Thursday, February 14, 2019

A connection of singularities of minimal surfaces with generation of Higgs vacuum expectation?

String world sheet appear as singularities of space-time surfaces as minimal surfaces. At string world sheets minimal surface equations fail and there is transfer of Noether charges associated with Kähler and volume degrees of freedom at string world. This has interpretation as analog for the interaction of charged particle with Maxwell field.

What about the physical interpretation of the singular divergences of the isometry currents JA of the volume action located at string world sheet?

  1. The divergences of JA are proportional to the trace of the second fundamental form H formed by the covariant derivatives of gradients ∂αhk of H-coordinates in the interior and vanish. The singular contribution at string world sheets is determined by the discontinuity of the isometry current JA and involves only the first derivatives ∂αhk.

  2. One of the first questions after ending up with TGD for 41 years ago was whether the trace of H in the case of CP2 coordinates could serve as something analogous to Higgs vacuum expectation value. The length squared for the trace has dimensions of mass squared. The discontinuity of the isometry currents for SU(3) parts in h=u(2) and its complement t, whose complex coordinates define u(2) doublet. u(2) is in correspondence with electroweak algebra and t with complex Higgs doublet. Could an interpretation as Higgs or even its vacuum expectation make sense?

  3. p-Adic thermodynamics explains fermion masses elegantly (understanding of boson masses is not in so good shape) in terms of thermal mixing with excitations having CP2 mass scale and assignable to short string associated with wormhole contacts. There is also a contribution from long strings connecting wormhole contacts and this could be important for the understanding of weak gauge boson masses. Could the discontinuity of isometry currents determine this contribution to mass. Edges/folds would carry mass.

  4. The non-singular part of the divergence multiplying 2-D delta function has dimension 1/length squared and the square of this vector in CP2 metric has dimension of mass squared. Could the interpretation of the discontinuity as Higgs expectation make sense? If so, Higgs expectation would vanish in the space-time interior.

    Could the interior modes of the induced spinor field - or at least the interior mode of right-handed neutrino νR having no couplings to weak or color fields - be massless in 8-D or even 4-D sense? Could νR and νbarR generate an unbroken N=2 SUSY in interior whereas inside string world sheets right-handed neutrino and antineutrino would be eaten in neutrino massivation and the generators of N=2 SUSY would be lost somewhat like charged components of Higgs!

    If so, particle physicists would be trying to find SUSY from wrong place. Space-time interior would be the correct place. Would the search of SUSY be condensed matter physics rather than particle physics?

    Remark: There is an interesting delicacy involved. Consider an edge at 3-surface. The divergence from the discontinuity contains contributions from two normal coordinates proportional to a delta function for the normal coordinate and coming from the discontinuity. The discontinuity must be however localized to the string rather than 2-surface. There must be present also a delta function for the second normal coordinate. Hence the value of normal discontinuity must be infinite along the string. One would have infinitely sharp edge. A concrete example is provided by function y= |x|α, α<1. This kind of situation is encountered in Thom's catastrophe theory for the projection of the catastrophe: in this case one has α=1/2. This argument generalizes to 3-D case but visualization is possible only as a motion of infinitely sharp edge of 3-surface.

    Kähler form and metric are second degree monomials of partial derivatives so that an attractive assumption is that gαβ, Jαβ and therefore also the components of volume and Kähler energy momentum tensor are continuous. This would allow ∂nihk to become infinite and change sign at the discontinuity as the idea about infinitely sharp edge suggests. This would reduce the continuity conditions for canonical momentum currents to rather simple form

    TninjΔ ∂njhk=0 .

    which in turn would give


    stating that canonical momentum is conserved but transferred between Kähler and volume degrees of freedom. One would have a condition for a continuous quantity conforming with the intuitive view about boundary conditions due to conservation laws. The condition would state that energy momentum tensor reduces to that for string world sheet at the singularity so that the system becomes effectively 2-D. I have already earlier proposed this condition.

    The reduction of 4-D locally to effectively 2-D system raises the question whether any separate action is needed for string world sheets (and their boundaries)? The generated 2-D action would be similar to the proposed 2-D action. By super-conformal symmetry similar generation of 2-D action would take place also in the fermionic degrees of freedom. I have proposed also this option already earlier.

See the chapter The Recent View about Twistorialization in TGD Framework or the article More about the construction of scattering amplitudes in TGD framework.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


Ulla said...

The non-singular part of the divergence multiplying 2-D delta function has dimension 1/length squared and the square of this vector in CP2 metric has dimension of mass squared.

How is that possible?

Matti Pitkänen said...

This is just dimensional analysis. delta function means that the divergence is nonvanishing only at the string world sheet. This is due to the discontinuity of some partial derivatives of imbedding space coordinates. Geometric manner to understand this is simple: the 3-surface as edge/fold at which derivatives are discontinuous.

Ulla said...

Thanks for your answer, and for your patience with me. I don't want to be angry at you.