Technical problems of the holography= holomorphy vision

The realization of holography= holomorphy vision has 3 problems.

1. Hyper complex conjugation u↔ v 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.

   If u↔ v represents hypercomplex conjugation, the functions f_i and their generalized complex conjugates involving the replacement of hypercomplex coordinate u with v need not define the same surface. Their intersection is in general a 3-dimensional surface X³ with u=v. Should one fuse these surfaces so that hypercomplex conjugation by geometric symmetry? What is the physical interpretation of the two regions?

2. It is far from obvious whether CP₂ type extremals and their deformation allow a realization in terms of holography= holomorphy vision. It turns out Wick rotation provides a solution to the problem.
3. The surfaces of u-type and v-type do not allow light-like partonic orbits at which the induced metric changes the signature. It seems that the space-time surfaces must have regions in which the dynamic embedding space coordinates depend on both u and v. Also in this case Wick rotation solves the problem.

1. Definition of hypercomplex conjugation

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 and there is only the contribution of M⁴ metric to g_uv. Partonic orbits are not possible.
2. For option b), the roots of the conjugate f do not coincide with the roots of f unless symmetries exist. Since one can transform the hypercomplex and complex cases to each other by a Wick rotation, it seems necessary to assume that the union of the u-type and v-type regions defines the space-time surface. Hypercomplex conjugation would be a non-local symmetry transforming to each other two parts of the space-time surface. The 3-surface u=v would be a 3-dimensional surface along which the two space-time regions would be glued together,

Consider the option b) in more detail.

1. How to identify the u- and v-type regions? In the model for elementary particles, Euclidian regions as deformations of CP₂ extremals connect two Minkowskian space-time sheets, which are extremely near to each other having a distance of order CP₂ radius. Could the two Minkowskian space-time sheets correspond to u- and v-type regions and could generalized complex conjugation (u,w,ξ₁,ξ₂)↔ (v,w, ξ₁, ξ₂) transform then to each other.
2. Could the 3-surface X³ with u=v correspond to a surface in the interior of CP₂ extremal which along which the sheets are glued together? The condition u=v makes the two conditions equivalent as ordinary complex conjugates of each other. For the simplest option u=m⁰+m³, v= m⁰-m³ the condition u=v gives m³=0 and so that the intersection of the u and v-type regions would be 3-D surface inside the wormhole contact and analogous to a particle at rest.
3. In complex analysis, the counterpart of the 3-surface X³ defined by the u=v condition is the real axis. Could u- and v type solutions represent an analog of 2-valued analytic functions, such as z^1/2 having a cut discontinuity for the imaginary part along the positive real axis, effectively replacing the complex plane with its 2-fold covering.

Now the cut would be associated with the values of the dynamical embedding space coordinates and the two sides of X³ would correspond to the two Minkowskian space-time sheets. At X³ the u- and v-type time evolutions would be glued together. Discontinuity for dynamical H coordinates would mean that both sheets haave a hole and are actually separate. If only derivatives are discontinuous, X³ represents a 3-D edge.

Could X³ could be interpreted in terms of an exotic smooth structure allowing an interpretation as the standard smooth structure with defects? The u- lines would transform to v-lines at X³ and give rise to edges violating the standard smoothness.

Also the partonic orbits could define analogous defects since the u- resp. v-lines could have an edge. The identification of fermion lines as these kinds of lines allow the interpretation of defects as vertices for the creation of fermion-antifermion pair as turning of fermion line backwards in time (see this and this)?

2. How to represent CP₂ type extremals and their deformations?

CP₂ type extremals represent basic solutions of field equations. For them the M⁴ projection is a light-like curve and does not contribute to the induced metric. But how to represent this surface in holography= holomorphy vision. What is required is that the curve parameter u of the light-like curve is a function of a single real CP₂ coordinate. For instance, the function f₁= g(ξ₁)+u would formally give either u=g(ξ₁), which makes no sense. The second option is Im(g(ξ₁))=0, which violates holomorphy and reduces the dimension of the CP₂ type extremal to 3.

Here Wick rotation suggests an elegant solution.

1. One can replace the hypercomplex coordinate u and its conjugate v with a complex coordinate z and its conjugate z. For instance, for (u,v)= (m⁰+m³,m⁰-m³) this would change m³ to -m³ and if one uses m⁰+im³ and m⁰-im³ as Wick rotate coordinates (z,z) this corresponds to complex conjugation. One can solve the roots of f_i for their Wick rotated form and perform Wick rotation to the roots to get the solution. The problem posed by the CP₂ type extremals disappears.

Note that one can also interpret the metric of M² for complex coordinates (z,z) as a number theoretic metric introduced in M⁸ in the context of M⁸-H duality (see this). The metric in (u,v) ((m⁰,m³)) coordinates corresponds to the imaginary (real ) part of z².

Wick rotation predicts that the solved embedding space complex coordinates depend on both u and v in some region, in which the metric is Euclidean. It transforms to Minkowskian metric at a 3-D light-like interface identifiable as partonic orbit. This is just what is wanted. The reason is that for pure u and v type regions with 4-D M⁴ projection the component g_uv of the induced metric receives no contribution from CP₂ and corresponds to the metric of the empty Minkowski spaces. Partonic surfaces would not be possible. u- resp. v-type regions correspond to regions, where it is possible to regard u resp. v as a parameter appearing in f_i rather than a dynamical variable to be solved.

Outside the partonic orbit assignable to a given Minkowskian space-time sheet, one has a u- or v-type solution. Wick rotation implies that in the Euclidean region the metric receives a contribution from M⁴. This in turn makes possible the presence of a light-like 3-D interface at which the 4-metric becomes degenerate and actually metrically 2-dimensional so that it can be glued to the induced metric of a Minkowskian space-time sheet as u- or v-type solution. It seems that the pairs of space-time sheets connected by wormhole contacts represent a basic solution type for holography= holomorphy vision. The wormhole contacts generalize also tothe solutions which represent cosmic strings and their deformations. Even the vertices for the creation of fermion-antifermion pairs seem to emerge as defects of exotic smooth structures.

This picture would relate several key ideas of TGD: holography= holomorphy vision involving hypercomplex numbers, the notion of light-like partonic orbit, the idea that exotic smooth structures make possible non-trivial scattering theory in 4 dimensional space-time. One can compare this picture with the intuitive phenomenological picture.

1. The doubling and partonic seems to be an inherent property of the holography= holomorphy vision and of hypercomplex analyticity. u-v pairing is true also in CP₂-type regions. The intersection of u and v regions has a well-defined and in general non-vanishing g_uv, which does not reduce to mere Minkowskian contribution to it. The condition g_uv=0 is true at the partonic orbit defining the interface between Euclidean wormhole contact as a piece of CP₂ type extremal and Minkowskian space-time sheets.
2. Could the u- and v-type regions correspond to a pair of Minkowskian space-time and also Euclidean space-time regions? At the partonic orbit the u- and v-sheets intersect. This does not conform with the intuitive view about the wormhole contacts. Could this mean that there is no wormhole contact such that its throats carry opposite homology charges? This does not conform with the physical intuitions. If one wants to keep the view about parallel space-time sheets, both Minkowskian space-time sheets must be u-v pairs and both wormhole throats must be intersections of the u and v sheets?

See the article Holography= holomorphy vision and a more precise view of partonic orbits or the chapter Holography= holomorphy vision: analogues of elliptic curves and partonic orbits .

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.

Source: https://matpitka.blogspot.com/2025/07/technical-problems-of-holography.html