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Showing posts from July, 2019

About the mathematics needed in TGD

The following is a comment from FB discussion. Since the answer developed to a summary of the mathematics needed in TGD, I decided to make it blog post.

I think that TGD is a "problem" for anyone in the sense that it is very difficult to get graps of what it really is. The reason is that TGD have been silenced for 4 decades - censorship in archives started around 93 or so and has had fatal consequences. The idea about the hegemony of M-theory estabilished by censorship was terribly silly.

TGD is an outcome of concentrated effort lasted more than 4 decades and involves 24 books. I guess that the minimum time to get some perspective to TGD is one year. For instance, I have worked with twistors for about 10 years and gradually begin to really understand the twistorialization of TGD. I am skeptic about communication of TGD without a long series of lectures and personal face to face discussions. This has not been possible, and now also my age poses strong limitations…

What extensions of rationals could be winners in the fight for survival?

It would seem that the fight for survival is between extensions of rationals rather than individual primes p. Intuition suggests that survivors tend to have maximal number of ramified primes. These number theoretical speciei can live in the same extension - to "co-operate".

Before starting one must clarify for myself some basic facts about extensions of rationals.




Extension of rationals are defined by an irreducible polynomial with rational coefficients. The roots give n algebraic

numbers which can be used as a basis to generate the numbers of extension ast their rational linear combinations. Any number of extension can be expressed as a root of an irreducible polynomial. Physically it is is of interest, that in octonionic picture infinite number of octonionic polynomials gives rise to space-time surface corresponding to the same extension of rationals.



One can define the notion of integer for extension. A precise definition identifies the integers as ideals. Any i…

Trying to understand why ramified primes are so special physically

Ramified primes (see this and this) are special in the sense that their expression as a product of primes of extension contains higher than first powers and the number of primes of extension is smaller than the maximal number n defined by the dimension of the extension. The proposed interpretation of ramified primes is as p-adic primes characterizing space-time sheets assignable to elementary particles and even more general systems.

In the following Dedekind zeta functions (see this) as a generalization of Riemann zeta are studied to understand what makes them so special. Dedekind zeta function characterizes given extension of rationals and is defined by reducing the contribution from ramified reduced so that effectively powers of primes of extension are replaced with first powers.

If one uses the naive definition of zeta as analog of partition function and includes full powers Piei, the zeta function becomes a product of Dedekind zeta and a term consisting of a finit…

Libet's paradoxical findings and strange findings about state function reduction in atomic scales

Perceiving is basically quantum measuring, More precisely, perceptions correspond to the counterparts of so called weak measurements in TGD (zero energy ontology) analogous to classical measurements. The observables measured in weak measurements are such that they commute with the observables whose eigenstate is the permanent part of self, the "soul". Big ( that is ordinary) state function reductions mean the death of self and its reincarnation with opposite arrow of time. This holds universally in all scales.

For the change of the arrow of time the recent findings gave direct support in atomic scales (see this). Effectively there is a deterministic process leading to the final state of reduction. This is an illusion: reduction produces superposition of deterministic classical time evolutions beginning from the final state but backwards in time of observer. Experimenters misinterpreted this as time evolution with standard arrow of time leading to the final …

M8-H duality and twistor space counterparts of space-time surfaces

It seems that by identifying CP3,h as the twistor space of M4, one could develop M8-H duality to a surprisingly detailed level from the conditions that the dimensional reduction guaranteed by the identification of the twistor spheres takes place and the extensions of rationals associated with the polynomials defining the space-time surfaces at M8- and twistor space sides are the same. The reason is that minimal surface conditions reduce to holomorphy meaning algebraic conditions involving first partial derivatives in analogy with algebraic conditions at M8 side but involving no derivatives.




The simplest identification of twistor spheres is by z1=z2 for the complex coordinates of the spheres. One can consider replacing zi by its Möbius transform but by a coordinate change the condition reduces to z1=z2.



At M8 side one has either RE(P)=0 or IM(P)=0 for octonionic polynomial obtained as continuation of a real polynomial P with rational coefficients giving 4 conditions (RE/IM de…