Sunday, May 19, 2024

Antipodal duality from the TGD point of view: answer to a question by Avi Shrikumar

Avi Shrikumar asked about the antipodal duality (see this), which has been discovered in QCD but whose origin is not well-understood.

Antipodal duality implies connections between strong and electroweak interactions, which look mysterious since in the standard model these interactions are apparently independent. This kind of connections were discovered long before QCD and expressed in terms of the conserved vector current hypothesis (CVC) and partially conserved axial current PCAC hypothesis for the current algebra.

I looked at the antipodal duality as I learned of it (see this) but did not find any obvious explanation in TGD at that time. After that I however managed to develop a rather detailed understanding of how the scattering amplitudes emerge in the TGD framework. The basic ideas about the construction of vertices (see this and this) are very helpful in the sequel.

  1. In TGD, classical gravitational fields, color fields, electroweak fields are very closely related, being expressed in terms of CP2 coordinates and their gradients, which define the basic field like variables when space-time surface 4-D M4 projection. TGD predicts that also M4 possesses Kähler structure and gives rise to the electroweak U(1) gauge field. It might give an additional contribution to the electroweak U(1) field or define an independent U(1) field.

    There is also a Higgs emission vertex and the CP2 part for the trace of the second fundamental fundamental form behaves like the Higgs group theoretically. This trace can be regarded as a generalized acceleration and satisfies the analog of Newton' s equation and Einstein's equations. M4 part as generalized M4 acceeration would naturally define graviton emission vertex and CP2 part Higgs emission vertex.

    This picture is bound to imply very strong connections between strong and weak interactions and also gravitation.

  2. The construction of the vertices led to the outcome that all gauge theory vertices reduce to the electroweak vertices. Only the emission vertex corresponding to Kähler gauge potential and photon are vectorial and can contribute to gluon emission vertices so that strong interactions might involve only the Kähler gauge potentials of CP2 and M4 (something new).
  3. The vertices involving gluons can involve only electroweak parity conserving vertices since color is not a spin-like quantum number in TGD but corresponds to partial waves in CP2. This implies very strong connections between electroweak vertices and vertices involving gluon emission. One might perhaps say that one starts the U(1) electroweak vertex and its M4 counterpart and assigns to the final state particles as a center of mass motion in CP2.

    If this view is correct, then the standard model would reflect the underlying much deeper connection between electroweak, color and gravitational interactions implied by the geometrization of the standard model fields and gravitational fields.

See the article Antipodal duality and TGD or the chapter About TGD counterparts of twistor amplitudes .

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.

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