Tuesday, November 03, 2009

Why SUSY would not allow fields with spin higher than two?

The recent progress in understanding QFT limit of TGD led to a question, which I would be happy to find an answer. The standard wisdom says that N=8 is absolute upper bound for the super-symmetry (spins larger than 2 are not regarded as physical). In TGD N=8 emerges naturally for space-time surfaces due to the dimension D=8 of imbedding space and the fact that imbedding space spinors with given H-chirality (quarks and leptons which color appearing as partial waves in CP2 have 8 complex components. One obtains N=8 if one considers only the super-algebra defined by the oscillator operators associated with the lowest modes of these spinor fields at light-like 3-surfaces obtained as a solutions of the modified Dirac equation with measurement interaction term.

It is also possible to consider the supersymmetry generated by all modes of the induced spinor fields and thus with a quite large (even infinite for string like objects) number N of super generators. This supersymmetry is broken as all supersymmetries in TGD framework. This means that rather high spins are present in the analogs of scalar and vector multiplets and the Kähler potential (expected to be closely related to the Kähler function of the world of the classical worlds (WCW)) describing interaction of chiral multiplet with a vector multiplet can be constructed also for any value of N - at least formally. If one believes on the generalization of the bosonic emergence, one expects that bosonic part of the action is generated radiatively as one functionally integrates over the fields appearing in the chiral multiplet.

I tried to find material from web about possibly existing proposals for N>8 SUSY theories or D>12 SUSY theories containg higher spin fields. I found proposals for higher spin theories with N=1 for instance, but nothing else. Superstring thinking has really made its way through: D=12 (F-theory) and N=8 are the absolute upper bounds! It seems that my colleagues enjoying a monthly salary are maximally rational career builders.

The standard wisdom says that is is not possible to construct interactions for higher spin fields. Is this really true? Why wouldn't the analogs of scalar (chiral/hyper) and vector multiplets make sense for higher values of N? Why would it be impossible to define an spin 1/2 chiral super-field associated with the vector-multiplet and therefore the supersymmetric analog of YM action using standard formulas? Why the standard coupling to chiral multiplet would not make sense? Could some-one better-informed tell me the answer?

One objection against higher spins is of course the lack of the geometric interpretation. Spin 1 and Spin 2 fields allow it. Can one then imagine any geometric interpretation for higher spin components of super-fields? John Baez and others are busily developing non-Abelian generalizations of group theory, categories and geometry and speak about things that they call n-groups, n-categories, and n-geometries. Could the generalization of ordinary geometry to n-geometry in which parallel translations are performed for higher dimensional objects rather than points provide a natural interpretation for gauge fields assigned to higher spins? One would have natural hierarchy. Parallel translations of points would give rise curves, parallel translations of curves would give rise to surfaces, and so on. As as a special case the entire hierarchy of these parallel translations would be induced by ordinary parallel translation as I suggested in this blog for years ago.

Addition: At this moment one can make only guesses concerning the super-fields describing wormhole throats and contacts as particles.

  1. The physical picture suggested by the notion of emergence is that kinetic terms behave negative powers of Dirac operator since wormhole throats carry a collection of collinearly moving fermions with momentum appearing in the measurement interaction term identified as the total momentum.

  2. This suggests a construction of a super-field from any finite polynomial P(θ,x) of theta parameters by assigning to each monomial appearing in it the monomial P(∂θc σc kk →,x) and applying it to the conjugate of X. Here → tells that the derivative is applied to P itself. All theta parameters can be included and c denotes conjugation denoted by overline usually. By restricting the degree of the monomial to one half of the maximal the construction works also for a finite value of N.

  3. In the analog of the chiral action monomials and their conjugates would combine to form a term involving a power of Dirac operator equal to the degree of the monomial of thetas so that kinetic terms would come as powers of σkk.

  4. Only the spinor and vector terms would behave in the expected manner and scalar term would vanish. In particular, for spin 2 - the propagator would behave like p-4 for large momenta. This conforms with the view that graviton must correspond to a string like object consisting of a superposition of pairs of wormhole contacts and of wormhole throats rather than single wormhole throat. If this expansion makes sense, higher spin propagators would behave as increasingly higher inverse powers of momentum and would not contribute much to the high energy physics. At energies much smaller than mass scale they would give rise to contact terms proportional to a negative power of mass dictated by the number of thetas.

  5. This is certainly not all that is needed since interactions must be included too. Here one might consider a generalization of Dirac action as a trilinear interaction term formed from similar "chiral field" assignable to bosons described as wormhole contacts with negative and positive energy thetas and from positive and negative energy fermionic super fields. Generalization of bosonic emergence would give purely bosonic part of action as radiative corrections. More conventional approach would add the bosonic kinetic term also the action.

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

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