_{G,n}=(1+i)

^{n}-1 in cosmology, biology, and nuclear and particle physics. In particle physics paragraph appears the following line about new ultra high energy physics - perhaps scaled up copies of hadron physics.

* n∈{2, 3, 5, 7, 11, 19, 29, 47, 73} correspond to energies not accessible at LHC. n= 79 might define new copy of hadron physics above TeV range - something which I have not considered seriously before. The scaled variants of pion and proton masses (M _{107} hadron physics) are about 2.2 TeV and 16 TeV. Is it visible at LHC is a question mark to me.*

Today I saw the posting of Lubos suggesting that M_{G,79} pion might have been already seen!! Lubos tells about a bump around 2(!)TeV energy observed already earlier at ATLAS and now also at CMS. See the article in Something goes bump in Symmetry Magazine. The local signficance is about 3.5 sigma and local significance about 2.5 sigma. Bump decays to weak bosons.

Many interpretations are possible. An interpretation as new Higgs like particle has been suggested. Second interpretation - favored by Lubos - is as right-handed W boson predicted by left-right- symmetric variants of the standard model. If this is correct interpretation, one can forget about TGD since the main victory of TGD is that the very strange looking symmmetries of standrad model have an elegant explanation in terms of CP_{2} geometry, which is also twistorially completely unique and geometrizes both electroweak and color quantum numbers.

Note that the masses masses of M_{G,79} weak physics would be obtained by scaling the masses of ordinary M_{89} weak bosons by factor 2^{(89-79)/2)}= 512. This would give the masses about 2.6 TeV and 2.9 TeV.

There is however an objection. If one applies p-adic scaling 2^{(107-89)/2}=2^{9} of pion mass in the case of M_{89} hadron physics, M_{89} pion should have mass about 69 GeV (this brings in mind the old and forgotten anomaly known as Aleph anomaly at 55 GeV). I proposed that the mass is actually an octave higher and thus around 140 GeV: p-adic length scale hypothesis allows to consider octaves. Could it really be that a pion like state with this mass could have slipped through the sieve of particle physicists? Note that the proton of M_{89} hadron physics would have mass about .5 TeV.

I have proposed that M_{89} hadron physics has made itself visible already in heavy ion collisions at RHIC and in proton- heavy ion collisions at LHC as strong deviation from QCD plasma behavior meaning that charged particles tended to be accompanied by particles of opposite charged in opposite direction as if they would be an outcome of a decay of string like objects, perhaps M_{89} pions. There has been attempts - not very successful - to explain non-QCD type behavior in terms of AdS/CFT. Scaled up variant of QCD would explain them elegantly. Strings would be in D=10. The findings from LHC during this year probably clarify this issue.

See the chapter New particle physics predicted by TGD: part I of "p-Adic Length Scale Hypothesis" or the article What is the role of Gaussian Mersennes in TGD Universe?

For a summary of earlier postings see Links to the latest progress in TGD.

## 5 comments:

Matti, why Low Gaussian primes? your list of primes is a subset of the factors of the dimension of the friendly giant group.

https://en.m.wikipedia.org/wiki/Monstrous_moonshine

The monster group was investigated in the 1970s by mathematicians Jean-Pierre Serre, Andrew Ogg and John G. Thompson; they studied the quotient of the hyperbolic plane by subgroups of SL2(R), particularly, the normalizer Γ0(p)+ of Γ0(p) in SL(2,R). They found that the Riemann surface resulting from taking the quotient of the hyperbolic plane by Γ0(p)+ has genus zero if and only if p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71. When Ogg heard about the monster group later on, and noticed that these were precisely the prime factors of the size of M, he published a paper offering a bottle of Jack Daniel's whiskey to anyone who could explain this fact (Ogg (1974)).

--crow

Gaussian primes are really big but the primes defining them are logarithmically smaller. k=379 defines scale slightly large than that defined by the age of the Universe.

Larger ones exist but are not terribely interesting for human physicists for a long time;-)

The prime k above defines Gaussian Mersenne as (1+i)^k-1 and the associated real prime defined by its norm is rather large - rather near to 2^k and for k= 79 this is already quite big. k=113 characterises muon and nuclear physics, k=151,157,163,167 define a number theoretical miracle in the range cell membrane thickness- size of cell nucleus. Then there are astro- and cosmophysically important Gaussian Mersennes: http://matpitka.blogspot.fi/2015/06/gaussian-mersennes-in-cosmology-biology.html .

The lowest Gaussian Mersennes correspond to k=2, 3, 5, 7, 11, 19, 29, 47, 73. This list is indeed contained by the list of lowest monster primes p= 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 or 71 up to 47 . 73 does not appear as a factor of Monster anymore.

Gamma_0( p) is the kernel of the modulo homomorphism mapping SL(2,Z) to Sl(2,Z/pZ). The points of hyperbolic plane at the orbits of this group are identified. The outcome is sphere for p==k associated with Gaussian Mersennes.

If I have understood correctly, the order of Monster is product of these primes and therefore Monster contains a subgroup with order, which is product of the primes associated with Gaussian Mersennes, which are definitely outside the reach of LHC. This group could be christened as Gaussian Monster or Particle Physics Monster;) Number theory and particle physics meet each other!;-).

Speaking seriously, could this mean that the high energy physics above M_G,79 energy is somehow different from that below in TGD Univese? Is k=47 somehow special: it correspond to energy scale 17.6*10^3 TeV=17.6 PeV (P==Peta). Pessimistic would argue that this scale is the Monster energy scale never reached by human particle physicists;-).

Additional comment about discrete subgroups of SL(2,C). The modding operation modulo p or p^n for SL(2,C) , p prime, or even modulo Gaussian prime gives a sub-group allowing to construct tessellation (lattice) of 3-D hyperbolic space by hyperbolic manifolds. Modding is also possible with respect to integer or Gaussian integer. What could be the physical interpretation?

This could relate directly to p-adicity! Also in cosmological context: http://matpitka.blogspot.fi/2015/06/transition-from-flat-to-hyperbolic.html http://matpitka.blogspot.fi/2015/06/gaussian-mersennes-in-cosmology-biology.html

Sorry for a typo. SL(2,C) should read SL(2,Z) or SL(2,Z+iZ). Z could be extended to some algebraic extension of rationals and prime p with corresponding prime.

http://eclass.uoa.gr/modules/document/file.php/MATH154/Σεμινάριο%202011/kulkarni-1991.pdf

it is mentioned that the congruence subgroups associated with primes are very special.

Still a correction. I used the term modding loosely. Congruence groups are subgroups of SL(2,Z) which consist of identity matrices mod n. The notion generalises to integers of any algebraic extension of rationals. Congruence subgroup associated with n is subgroup of that associated with any of its factors.

The special role of primes is due to the fact that these groups can be extended to the p-dic counterparts of SL(2,Z) by replacing Z with p-adic integers. These subgroups would be especially interesting concerning

the construction hyperbolic manifolds of 3-D hyperbolic

space defining lattice cell of its tessellation.

e^p is p-adic number so that roots of e define finite-D algebraic extensions of p-adics. The special role of e in real context can be seen as a convention but in p-adic context it is not: p-adic variant of Lie group is obtained by exponentiation of elements of Lie algebra which are proportional to p (one obtains hierarchy of sub-Lie groups in powers of p). These subgroups and algebraic groups

generate p-adic variants of Lie groups.

What is nice that in algebraic extension of p-adics its primes P also have e( P) as number in extension. Also now finite roots of e(P/n) exist and one can ask how large is the number of real transcendentals having this kind of representation?

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