## About the SUSY generated by covariantly constant right-handed neutrinos
The interpretation of covariantly constant right-handed neutrinos (briefly ν - 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
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
- 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
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.
Concerning the quantization of ν
As discussed, M
- 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 or the article with the same title. |