Space-Time Super-Symmetry and TGD

Contrary to the original expectations, TGD seems to allow a generalization of the space-time super-symmetry. This became clear with the increased understanding of the modified Dirac action. The introduction of a measurement interaction term to the action allows to understand how stringy propagator results and provides profound insights about physics predicted by TGD (see the new chapter Does the Modified Dirac Equation Define the Fundamental Action Principle of "TGD: Physics as Infinite-Dimensional Geometry").

The appearance of the momentum and color quantum numbers in the measurement interaction couples space-time degrees of freedom to quantum numbers and allows also to define SUSY algebra at fundamental level as anti-commutation relations of fermionic oscillator operators. Depending on the situation a finite-dimensional SUSY algebra or the fermionic part of super-conformal algebra with an infinite number of oscillator operators results. The addition of a fermion in particular mode would define particular super-symmetry. Zero energy ontology implies that fermions as wormhole throats correspond to chiral super-fields assignable to positive or negative energy SUSY algebra whereas bosons as wormhole contacts with two throats correspond to the direct sum of positive and negative energy algebra and fields which are chiral or antichiral with respect to both positive and negative energy theta parameters. This super-symmetry is badly broken due to the dynamics of the modified Dirac operator which also mixes M⁴ chiralities inducing massivation. Since righthanded neutrino has no electro-weak couplings the breaking of the corresponding super-symmetry should be weakest.

The question is whether this SUSY has a realization as a SUSY algebra at space-time level and whether the QFT limit of TGD could be formulated as a generalization of SUSY QFT. There are several problems involved.

1. In TGD framework super-symmetry means addition of fermion to the state and since the number of spinor modes is larger states with large spin and fermion numbers are obtained. This picture does not fit to the standard view about super-symmetry. In particular, the identification of theta parameters as Majorana spinors and super-charges as Hermitian operators is not possible.

2. The belief that Majorana spinors are somehow an intrinsic aspect of super-symmetry is however only a belief. Weyl spinors meaning complex theta parameters are also possible. Theta parameters can also carry fermion number meaning only the supercharges carry fermion number and are non-hermitian. The the general classification of super-symmetric theories indeed demonstrates that for D=8 Weyl spinors and complex and non-hermitian super-charges are possible. The original motivation for Majorana spinors might come from MSSM assuming that right handed neutrino does not exist. This belief might have also led to string theories in D=10 and D=11 as the only possible candidates for TOE after it turned out that chiral anomalies cancel.

3. The massivation of particles is basic problem of both SUSYs and twistor approach. The fact that particles which are massive in M⁴ sense can be interpreted as massless particles in M⁴×CP₂ suggests a manner to understand super-symmetry breaking and massivation in TGD framework. The octonionic realization of twistors is a very attractive possibility in this framework and quaternionicity condition guaranteing associativity leads to twistors which are almost equivalent with ordinary 4-D twistors.

4. The first approach is based on an approximation assuming only the super-multiplets generated by right-handed neutrino or both right-handed neutrino and its antineutrino. The assumption that right-handed neutrino has fermion number opposite to that of the fermion associated with the wormhole throat implies that bosons correspond to N=(1,1) SUSY and fermions to N=1 SUSY identifiable also as a short representation of N=(1,1) SUSY algebra trivial with respect to positive or negative energy algebra. This means a deviation from the standard view but the standard SUSY gauge theory formalism seems to apply in this case.

5. A more ambitious approach would put the modes of induced spinor fields up to some cutoff into super-multiplets. At the level next to the one described above the lowest modes of the induced spinor fields would be included. The very large value of N means that N > 32 SUSY cannot define the QFT limit of TGD for higher cutoffs. One must generalize SUSYs gauge theories to arbitrary value of N but there are reasons to expect that the formalism becomes rather complex. More ambitious approach working at TGD however suggest a more general manner to avoid this problem.
   1. One of the key predictions of TGD is that gauge bosons and Higgs can be regarded as bound states of fermion and antifermion located at opposite throats of a wormhole contact. This implies bosonic emergence meaning that it QFT limit can be defined in terms of Dirac action. The resulting theory was discussed in detail in and it was shown that bosonic propagators and vertices can be constructed as fermionic loops so that all coupling constant follow as predictions. One must however pose cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops in order to obtain finite theory and to avoid massivation of bosons. The resulting coupling constant evolution is consistent with low energy phenomenology if the cutoffs in hyperbolic angle as a function of p-adic length scale is chosen suitably.

   2. The generalization of bosonic emergence is natural in the sense that the TGD counterpart of SUSY is obtained by the replacement of Dirac action with action for chiral super-field coupled to vector field as the action defining the theory so that the propagators of bosons and all their super-counterparts would emerge as fermionic loops.

   3. The huge super-symmetries give excellent hopes about the cancelation of infinities so that this approach would work even without the cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops. Cutoffs have a physical motivation in zero energy ontology but it could be an excellent approximation to take them to infinity. Alternatively, super-symmetric dynamics provides cutoffs dynamically.

6. The intriguing formal analogy of the Kähler potential and super-potential with the Kähler function defining the Kähler metric of WCW and determined up to a real part of analytic function of the complex coordinates of WCW. This analogy suggests that the action defining the SUSY-Kähler potential- is identifiable as the Kähler function defining WCW Kähler metric at its maximum. Super-potential in turn would correspond to a holomorphic function defining the modification of Kähler function due and the space-time sheet due to measurement interaction. This beautiful correspondence would make WCW geometry directly visible in the properties of QFT limit of TGD.

To sum up, the new chapter fuses three ideas developed during this year. The generalization of the twistor formalism via the induced octonionic twistor structure and masslessness in 8-D sense as a prerequisite for twistorialization and higher N super-symmetry, bosonic emergence, and the possibility to realize space-time super-symmetry algebra via the introduction of the measurement interaction term in the modified Dirac action. It seems that all basic prerequisite for developing quantum TGD to a calculable theory exist but a collective effort is of course needed to achieve this.

For the details see the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".