Holography= holomorphy vision and a more precise view of partonic orbits

Partonic orbits are a central piece of the TGD view of elementary particles. In the sequel a more precise identification of these surfaces is considered.

1. The roles of fermionic and geometric degrees of freedom

It is useful to clarify some basic aspects of dynamics in the TGD Universe.

1. There are two kinds of degrees of freedom in TGD: geometric, i.e. degrees of freedom of the space-time surface, and fermionic. All elementary particles are made up of fermions and antifermions: bosons emerge. There are no bosonic primary quantum fields.
2. The basic result from the solution of the Dirac equation for H spinor fields, assuming that M⁴ has a non-trivial Kähler structure (see this), is that the mass scale of colored partial waves of fermions is given by CP₂ mass scale and there are no free massless gluons or quarks. However, assless color singlets for which the difference in the numbers of quarks and antiquarks is a multiple of three, are possible. This gives baryons and mesons.

Here comes a crucial difference between QCD and TGD. In lattice QCD there would be no g-2 anomaly whereas the approach based on the information given by physical hadrons imply the anomaly. In TGD, the anomaly would be real and the new physics TGD predicts it. For example, copies of hadron physics at larger mass scales are predicted. Also the higher color partial waves give rise to new hadrons and also leptons. It will be exciting to see whether QCD or TGD is right.

The arguments of the n-point functions of the second quantized free fermion fields of H (scattering amplitudes) are points of the spacetime surface so that the dynamics of the spacetime surface affects the scattering amplitudes. Effectively, the spacetime surface defines the classical background in terms of the induced fields: induced metric, spinor connection, etc… A particle can be seen in two ways:

1. Particle as a 3-surface and its Bohr orbit as a four-surface X⁴.
2. Particle as a fermion and its orbit, the fermion line is a light-like curve, maybe even a geodesic line in M⁴× CP₂.

The spacetime surface has an anatomy.

1. X⁴ has internal structure and the 3-D partonic orbits define light-like surfaces X³ at which the Minkoski signature of the surface becomes Euclidean so that the metric determinant vanishes.
2. A fermion line would be an intersection of 2-D string world sheet and a 3-D light-like partonic orbit. A string world sheet can naturally be obtained as the intersection of two spacetime surfaces if they have the same Hamilton-Jacobi structure, i.e. the same H coordinates u,w, ξ₁, ξ₂ and their conjugates (uv).

One can ask whether the mutual interactions of particles as space-time surfaces occur only when they have the same Hamilton-Jacobi structure. If so the interactions can be described in terms of their intersections consisting of string world sheets and fermion lines at their boundaries. If so, a strong analogy with string models would emerge.

Could also the self-interactions could be described by considering infinitesimal deformation of the space-time surface preserving H-J structure and finding the string world sheets in this case.

The genus of the parton surface is an important topological quantum number. The genera g=0,1,2 corresponds to the observed fermion generations. g=2 allows a bound state for the handles of the sphere that are like particles. This is because g ≤ 2 allows global conformal symmetry. In the gge 2 topology, g handles are like particles in a multiparticle state, and the mass spectrum of the states is continuous, unlike for elementary particles.

Also the homological charge of the partonic 2-surface, identifiable as Kähler magnetic charge of the space-time surface is an important topological quantum number.

2. How to find the 3-D light-like trajectories of parton surfaces

The trajectories of partonic 2-surfaces are singularities at , the Euclidean induced 4-geometry transforms into Minkowskian. The light-like dimension implies sqrt{|det(g)|}=0. The challenge is to derive the partonic orbits from this.

What conditions are used.

1. The space-time surface X⁴ ids defined by the conditions (f₁,f₂)=(0,0), where f₁ and f₂ are analytic functions H= M⁴× CP₂→ C² depending only on the hypercomplex coordinate u (or v) with light-like coordinate curves and complex coordinates w, ξ₁ and ξ₂ of H but not on the coordinates v as hypercomplex conjugate of u (or u) and the conjugates w, ξ₁, ξ₂.

As a special case, f_i are polynomials or rational functions. Additional restrictions can be posed on the coefficients of the polynomials. The conditions (f₁,f₂)=(0,0) have been studied in some cases (see this).

(det(g)^1/2=0 gives an additional condition and gives a 3-D light-like partonic orbit X³.

The rows of the induced metric g can be written as a matrix in the general case in the form

(g_uu, g_uv , g_uw, g_uw)

(g_vu, g_vv , g_vw, g_vw)

(g_wu, g_wv , g_ww, g_ww)

(g_wwu, g_wv , g_ww, g_ww)

Holomorphy implying that the embedding space metric and induced metric are tensors of type (1,1) implies the vanishing of a large fraction of elements. This gives

(0,g_uv , 0, g_uw)

(g_vu,0 , g_vw, 0)

(0,g_wv ,0, g_ww)

(g_wu,0 , g_ww, 0)

The symmetry g_αβ=g_αβ leaves only 4 independent matrix elements. g_uv, g_uw, g_vw, g_ww.

The determinant det(g) is obtained by expanding the first row in the relation.

det(g)= -g_uvcof_uv- g_uwcof_uw .

For example, cof_uv is obtained by dropping the row and column passing through uv from the original matrix, resulting in a 3×3 matrix. cof_uv is its determinant. cof_uv is calculated with the same algorithm. This is all elementary matrix algebra.

The vanishing condition for the determinant is as follows:

det(g)= -g_uvcof_uv- g_uwcof_ww=0 .

This gives one additional real condition defining the 3-D surface as the light-like trajectory of the partonic 2-surface.

3. det(g)=0 conditions as a generalization of Virasoro conditions

The determinant condition has an interpretation as a generalization of the Virasoro conditions of string models to the 4-D context.

1. If the situation were 2-dimensional instead of 4-D, the det(g)=0 condition would give a light-like curve and the light-likeness would give rise to the Virasoro conditions. This was actually one of the first observations as I discovered CP₂ extremals, whose M⁴ projection is a light-like curve for the Kähler action (see this). The conditions as such are not Virasoro conditions. It is the derivative of the conditions with respect to the curve parameter, which gives the Virasoro conditions. By taking Fourier transform one obtains the standard form of the Virasoro conditions.

The Virasoro conditions can fail at discrete points and these singularities have an interpretation as vertices and also as points at which the generalized holomorphy fails. The poles and zeros of the ordinary analytic function are analogs for this.

For the 4-D generalization, the light-like curve is replaced by a 3-D light-like parton trajectory. The analogs of Virasoro conditions would be very natural also now because 2-D conformal invariance is generalized to 4-dimensional one. The Virasoro conditions have one integer, the conformal weight. Now the Fourier transform with respect to the coordinates of X⁴, say u and w gives conditions labelled by two integers having interpretation as conformal weights.

This suggests that conditions can be seen as analogs of Virasoro conditions. Their generalization gives rise to analogs of the corresponding gauge conditions for the Kac-Moody algebra, just like in the string model. A lot of physics would be involved.

A new element brought by TGD is that algebras would have non-negative conformal weights meaning that an entire fractal hierarchy of isomorphic algebras is predicted such that subalgebra and its commutator with the entire algebra annihilate the physical states (see this). This makes possible a hierarchy of gauge symmetry breakings in which a subspace of the entire algebra transforms from a gauge algebra to a dynamical algebra.

4. How to solve the det(g)=0 condition?

1. We need to solve the induced metric. This means moving from algebraic geometry to differential geometry because the induced metric is of the form

g_αβ = h_kl ∂_αh^k ∂_βh^l .

Here α and β refer to u,v,w,w and k and l refer to u,v,w,w, ξ₁,ξ₂. The metric of H in these coordinates can be written easily. From this, we need to calculate the induced metric.

Analyticity drops some of the derivatives because u, w, ξ₁, ξ₂ is a function only of the coordinates u,w whereas v, w, ξ₁,ξ₂ is a function only of the coordinates v,w.

If u,w can serve as coordinates of X⁴, we only obtain the trivial conditions u=u and w=w, but ξ₁=ξ₁(u,w) and ξ₂=ξ₂(u,w) require an analytical solution for f₁=f₂=0. Of course, we also have to calculate the partial derivatives of the conjugate variables, but they are obtained by conjugating the already calculated ones.

If f_i are polynomials of degree n<5 with respect to w, analytic expressions for ξ_i(u,w) are possible and the analytic calculation of the partial derivatives can be considered. Otherwise, we have to use numerical methods. One could hope that a symbolic program for calculating partial derivatives could be found .

Numerical calculation of partial derivatives requires that the variables h^k are examined at nearby points u,w and the difference quotient is calculated. If the discretization is sufficiently dense, the partial derivatives can be estimated using neighboring points.

Then we just have to calculate the components of the induced metric and from these det(g) using the formula given above and find the points |det(g)|=0, i.e. the partonic orbit, by minimizing |det(g)|.

In this way, we get from each point of the discretized space-time surface X⁴ to a new point with a smaller value of |det(g)|. Is it necessary to choose a sufficiently large, 3-dimensional discrete subset of X⁴ and proceed in this way. The calculation would boil down to a repetition of the following steps to get |det(g)|=0 in a given accuracy.

1. Calculate f₁=f₂=0 giving a 4-D surface X⁴.
2. Calculate the partial derivatives at the point of X⁴. If the discretization is dense enough, there is no need for a deformation.
3. Calculate the induced metric at each point.
4. Calculate |det(g)| for each point.
5. Go the neighboring point at which the decrease of |det(g)| is largest.

See the article Holography = holomorphy vision and elliptic functions and curves in TGD framework or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?: part III.

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/holography-holomorphy-vision-and-more.html