N=4 SYM is definitely a completely exceptional theory and one cannot avoid the question whether it could in some sense be part of fundamental physics. In TGD framework right handed neutrinos have remained a mystery: whether one should assign space-time SUSY to them or not. Could they give rise to N=2 or N=4 SUSY with fermion number conservation?
My latest view is that fully covariantly constant right-handed neutrinos decouple from the dynamics completely. I will repeat first the earlier arguments which consider only fully covariantly constant right-handed neutrinos.
- N=1 SUSY is certainly excluded since it would require Majorana property not possible in TGD framework since it would require superposition of left and right handed neutrinos and lead to a breaking of lepton number conservation. Could one imagine SUSY in which both MEs between which particle wormhole contacts reside have N=2 SUSY which combine to form an N=4 SUSY?
- Right-handed neutrinos which are covariantly constant right-handed neutrinos in both M4 degrees of freedom cannot define a non-trivial theory as shown already earlier. They have no electroweak nor gravitational couplings and carry no momentum, only spin.
The fully covariantly constant right-handed neutrinos with two possible helicities at given ME would define representation of SUSY at the limit of vanishing light-like momentum. At this limit the creation and annihilation operators creating the states would have vanishing anticommutator so that the oscillator operators would generate Grassmann algebra. Since creation and annihilation operators are hermitian conjugates, the states would have zero norm and the states generated by oscillator operators would be pure gauge and decouple from physics. This is the core of the earlier argument demonstrating that N=1 SUSY is not possible in TGD framework: LHC has given convincing experimental support for this belief.
Could massless right-handed neutrinos covariantly constant in CP2 degrees of freedom define N=2 or N=4 SUSY?
Consider next right-handed neutrinos, which are covariantly constant in CP2 degrees of freedom but have a light-like four-momentum. In this case fermion number is conserved but this is consistent with N=2 SUSY at both MEs with fermion number conservation. N=2 SUSYs could emerge from N=4 SUSY when one half of SUSY generators annihilate the states, which is a basic phenomenon in supersymmetric theories.
- At space-time level right-handed neutrinos couple to the space-time geometry - gravitation - although weak and color interactions are absent. One can say that this coupling forces them to move with light-like momentum parallel to that of ME. At the level of space-time surface right-handed neutrinos have a spectrum of excitations of four-dimensional analogs of conformal spinors at string world sheet (Hamilton-Jacobi structure).
For MEs one indeed obtains massless solutions depending on longitudinal M2 coordinates only since the induced metric in M2 differs from the light-like metric only by a contribution which is light-like and contracts to zero with light-like momentum in the same direction. These solutions are analogs of (say) left movers of string theory. The dependence on E2 degrees of freedom is holomorphic. That left movers are only possible would suggest that one has only single helicity and conservation of fermion number at given space-time sheet rather than 2 helicities and non-conserved fermion number: two real Majorana spinors combine to single complex Weyl spinor.
- At imbedding space level one obtains a tensor product of ordinary representations of N=2 SUSY consisting of Weyl spinors with opposite helicities assigned with the ME. The state content is same as for a reduced N=4 SUSY with four N=1 Majorana spinors replaced by two complex N=2 spinors with fermion number conservation. This gives 4 states at both space-time sheets constructed from νR and its antiparticle. Altogether the two MEs give 8 states, which is one half of the 16 states of N=4 SUSY so that a degeneration of this symmetry forced by non-Majorana property is in question.
Is the dynamics of N=2 or N=4 SYM possible in right-handed neutrino sector?
Could N=2 or N=4 SYM be a part of quantum TGD? Could TGD be seen a fusion of a degenerate N=4 SYM describing the right-handed neutrino sector and string theory like theory describing the contribution of string world sheets carrying other leptonic and quark spinors? Or could one imagine even something simpler?
What is interesting that the net momenta assigned to the right handed neutrinos associated with a pair of MEs would correspond to the momenta assignable to the particles and obtained by p-adic mass calculations. It would seem that right-handed neutrinos provide a representation of the momenta of the elementary particles represented by wormhole contact structures. Does this mimircry generalize to a full duality so that all quantum numbers and even microscopic dynamics of defined by generalized Feynman diagrams (Euclidian space-time regions) would be represented by right-handed neutrinos and MEs? Could a generalization of N=4 SYM with non-trivial gauge group with proper choices of the ground states helicities allow to represent the entire microscopic dynamics?
Irrespective of the answer to this question one can compare the TGD based view about supersymmetric dynamics with what I have understood about N=4 SYM.
- In the scattering of MEs induced by the dynamics of Kähler action the right-handed neutrinos play a passive role. Modified Dirac equation forces them to adopt the same direction of four-momentum as the MEs so that the scattering reduces to the geometric scattering for MEs as one indeed expects on basic of quantum classical correspondence. In νR sector the basic scattering vertex involves four MEs and could be a re-sharing of the right-handed neutrino content of the incoming two MEs between outgoing two MEs respecting fermion number conservation. Therefore N=4 SYM with fermion number conservation would represent the scattering of MEs at quantum level.
- N=4 SUSY would suggest that also in the degenerate case one obtains the full scattering amplitude as a sum of permutations of external particles followed by projections to the directions of light-like momenta and that BCFW bridge represents the analog of fundamental braiding operation. The decoration of permutations means that each external line is effectively doubled. Could the scattering of MEs can be interpreted in terms of these decorated permutations? Could the doubling of permutations by decoration relate to the occurrence of pairs of MEs?
One can also revert these questions. Could one construct massive states in N=4 SYM using pairs of momenta associated with particle with label k and its decorated copy with label k+n? Massive external particles obtained in this manner as bound states of massless ones could solve the IR divergence problem of N=4 SYM.
- The description of amplitudes in terms of leading singularities means picking up of the singular contribution by putting the fermionic propagators on mass shell. In the recent case it would give the inverse of massless Dirac propagator acting on the spinor at the end of the internal line annihilating it if it is a solution of Dirac equation.
The only way out is a kind of cohomology theory in which solutions of Dirac equation represent exact forms. Dirac operator defines the exterior derivative d and virtual lines correspond to non-physical helicities with dΨ ≠ 0. Virtual fermions would be on mass-shell fermions with non-physical polarization satisfying d2Ψ=0. External particles would be those with physical polarization satisfying dΨ=0, and one can say that the Feynman diagrams containing physical helicities split into products of Feynman diagrams containing only non-physical helicities in internal lines.
- The fermionic states at wormhole contacts should define the ground states of SUSY representation with helicity +1/2 and -1/2 rather than spin 1 or -1 as in standard realization of N=4 SYM used in the article. This would modify the theory but the twistorial and Grassmannian description would remain more or less as such since it depends on light-likeneness and momentum conservation only.
3-vertices for sparticles are replaced with 4-vertices for MEs
In N=4 SYM the basic vertex is on mass-shell 3-vertex which requires that for real light-like momenta all 3 states are parallel. One must allow complex momenta in order to satisfy energy conservation and light-likeness conditions. This is strange from the point of view of physics although number theoretically oriented person might argue that the extensions of rationals involving also imaginary unit are rather natural.
The complex momenta can be expressed in terms of two light-like momenta in 3-vertex with one real momentum. For instance, the three light-like momenta can be taken to be p, k, p-ka, k= apR. Here p (incoming momentum) and pR are real light-like momenta satisfying p⋅ pR=0 with opposite sign of energy, and a is complex number. What is remarkable that also the negative sign of energy is necessary also now.
Should one allow complex light-like momenta in TGD framework? One can imagine two options.
- Option I: no complex momenta. In zero energy ontology the situation is different due to the presence of a pair of MEs meaning replaced of 3-vertices with 4-vertices or 6-vertices, the allowance of negative energies in internal lines, and the fact that scattering is of sparticles is induced by that of MEs. In the simplest vertex a massive external particle with non-parallel MEs carrying non-parallel light-like momenta can decay to a pair of MEs with light-like momenta. This can be interpreted as 4-ME-vertex rather than 3-vertex (say) BFF so that complex momenta are not needed. For an incoming boson identified as wormhole contact the vertex can be seen as BFF vertex.
To obtain space-like momentum exchanges one must allow negative sign of energy and one has strong conditions coming from momentum conservation and light-likeness which allow non-trivial solutions (real momenta in the vertex are not parallel) since basically the vertices are 4-vertices. This reduces dramatically the number of graphs. Note that one can also consider vertices in which three pairs of MEs join along their ends so that 6 MEs (analog of 3-boson vertex) would be involved.
- Option II: complex momenta are allowed. Proceeding just formally, the (g4)1/2 factor in Kähler action density is imaginary in Minkowskian and real in Euclidian regions. It is now clear that the formal approach is correct: Euclidian regions give rise to Kähler function and Minkowskian regions to the analog of Morse function. TGD as almost topological QFT inspires the conjecture about the reduction of Kähler action to boundary terms proportional to Chern-Simons term. This is guaranteed if the condition jKμAμ=0 holds true: for the known extremals this is the case since Kähler current jK is light-like or vanishing for them. This would seem that Minkowskian and Euclidian regions provide dual descriptions of physics. If so, it would not be surprising if the real and complex parts of the four-momentum were parallel and in constant proportion to each other.
This argument suggests that also the conserved quantities implied by the Noether theorem have the same structure so that charges would receive an imaginary contribution from Minkowskian regions and a real contribution from Euclidian regions (or vice versa). Four-momentum would be complex number of form P= PM+ iPE. Generalized light-likeness condition would give PM2=PE2 and PM⋅PE=0. Complexified momentum would have 6 free components. A stronger condition would be PM2=0=PE2 so that one would have two light-like momenta "orthogonal" to each other. For both relative signs energy PM and PE would be actually parallel: parametrization would be in terms of light-like momentum and scaling factor. This would suggest that complex momenta do not bring in anything new and Option II reduces effectively to Option I. If one wants a complete analogy with the usual twistor approach then PM2=PE2≠ 0 must be allowed.
Is SUSY breaking possible or needed?
It is difficult to imagine the breaking of the proposed kind of SUSY in TGD framework, and the first guess is that all these 4 super-partners of particle have identical masses. p-Adic thermodynamics does not distinguish between these states and the only possibility is that the p-adic primes differ for the spartners. But is the breaking of SUSY really necessary? Can one really distinguish between the 8 different states of a given elementary particle using the recent day experimental methods?
- In electroweak and color interactions the spartners behave in an identical manner classically. The coupling of right-handed neutrinos to space-time geometry however forces the right-handed neutrinos to adopt the same direction of four-momentum as MEs has. Could some gravitational effect allow to distinguish between spartners? This would be trivially the case if the p-adic mass scales of spartners would be different. Why this should be the case remains however an open question.
- In the case of unbroken SUSY only spin distinguishes between spartners. Spin determines statistics and the first naive guess would be that bosonic spartners obey totally different atomic physics allowing condensation of selectrons to the ground state. Very probably this is not true: the right-handed neutrinos are delocalized to 4-D MEs and other fermions correspond to wormhole contact structures and 2-D string world sheets.
The coupling of the spin to the space-time geometry seems to provide the only possible manner to distinguish between spartners. Could one imagine a gravimagnetic effect with energy splitting proportional to the product of gravimagnetic moment and external gravimagnetic field B? If gravimagnetic moment is proportional to spin projection in the direction of B, a non-trivial effect would be possible. Needless to say this kind of effect is extremely small so that the unbroken SUSY might remain undetected.
- If the spin of sparticle be seen in the classical angular momentum of ME as quantum classical correspondence would suggest then the value of the angular momentum might allow to distinguish between spartners. Also now the effect is extremely small.