^{8}× E

^{8}group of symmetries.

The problem of these scenarios is that they have no connection to experimenal reality. Heterotic string is extremely ugly mathematically, probably the ugliest object created by my respected colleagues in the history of the theoretical physics. If nomen is omen its Monster group based relative is at least equally ugly;-).

Speaking more seriously, Monster group ) is a fascinating creature. In particular, there is no known way to represent its elements (what does this strictly mean is not quite clear to me). In any case, this looks strange! It would be nice to find a place also for the Monster in physics and also its physical representation to see what it really looks like;-).

If I remember correctly, Galois groups acting as automorphisms of algebraic extensions of rationals involve all finite groups, also Monster as one learns from the Wikipedia article. This is why Monster - democratically together with all other Galois groups - could have place in the number theoretical physics of TGD Universe.

- In TGD this would also mean the existence of a concrete representation. Galois group - in special case Monster - would transform to each other number theoretic discretizations -"spines" - of the space-time surface characterizing it as a monadic sub-manifold and give rise to an n-fold covering space, n the order of Monster group. The "spine" of the space-time surface is a number theoretic discretization with points in preferred coordinates having values in an extension of rationals (algebraic extension involving also powers of some root of e).

- Strong holography (SH) conjectures that one can construct preferred extremal from data given at 2-D string world sheets and partonic 2-surface provided one also fixes these discretizations with finite pinary cutoff so that they do not have "too many" points. Number theoretic discretization would break SH meaning that TGD does not fully reduce to string theory like theory with everything coded by data at 2-surfaces.

- Intriguingly, also effective Planck constant n= h
_{eff}/h labelling dark matter as phases of ordinary matter in TGD Universe corresponds to the number for the sheets of space-time surface regarded as covering space!

Could h_{eff}/h correspond to the order of Galois group of the extension? If so, the physics of dark matter would be number theoretical physics and could reduce to a theory of finite groups appearing as Galois groups! This would be extremely elegant climax for the story starting that radiation at EEG frequencies has quantal effect on vertebrate brain although the EEG photons energies in the standard quantum theory are too small by 10 orders of magnitude!

- Monster group would correspond to one particular phase of dark matter is the largest sporadic finite simple group (there are 26 (amazing!) sporadic groups) with n= 2
^{46}× 3^{20}× 5^{9}× 7^{6}× 11^{2}× 13^{3}× 17 × 19 × 23× 29× 31 × 41 × 47 × 59 ×71 ∼ 8× 10^{53}.

- The values of n= h
_{eff}/h would be quite generally orders of finite groups or of factors of these. For the orders of finite groups see this and this. I vaguely remember that powers of 2: n= 2^{k}, are very strongly favoured as orders of finite groups.Note that also the order of Monster has large power of 2 as a factor.

- Monster appears as an automorphism group of a Kac-Moody algebra. Is this true more generally for finite groups or Galois groups? In TGD Kac-Moody algebras could emerge dynamically for preferred extremals for which sub-algebra of super-symplectic algebra and its commutator with the full algebra give rise to vanishing classical Noether charges (and annihilate the quantum states). The remnant of the super-symplectic symmetry would be Kac-Moody algebra acting on the induced spinor fields and string world sheets.

Could these geometrically realized Galois groups appear as automorphism groups of these Kac-Moody algebras? The action of Galois group on the spine of monadic manifold must induce an automorphism of the dynamical Kac-Moody algebra so that this seems to be the case. If true, this would allow to say something highly non-trivial about the relationship between Galois group and corresponding dynamical Kac-Moody algebra. A real mathematician would be however needed to say it;-).

- If this intuition is correct, the subgroups of Galois groups should play a key role in the rules of phase transitions changing the value of n= h
_{eff}/h. I have proposed that n can be reduced in phase transition only to a factor of n. This would correspond to the breaking of Galois symmetry group to its subgroup: spontaneous symmetry breaking number theoretically!

## 4 comments:

it is not true that there is no known way to represent its elements...there is, just not an "easy" way

Your are of course right. What I have mind is concrete geometric representation.

the wikipedia page has some interesting info about the monster.. i enjoyed Gannons book moonshine beyond the monster.. https://en.m.wikipedia.org/wiki/Monster_group

Let V be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 31+12.2.Suz.2, where Suz is the Suzuki group. Elements of the Monster are stored as words in the elements of H and an extra generator T. It is reasonably quick to calculate the action of one of these words on a vector in V. Using this action, it is possible to perform calculations (such as the order of an element of the Monster). Wilson has exhibited vectors u and v whose joint stabilizer is the trivial group.

i remember this stuff from when inwas studying Clifford algebras.

i had a funny experience last night.. standing in a darkened house that is being renovated/under construction late at night.. quiet.. just a light wind. i had the feeling of being within the house and then looking at the windows thru the night sky which was undergoing some sort of dynamical tesselation as if the boundary of the universe was some sort of hyperbolic tesselation. with extraordinaryly complex and beautiful flows. the representations were somewhat stable and as i moved my body and head around it would slightly displace the flows as if i was in a stream of some sort. there is no way i could ever capture the experience in rendering or pictures. just a dusty old house yet i had the feeling this 4d spacetime is just a subset of something absurdly larger than what is conceivable.. maybe it was a glimpse of the Monster group.. made me want to get the book out and study the topic again but i am kind of burned out since i hit a roadblock on the riemann hypothesis... some of the "approximation zeros" are on the wrong side of derivates of Z and hence i cant figure out which mapping function would map them to their "homes" at the zeta zeros.. only managed to show that about 99% map to their homes.. maybe ill put the pdfs on vixra

I remember that it was you who sent this book to me. Sad that I am so lazy reader. At this age preparation of own books is more does not leave time to reading.

How Galois goup acts on spinors is an inteteresting question. Is there also the analog of spin rotation involved besides rotation of argument.

These mathematical dreams are the best. Deep calmness of mind and profound sense of mystery.

It seems now clear that both h_eff/h=n and preferred p-adic primes reduce to number theory. n is number of sheets for some kind of covering space and Galois group defines a covering of the "spine" of monadic manifold with descritization consisting of points in extension of rationals. n is number of points for Galois group!

Preferred primes naturally correspond to ramified primes for given extension. Their decomposition to primes of extension contains some higher powers of prime of extension. They are like multiple roots of polynomial.

This is not yet enough. For some extensions especially large number of discretization the extension of string world sheets and partonic 2-surfaces plus 4-D discretization to real space-time surface exists. Very large number of imaginations/intensions are realizable. These extensions and corresponding ramified primes are survivers in the fight for number theoretical survival as especially capable realizers of intentions!

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