Friday, September 29, 2023

A proposal for an explicit realization of the Hamilton-Jacobi structures

I have not been able to propose an explicit realization of the Hamilton-Jacobi structures hitherto. The Hamilton-Jacobi structure should define a slicing of the space-time surface, not necessarily orthogonal, partonic 2-surfaces and 2-D string worlds sheets. The Hamilton-Jacobi structure could be seen as a 4-D generalization of 2-D complex structure. Could Hamilton-Jacobi structures be convolutions of 2-D complex structures with their Minkowskian analogs such that the conformal moduli of the partonic 2-surfaces and strings world sheets (possibly making sense) depend on the point space-time surface?
  1. The conformal structures of partonic 2-surfaces are classified by their conformal moduli expressible in terms of Teichmueller parameters (see this). Could one consider some kind of analytical continuation of the moduli spaces of the partonic 2-surfaces with different topologies to moduli spaces of time-like string world sheets?

    This would give a direct connection with p-adic mass calculations in which the (p-adic counterparts of) these moduli spaces are central. In p-adic mass calculations (see this and this), one assumes only partonic 2-surfaces. However, the recent proposal for the construction of preferred extremals of action as minimal surfaces, realizing holography in terms of a 4-D generalization of the holomorphy of string world sheets and partonic 2-surfaces, assumes a 4-D generalization of 2-D complex structure and this generalization could be just Hamilton-Jacobi structure.

  2. The boundaries of a string world sheet can also have space-like portions and they would be analogous to the punctures of 2-D Euclidian strings (now partonic 2-surfaces are closed).
  3. Can Minkowskian string world sheets have handles? What would the handle of a string world sheet look like? String world sheets have ends at the boundaries of the causal diamond (CD) playing a central role in zero energy ontology (ZEO).

    If the 1-D throat of the handle is a smooth curve, it must have portions with both space-like and time-like normal: this is possible in the induced metric but the portion with a time-like normal should carry vanishing conserved currents. For area action this is not possible. Intuitively, the throat of the handle would look physically to a splitting of a planar string to two pieces such that the conserved currents go to the handle. If this is the case, the 1-D throat would have two corners at which two time-like halves of the throat meet.

    The handle would be obtained by moving the throat along a space-like curve such that its Minkowskian signature and singularity are preserved. CP2 contribution to the induced metric might make this possible.

See the article Hamilton-Jacobi Structure as a 4-D generalization of 2-D complex structure 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.

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