If space-time means space-time surface, standard space-time SUSY extends to infinite-dimensional super-conformal symmetry generalizing in the case of right-handed neutrinos to generalization of 2-D conformal symmetry to 4-D context. For other fermions states localized to 2-D surfaces by the conservation of electric charge, one has 2-D super-conformal symmetry. Space-time super symmetry in ordinary sense means super symmetry at imbedding space level. The progress in the understanding of preferred extremals of Kähler action and of solutions of the modified Dirac equations were the decisive steps together with the results from LHC which suggests that space-time SUSY in standard sense is absent and that Higgs is not quite it is expected to be.
LHC suggests that one does not have N=1 SUSY in standard sense and TGD view about gauge boson massivation in terms of pseudoscalar coupling to instanton density does not need SUSY to stabilize the radiative corrections to the tachyonic mass term which is now absent from the beginning since it would break conformal invariance. The mexican hat is not there!
Why one cannot have standard space-time SUSY in TGD framework? Let us begin by listing all arguments popping in mind.
- Could covariantly constant νR represents a gauge degree of freedom? This is plausible since the corresponding fermion current is non-vanishing.
- The original argument for absence of space-time SUSY years ago was indirect: M4× CP2 does not allow Majorana spinors so that N=1 SUSY is excluded.
- One can however consider N=2 SUSY by including both helicities possible for covariantly constant νR. For νR the four-momentum vanishes so that one cannot distinguish the modes assigned to the creation operator and its conjugate via complex conjugation of the spinor. Rather, one oscillator operator and its conjugate correspond to the two different helicities of right-handed neutrino with respect to the direction determined by the momentum of the particle. The spinors can be chosen to be real in this basis. This indeed gives rise to an irreducible representation of spin 1/2 SUSY algebra with right-handed neutrino creation operator acting as a ladder operator. This is however N=1 algebra and right-handed neutrino in this particular basis behaves effectively like Majorana spinor. One can argue that the system is mathematically inconsistent. By choosing the spin projection axis differently the spinor basis becomes complex. In the new basis one would have N=2 , which however reduces to N=1 in the real basis.
- Or could it be that fermion and sfermion do exist but cannot be related by SUSY? In standard SUSY fermions and sfermions forming irreducible representations of super Poincare algebra are combined to components of superfield very much like finite-dimensional representations of Lorentz group are combined to those of Poincare. In TGD framework νR generates in space-time interior generalization of 2-D super-conformal symmetry but covarianlty constant νR cannot give rise to space-time SUSY.
This would be very natural since right-handed neutrinos do not have any electroweak interactions and are delocalized into the interior of the space-time surface unlike other particles localized at 2-surfaces. It is difficult to imagine how fermion and νR could behave as a single coherent unit reflecting itself in the characteristic spin and momentum dependence of vertices implied by SUSY. Rather, it would seem that fermion and sfermion should behave identically with respect to electroweak interactions.
The third argument looks rather convincing and can be developed to a precise argument.
- If sfermion is to represent elementary bosons, the products of fermionic oscillator operators with the oscillator operators assignable to the covariantly constant right handed neutrinos must define might-be bosonic oscillator operators as bn= ana and bn†= an†a† One can calculate the commutator for the product of operators. If fermionic oscillator operators commute, so do the corresponding bosonic operators. The commutator [bn,bn†] is however proportional to occupation number for νR in N=1 SUSY representation and vanishes for the second state of the representation. Therefore N=1 SUSY is a pure gauge symmetry.
- One can however have both irreducible representations of SUSY: for them either fermion or sfermion has a non-vanishing norm. One would have both fermions and sfermions but they would not belong to the same SUSY multiplet, and one cannot expect SUSY symmetries of 3-particle vertices.
- For instance, γ FF vertex is closely related to γ Fs Fs in standard SUSY (subscript "s" refers to sfermion). Now one expects this vertex to decompose to a product of γ FFs vertex and amplitude for the creation of νRνsR from vacuum so that the characteristic momentum and spin dependent factors distinguishing between the couplings of photon to scalar and and fermion are absent. Both states behave like fermions. The amplitude for the creation of νRνsR from vacuum is naturally equal to unity as an occupation number operator by crossing symmetry. The presence of right-handed neutrinos would be invisible if this picture is correct. Whether this invisible label can have some consequences is not quite clear: one could argue that the decay rates of weak bosons to fermion pairs are doubled unless one introduces 2-1/2 factors to couplings.
Addition: Lubos tells in his postings about SUSY theoreticians Li, Maxin, Nanopoulos, and Walker. They promise the discovery of proton decay, stop squark, and lightest supersymmetric particle from 2012 LHC data, perhaps even within few months! I cannot promise any of these sensational events. Neither stop nor neutralino has place in TGD Universe, and even proton refuses to decay;-). Even worse, there is no purpose of life for standard SUSY in TGD Universe: in its fanatic ontological minimalism TGD Universe does not tolerate free lunchers;-).
For details and background see the new chapter SUSY in TGD Universe of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy", and the article The recent vision about preferred extremals and solutions of the modified Dirac equation.