Thursday, January 25, 2024

Modified Dirac equation and the holography=holomorphy hypothesis

The understanding of the modified equation as a generalization of the massless Dirac equation for the induced spinors of the space-time surface X4 is far from complete. It is however clear that the modified Dirac equation is necessary.

Two problems should be solved.

  1. It is necessary to find out whether the modified Dirac equation follows from the generalized holomorphy alone. The dynamics of the space-time surface is trivialized into the dynamics of the minimal surface thanks to the generalized holomorphy and is universal in the sense that the details of the action are only visible at singularities which define the topological particle vertices. Could holomorphy solve also the modified Dirac equation? The modified gamma matrices depend on the action: could the modified Dirac equation fix the modified gamma matrices and thus also the action or does not universality hold true also for the modified Dirac action?
  2. The induction of the second quantized spinor field of H on the space-time surface means only the restriction of the induced spinor field to X4. This determines the fermionic propagators as H-propagators restricted to X4. The induced spinor field can be expressed as a superposition of the modes associated with X4. The modes should satisfy the modified Dirac equation, which should reduce to purely algebraic conditions as in the 2-D case. Is this possible without additional conditions that might fix the action principle? Or is this possible only at lower-dimensional surfaces such as string world sheets?
In the article Modified Dirac equation and the holography=holomorphy hypothesis a proposal for how to meet these challenges is proposed and a holomorphic solution ansatz for the modified Dirac equation is discussed in detail.

See the article Modified Dirac equation and the holography=holomorphy hypothesis or the chapter Symmetries and Geometry of the ”World of Classical Worlds”.

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