Confession of a very stupid mistake

The most painful events in the life of theorists are the sudden realizations that one has made a stupid mistake a year or two ago and must start charting the consequences. How to correct them and how to preserve some respectability. Should I edite my history after the error? Can anyone take seriously the mess that I have produce during almost half a century?

At this time I noticed a really stupid mistake in my attempts to understand the generalized complex structure of M⁴ combining complex structure for Euclidian E² and hypercomplex structure for Minkowski M². This generalizes to Hamilton-Jacobi structure which corresponds to a integrable local decomposition M²times E².

1. I had interpreted the hypercomplex conjugation for some mysterious reason completely wrong. The correct form is of course (u,v)→ (u,-v) as in the complex case and satisfies the and commutes with arithmetic operations. The first half of me had however interpreted hypercomplex conjugation as uleftrightarrow v although second half of me had of course understood that it corresponds to a multiplication by the hypercomplex unit e, e²=1 (by an imaginary unit for the ordinary complex numbers).

This led to a wrong identification of u and v as a hypercomplex number h and its conjugate h}. The correct identification is as h=(u,v) and its conjugate h=(v,u). The important implication is that hypercomplex analytic functions involving powers of (u,v) must be defined using hypercomplex arithmetics.

This implies that the identification of functions f(u,w,ξ₁,ξ₂) resp. f =(f₁,f₂)(v,w,ξ₁, ξ₂) identified as generalized analytic (antianalytic) functions is wrong. The same applies to their conjugates in which v appears.

This challenges the picture of holography= holomorphy vision developed hitherto. The general solution ansatz is not lost but the surfaces constructed as roots of f=(f₁,f₂)(u,w,ξ₁,ξ₂) are not 4-dimensional having v as a passive degree of freedom not visible in dynamics and therefore being effectively 3-dimensional. They are genuinely 3-dimensional. The only sensible looking interpretation is in terms of 3-D surfaces serving as holographic data.

The identification of these 3-surfaces as roots of (f₁,f₂) makes sense since if h=(u,v) is real h=(u,0) or imaginary h=(0,v), the hypercomplex arithmetics reduces to ordinary arithmetics. The earlier solutions therefore describe only the 3-D holographic data and the intersection of the 3-D roots of f=(f₁,f₂)(u,w,ξ₁,ξ₂) and its conjugate f=(f₁,f₂(v,w,ξ₁, ξ₂) has an interpretation as a 2-D surface identifiable as a partonic 2-surface.

This interpretation is consistent with the ordinary complex analysis. One can construct ordinary complex analytic functions from real analytic functions defined by the data provided by their poles. These data serve as holographic data. The continuation need not be unique and cuts can be chosen in several ways (consider only the function sqrt{z}) but real is number-theoretically preferred since complex arithmetics reduces to real arithmetics at it. In the recent situation both u and v are real and this means that they both serve as analogs of the real axis. The different choices of H-J structure would correspond to different continuations. One could interpret the partonic orbits as counterparts of cuts of an analytic function whereas partonic 2-surfaces might serve as analogs of poles.

The physical interpretation is that the 3-D partonic orbits as bundles of v=0- u=0 light-like curves intersect at the partonic 2-surface representing the vertex. At the vertex u=v=0 analogous to a tip of the light-cone the branching has interpretation in terms of the defect of the standard smooth structure and having interpretation as exotic smooth structure. The two branches emerging in u and v directions correspond physically to a creation of a fermion-antifermion pair in classical fields having interpretation as induced fields. The question is how to construct the full 4-D space-time surfaces determined by the generalized analytic functions. Wick rotation comes to rescue here.

1. The analytic functions of h=(u,v) can be defined by using hypercomplex arithmetics to define powers of (u,v) The problem is how to define the products of hyper-complex numbers with complex numbers appearing as arguments of the functions f_i. This is not a problem for the restrictions to 3-D holographic data.
2. Wick rotation is an obvious guess for how to construct the space-time surfaces. The Wick rotation (u,v)→ (x+iy)=(u+iv) transforms f(h)=(u,v),w,ξ₁,ξ₂) to an analytic function of 4 complex coordinates. One can find the roots of f=(f₁,f₂) and map the roots to space-time surfaces by the inverse map (x+iy)→ (u,v).

The corrected view challenges the suggestions made on basis of the earlier picture.

1. u or v is not a passive dynamical variable anymore. The conjugate of hyper complex conjugate h= (u,-v) of h=(u,v) can be regarded as a passive variable only in the sense that the space-time surface is determined by f and the vanishing of f=0 implies the vanishing of f.
2. This means that some suggestions that follow from the passive role of the v-variable are wrong. One such suggestion was that the intersection of the surfaces X⁴ and Y⁴ with the same H-J structure is 2-D since the intersection is effectively an intersection of 2 3-D surfaces. The proposed interpretation was as a string world sheet.

If one believes on a generic topological argument holding true in the absence of symmetries, the intersection of surfaces X⁴ and Y⁴ and self intersection of X⁴ with its infinitesimal deformation is discrete rather than 2-D. It would corresponds to the intersection form for the two surfaces playing a key role in 4-dimensional topology and in knot theory.

Here one must be very cautious. String world sheets are physically very attractive and holography is a symmetry reducing space-time surfaces to effectively 3-dimensional objects. Could this imply that the intersection is actually a 2-D string world sheet as a kind of dual for the partonic 2-surface.

The form of the metric deduced for the space-time surface does not hold true. The induced metric has the general form dictated by the H-J structure and reduces to the proposed simple form only at the 3-D holographic data possibly having interpretation as partonic orbits.

What happens to the 3-surfaces det(g₄)=0 serving as candidates for the interfaces of Minkowskian and Euclidean space-time regions. Could they correspond to the partonic orbits?

Why would the condition det(g₄)=0 be necessary at the intersection of the two branches with u=v=0 as the analog of a tip of the light-cone. For the ordinary light-cone, det(g₄) vanishes at the tip for the Robertson-Walker coordinates. If det(g₄) is non-vanishing and differs for the branches, the definition of 4-D volume element becomes problematic. If the volume elements are identical, this problem disappears but in the case of Robertson Walker metric the tip would correspond to a genuine hole for g₄ ≠ 0. Note that one can consider the H-J structure with coordinates analogous to Robertson-Walker coordinates. The complex coordinate w would correspond to that for sphere r_M= constant and u and v would correspond to t-r and t+r.

The analogy with Robertson-Walker metric suggests an interpretation of the condition g_uv=0, when the partial derivatives of complex coordinates correspond to the two holographic continuations. CP₂ type extremals having 1-D light-like curve as M⁴ projection provide a tests for the holography= holomorphy vision.

1. For the wrong proposal one finds that one obtains only 3-D sub-manifolds of CP₂ rather than the full CP₂ or its deformation. The correct view of generalized holomorphy explains this: the 3-D section that was obtained represents only the holographic data.
2. But is it possible to obtain CP₂ type extremals and their deformation from the correct formulation? Wick rotation of a solution should give the CP₂ extremal from its Wick rotate version. One should understand the emergence of light-like CP₂ geodesic emerging at the 3-D throats of the CP₂ type extremal as a wormhole contact. The M⁴ metric contributes to the induced metric of CP₂ and it should be possible to choose (u,v) pairs as one half of coordinates. The induced Euclidean metric would reduce to a metrically 2-D form as the throat is approached. Thi could make sense but means that the M⁴ metric contributes to the induced metric unlike in the case of CP₂ type extremals.
3. Should one give up CP₂ extremals as unrealistic because the gluing to Minkowskian space-time sheets is not taken into account. One can drill two holes to CP₂ deformed to have a 1-D light-like M⁴ projection but one cannot satisfy the boundary conditions at the resulting boundaries. For a realistic solution the 1-D light-like projection would be replaced with 3-D light-like partonic orbit.

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/2025/06/confession-of-very-stupid-mistake.html