea said...

Thanks, Tommaso! All I know is that I couldn't hack it as an experimentalist. Kudos to you guys.

March 22, 2007 10:40 AM  
Blogger Matti Pitkanen said...

The time parameter emerges as a parameter of an outer automorphism Delta^(it)= exp(iHt) (or much more suggestively Delta^(it)=p^(iHt) from the p-adic point of view) associated with a factor of type III. There is a non-uniqueness due to the presence of unitary inner automorphism having interpretation as an analog for change of basis. An attractive idea is that inner automorphisms act as gauge transformations so that everything would be unique physically.


In TGD framework this parameter would correspond to the lightlike time parameter associated with the orbit of partonic 2-surface (the size of parton can be arbitrarily large).

I have considered here the possibility that at level of operators one has also in TGD factors of type III but that at the level of states everything reduces to factors of type II_1. This would be analogous to the reduction of the oscillator operator algebra to that spanned by creation operators and enough for the state construction. This would fix the time evolution uniquely if inner automorphisms act as gauge transformations.

This time evolution, which determines coupling constant evolution as a function of p-adic prime p, is fixed also by the time evolution assignable to the generalized eigen modes of the modified Dirac operator but the hope is that reduction to something much more general might allow an explicit calculation of p-adic coupling constant evolution in some remote future by using those mysterious methods of mathematicians;-).

March 22, 2007 5:50 PM  
Blogger Matti Pitkanen said...

Still a comment. The difference between *outer* and *inner* automorphisms is very delicate and highly interesting physically if one wants to develop in detail physics as a generalized number theory vision.

For hyper-finite factors of type II_1 the action of inner automorphisms is a-->uau^(-1) and non-trivial only for a finite number of tensor factors in a representation as an infinite tensor power of 2x2 matrix algebras (Clifford algebras). Thus *hyper-finiteness*. For outer automorphisms you have *infinite-fold copy* of similar action.

In the representation as group algebra a subset of S_infty outer automorphisms are induced by a permutation which is of form PxPxPx.... ad infinitum. P is S_n element and infinite product does not belong to S_infty so that inner automorphism is not in question.

There is a direct analog with a global gauge transformation which does not have a compact support and acts also as an outer automorphism and this brings immediately in mind a number theoretical version of local gauge invariance, spontaneous symmetry breaking, etc.. at the level of Galois group for the closure of rationals. Local gauge invariance would be replaced with the invariance of states with respect to S_infty element and *outer* automorphisms of above kind would define analogs of *global* gauge transformations.

Also the generalization of local Galois invariance to continuous transformations is suggested by McKay correspondence. Their action would be non-trivial in a finite number of 2x2 matrix algebra factors. This would be huge gauge invariance: super-conformal symmetries would be the most natural identification.

In braid picture at space-time level you would have an infinite braid and infinite-fold copy of basic B_n braiding: this makes possible quantum classical correspondence since the B_nxB_nx... ad infinitum can be represented by a finite n-braid at space-time level without loosing information. The mere representability at space-time level would force this kind of picture. The interpretation could be in terms of spontaneous symmetry breaking for the representations of Galois group of algebraic closure of rationals.

March 22, 2007 5:54 PM  
Anonymous Anonymous said...

Matti Pitkanen mentions "... hyper-finite factors of type II_1 ..." which are based on "... infinite tensor power of 2x2 matrix algebras (Clifford algebras) ...".

Those 2x2 matrix algebras are Complex Clifford algebras Cl(N,C), and, since Complex Clifford algebras have periodicity 2, for any large N, periodicity 2 means that
Cl(2N,C) = Cl(2,C) x ...(N times tensor product)... x Cl(2,C)
so that when you construct the hyperfinite factor of type II_1,
the completion of the union of all the tensor products of Cl(2,C) has a nice consistent structure and the hyperfinite II_1 factor is a nice consistent algebra.

If you look at Real Clifford algebras Cl(N,R), you see periodicity 8,
and the basic building block for an analogous construction is Cl(8,R),
so that
Cl(8N,R) = Cl(8,R) x ...(N times tensor product)... x Cl(8,R)
so you get what I call a generalized hyperfinite von Neumann factor.
Since Cl(8,R) has 16 primitive idempotents,
which is equivalent to saying that Cl(8) has a 16-dim spinor space
that is reducible to two mirror image 8-dim half-spinor spaces,
Cl(8,R) is not only the fundamental building block for a generalized hyperfinite II_1 factor,
it has a natural structure to carry the fundamental first-generation fermions,
as described in Carl Brannen's work.

Tony Smith

March 24, 2007 10:34 PM  

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