TGD diary: Considerations inspired by LLM summaries of TGD articles

Considerations inspired by LLM summaries of TGD articles

Tuomas Sorakivi prepared LLM summaries about some articles related to TGD, in particular the article (see this) in which the relation of the holography = holomorphy vision to elliptic surfaces and the notion of partonic orbits are considered.

The discussions and the LLM summaries inspired considerations related to the general view about the definition of the partonic orbits involving the conditions g₄^1/2=0 assuming generalized holomorphy and to the details related to the model for the pairs of space-time sheets connected by wormhole contacts.

What conjugation means for generalized complex coordinates?

Generalized complex structure involves hypercomplex coordinates and this involves non-trivial delicacies related to the counterpart of generalized complex conjugation.

The expression for g_uv involves conjugation of CP₂ cooordinates ξ^k. It is important to note that conjugation means that means
ξ^k(u,w)→ ξ^k(v,w) .
This is because v is the hypercomplex conjugate of u. In the conditions f_i=0, only the hypercomplex and therefore real u coordinate occurs in the functions f_i(u,w,ξ¹,ξ²), i=1,2.
What is the interpretation of the fact that the hypercomplex conjugation u→ v is involved? The presented model for a pair of spacetime sheets is that, for example, the upper sheet has an active coordinate u and the lower one has v. Conjugation would take from the "upper" spacetime sheet to the "lower" one if both are involved. This would indicate that the sheets are the relations of generalized complex conjugation. This is not a necessary assumption, but it is possible and I have suggested it.
This formal interpretation seems strange, but in ordinary complex conjugation it is like this. x+iy, y≥ 0 corresponds to the upper half plane and x-iy, y≥ 0 to the lower half plane. Conjugation takes from the upper half plane to the lower one. On the real axis y=0 the planes meet.
So two 4-D Minkowski spacetime sheets would be generalizations of the half planes. The real axis would be the Euclidean 3-D CP₂ inside the extremal: it is not the same as the parto orbit: the language model had mixed them up. In the used H-J coordinates, u=t-z=v=t+z, that is z=0, would hold. This 3-surface in the direction of time would correspond to the world line of a particle at rest in M⁴.

Connection with particle massivation and ideas of Connes

The fact that this 3-surface inside the CP₂ type extremal is like a particle at rest necessarily means that there are 2 space-time sheets and they are connected by a wormhole contact. Massification has necessarily occurred.

If only one space-time sheet is involved, it is a half-plane equivalent of one of the two. Is this possible? Could the light-like 3-D orbit of the parton surface be a track edge in the Minkowski region? Is such a solution possible or are wormhole contacts and a pair of space-time sheets necessarily needed. In any case, the fermion lines would be on partonic 2-surfaces, so a partonic surface is needed.
Interestingly, a top French mathematician Connes ended up proposing that the Higgs mechanism in non-commutative geometry would correspond to the Minkowski space doubling in the same way. Also in TGD framework the massivation would occur in the same way!
I have been in Schrödinger's cat-like state regarding this question: it would seem that the boundary conditions do not allow boundaries at all. On the other hand, I have also considered the possibility allowing light-like boundaries.
The fact that only the coordinate u or v appears in the generalized analytic functions f₁ and f₂ means that an analogy is made between the wave motion at the speed of light t-z or t=z and the coordinate on which the wave depends. In the string model, the terms left mover and right mover are used.
The situation in which both space-time sheets are involved would correspond in the string model to the fact that a wave coming along one space-time sheet is reflected back on this three-surface of CP₂ type extremal and returns along the other space-time sheet.
If a single sheet with light-like boundaries are possible, it would correspond to massless particles. Either a left-mover or a right-mover, but not both. On the other hand, p-adic thermodynamics predicts that photons and gravitons also have a small mass.

Testing whether the conditions g_uv=0 allow solutions

Tuomas had, using the language model, come up with a proposal to investigate whether there are analytical solutions to the condition g_uv=0 on a partonic surface. If there are, then we can be satisfied. On the other hand, it could happen that there are none. I thought about it at night and found out that such solutions really do exist. The task is to find such a simple situation that numerical calculations are not needed.

I already made a simplifying assumption earlier that f₂ is of the form f₂= ξ²-wⁿ. There would be no u-dependence at all. f₂=0 would give ξ₂= wⁿ. There would be no need to find the roots either.
A more general solution would be f₂= P₂(ξ²,w) without u-dependence. Now the roots of the polynomial must be solved. This does not change the situation.
We could make a similar assumption for f₁, but assume u-dependence.
f₁= f₁(ξ₁,w,u) = ξ₁- g(w,u) .
We can simplify it even further by assuming
g(w,u)= u h(w) .
So we can solve ξ₁ as
ξ₁= uh(w) .
Now we have everything we need to solve the condition g_uv=0.
1. The CP₂ metric s_{ξⁱ ξ^j} is known. Here we must remember that conjugation means u→ v!
2. The vanishing condition g_uv= 0 gives
  s_kl ∂_uξ^k ∂_v ξ^l =-1 .
3. The non-vanishing partial derivatives are
  ∂_uξ¹=h(w)
  ∂_v ξ¹ =h(w) .
  This gives
  h(w)h(w) s₁₁ = -1 .
4. The component of the CP₂ metric s₁₁ ≤ 0 appears in the formula (the CP₂ metric is Euclidean) and is known and is proportional to 1/(1+r²) (see this),
  r²= ξ¹ξ¹+ ξ²ξ²
  and depends on the uv via ξ¹ξ¹ = uvg(w)g(w) . The equation can be solved for the uv function in terms of a function k(w,w) deducible from the condition:
  uv= k(w,w).
  In the (u,v) plane, this is a hyperbola for the given values of w. So there are solutions. We can breathe a sigh of relief.