### About the SUSY generated by covariantly constant right-handed neutrinos

The interpretation of covariantly constant right-handed neutrinos (briefly ν

_{R}in what follows) in M

^{4}× CP

_{2}has been a continual head-ache. Should they be included to the spectrum or not. If not, then one has no fear/hope about space-time SUSY of any kind and has only conformal SUSY. First some general observations.

- In TGD framework right-handed neutrinos differ from other electroweak charge states of fermions in that the solutions of the modified Dirac equation for them are delocalized at entire 4-D space-time sheets whereas for other electroweak charge states the spinors are localized at string world sheets (see this).

- Since right-handed neutrinos are in question so that right-handed neutrino are in 1-1 correspondence with complex 2-component Weyl spinors, which are eigenstates of γ
_{5}with eigenvalue say +1 (I never remember whether +1 corresponds to right or left handed spinors in standard conventions).

- The basic question is whether the fermion number associated with covariantly constant right-handed neutrinos is conserved or conserved only modulo 2. The fact that the right-handed neutrino spinors and their conjugates belong to unitarily equivalent pseudoreal representations of SO(1,3) (by definition unitarily equivalent with its complex conjugate) suggests that generalized Majorana property is true in the sense that the fermion number is conserved only modulo 2. Since ν
_{R}decouples from other fermion states, it seems that lepton number is conserved.

- The conservation of the number of right-handed neutrinos in vertices could cause some rather obvious mathematical troubles if the right-handed neutrino oscillator algebras assignable to different incoming fermions are identified at the vertex. This is also suggested by the fact that right-handed neutrinos are delocalized.

- Since the ν
_{R}:s are covariantly constant complex conjugation should not affect physics. Therefore the corresponding oscillator operators would not be only hermitian conjugates but hermitian apart from unitary transformation (pseudo-reality). This would imply generalized Majorana property.

- A further problem would be to understand how these SUSY candidates are broken. Different p-adic mass scale for particles and super-partners is the obvious and rather elegant solution to the problem but why the addition of right-handed neutrino should increase the p-adic mass scale beyond TeV range?

If the ν_{R}:s are included, the pseudoreal analog of * N*=1 SUSY assumed in the minimal extensions of standard model or the analog of * N*=2 or even * N*=4 SUSY is expected so that SUSY type theory might describe the situation. The following is an attempt to understand what might happen. For an earlier attempt see this.

**1. Covariantly constant right-handed neutrinos as limiting cases of massless modes **

For the first option covariantly constant right-handed neutrinos are obtained as limiting case for the solutions of massless Dirac equation. One obtains 2 complex spinors satisfying Dirac equation n^{k}γ_{k}u=0 for some momentum direction n^{k} defining quantization axis for spin. Second helicity is unphysical: one has therefore one helicity for neutrino and one for antineutrino.

- If the oscillator operators for ν
_{R}and its conjugate are hermitian conjugates, which anticommute to zero (limit of anticommutations for massless modes) one obtains the analog of*N*=2 SUSY.

- If the oscillator operators are hermitian or pseudohermitian, one has pseudoreal analog of
*N*=1 SUSY. Since ν_{R}decouples from other fermion states, lepton number and baryon number are conserved.

Note that in TGD based twistor approach four-fermion vertex is the fundamental vertex and fermions propagate as massless fermions with non-physical helicity in internal lines. This would suggest that if right-handed neutrinos are zero momentum limits, they propagate but give in the residue integral over energy twistor line contribution proportional to p^{k}γ_{k}, which is non-vanishing for non-physical helicity in general but vanishes at the limit p^{k}→ 0. Covariantly constant right-handed neutrinos would therefore decouple from the dynamics (natural in continuum approach since they would represent just single point in momentum space). This option is not too attractive.

**2. Covariantly constant right-handed neutrinos as limiting cases of massless modes **

For the second option covariantly constant neutrinos have vanishing four-momentum and both helicities are allowed so that the number of helicities is 2 for both neutrino and antineutrino.

- The analog of
*N*=4 SUSY is obtained if oscillator operators are not hermitian apart from unitary transformation (pseudo reality) since there are 2+2 oscillator operators.

- If hermiticity is assumed as pseudoreality suggests,
*N*=2 SUSY with right-handed neutrino conserved only modulo two in vertices obtained.

- In this case covariantly constant right-handed neutrinos would not propagate and would naturally generate SUSY multiplets.

**3. Could twistor approach provide additional insights?**

Concerning the quantization of ν_{R}:s, it seems that the situation reduces to the oscillator algebra for complex M^{4} spinors since CP_{2} part of the H-spinor is spinor is fixed. Could twistor approach provide additional insights?

As discussed, M^{4} and CP_{2} parts of H-twistors can be treated separately and only M^{4} part is now interesting. Usually one assigns to massless four-momentum a twistor pair (λ^{a}, ξ^{a'}) such that one has p^{aa'}= λ^{a}ξ^{a'} ( ξ denotes for "\hat(\lambda)" which html does not allow to express). Dirac equation gives λ^{a}= +/- (ξ^{a'})^{*}, where +/- corresponds to positive and negative frequency spinors.

- The first - presumably non-physical - option would correspond to limiting case and the twistors λ and ξ would both approach zero at the p
^{k}→ 0 limit, which again would suggest that covariantly constant right-handed neutrinos decouple completely from dynamics.

- For the second option one can assume that either λ or ξ
^{a'}vanishes. In this manner one obtains 2 spinors λ_{i}, i=1,2 and their complex conjugates ξ^{a'}_{i}as representatives for the super-generators and could assign the oscillator algebra to these. Obviously twistors would give something genuinely new in this case. The maximal option would give 4 anti-commuting creation operators and their hermitian conjugates and the non-vanishing anti-commutators would be proportional to δ_{a,b}λ^{a}_{i}(λ^{b})_{j}^{*}and δ_{a,b}ξ^{a'}_{i}(ξ^{a'}_{j})^{*}.

If the oscillator operators are hermitian conjugates of each other and (pseudo-)hermitian, the anticommutators vanish.

An interesting challenge is to deduce the generalization of conformally invariant part of four-fermion vertices in terms of twistors associated with the four-fermions and also the SUSY extension of this vertex.

For details see the new chapter Some fresh ideas about twistorialization of TGD of "Towards M-matrix" or the article with the same title. The homepage is off just now as I told in previous posting but I hope that the situation changes within week or two.

## 8 Comments:

http://motls.blogspot.com/2013/09/16-out-of-half-billion-elite-calabi-yau.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+LuboMotlsReferenceFrame+%28Lubos+Motl%27s+reference+frame%29

5 hrs after your post I see this one of Lubos... what do you make of it - I suspect it comes closer to your and my ideas of a wider physics... a matter of simple counting and non-Lie groups? Why did you pick this topic today - something in the science news I did not come across?

To Edgar Otto:

I do not believe in super-string approach as I have explained many times. The evolution of particle physics during last four decades is a sequence of fatal mistakes due to sloppy thinking known as pragmatism.

The fundamental mistake of string theory was made in the beginning. Giving up 4-D space time and considering strings and fundamental objects is the basic mistake. After that came spontaneous compactification, brane non-sense, and finally landscape catastrophe. Lubos's last hope is that one could find the correct variant of string model by just studying Calabi-Yaus. This does not work since physical input is wrong extrapolation of standard model.

The mistakes at the level of phenomenology (just fits by choosing suitable Calabi-Yau or M-theory variant of it) are based on GUT philosophy and erratic view about SUSY as N=1 SUSY: already now LHC has in practice excluded this option.

There is also a mistake at the level of QCD. Also the interpretation of color symmetries and electroweak symmetries as completely comparable gauge symmetries is mistake if TGD is correct and I have strong reasons to believe that it is.

In TGD ew and color symmetries correspond to holonomies and isometries of CP_2. One should return back to the level of QCD if one wants to get out of the recent dead alley whose existence is now admitted even by particle theorists.

I one wants to get out of this alley, one should start by answering questions like "What is behind conservation of B and L?", "What selects standard model symmetries as special", "How to overcome the problems of general relativity: in particular energy problem", etc...

But all that I say - although correct - does not help: it is like trying to stop a Pendolino travelling 200 km/h towards a stone wall by bravely standing on the rails.

Thanks for the reply - what if the Pendolino is going as fast as it can or maybe if it reaches the wall tunnel thru it?

After all this new reading and debates on line, new experiments and so on I do feel a little lost as you say the particle physicists admit they are. I guess I asked you some questions in the wrong framework- are not p-adic ideas rather higher in space and symmetry concepts than our normal idea of matter? What would dark matter be but the assumption all things (on some scale) have a sort of shadow thus somewhere no half spins to balance everything (can we not get ahead or travel behind a right handed neutrino at minus light speed?)

QED...QCD...QxD...QyD... ? How far can the levels, generations, of this idea go - do you say it ends in the nucleus of atoms? Do you think mass may not be a continuous value but only discrete corpuscular in nature?

Do you imagine the universe to be finite or infinite in extent or maybe something different or in between? As strings and LQG do not seem to express it wall, I ask what the hell is gravity? What do you think it is if your system looks beyond its particle insights?

If we limit it to TGD QCD what is the next level of things but half values exceptional and with asymmetries? Sure, a fifth place may thus vibrate -or do we just reach a plateau at the top that obeys the settled influences of angles and complexity we call chaos science principles?

In fact nature has the limited vision we do in the "spontaneous compactification" or my condensing ideas of such higher dimensional polytopal shadows as a sensible evolving view laws given. So that is why I wish you would list the arithmetic so I can match it better or is this just an intuitive thing "phenomenological" or "pragmatic" depending on how we see those terms applying to science?

All of us eventually reach the generalize theory (hopefully there can be still unexpected new mutations of it that surprise and arise) even Lubos trudges ahead into the unknown with contradictions (see last post of his on the right handed neutrino) and where it becomes more sensible it looks more and more like the general theories of us all, alternative and standard.

To Edgar Otto:

I am not saying that the scale hierarchy ends with nucleus or atoms. Just the opposite. p-Adic length scale hierarchy continues down to CP_2 scale which is 10^4 Planck lengths. No desert from TeV to Planck mass as GUTs assume.

TGD strongly suggests copies of hadronic physics at Mersenne primes. M_107 corresponds to ordinary hadron physics. There are several indications for M_89 at TeV scale. Next hadron physics would be associated with M_61 and M_31. New data from LHC around 2015 might provide more evidence for M_89. At least strongest evidence comes from the strange events at RHIC and LHC suggesting decay of string like objects to hadrons. Perturbative QCD is not consistent with them but it is not politically correct to say this aloud: better to speak about AdS/CFT correspondence.

TGD is consistent with standard model gauge symmetries but interpreted geometrically or number theoretically: both interpretations are possible. A lot of new physics is however predicted and SUSY assignable to right handed neutrino is part of this new physics and differs from the SUSY hunted at LHC.

TGD predicts that fundamental objects are M^4xCP_2 spinors of two chiralities: lepton and baryon number conservation. This together with space-time as a 4-surface property means a huge reduction in degrees of freedom: space-time topology is the compensating factor. Bosons are fermion antifermion bound states. There are definite differences from QCD since these spinors do not carry color as spin like quantum number but as CP_2 "angular momentum".

In a higher dimension (the 480 quark like possibilities are enumerated appear in the coordinate formulas of the usual physics. In a sense then ten times 48, 2^2 x 3!, or 4 x 120 and so on... where does 89 come into it, the steps to suggest these things or the reasoning behind it?) Can the antiparticle-particles totally cancel out to zero? Why can there not be a higher analog to color when things are not differentiation of manifolds or in a sense self dual? Yet they imagine higher knots beyond curves in three space.

I agree there are more than one way to reduce degrees of freedom but I do not agree that four space is a simpler and not more complicated in symmetry. Nor that the distinctions of mixed or contravariant and covariant bundles or directions are distinct when we consider deeper forms of high or low dimensions as to the double reduction or increases of symmetry.

I so not know some of the terms in your reply B? L? Pertubative QCD? but surely hadrons can be derived from 2^n paths- after all what is the manifolds of 6 dimensions in a plane as if 3 + 3 fold chirality? In which case to adopt the literal idea of angular momentum as if a model of spin or notation does not make a complete space (8 dimensions would do better, or better yet 16 if we depend on just complex analysis) that is where flat spaces of 8D exactly fill spheres of 8D in close packing.) not to say the idea is not useful, just not complete for a balanced and dynamic physics. In fact such fanciful structures are beautiful and abstract- so are the seven manifolds of Riemann Rene Thom envisioned in his catastrophe theory of Universal Topology ub 74 or so. All this about how we put square things into round holes...or about deeper ideas of absolute values squared and so on... Did you include Weyl's gauge ideas in your objections?

Matti, thought you might have some ideas about this recent science news

http://news.discovery.com/space/astronomy/weird-planetary-nebula-alignment-discovery-hubble-eso-130904.htm

To Stephen:

Looks very interesting and fits nicely with general vision about cosmic strings and their decay remnants (magnetic flux tubes increasing in thickness) as basic structural elements of TDG Universe. Galaxies would be like pearls in necklace along long cosmic string: for this there is a lot of support. By fractality, also stars could be organized in the same manner and in the case of solar system there is evidence for this.

The results in the link give additional support for this picture. The direction along which the nebulas are aligned corresponds to big cosmic string and they move freely along it. This link contains the article

http://tgdtheory.com/articles/inflatgd.pdf .

The link does not work now since the webhotel service provider turned out to be a criminal and just closed his web hotel ( http://ilmainenwebhotelli.com ). It is not possible to get a contact to the fellow.

There are many people who have lost their money and are bitter. For me the loss is bigger: my entire life work virtually disappears from existence. I have my own grave suspicions about who has paid, to whom, and for what. The academic circles of Finland hate me really bitterly.

Matti,

I think you can go to Google and they will send whatever you have posted in relation to blogger... check it out.

It is very difficult to lose work and that is true of manuscripts as well the problems with digital copies. It is the wonder you can still think clearly. I doubt this is a deliberate event - just a case of poor or mindless evolving technical design.

That said, the alignment of galaxies is not a recent story and a shared computer project was undertaken (I do not know the results) to find why some are aligned like tops in the same direction - or recently that of local nebula perpendicular to such lines or strings.

We already know something is different in the galactic core where even the bars may rotate oppositely than the main bar (again many assume merged galaxies and so on).

I do not think the idea of basing things on just a corner of a structure, the coming to a vertex, is a complete picture where there can be wider issues of symmetry on the changes of dimension beyond 4D we have to consider all the geometric substructures.

Now, some force idea say G, while this is handled in the equations of physics as a compliment G^-1 with the assumption of some minimum non negative vacuum scale differentiable or not it is held that there is not such mathematical object as 1/G to represent this- but of such supersymmetric mirrors which side of it does nature choose rather than our defined conventions? How would you represent a complex number by addition of coordinates if part of it were on the other side of such dark mirrors> And yes when these are actually added it seems trivial that they eventually approach the golden section - not a plan but an accident some think in the Cheops pyramid measures and angles.

keep working as if new explorations will make up in the event of setbacks and losses... maybe even equations written in the sand at low tide will not be seen- save the proud representative of humanity who as at least one soul seen it and wrote it down...

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