### Scattering amplitudes in positive Grassmannian: TGD perspective

The quite recent but not yet published proposal of Hamed and his former student Trnka has gained a lot of attention. There is a popular article in Quanta Magazine about their work. There is a video talk by Jaroslav Trnka about positive Grassmannian (the topic is actually touched at the end of the talk but it gives an excellent view about the situation) and a video talk by Nima Arkani-Hamed. One can also find the slides of Trnka .

The basic claim is that the Grassmannian amplitudes reduce to volumes of positive Grassmannians determined by external particle data and realized as polytopes in Grassmannians such that their facets correspond to logarithmic singularities of a volume form in one-one correspondence with the singularities of the scattering amplitude. Furthermore, t the factorization of the scattering amplitude at singularities corresponds to the singularities at facets. Scattering amplitudes would characterize therefore purely geometric objects. The crucial Yangian symmetry would correspond to diffeomorphisms preserving the positivity property. Unitarity and locality would be implied by the volume interpretation. Nima concludes that unitarity and locality, gauge symmetries, space-time, and even quantum mechanics emerge. One can however quite well argue that its the positive Grassmannian property and volume interpretation which emerge. In particular, the existence of twistor structure possible in Minkowskian signature only in M^{4} is absolutely crucial for the beautiful outcome, which certainly can mean a revolution as far as calculational techniques are considered and certainly the new view about perturbation theory should be important also in TGD framework.

The talks inspired the consideration of the possible Grassmannian formulation in TGD framework in more detail and to ask whether positivity might have some deeper meaning in TGD framework.

The vision about what BCFW approach to generalized Feynman diagframs could mean has been fluctuating wildly during last months. The Grassmannian formalism for scattering amplitudes is expected to generalize for generalized Feynman diagrams: the basic modification is due to the possible presence of CP_{2} twistorialization and the fact that 4-fermion vertex - rather than 3-boson vertex - and its super counterparts define now the fundamental vertices. Both QFT type BFCW and stringy BFCW can be considered. The recent vision is as follows.

- Fermions of internal lines are massless in real sense and have unphysical helicity. Wormhole contacts carrying fermion and antifermion at their opposite throats correspond to basic building bricks of bosons. For fermions second throat is empty. The residue integral over the momenta of internal lines replaces fermionic propagator with its inverse and replaces 4-D momentum integration with integration over light-cone using the standard Lorentz invariant integration measure.

- 4-fermion vertex defines the fundamental vertex and contains constant proportional to length squared. This is definitely a problem. If all 4-fermion vertices contain one bosonic wormhole contact, one can replace regard the verties effectively as BFF or BBB vertices. The four-fermion coupling constant L
^{2}having dimensions length squared can be replaced with 1/p^{2}for the bosonic line: this is the new ingredient allowing to overcome the difficulties assignable to four-fermion vertex.

- For QFT type BFCW BFF and BBB vertices would be an outcome of bosonic emergence (bosons as wormhole contacts) and 4-fermion vertex is proportional to factor with dimensions of inverse mass squared and naturally identifiable as proportional to the factor 1/p
^{2}assignable to each boson line. This predicts a correct form for the bosonic propagators for which mass squared is in general non-vanishing unlike for fermion lines. The usual BFCW construction would emerge naturally in this picture. There is however a problem: the emergent bosonic propagator diverges or vanishes depending on whether one assumes SUSY at the level of single wormhole throat or not. By the special properties of the analog of N=4 SUSY SUSY generated by right handed neutrino the SUSY cannot be applied to single wormhole throat but only to a pair of wormhole throats.

- This as also the fact that physical particles are necessarily pairs of wormhole contacts connected by fermionic strings forces stringy variant of BFCW avoiding the problems caused by non-planar diagrams. Now boson line BFCW cuts are replaced with stringy cuts and loops with stringy loops. By generalizing the earlier QFT twistor Grassmannian rules one ends up with their stringy variants in which super Virasoro generators G, G
^{†}and L bringing in CP_{2}scale appear in propagator lines: most importantly, the fact that G and G^{†}carry fermion number in TGD framework ceases to be a problem since a string world sheet carrying fermion number has 1/G and 1/G^{†}at its ends. The general rules is simple: each line emerging from 4-fermion vertex carries 1/G and 1/G^{†}as vertex factor. Twistorialization applies because all fermion lines are light-like.

- A more detailed analysis of the properties of right-handed neutrino demonstrates that modified gamma matrices in the modified Dirac action mix right and left handed neutrinos but that this happens markedly only in very short length scales comparable to CP
_{2}scale. This makes neutrino massive and also strongly suggests that SUSY generated by right-handed neutrino emerges as a symmetry at very short length scales so that spartners would be very massive and effectively absent at low energies. Accepting CP_{2}scale as cutoff in order to avoid divergent gauge boson propagators QFT type BFCW makes sense. The outcome is consistent with conservative expectations about how QFT emerges from string model type description.

- The generalization to gravitational sector is not a problem in sub-manifold gravity since M
^{4}- the only space-time geometry with Minkowski signature allowing twistor structure - appears as a Cartesian factor of the imbedding space. A further finding is that CP_{2}and S^{4}are the only Euclidian 4-manifolds allowing twistor space with Kähler structure. Since S^{4}does not allow Kähler structure, CP_{2}is completely unique just like M^{4}. Stringy picture indeed treats gravitons and other elementary particles completely democratically.

- The analog of twistorial construction in CP
_{2}degrees of freedom based on the notion of flag manifold and geometric quantization is proposed. Light-likeness in*real*sense poses a powerful constraint analogous to constraints posed by moves in the case of SYMs and if volume of a convex polytope dictated by the external momenta and helicities provides a representation of the scattering amplitude, the tree diagrams would give directly the full volume.

As both Arkani-Hamed and Trnka state "everything is positive". This is highly interesting since p-adicization involves canonical identification, which is well defined only for non-negative reals without further assumptions. This raises the conjecture that positivity is necessary in order to achieve number theoretical universality.

**Addition**: Lubos wrote again about amplituhedrons and managed to write something about which I can agree almost whole-heartedly. Not a single mention of superstrings or Peter Woit! I also agree with Lubos about Scott Aaronson's parody: Scott was entertaining but not deep. The twistors and Grassmannians will revolutionize theoretical physics in many manners: my basic bet is that the uniqueness of M^{4}×CP_{2} from twistorial considerations and positivity conditions (whatever they really mean Minkowskian signature!) as a prerequisite for p-adicization will be at the core of the revolution and make TGD a mathematical "must". Mathematicians might be able to generalize the Grassmannian approach to CP_{2} degrees of freedom without much effort.

For details see the chapter Some fresh ideas about twistorialization of TGD or the article with the same title.

## 57 Comments:

Matti,

Was this not the Hammed whom you had extensive comment dialog on the nature of TGD? I did not exchange with him direct talk but I did post several things between your conversation. The quasic notation of course is about a positive diagram method and a more general concept of regions involving singularity.

I saw the slides and did not think these were so deep but I agree this direction is little understood by the complaints of overly cautious physicists in comments and the impact is vastly underestimated.

By chance I first heard of it thru Lubos who had an interesting response... to your ideas perhaps as not so out of the mainstream after all...humorous. But why do you want to go back to analyze how TGD fits in when part of the theory seems based on your and other arithmetical geometry bloggers as the long standing source?

My Quasic view makes other philosophical stances critical to a wider generalization that is a new physics all documented in my blog and not proposed in papers on or off line. The dynamic action of the boundary or of the area can be of different senses which is a force or which a falling.

The problem is how we better define the idea of dimensions and groups to which not only are we limited by the old quantum terminology and complicated formuli but by the range of standard theory and even string theory limited in its reduction methods of geometry- scattering angles are just one way to interpret deeper things that amount to ideas of complex analysis including octonions. We can see this in partial theories where they invole 8 natural dimensions, the Monster groups and all that.

It is also not to be assumed in his theory that this involves a sort of emergence- nor a specific origin in cosmology or particle physics. That is but half the view. We see this in the realization that the Big Bang alternative may not exist as such or the expanding universe is but an illusion (better the lack of understanding the symmetry of the idea of prime uniqueness and the use of e^n notations.

It is critical that we understand, as you maintain and see in further depth of TGD the role of cyclic groups and continuum power sets of two... half infinity or more is the issue so to speak that complexity is as well as arises and moves in the positive direction of time as a probability asymmetric average.

The PeSla

At last Lubos has a posting on this idea of a structure (which I have also stated as beyond twistor theory or a generalization of it) which is reasonable and sensible.

Welcome, well depending on where the winds of spin shifts, to our revolution our humble correspondent.

Maybe if you applied this to some form of string theory you just might add to a significant vision that surpasses the quantum theory after all in its scope...

To Otto:

String theory cannot be "applied" to TGD. It would be a horrible mistake to try this since string theory has degenerated to a physical nonsense due to the landscape stuff coming basically from an attempt to deduce four-dimensional space-time although theory describes 2-D space-times as basic objects. String theory would be a fantastic theory if space-times were 2-D surfaces 2-D in 10-D space.

TGD shares however a lot of good aspects of string theory.

*Super-conformal invariance in 2-D sense is generalized to 3-D light-like case. Strings appear as string like 4-surfaces - cosmic strings and the whole TGD inspired cosmology relies on them. Yangian generalizaton of these symmetries could be the ultimate symmetry.

*String world sheets appear as loci of solutions of modified Dirac equation for spinor modes with right-handed neutrino as exception.

* AdS/CFT duality is replaced with a duality in which space-time surface replaces AdS and corresponds to strong form of holography reducing to strong form of general coordinate invariance. Note again how the conceptual economy eliminates works.

I have of course nothing against string as a mathematical and physical object. I am only against M-theory, F-theory, and whatever containing all this ad hoc stuff generated by spontaneous compactification idea, certainly one of the ugliest ideas in the history of theoretical physics. Spontaneous compactification makes string theory a cognitive monster.

Matti,

That was a helpful and informative reply as to your take on the current debates on physics.

I hate to bring it up but Lubos has a guest on his blob today by Andreas Karch which seems to be written well presenting his stance on holography and so on... I like his understanding of what is called Toy Models as in your concluding interpretation of "spontaneous compactification" as philosophically reductionist physics.

I am conflicted with soul searching as to these Theories Of (TOE) Yiddish Physics that sustains the old order of things fine tuning the theoretical terrain against any possibility of factual alternative systems... Einstein was Wrong! Economics is not harder than physics it is physics, and gives the substance of meaning to the prevailing myths of our time as if a quasi religion or inflation idea that rules the world by the mathematical model of compound interest that eventually justifies the elimination of anything higher or spiritual. We are victims of this cultural war which in the LHC is centered in Europe at this time.

If various theoreticians know where the problems may be then why can they not imagine a wider answer after all this time... the slow molasses like tar drop experiment has what to do exactly with Higgs and other Strong mass and all that? Is this simplest model more instructive than our colliders? Or is it just living on the surface of our simple physics familiar in our life scales and time?

The sad thing is that even with these unanswered questions the live and breathe as if they are right and feel others a threat or potential one so defame them as crackpots, even to there own natural allies or poor relatives.

What do you think of this article if you bother to review theory 101?

Maybe its connection to Uni Wisconsin and NPR is an emotional all poetry is about the holocaust victim mentality, all the cats in the Big Bang comedy have Jewish names (as in that episode). Or that 2% of the songwriters form 80% of what is American culture heritage. Another fund social and political issues as science from NPR.

Hey I like Barbara Streisand and Dylan's Bible moral issues down highway 61 - it was my era and he played like me (like music hundreds of years ago before jazz and swing claimed it) math combinations really. Except I had no part of the hallucination LSD revolution.

But in our cultural experiences I find it hard to find a common language to discuss the issues objectively- but still work alone not much of a joiner after all.

Nor should we beat each other up in the admiration of the themes of the movie Fight Club.

I wonder how T. Banks approach is? He evidently tries to imitate TGD but with the stringy concept. Does he also fail as Witten once did with geometrodynamics, due to the frame from Einsteinian view?

Matti

in a flurry of exchanges on google plus with David Chako... who seems to share many ideas... I mentioned you in a post... some of which is on my current pesla blogspot where I posted some of the conversation from my side.

p-adics I used casually for a brief point but knew nothing about them until I visited your pages and then elsewhere on the web.

http://arxiv.org/abs/1208.5715 The Top 10^{500} Reasons Not to Believe in the Landscape

T. Banks

To Ulla:

superstring hegemony is finally beginning to crunch down.

Banks has been one of the top figures and as the title of the paper tells has finally realized that the landscape makes no sense. Susskind admitted recently that string theory is not complete: a very politically manner to say that it is physical nonsense.

Also Polchinski have become skeptic. Even the cherished AdS/CFT dogma have been questioned.

Neil Turok talked to new students some time ago and made it clear that last decades in theoretical physics have produced endlessly models which do not work.

This raises my cautious optimism: theoretical physics colleagues are certainly slow minded and shallow thinkers, a crowd running behind the latest fashion, but they are not complete idiots;-).

To Ulla:

Tom Banks is teacher of Lubos and becoming now ex string theorist. Tom Banks "independently" discovered hyperfinite factors of type II_1 and also causal diamonds. Now he has discovered about string theory what I have been telling since 1984!;-)

Dear Matti,

1-there is a basic difference between classical TGD and classical standard QFT: Dirac spinor in classical TGD is induced to a very small 3-surface. It’s argument is just over the 3-surface.

But in classical standard QFT, it has extended over large region of space-time. It’s argument is over the space-time.

If this is correct, I had not been realized this very simple thing before and was made some confusions.

I try to understand better by making simple questions:

2-Suppose there is a moving electron. It produces electromagnetic potential around it. The electromagnetic potential is induced spinor connection of CP2 to the 3-surface of photon. But in this case, the components of weak potential vanish. By just classical TGD, one cannot describe interaction of the electron by other electrons? It needs quantum TGD and there is space of 3-surfaces(of photons) between the electrons So that if the 3-surfaces supposed to be points, Quantum TGD is reduced to classical electrodynamics?

Dear Hamed,

excellent questions going to the heart of TGD;-). I explain first various meanings of "spinor" in TGD framework and then answer your questions.

1. There are three kinds of spinors in TGD.

* WCW spinors which correspond to Fock states created by fermionic oscillator operators assignable to second quantized induced spinor fields. WCW gamma matrices are expressible in terms of fermionic oscillator operators and have well-defined quark or lepton number and anticommute to WCW Kaehler metric. Physical states correspond to WCW spinor fields which in ZEO ontology correspond pairs of states assignable to the partonic 2-surfaces at the opposite boundaries of CD. Also 4-D tangent space-data of partonic 2-surfaces are included in the definition of states.

*Imbedding space spinors are essentially associated with *cm* degrees of freedom at imbedding space level and characterized the ground states of representations of super conformal algebras and associated Kac-Moody algebras and supersymplectic algebras.

They are tensor products of M^4 and CP_2 spinors: chirality corresponds to whether lepton or quark is in question and different coupling to Kaehler gauge potential guarantees that quark like spinors correspond triality=1 (-1 for antiquarks) representations and leptonic to triality =0 representations. The CP_2 spinor harmonics however do not have correct correlation between ew and color quantum numbers (or rather there is correlation depending on em charge). For representations of Super Virasoro algebra for which generators can carry color quantum numbers the physical states are either color triplets or singlets as required. This is gone through in p-adic mass calculations. It would be nice if someone would go the arguments through some day again.

*Induced spinors are imbedding space spinors at space-time surfaces. For quark chirality they differ from ordinary quarks spinors in that color do not appear as spinlike quantum number as in QCD. The condition that the spinor modes have well-defined em charge is non-trivial since induced electroweak gauge potential contains W part and mixes different fermionic charge states. The proposed solution of the problem is that spinor modes are localized at 2-D string world sheets. Therefore fermionic string model would become part of TGD.

This condition would not apply to right-handed neutrino which has no ew couplings Right handed neutrinos are delocalized at entire space-time sheets and genuinely 4-D. Right-handed neutrino is also in a special role since it allows covariantly constant mode because electroweak couplings vanish. The covariantly constant mode could generate N=2 SUSY and the degeneracy of states for fermion and its sparterns would be same as for N=4 SUSY multiplet which together with possibility to twistorizalize in M^4 and CP_2 degrees of freedom suggests very strong similarity with N=4 SUSY. I have developed a view what the Grassmannian picture in TGD would look like. What is remarkable that M^4 and CP_2 are completely unique: M^4 is the only 4-D geometry with Minkowskian signature allowing twistor structure and CP_2 the only Euclidian 4-D Kaehler geometry allowing also Kahler structure in twistor space. Hence TGD is twistorially unique. The challenge would be to understand twistorialization in CP_2 degrees of freedom. Twistorialization relies on 4-fermion vertex instead 3-vertex as in N=4 SUSY. Virtual fermions have light-like momenta but non-physical helicity so that propagator line proportional to inverse of propagator does not vanish. In SUSY approach one allows light-likeness in complex sense.

To be continued....

Dear Hamed,

a continuation giving answers to your questions.

Answers to your questions require distinction between two different levels of theory. QFT limit associated with cm degrees of freedom of 3-surface and microscopic theory defined in terms of space-time sheets and analogous to string theory.

1. Modified Dirac equation for induced spinors is 4-dimensional. Solutions are restricted to string world sheets except for right-handed neutrino. These spinors correspond to truly microscopic level relating to the QFT level somewhat like string theory to its QFT limits.

2. Induced spinors do *not* correspond directly to the spinors of QFT:s. At pointlike limit 3-surfaces become points and only cm degrees of freedom - M^4xCP_2 and corresponding spinors representing ground states of conformal algebra representations remain. The massless states of these representations with vanishing conformal weight correspond to low energy world. This step is like Kaluza-Klein theory and dropping higher modes in CP_2 degrees of freedom gives QFT limit. Also twistor Grassmannian approach has already integrated over the microscopic details and fermionic strings are not visible anymore.

3. Classical TGD with induced spinor fields is analogous to string model description, very microscopic. I already mentioned the necessity to assumed that spinors modes with well-defined em charge are localized to 2-D string world sheets (again an argument to be checked!). Microscopically can identify elementary particles as pairs of wormhole contacts with Kaehler magnetic monopole fluxes between them along the space-time sheets connected by contacts.

Fermionic string is closed string associated with this structure and this would provide the string description of elementary particle: this includes also gauge boson quanta, which correspond to wormhole contacts with fermion and antifermion at opposite throats. Closed string interactions would be the microscopic description of the interactions at space-time level and would describe all interactions.

At QFT limit one would obtain twistor Grassmann description using fermions and identifying gauge bosons as bound states of massless fermions assigned to opposite throats of wormhole contact. On mass shell gauge bosons would be bound states of massless fermions. Amusingly, just now condensed matter physics reported the discovery bound states of massless photons! Mechanism forming bound states is of course not the same.

As a consequence, standard model results at point like limit, not only QCD. This involves idealization of color of partial waves with spin-like color and genus characterizing particle families as additional degeneracy index. At hadronic scales these idealizations are excellent just as idealization of Sun and planets as point like objects is excellent in solar system gravitational theory.

4. One very important question that you did not ask: how does one obtain perturbation theory, that is linear superposition for various gauge fields treated as independent degrees of freedom if all gauge fields and also metric of space-time are expressible using only four suitably chosen coordinates of imbedding space?

The answer at microscopic level is provided by many-sheeted space-time. Each system carries its classical fields at space-time sheets belonging to it. Test particle develops touches all these space-time sheets and thus interacts. The wormhole contact in question is not stable since it does not carry Kahler magnetic flux. The effects of various space-time sheets on particle are additive and this is enough for linear superposition.

At QFT limit the answer is provided by the fact that the fields of QFT correspond to massless ground states of Super conformal representations characterized by spinor harmonics of M^4xCP_2 and momentum, color, and ew quantum numbers. For these linear superposition holds true in perturbation theory.

Matti

Dear Matti,

Thanks,

There is a canonical momentum associated with imbedding space coordinates of cm of a 3-surface. The momentum is derived from partial derivative of lagrangian with respect to the coordinates.

In other hand, there is another momentum:

If we write the Fourier series for the induced spinor of the 3-surface, there are momentums associated with every mode as p=hbar*k. the momentums are not time derivatives of any coordinate.

What is interpretation of TGD about the last momentum?

What is relation between the first momentum and the last?

Dear Hamed,

there are canonical momentum densities and in terms of these one can assigned conserved four momentum densities (also angular momentum densities and color charge densities) to Kaehler action. Integral over 3-surface gives the classical four momentum.

There is also Fourier series expansion making sense in open region of space-time with trivial topology- and many other expansions such as partial wave expansion. Since field equations are extremely non-linear, all wave vectors are present. One can speak only about wave vectors since hbar does not appear in classical field equations and they do not correspond to physical four-momentum values. In linear free field theories there would be interpretation as real momenta but not now.

Note that canonical quantization of field theories does not generalise since the correspondence between canonical momentum densities and time derivatives of imbedding space coordinates is not 1-to-1 and one cannot solves time derivatives in terms of canonical momentum densities. This problem led to the idea about WCW geometry.

An interesting question is whether classical four momentum corresponds to quantal four-momentum as quantum classical correspondence would suggest. More generally, is this correspondence true for Cartan algebra. Preferred extremal property strongly suggests the analog of Bohr quantisation for various classical charges.

Very intriguing discussion here. I noticed this will broswing the web, interesting stuff anyway... http://mobile.theverge.com/2013/10/4/4799326/day-at-genius-camp-the-institute-for-advanced-study the silly thing is I am fairly familiar with the stuff discussed by this author.. stumbled across them working on the fruitless Riemann Zeta hypothesis research a few years ago... anyway, peace out

For some reason I find this very entertaining

http://en.wikipedia.org/wiki/Kelvin_transform

Also, you guys should install retroshare at http://retroshare.sf.net and add me, here are 3 of my keys

https://docs.google.com/uc?export=download&id=0B5kp8BrW_9rdc2hTNld2M2ZkOW8

https://docs.google.com/uc?export=download&id=0B5kp8BrW_9rdMFhNRWVjOTNYSEE

https://docs.google.com/uc?export=download&id=0B5kp8BrW_9rdeXhmOG5uZmNuTlU

I've even mirrored tgdtheory.fi on my node :)

I know Matti won't join due to some nebulous "info flux" :(

http://www.aeonmagazine.com/being-human/will-we-ever-get-our-heads-round-consciousness/

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What bliss! What joy! In these troubled times, we can take comfort in knowing that Mary Worth is somewhere being thanked for her alleged good deeds.

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Matti,

I wondered where you had been as I had not seen a blog post for awhile- these times are generating wider ideas to explain recent data. You have been busy with dialog here in the comments... the odd thing is we have reached some same conclusions - I think you see me in just a string and supersymmetry camp. It does not work as you so describe but it is hardly dying out for it stands on what to me are incomplete or fatal assumptions not deep enough to show where it fits in. Why is there only the right hand of things like neutrinos? OK M4 CP2 sym breaking spontanously and all that, yang mills compactification and all that and anything that in matters of gravity or particle radiation or spin in motion that depends simply on angle or complex analysis (quaternions) simply confines our exporations. So let us try this - another blogger I follow but comment rarely to Hogg. what do you think of this which is an older theory that suggests pulsars are four dimensional processes (but the truth is a little more complicated than that): http://hoggresearch.blogspot.com/2013/10/pulsars-eccentricities.html

You either have been deeper into thoughts and beakthru's or it has not been understood or communicated. Also we see that the number of jobs in the field is fewer than the next Fibonacci number of PhD's the career path worm hole empty or not but for deeper reasons. Is Hammed the same person I speculated was the one with the amplitudeohedron? No matter, as in an open society science thrives only with our open minds and a closed one can only make copies of a superior source as we forget earlier history and go down dead end paths. Empty worm holes... perhaps, but not by the Fermi terminology of the past present.

And hello Steven, I posted a couple of things on viXra blog again...

Dear Otto,

I spent a week in travel and could not comment on anything. I have had also other activities preventing blog activity.

There are two Hameds! The Hamed here is considerably younger than the Hamed there;-).

Some comments inspired by you question about right handed neutrinos.

1. There are also left handed neutrinos. They couple to weak interactions whereas righthanded ones have no weak interactions. This is crucial. As a consequence righthanded neutrinos behave quite differently as compared to other fermions: they are delocalized to entire space-time sheet rather than being localized to string world sheets by the condition that the mode has well-defined em charge.

2. Right-handed neutrino allows covariantly constant modes giving rise to SUSY which at the combinatorial level (number of spartners) corresponds to N=4 SUSY.

3. My educated guess is that this together with masslessness of also virtual fermions (helicity unphysical) implies that loop corrections to self energy and to vertex vanish. An even stronger implication would be vanishing of all loop corrections. The reason would masslessness of virtual fermions.

Virtual bosons identified as wormhole contacts with fermion and antifermion at the opposite ends would have continuous mass spectrum and this would make BFCW non-trivial: cut would be associated with virtual boson line.

4. What is especially beautiful is that the breaking of SUSY with identical mass formulas but different p-adic mass scales for physical fermions would not affect this cancellation mechanism since it takes place only for on external on mass shell fermions.

5. BRCW recursion would reduce to that for tree diagrams. There would be no non-planar diagrams so that also this problem would be solved. Theory would be very much like gauge theory with fermions and gauge bosons consisting of fermion-antifermion pairs but with vanishing radiative corrections.

6. p-Adic coupling constant evolution would give rise to discretized coupling constant evolution. Different p-adic primes (possibly with restriction to primes near powers of 2) would correspond to different length scales as different phases of matter.

7. This is the TGD counterpart for standard model. To include gravitons one must replace elementary particles as wormhole contacts with pairs of wormhole contacts connected by Kahler magnetic monopole flux tubes giving rise to closed magnetic flux loops with associated string so that string model like description emerges.

The fermionic lines of diagrams of a theory with BFF and BBB vertices with B identified as wormhole contact are replaced string world sheets. Bosonic lines correspond to pairs of string world sheets connected by wormhole contacts at ends of string. You get physical mechanism for the massivation of weak bosons as screening of weak charges, analogs of hadrons, and gravitation.

This is the overall view now and I believe that it begins to be rather near to the correct one now. Twistor Grassmannian approach has been crucial in inspiring the evolution of this picture and it is interesting to see what amplituhedron in this framework means.

Matti

Thank you Matti, I did like they young Hammed's questions and your replies to him...

You item 4 is one of which I agree but not sure we see the same alternative thing or the usual physics does not see something similar also if they would dare see more.

I wonder how he amplitudhedron and the associahedron compare?

number 6 does relate to my recent informal drawing on my last post blackboard as we can do more with these almost invisible ideas at the arithmetical and topological foundations...

I said I think the N=4 SUSY does have a variety of symmetry breaking concepts... as in number 2. but co-variance is a relative thing that if this applies to mirrors that seem like dark matter, in an absolute value structure (quason) finite large or small models need not diverge as a unified theory. One that merges holographic and fractal concepts, string and loop theories, spins and twists and so on... emerging and steady states...two worm hole mouths with nothing in between could be a continuum of mass per my quasic principle of what is different or the same even if the tunnel is empty, full or multiple.

I just want to get things clearer in all our researches... to see in the discernible of indiscernibles ( Leibniz ) I agree near the correct answers... with a few more simple steps and hopefully counting in a quasi-finite universe.

Dear Edgar,

Amplituhedron is essentially dynamical object. If I have understood correctly, associahedrons are purely combinatorial objects so that as such they are not enough.

Nima et al claims that the postive and real volume of amplituhedron gives the entire scattering amplitude. This cannot be true since unitarity requires complex scattering amplitudes. It is a pity that they use so sloppy language. From some source I got the impression that the full amplitudes are products of complex MHV amplitude (maximal helicity violating amplitude) and volume of a polyhedron.

By the way, there is an excellent peddagogical article about twistor Grassmannian approach in arXiv:

http://arxiv.org/pdf/1308.1697.pdf .

Thanks for the link, sort of a reference for old ideas as if the usual hieroglyphics to help us learn a dead language like Hebrew or Ancient Greek then resurrect it with its difficult and forgotten tenses in the name of national pride...

The language may sound a certain way in a theory but still be gibberish. It is ok for its time. Why bother to speak tongue tied in Feynmanese or Grassmaian when we may not have clear meaning in what is conformal complex numbers or some vague named God as supper gravity...? I thought your approach transcended this but like Newton presented the ideas in the old geometry without the benefit of his calculus. Even then the gist of this article tells me the associahedron and ampltudohedron are the same animal if they exist, that difference I suppose in what is dynamic or not, majorana or not with respect to neutral particles. It all starts with a narrow dimensional view of symmetry and conservation. Oh well,I will still relish your posting but the language barrier between our ideas are more difficult than that of Finnish and English... Cheers, and maybe check out viXra some more.

https://www.simonsfoundation.org/quanta/20131010-neutrino-experiment-intensifies-effort-to-explain-matter-antimatter-asymmetry/

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Sorry for this mad brad. This is the third similar "comment".

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Mad Brad has many names….;-)

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Mad Brad has really many names. For God's sake Brad: go to psychiatrist.

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Mad Brad again!

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I am sorry for Mad Brad. He has lost totally his mental health. The sends these idiotic messages on daily basis now. He should get professional help as soon as possible.

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Really sorry for this imbecile sending these idiotic posts. I hope that he could find help to this problems.

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Dear Samad/Tarir/ Abdullah/.…

A friendly advice of an older man.

Please, go to talk with some professional of psychiatry if it is possible. You are badly in need of psychiatric help. No one in his right mind spends his time time to this kind of stupid trolling. You have nothing to lose, you can only win. Just talking about your problems could help to get wider perspective to them.

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Please Hassan/etc… try to find help. You have bad personal problems. You are wasting my time and most importantly yours. By finding help in time you could live a good life instead of wasting it to this stupid warfare. With all sympathy and realizing that

you really are in need of help.

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Hopeless case would doctor say.

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This Ahsan Afsar alias Adnan Khan isa sad case. He is sending advertisements with average rate of 2 per week. Has done this for a couple of years. Doesn't this poor fellow have anything else to do?

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