Thursday, January 11, 2007

About the correspondence between infinite primes, points of world of classical worlds, and configuration space spinor fields

The idea that configuration space CH of 3-surfaces, "the world of classical worlds", could be realized in terms of number theoretic anatomies of single space-time point using the real units formed from infinite rationals, is very attractive.

The correspondence of CH points with infinite primes and thus with infinite number of real units determined by them realizing Platonia at single space-time point, can be understood if one assume that the points of CH correspond to infinite rationals via their mapping to hyper-octonion real-analytic rational functions conjectured to define foliations of HO to hyper-quaternionic 4-surfaces inducing corresponding foliations of H.

The correspondence of CH spinors with the real units identified as infinite rationals with varying number theoretical anatomies is not so obvious. It is good to approach the problem by making questions.

  1. How the points of CH and CH spinors at given point of CH correspond to various real units? Configuration space Hamiltonians and their super-counterparts characterize modes of configuration space spinor fields rather than only spinors. Does this mean that only ground states of super-conformal representations, which are expected to correspond elementary particles, correspond to configuration space spinors and are coded by infinite primes?

  2. How do CH spinor fields (as opposed to CH spinors) correspond to infinite rationals? Configuration space spinor fields are generated by elements of super-conformal algebra from ground states. Should one code the matrix elements of the operators between ground states and creating zero energy states in terms of time-like entanglement between ground states represented by real units and assigned to the preferred points of H characterizing the tips of future and past light-cones and having also interpretation as arguments of n-point functions?

The argument represented in detail in TGD as a Generalized Number Theory III: Infinite Primes is in a nutshell following.

  1. CH itself and CH spinors are by super-symmetry characterized by ground states of super-conformal representations and can be mapped to infinite rationals defining real units Uk multiplying the eight preferred H coordinates hk whereas configuration space spinor fields correspond to discrete analogs of Schrödinger amplitudes in the space whose points have Uk as coordinates. The 8-units correspond to ground states for an 8-fold tensor power of a fundamental super-conformal representation or to a product of representations of this kind.

  2. General states are coded by quantum entangled states defined as entangled states of positive and negative energy ground states with entanglement coefficients defined by the product of operators creating positive and negative energy states represented by the units. Normal ordering prescription makes the mapping unique.

  3. The condition that various symmetries have number theoretical correlates leads to rather detailed view about the map of ground states to real units. As a matter fact one ends up with a detailed view about number theoretical realization of fundamental symmetries of standard model.

  4. It seems that quantal generalization of the fundamental associativity and commutativity conditions might be needed in the sense that quantum states are superpositions over all possible associations associated with a given hyper-octonionic prime. Only infinite integers identifiable as many particle states would reduced to infinite rational integers mappable to rational functions of hyper-octonionic coordinate with rational coefficients. Infinite primes could be genuinely hyper-quaternionic. This would imply automatically color confinement but would allow colored partons.
For more details see the chapter TGD as a Generalized Number Theory III: Infinite Primes. of "TGD as a Generalized Number Theory".


At 1:01 AM, Blogger Matti Pitkanen said...

For some reason the comments sent to my blog arrive as emails to me but are not visible in the blog itself. I do not know the reason nor cure yet.
Below is Robert Paster's recent comment.

"For me, the main importance of p-adic mathematics as a tool for consciousness study is the great efficiency of p-adic mathematics at capturing a memory or a thought or a dream.

To see this efficiency, first think four-dimensionally, that is think of an organism's three-dimensional spatial extent varying over the fourth dimension of time. p-Adic mathematics, the mathematics of enclosure, straightforwardly captures this four-dimensional record (of a memory or a thought or a dream) as a p-adic number field, which is a record of enclosures.

Practice this technique on your own thoughts as you think them, or alternatively on a conversation as you participate in it. You will see added details enlarging the thought enclosure, or you will see shifts to related enclosures that encompass or are encompassed by the previous thought, or that have some p-adic strand in common.

TGD's four-dimensional enclosures link to each other in eight-dimensional space (M4+ x CP2), guided by maximization of information content (negentropy maximization principle).

This same process is called equilibration by the great epistemologist and psychologist Jean Piaget, for whom assimililation and accommodation processes (which have exact p-adic analogs) result in the achievement of higher and higher levels of equilibration as life's central process.

--Posted by Robert Paster to TGD diary at 1/15/2007 03:02:50 PM


Dear Robert,

I am sorry for the problems with the blog and hope that they could be cured. The basic aspect of p-adic topology is indeed this natural decomposition of world into objects(compact-open topology).

Equally important aspect is the fact that rationals and subset of algebraics can be regarded as common to reals and p-adics. This is what makes possible to unify real and p-adic physics together and would naturally be also responsible for the discreteness of real world cognitive representations.

Best Regards,

At 1:43 PM, Blogger Mahndisa S. Rigmaiden said...

01 17 07

Hello Matti:
Here is a test comment. I have been busy, so forgot to mention that the emails you got from me were sent as emails and NOT posted as comments! I shoulda told you sooner. Whatever the case, I hope the issues you are having with blogger get fixed. And I think p-adics are a great way to represent consciousness...


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