Analog with Higgs mechanism and 3-D singular surfaces as analogs of soap bubbles

The vertices for the fermion pair creation are located 3-D singular surfaces X³ at which the conformal invariance fails. Can one say anything interesting about these 3-surfaces or even find a possible analogy with the existing physics? The idea that the trace of the second fundamental form which becomes singular at X³ relates closely to Higgs has been around from the beginning of TGD. In the following I want to show that this idea has finally found a precise form.

1. The trace of the 8-D second fundamental form defines a generalized local acceleration orthogonal to the space-time surface X⁴, which vanishes almost everywhere by the minimal surface property. It is non-vanishing only at the 3-D singularities X³ representing edges of X⁴. The vertices for fermion pair creation as edges of fermion lines are assigned to intersections of string world sheets with X³ in the intersection of intersecting space-time surfaces with the same Hamilton-Jacobi structure.
2. The second fundamental form is analogous to generalized Higgs, call it H, with the CP₂ part being group theoretically like the ordinary Higgs field and indeed causing a violation of the conservation of M⁴ chirality. The M⁴ part is identifiable as ordinary local acceleration. The generalized acceleration has the dimension of inverse length, that is the dimension of mass divided by Planck constant. Higgs vacuum expectation corresponds to the fermion mass in the standard model. Is this true also in TGD?
3. By both Dirac equation in H and Dirac equation with M⁴ Kähler form for CD the fermions are massless in 8-D sense. If the additivity of M⁴ mass squared, identified as a conformal weight, is assumed, all many-fermion states have a vanishing mass squared in 8-D sense so that the total M⁴ mass squared equals to CP₂ mass squared proportional to color Casimir. It vanishes for color singlets but has a CP₂ mass scale as a natural unit for colored states.
4. If M⁴- and CP₂ parts of the generalized Higgs have magnitude equal to the mass squared of the particle, quantum-classical correspondence is realized. p-Adic thermodynamics would predict M⁴ mass squared (see this). At X³, the condition a_M⁴ = m/h_eff for the M⁴ acceleration, where m the mass of the particle, would be satisfied. Conformal invariance would fail X³ but the 8-D masslessness would remain true. Local mass value depends on the point of X³ unless the magnitudes of a_M⁴ and a_CP2 are constant. The direction of 8-D acceleration is orthogonal to X⁴ and also X³.
5. What could be the physical interpretation for the possible constant magnitude of M⁴ and CP₂ accelerations? Isometrically embedded 3-sphere in E⁴ represents a basic example of this kind of surface. Also a soap bubble is an example of a surface with constant value of local acceleration. The surface tension corresponds to the local acceleration and is proportional to the pressure difference. This suggests that the 3-D singularities are analogous to the surfaces of 3-D soap bubbles.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Explicit formulas for the vertices in TGD

It is now intuitively clear that the vertices at the fundamental level should be just the standard model vertices assignable to the 3-D singularities of the divergence of the fermionic current. Also the idea that the vertices should have an interpretation as defects of standard smooth structures to which exotic smooth structure (see this, this, and this) can be assigned (see this, this, this). The difficult challenge (to me at least) is to deduce the vertices in a convincing way. The option discussed below seems to be the most promising of the options considered hitherto.

Are the vertices due to the non-conservation of fermion current associated with the induced/modified Dirac action

One can start from the standard model view of vertices as an intuitive guide line.

1. The singularities should give the emission vertices for Higgs and electroweak gauge bosons. What is new is that electroweak gauge bosons have an interpretation as gluons with the weak gauge group identified as the holonomy group U(2) identifiable as a subgroup of color group SU(3). Strong interactions correspond to the isometries of SU(3) and electroweak interactions to the holonomies assignable to the CP₂ spinor degrees of freedom. Generalized Higgs corresponds to the trace of the second fundamental form having a 3-D singularity at the vertex having interpretation as 8-D local acceleration. This means that the classical action has no role as far as vertices are considered.
2. The interaction vertices emerge from the induced/modified Dirac action. By the modified Dirac equation, this action vanishes almost everywhere. The Dirac action reduces to a divergence of the fermion current and this suggests that at the singularity this divergence is non-vanishing. This conforms with the view that a fermion pair is created and fermion line terms backwards in time. At the 3-D singularity the divergence of the fermion current should have a delta function divergence making possible non-vanishing various vertices.
3. The vertex for the generalized Higgs is not a problem. The trace of the quantity D_μ(Γ^μg₄^1/2) is proportional to the trace of the seocnd fundamental form and gives 3-D delta function for the vertex of the annihilation of Higgs to fermion-antifermion pair.
4. How to obtain electroweak vertices and the possible vertex related to the M⁴ Kähler gauge potential? The electroweak vertices should come from a 3-D delta function singularity X³ of the Dirac action density

   Ψ(-D^←_μ Γ^μ -Γ^μD_μ)Ψg₄^1/2).

   The problem is that the components of the induced spinor connection A have only a step function like discontinuity rather than the desired delta function singularity. The desired singularity should be equal to the difference A₊-A_- of the gauge potentials at the two sides of the singularity, which is invariant under gauge transformations if their action is the same at the two sides. Vertex would be defined by this difference rather than vector potential as in perturbative gauge theories.

5. This can be achieved if the spinor fields have a discontinuity at X³, which gives rise to the difference A₊-A_- under the action of derivatives to the spinor field at X³. This is achieved if the gauge potentials A₊ and A_- are related by a gauge transformation g in the holonomy group. This gives A₊= A_- +dgg^-1 giving dg= (A₊-A_-)g. The spinor fields are discontinuous at the singularity and related by Ψ₊= gΨ_-. The derivative dΨ₊ is given by dΨ₊ = d(gΨ_-)= dgΨ_- + gdΨ_- = (A₊-A_-)gΨ_-+ gdΨ_-. The difference of gauge potentials at the singularity gives rise to the desired vertex.
6. For this option, the vertices are universal and do not depend on classical action for the induced Dirac equation. Also for the modified Dirac equation the electroweak vertices are determined by a gauge transformation in the holonomy group of CP₂. The conservation of the isometry currents for the classical action implies that the Higgs vertex as the divergence of the induced/modified Gamma matrices are expressible in terms of current assignable to the remaining parts of the action. If the classical action is a mere volume action, the conservation of classical isometry currents does not allow singularities. Therefore Kähler action plus volume term strongly suggested by the twistor lift is the minimal option.

The relation to the exotic smooth structures

The vertices should be non-trivial for the 3-D singularities at which the minimal surface property of the space-time surface fails and which corresponds to the defects for the standard smooth structure, which transform it to exotic smooth structure (see this, this, and this). What does this really mean, is far from clear. I have discussed this problem already earlier (see this, this, this) but I am not satisfied with the view.

The edge of the fermion line at the singularity means the breaking of standard smooth structure at which the light-like hypercomplex coordinate changes from u to its hypercomplex conjugate v. The derivatives of the embedding space coordinates as functions of u resp. v at the singularity are infinite for the standard smooth structure. Also the induced Dirac spinors are discontinuous and related by a gauge transformation at the two sides of the singularity if the above argument is correct.

For the exotic smooth structure, the derivatives are continuous at the singularity. Also the vertices should remain the same in the exotic smooth structure. The only reasonable identification of the vertices is as regions at which the exotic smooth structure fails to reduce to the standard smooth structure. Could one introduce a 3-D term to the Dirac action additional term localized to the singular surfaces of the standard smooth structure to guarantee that the non-vanishing divergence of the ordinary Dirac action is compensated by this additional term giving rise to the standard model vertices for the Dirac action.

I have proposed the assignment of Chern-Simons-Kähler action and its fermionic counterpart to the 3-D light-like partonic orbits and a similar term can be considered also now. The modified Dirac action for the Kähler Chern-Simons term would contain the standard model couplings to spinors.

See the article Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The magnetosphere of Jupiter as a seat of plasma life and its moon Europa as a seat of chemical life?

The FM signals  in radio frequency range 12-40 MHz from Jupiter's moon Ganymede have been reported 4 years ago. The claimed FM signals from the moon Europa have received a lot of attention.  In human  radio wave communications in the frequency range  87.5-  108.0 MHz frequency modulation is used to code  information. This  has raised  speculations about intelligent life  under the ice cover of  Europe and raised the question whether  the radiation could code for information  and even represent   a response  to the signals from the Earth.  

This finding is especially interesting from the TGD point of view since frequency modulation is a basic vay to code information in TGD inspired quantum biology. Also the phenomenological notion of magnetic flux tube is central in the description of the magnetic field of Jupiter: in the TGD framework this notion is not phenomenological and is the basic notion in  practically all applications of TGD based on the notion of field body and the hierarchy of phases of ordinary matter labelled by effective Planck constant $h_{eff}$. Gravitational and electric Planck constants associated with long range classical gravitational and electric fields are of special importance.  Frequency modulation is the key mechanism of communications and control in TGD based quantum biology.

This motivates the  development of the TGD based models for the magnetospheres of Jupiter and its moons and   also an analog for the maser cyclotron instability. The ensuing model for Jupiter's auroras is consistent with the empirical facts. Also the possibility of plasma life at the field bodies associated with Jupiter as well as primitive life in the interior of Europa can be considered.

See the article The magnetosphere of Jupiter as a seat of plasma life and its moon Europa as a seat of chemical life? or the chapter Quantum gravitation and quantum biology in TGD Universe

The description of E and B modes of CMB in the TGD framework

There are two key differences between inflation theory and TGD. In TGD, the almost constant value of the CMB temperature is due to quantum coherence in arbitrarily long scales rather than exponential expansion which does not look plausible. There is however expansion in the transition from the primordial cosmic string dominated phase in which space-time surfaces have 2-D string world sheets as M^4 projection, to monopole flux tubes liberating energy and leading to a radiation dominated cosmology. An entire sequence of phase transitions leading to the thickening of the monopole flux tubes is predicted, and interpreted as transitions between copies of standard model physics labelled by different p-adic mass scales predicted by TGD. If there is an exponential expansion, it is associated with this sequence.

So called B modes are the key prediction of the inflation theory. They are generated already in the inflationary period in the exponential expansion amplifying quantum fluctuations to cosmic scales. The primordial B modes are caused by gravitational waves and leave an imprint of primordial quantum fluctuation in cosmic scales. Their observation would be a victory of inflationary cosmology but their observation is extremely difficult due to the fact that also gravitational lensing transforms E modes to B modes.

The massless extremals (MEs) as counterparts of classical radiation fields provide a TGD based model for E and B modes. The prediction of the holography = holomorphy principle is that these modes are interchangeable locally. The local polarization vector for MEs is a holomorphic vector for which curl and divergence vanish apart from singularities, where the holomorphy fails. E and B modes differ only globally: for the B modes the coordinate lines of the polarization vector are closed curves around singularities. For E they connect singularities. The detected intensity vanishes outside the singular points,where it has a delta function type singularity. A natural interpretation as a vertex for photon emission/absorption.

See the article The description of E and B modes of CMB in the TGD framework or the chapter TGD and cosmology .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.