Could vacuum expectation value of Higgs have TGD counterpart?

Although it seems that Higgs mechanism is replaced by p-adic thermodynamics in TGD framework, one can ask whether the analog of Higgs vacuum expectation somehow emerges. This is expected to be the case if one believes that QFT theory limit for TGD makes sense (for CMAP representations about TGD see this).

1. TGD allows Higgs like particles and it would be very difficult to develop a convincing arguments for their non-existence. The model for elementary particles as string like objects consisting of "ur-fermions" with electro-weak quantum numbers of electron and quark allows also scalar particles.
   
   
2. The notion of Higgs vacuum expectation as fundamental mechanism of particle massivation is not supported by the following argument.
   
   
   
   1. Higgs would naturally correspond to CP₂ vector field most naturally in complex coordinates for CP₂ behaving like electroweak doublet.
      
      
   2. This field should be covariantly constant but CP₂ does not allow covariantly constant vector fields. This could conform with the view that p-adic thermodynamics describes particle massivation.
      
      
3. One can consider a loophole to this argument. Could it happen that Higgs field is covariantly constant only with respect to the induced spinor connection? For instance, at the string world sheets associated with fermions or at the orbits of string ends? Or is there some other way for Higgs vacuum expectation to creep in.
   

Quite recent little progress in the understanding of Kähler-Dirac equation this) suggests that the analog of Higgs vacuum expectation might make sense and could code for masses of the particles in space-time geometry, maybe in the geometry of space-time sheets at which particle is topologically condensed. Particle masses would follow from p-adic thermodynamics (see this and this).

1. The Kähler-Dirac action for the modes of induced imbedding space spinor field reduces to a vanishing contribution in the interior of space-time surface and by weak form of electric magnetic duality (WFEMD) to a boundary term given by Chern-Simons action.
   
   
2. For the light-like partonic orbits associated with wormhole contacts these terms must vanish by conservation of fermionic charges and one obtains just algebraic form of M⁴ Dirac action at the partonic 2-surface so that massless fermionic propagators are obtained. There is nothing Higgs like in propagators and this conforms nicely with the stringy version of twistor Grassmannian approach which applies in TGD.
   
   
3. At the space-like 3-surfaces defining the ends of space-time at boundaries of CDs one obtains algebraic massless Dirac plus Chern-Simons term.
   
   
4. Could this term play the the role of Higgs vacuum expectation? Does this term give only a small additional contribution to the masses of particles besides that given by p-adic thermodynamics for wormhole contacts and for string like object in electro-weak or even Compton length scale?
   
   
5. What happens at at the ends of string worlds sheets. Here the Higgs term would correspond to the normal (time-like) component of the Kähler-Dirac gamma matrix whose CP₂ part would give needed non-tachyonic contribution to mass. One would get algebraic form of M⁴ Dirac equation: p^kγ_k Ψ= ΓⁿΨ, where Γ^α=T^αkγ_k is modified gamma matrix defined as a contraction of the canonical momentum currents for Kähler action with imbedding space gamma matrices γ_k. Note that this Kähler-Dirac gamma matrix could be covariantly constant along the boundary of string world sheet and perhaps even inside string world sheet.
   
   
6. Note that Minkowskian part of Γⁿ would give a tachyonic contribution, which might relate to the still poorly understood question how the tachyonic ground states of super-conformal representations with half
   integer conformal weight emerge.
   
   

Although it seems that Higgs mechanism is replaced by p-adic thermodynamics in TGD framework, one can ask whether the analog of Higgs vacuum expectation somehow emerges. This is expected to be the case if one believes that QFT theory limit for TGD makes sense.

1. TGD allows Higgs like particles and it would be very difficult to develop a convincing arguments for their non-existence. The model for elementary particles as string like objects consisting of "ur-fermions" with electro-weak quantum numbers of electron and quark allows also
   scalar particles.
   
   
2. The notion of Higgs vacuum expectation as fundamental mechanism of particle massivation is not supported by the following argument.
   
   
   1. Higgs would naturally correspond to CP₂ vector field most naturally in complex coordinates for CP₂ behaving like electroweak doublet.
      
      
   2. This field should be covariantly constant but CP₂ does not allow covariantly constant vector fields. This could conform with the view that p-adic thermodynamics describes particle massivation.
      
      
3. One can consider a loophole to this argument. Could it happen that Higgs field is covariantly constant only with respect to the induced spinor connection? For instance, at the string world sheets
   associated with fermions or at the orbits of string ends? Or is there some other way for Higgs vacuum expectation to creep in.
   

Quite recent little progress in the understanding of Kähler-Dirac equation suggests that the analog of Higgs vacuum expectation might make sense and could code for masses of the particles in space-time geometry, maybe in the geometry of space-time sheets at which particle is topologically condensed. Particle mass itself would follow from p-adic thermodynamics.

1. The Kähler-Dirac action for the modes of induced imbedding space spinor field reduces to a vanishing contribution in the interior of space-time surface and by weak form of electric magnetic duality (WFEMD) to a boundary term given by Chern-Simons action.
   
   
2. For the light-like partonic orbits associated with wormhole contacts these terms must vanish by conservation of fermionic charges and one obtains just algebraic form of M⁴ Dirac action at the partonic 2-surface so that massless fermionic propagators are obtained. There is nothing Higgs like in propagators.
   
   
3. At the space-like 3-surfaces defining the ends of space-time at boundaries of CDs one obtains algebraic massless Dirac plus Chern-Simons term.
   
   
4. Could this term play the the role of Higgs vacuum expectation? Does this term give only a small additional contribution to the masses of particles besides that given by p-adic thermodynamics for wormhole contacts and for string like object in electro-weak or even Compton
   length scale?
   
   
5. What happens at at the ends of string worlds sheets. Here the Higgs term would correspond to the normal (time-like) component of the Kähler-Dirac gamma matrix whose CP₂ part would give needed non-tachyonic contribution to mass. One would get algebraic form of M⁴ Dirac equation: p^kγ_k Ψ= ΓⁿΨ, where Γ^α=T^{α k}γ_k is modified (Kähler-Dirac) gamma matrix defined as a contraction of the canonical momentum currents for Kähler action with imbedding space gamma matrices γ_k. Note that this Kähler-Dirac gamma matrix could be covariantly constant along the boundary of string world sheet and perhaps even inside string world sheet.
   
   
6. Note that Minkowskian part of Γⁿ would give a tachyonic contribution,
   which might relate to the still poorly understood question how the tachyonic ground states of super-conformal representations with half integer conformal weight emerge.