How to define hypercomplex conjugation?

It took a considerable time to realize that hyper complex conjugation u↔ w for the hypercomplex coordinates, which are real, might pose serious interpretational problems. Suppose that the roots of the generalized analytic maps f=(f₁,f₂)(u,w,ξ₁,ξ₂): H→ C² define the space-time surfaces.

What does one mean with the generalization of the complex conjugation when applied to the argument of f? Could it correspond a) to (u,w,ξ₁,ξ₂)→ (u,w,ξ₁,ξ₂) so that there is no hypercomplex conjugation or b) to (v,w,ξ₁,ξ₂)→ (u,w ,ξ₁,ξ₂) so that there is hypercomplex conjugation.

1. For option a), the roots of f and f represent the same surface. For the roots of f the contribution of complex coordinates to g_uv and g_vw is vanishing but the components g_uw.
2. For option b), the roots of the conjugate f do not coincide with the roots of f unless symmetries (possibly non-local) exist. The condition u=v makes the two conditions equivalent. For the simplest option u=m⁰+m³, v= m⁰-m³ the condition u=v gives m³=0 and so that the intersection of the two roots is 3-D surface analogous to a particle at rest. Could this surface be identified as partonic orbit. If one accepts both roots, one has a situation similar to the option a) for both branches and branches intersect at the 3-D surface.

The basic idea is that partonic orbits are light-like with respect to 4-metric g₄. Is it possible to realize this somehow for option b)? Could one assume that in the intersection of branches visualizable as intersection of two 1-D curves, the partial derivatives of complex H coordinates include both u and v derivatives so that the component g_uv of the induced metric receives an additional contribution from the complex coordinates. The partonic orbit would represent a violation of conformal symmetry in the sense that it would represent an intersection of analytic and anti-analytic region. In the case of ordinary violation of analyticity, the derivative of function f(z) is non-vanishing also with respect to z. The hypercomplex variant for the breaking of analyticity corresponds to the vanishing of the partial deritives of ξ_i with respect to both u and v.

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.