### Three dualities of the field equations of TGD

The basic field equations of TGD allow several dualities. There are 3 of them at the level of basic field equations (and several other dualities such as M

^{8}-M

^{4}× CP

_{2}duality).

- The first duality is the analog of particle-field duality. The spacetime surface describing the particle (3-surface of M
^{4}× CP_{2}instead of point-like particle) corresponds to the particle aspect and the fields inside it geometrized in terms of sub-manifold geometry in terms of quantities characterizing geometry of M^{4}× CP_{2}to the field aspect. Particle orbit serves as wave guide for field, one might say.

- Second duality is particle-spacetime duality. Particle identified as 3-D surface means that particle orbit is space-time surface glued to a larger space-time surface by topological sum contacts. It depends on the scale used, whether it is more appropriate to talk about particle or of space-time.

- The third duality is hydrodynamics-massless field theory duality Hydrodynamical equations state local conservation of Noether currents. Field equations indeed reduce to local conservation conditions of Noether currents associated with isometries of M
^{4}× CP_{2}. One the other hand, these equations have interpretation as non-linear geometrization of massless wave equation with coupling to Maxwell fields. This realizes the ultimate dream of theoretician: symmetries dictate the dynamics completely. This is expected to be realized also at the level of scattering amplitudes and the generalization of twistor Grassmannian amplitudes could realize this in terms of Yangian symmetry.

Hydrodynamics-wave equations duality generalizes to the fermionic sector and involves superconformal symmetry.

- What I call modified gamma matrices are obtained as contractions of the partial derivatives of the action defining space-time surface with respect to the gradients of imbedding space coordinate with imbedding space gamma matrices. Their divergences vanish by field equations for the space-time surface and this is necessary for the internal consistency the Dirac equation. The modified gamma matrices reduces to ordinary ones if space-time surface is M
^{4}and one obtains ordinary massless Dirac equation.

- Modified Dirac equation expresses conservation of super current and actually infinite number of super currents obtained by contracting second quantized induced spinor field with the solutions of modified Dirac. This corresponds to the super-hydrodynamic aspect. On the other hand, modified Dirac equation corresponds to fermionic analog of massless wave equation as super-counterpart of the non-linear massless field equation determining space-time surface.

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

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