Friday, November 24, 2023

Boundary conditions at partonic orbits and holography

TGD reduces coupling constant evolution to a number theoretical evolution of the coupling parameters of the action identified as Kähler function for WCW. An interesting question is how the 3-D holographic data at the partonic orbits relates to the corresponding 3-D data at the ends of space-time surfaces at the boundary of CD, and how it relates to coupling constant evolution.
  1. The twistor lift of TGD strongly favours 6-D Kähler action, which dimensionally reduces to Kähler action plus volume term plus topological ∫ J∧ J term reducing to Chern Simons-Kähler  action. The coefficients of these terms are proposed to be expressible in terms of number theoretical invariants characterizing the algebraic extensions of rationals and polynomials determining the space-time surfaces by M8-H duality.

    Number theoretical coupling constant evolution would be discrete. Each extension of rationals would give rise to its own coupling parameters involving also the ramified primes characterizing the polynomials involved and identified as p-adic length scales.

  2. The time evolution of the partonic orbit would be non-deterministic but subject to the light-likeness constraint and boundary conditions guaranteeing conservation laws. The natural expectation is that the boundary/interface conditions for a given action cannot be satisfied for all partonic orbits (and other singularities). The deformation of the partonic orbit requiring that boundary conditions are satisfied,  does not affect X3  but   the time derivatives ∂t hk at X3  are affected since the form of the holomorphic functions defining the space-time surface would change.   The interpretation would be in terms of duality of the holographic data associated with the partonic orbits resp. X3.  

     There can of course exist deformations, which require the change of the coupling parameters of the action to satisfy the boundary conditions. One can consider an analog of  renormalization group equations in which the deformation corresponds to a modification of the  coupling parameters of the action, most plausibly determined by the  twistor lift. Coupling parameters would label different regions of WCW and  the space-time surfaces possible for two different sets of coupling parameters would define interfaces between these regions.

 In order to build a more detailed view one must fix the details related to the action whose value defines the WCW  Kähler function.  
  1. If Kähler action is identified as Kähler action, the identification is unique. There is however the possibility that the imaginary exponent of the instanton term or the contribution from the Euclidean region is not included in the definition of Kähler function. For instance instanton term could be  interpreted as a phase of quantum state and would not contribute.
  2. Both Minkowskian and Euclidean regions are involved and the Euclidean signature poses problems. The definition of the  determinant as (-g4)1/2 is natural in Minkowskian regions but gives an imaginary contribution in Euclidean regions. (|g4|)1/2 is real in both regions. i(g4)1/2 is real in Minkowskian regions but imaginary in the Euclidean regions.

    There is also a problem related to the instanton term, which does not depend on the  metric determinant at all.  In QFT context the instanton term is imaginary and this is important for instance in QCD in  the definition of CP breaking vacuum functional. Should one include only the 4-D  or possibly only Minkowskian contribution to the Kähler function  imaginary coefficient for the instanton/Euclidian term would be possible?

  3.   Boundary conditions guaranteeing the conservation laws at the partonic orbits must be satisfied. Consider the  |g4| case.  Charge transfer between Euclidean and Minkowskian  regions. If the C-S-K term is real, also the  charge transfer between partonic orbit and 4-D  regions is possible.  The boundary conditions at the partonic orbit fix it to a high degree and also affect the time derivatives ∂thk at X3. This option looks physically rather attractive because classical conserved charges would be real.

    If the C-S-K term is imaginary it behaves like a free particle since charge exchange  with Minkowskian and Euclidean regions is not possible. A possible interpretation of the  possible M4 contribution to momentum could be in terms of decay width.  The symplectic charges do not however involve momentum. The imaginary contribution to momentum could therefore come only from the Euclidean region.

  4. If the Euclidean contribution is imaginary, it seems that it cannot be included in the Kähler function. Since in M8 picture the momenta of virtual fermions are in general complex, one could consider the possibility that  Euclidean contribution  to the momentum is imaginary and allows an  interpretation as a decay width.
See the article Symmetries and Geometry of the "World of Classical Worlds" or the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

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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