QFT limit of TGD and space-time supersymmetry

The understanding of the QFT limit of TGD is now a twenty year old challenge. How to feed information about classical physics characterized by Kähler action has been the basic question. The conflict with Poincare invariance destroying all hopes about practical calculations looks unavoidable. Zero energy ontology and the addition of measurement interaction depending on momenta and color charges to modified Dirac action led to a resolution of this dilemma. The point is that the momenta act on the tip of causal diamond rather than space-time coordinates, which therefore appear as external parameters like the couplings in Hamiltonian. QFT in infinitely slowly varying background fields is the counterpart in ordinary QFT but in TGD there is no need to pose this restriction. One obtains for each space-time point its own QFT limit. A weighted integral over amplitudes corresponding to these limits is performed in analogy with what is done in the theory of spin glasses at the level of statistical physics. As a matter fact, TGD Universe is 4-D quantum spin glass.

This led also to the realization that space-time supersymmetry can be realized at the fundamental level as anticommutation relations of the fermionic oscillator operators associated with the modes of the induced spinor field. The next task was to construct the counterpart of SUSY QFT limit for TGD. Here the problem was that the value of N for the super-symmetry in question is large or even infinite so that the standard notions of chiral and vector superfields fail. N=∞ limit how forced to find the correct formalism. It is considerably simpler than the standard one since chiral condition is replaced with Grassman analyticity. Bosonic emergence is unavoidable in this framework relying strongly on zero energy ontology and on the identification of fermions (bosons) and their super-partners in terms of wormhole throats (contacts). The finiteness of the theory follows by extending the standard argument stating that fermion and sfermion loops cancel each other in SUSY. One prediction is a hierarchy of exotic particles with propagators behaving like 1/pⁿ. For boson exchanges with n=2m the corresponding interaction potentials behave like exp(-mr)r^2m-3. For massless case n=4 gives linear confining force possibly highly relevant for QCD. Also the information about space-time surface (corresponds to maximum of Kähler function) can be feeded to the theory by using modified gamma matrices defined by Kähler action without losing Poincare invariance.

I dare to regard the resulting formalism combining the ideas about the generalization of twistors and about bosonic emergence with other basic ideas of TGD as the final breakthrough. The resulting formalism makes possible concrete calculational recipes. During last months I have often experienced a deep and to me strange feeling of relief. After 32 years of work my great mission has been realized to a high extent! The practical part of me of course starts to worry whether this 59 old me can survive this kind of strange feelings of relief without total collapse;-)? Probably so! There is so much to do and one thing to do is to look how much of this formalism generalizes to TGD proper.

I attach below also the abstract of the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry?, which can be found in the book "Towards M-Matrix".

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.

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 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 condition that N=∞ variants for chiral and vector superfields exist fixes completely the identification of these fields in zero energy ontology.
   1. In this framework chiral fields are generalizations of induced spinor fields and vector fields those of gauge potentials obtained by replacing them with their super-space counterparts. Chiral condition reduces to analyticity in theta parameters thanks to the different definition of hermitian conjugation in zero energy ontology (q is mapped to a derivative with respect to theta rather than to [`(q)]) and conjugated super-field acts on the product of all theta parameters.

   2. Chiral action is a straightforward generalization of the Dirac action coupled to gauge potentials. The counterpart of YM action can emerge only radiatively as an effective action so that the notion emergence is now unavoidable and indeed basic prediction of TGD.

   3. The propagators associated with the monomials of n theta parameters behave as 1/pⁿ so that only J=0,1/2,1 states propagate in normal manner and correspond to normal particles. The presence of monomials with number of thetas higher than 2 is necessary for the propagation of bosons since by the standard argument fermion and scalar loops cancel each other by super-symmetry. This picture conforms with the identification of graviton as a bound state of wormhole throats at opposite ends of string like object.

   4. This formulation allows also to use modified gamma matrices in the measurement interaction defining the counterpart of super variant of Dirac operator. Poincare invariance is not lost since momenta and color charges act on the tip of CD rather than the coordinates of the space-time sheet. Hence what is usually regarded as a quantum theory in the background defined by classical fields follows as exact theory. This feeds all data about space-time sheet associated with the maximum of Kähler function. In this approach WCW as a Kähler manifold is replaced by a cartesian power of CP₂, which is indeed quaternionic Kähler manifold. The replacement of light-like 3-surfaces with number theoretic braids when finite measurement resolution is introduced, leads to a similar replacement.

   5. Quantum TGD as a "complex square root" of thermodynamics approach suggests that one should take a superposition of the amplitudes defined by the points of a coherence region (identified in terms of the slicing associated with a given wormhole throat) by weighting the points with the Kähler action density. The situation would be highly analogous to a spin glass system since the modified gamma matrices defining the propagators would be analogous to the parameters of spin glass Hamiltonian allowed to have a spatial dependence. This would predict the proportionality of the coupling strengths to Kähler coupling strength and bring in the dependence on the size of CD coming as a power of 2 and give rise to p-adic coupling constant evolution. Since TGD Universe is analogous to 4-D spin glass, also a sum over different preferred extremals assignable to a given coherence regions and weighted by exp(K) is probably needed.

7. In TGD Universe graviton is necessarily a bi-local object and the emission and absorption of graviton are bi-local processes involving two wormhole contacts: a pair of particles rather than single particle emits graviton. This is definitely something new and defies a description in terms of QFT limit using point like particles. Graviton like states would be entangled states of vector bosons at both ends of stringy curve so that gravitation could be regarded as a square of YM interactions in rather concrete sense. The notion of emergence would suggest that graviton propagator is defined by a bosonic loop. Since bosonic loop is dimensionless, IR cutoff defined by the largest CD present must be actively involved. At QFT limit one can hope a description as a bi-local process using a bi-local generalization of the QFT limit. It turns out that surprisingly simple candidate for the bi-local action exists.