### Bosonic Emergence, Number Theoretic Universality, p-Adic Fractality, and Twistor Program

Mahndisa made some questions about p-adic fractalization of S-matrix. My reply had too many characters so that I decided to add it as a separate posting.

Dear Manhndisa,The problems are essentially number theoretical. And "ontological" as philosopher would say. I have a bundle of general ideas developed in various chapters of various books. I try to give impression about just the bare essentials.

This is the general picture now. I have the feeling that no big changes are needed anymore.

- In TGD framework M-matrix replaces S-matrix: zero energy ontology, etc. The S-matrix accompanying M-matrix is
notobtained as a unitary evolution operator as in QFT:s since M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state having as counterparts initial and final states of a physical event in positive energy ontology. Heisenberg and Schrodinger pictures relate to positive energy ontology and are therefore not terribly relevant here.

- Number theoretic universality. M-matrix of same functional form as a function of Lorentz invariants defines complex and p-adic valued variants. The strongest form of this universality is rationality of matrix elements. It should be also possible to algebraically continue this M-matrix to various number fields from the field of rationals (algebraics, reals, and p-adics are completions of rationals and the completion process generalizes to the completion of rational physics in various number fields). p-Adic physics
should be visiblealso at the real side at the level of matrix elements (besides mass spectrum) and here p-adic fractalization enters the stage.

- Bosonic emergence is essential for QFT limit and emerged from twistor related considerations, started then to look more or less independent of twistors so that I separated it to its own chapter, and finally turned out to be highly relevant for twistorizalition! The counterpart of bosonic YM action emerges from Dirac action via functional integral over fermion fields. In particular, the inverse of the bosonic propagator emerges as fermionic loop. Bosonic emergence in principle predicts all coupling constants and their evolution if one can fix the cutoffs involved with the loop integral over fermion momenta. Quantum criticality should determine the cutoff in hyperbolic angle. What quantum criticality means exactly at the level of MATLAB modules is the problem and with this problem I have worked last months and tested various hypothesis.

- After tedious and slow calculations it seems that the definition of criticality that I have worked for last month does not work (I managed to calculate yesterday and last night 30 first p-adic length scales using the proposal for criticality based on real physics: the resulting hyperbolic cutoff behaves in non-physical manner if its growth continues to say electron length scale). The proposal is that one should use essentially same definition but adding p-adic fractality. In this picture the real variant of bosonic propagator defines also p-adic propagator: the real Lorentz invariants appearing in matrix element are mapped to p-adic ones by a proper variant of canonical identification. The resulting p-adic sum of various contributions to the propagator from various p-adic mass scales is then mapped back to reals and you get p-adic fractal. This is just the visit from reality to p-adicity and successful return together with brand new p-adically fractal bosonic propagator!;-)

- What I realized yesterday is that internal consistency is achieved
only ifthe loops involving gauge bosons vanish. This has been one of the one thousand and one formulations of quantum criticality during years. Therefore onlytree diagramswith emerging bosonic propagators and free fermionic propagators are needed. One obtains non-trivial coupling constant evolutionandtree diagrams! Both real and p-adic versions of perturbation theory exist since the mathematically existence bosonic loop integrals are absent. Number theoretical universality and p-adic fractality are both obtained. That p-adicity is visible also at the level of real scattering amplitudes is highly satisfactory. This picture generalizes also to the quantum TGD proper.

- Whether the general definition of quantum criticality as vanishing of the bosonic loops is equivalent with definition of quantum criticality allowing to deduce hyperbolic cutoff and which I have studied during last month (whose technical definition I will not discuss here) after p-adic fractalization remains an open question.

- What is interesting that twistorialization, which was the starting point of the work one half year ago, works for
tree diagrams. Physics requires non-trivial coupling constant evolution and thus loops in standard framework but loops are the basic problem of twistor approach since particles in loops are massive and twistorialization for them is not elegant. All the fantastic results of twistorialization program (say this) are for tree diagrams. Bosonic loops would not be present p-adically (would vanish in real sense) in quantum critical TGD Universe.

The relevant text can be found from the last section of the new chapter Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".

## 8 Comments:

Thanks for your thoughtful response Matti. Now I must read the chapter so that I will have more questions for you. And to elucidate the comment I made in the previous post, I don't see relating the Schroedinger and Heisenberg pictures as problemmatic I simply used the example as an anecdote to say that these perspectives are connected via a simple relationship.

Obviously I know very little of how to connect the p-adic picture to either one of those conventional pictures. Therefore I am exceedingly excited to read about how you bridge these gaps. Thanks.

Dear Mahndiza,

in TGD framework zero energy ontology means that one leaves the framework of wave mechanics where these two pictures are natural (time evolution as evolution of operators or of states). In TGD framework one gives up the idea of time evolution as unitary process and defines M-matrix as timelike entanglement coefficients between positive and negative energy parts of zero energy state.

If one wants to apply p-adic picture to wave mechanics, one has to consider many number theoretical problems. For instance, does the notion of Schrödinger amplitude and its unitary evolution generalize to p-adic context? As I started to work with p-adic physics I looked whether systems like harmonic oscillator have p-adic counterparts. There are interesting number theoretic issues involved. For instance, p-adic variants of trigonometric functions do not have the physically desirable periodicity properties.

My recent view is that p-adic counterpart of Schrodinger equation is not physically interesting. p-Adic physics appears at much deeper level in TGD framework: for modified Dirac equation it makes sense to speak about its p-adic version. M-matrix is second example: one maps Lorentz invariants like mass squared to p-adic numbers but does not introduce p-adic momentum space or p-adic Minkowski space.

Yes, I was wondering about the unitary operators and whether or not they have p adic counterparts. Thanks for the explanation, now what you are saying makes more intuitive sense to me. Yes, the p-adic counterparts of the circular functions would be a bit strange. I say this from a peripheral standpoint, because I am just learning.

But since the prime number mesh isn't continuous and the spacing between primes isn't a constant, I would imagine that periodicity would be difficult or nearly impossible to define in the p-adic framework.

Thanks. I'll keep reading.

OK perhaps saying nearly impossible is a bit of hyperbole, especially because of disjoint nature of p adic mesh, it could be periodicity defined within one of the disjoint disks but certainly not periodic throughout all space, as we are accustomed to with sine and cosine.

Tell me if I am wrong on this and if so where I have erred, if you have the time. Thanks.

For instance cos(x)= 1+x^2/2!+... exists p-adically if x has p-adic norm smaller than on (that is norm 1/p^n, n>0). This function is not however periodic although basic trigonometric identifies are satisfied. Therefore Fourier analysis using cos(nx) and sin(nx) does not make sense. One can however imagine discrete generalizations of Fourier analysis and of periodic functions involving algebraic extensions of p-adic numbers containing phases exp(i2*pi/n) not existing as p-adic numbers. This requires pinary cutoff (finite measurement resolution).

S-matrices which which are unitary are typically not expressible in terms of time evolution operator. The simplest example is S=exp(i2*pi/n) for 1-state system. You cannot generate this operator by exponentiating Hamilton (say S=exp(iht), h=1, t=2*pi/n) since S belongs to extension of p-adics. S-matrix makes sense only for discrete values of time parameter. Quite generally, number theoretical constraints lead to quantization.

Matti, an enormous amount of progress has been made on the standard twistor theory scattering amplitudes in the last few years - check out the recent arxiv papers by Hodges et al. And it would be nice to finally make some contact with the results of these smart ex-string theorists.

I read a paper of Hodges but did not understand too much. The style of Arkadi-Hamed paper was more comprehensible to me. There is also nice article by Penrose about history of the program which I found from net. Probably you have all the papers below. I glue the bibliodata in any case.

R. Penrose (1999). The Central Programme of Twistor Theory. Chaos, Solitons and Fractals 10, 581-611.

E. Witten (2003). Perturbative Gauge Theory As A String Theory In Twistor Space. arXiv:hep-th/0312171v2.

N. Arkani-Hamed, F. Cachazo, C. Cheung, J. Kaplan (2009). The S-Matrix in Twistor Space. arXiv:hep-th/0903.2110v1.

Z. Bern, L. J. Dixon, D. C. Dunbar, and D.A. Kosower. One Loop N Point Gauge Theory Amplitudes, Unitarity And Collinear Limits. Nucl. Phys. B 425, 217 (1994).

See also Z. Bern et al (2008).

Recurrence, Unitarity, and Twistors.

One of the natural twistorial ideas in TGD framework is the twistorial counterpart for the representation of space-time as surface in M^4xCP_2. One can consider 12-D TxCP2 and 10-D PTxCP2. It turns out that dimension is 4 in perturbative phase a (QFT in M^4 is good approximation) and 6,8,10,12 in non-perturbative phases (string like objects, CP2 type extremals). Brane analogy is obvious and a connection with M-theory resp. F-theory is highly suggestive.

The chapter Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD of the book "Towards M-matrix" explains the recent speculative picture.

It was very interesting for me to read this post. Thanks for it. I like such themes and everything connected to them. I would like to read more soon.

Alex

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