About the justification for the holography = holomorphy vision and related ideas

The recent view of Quantum TGD (see this, this, this, this, this and this) has emerged from several mathematical discoveries.

1. Holography = holomorphy principle (HH) reduces classical field equations at the Minkowskian regions of the space-time surface to algebraic roots f=(f₁, f₂) = (0, 0) of two functions which are analytic functions of 4 generalized complex coordinates of H=M⁴× CP₂ involving 3 complex coordinates and one hypercomplex coordinate of M⁴.
2. Space-time surface as an analog of Bohr orbit is minimal surface, which means that it generalized the notion of geodesic line in the replacement of point-like particle with 3-surface and that the non-linear analogs of massless field equations are satisfied by H coordinates so that analog of particle-wave duality is realized geometrically.
3. Minimal surface property holds true independently of the classical action as long as it is general coordinate invariant and constructible in terms of the induced geometry. This strongly suggests the existence of a number theoretic description in which the value of action as analog of effective action becomes a number theoretic invariant.

4. The minimal surface property fails at 3-D singularities at which derivatives of the embedding coordinates are discontinuous and the components of the second fundamental form have delta function divergences so that its trace as local acceleration and an analog of the Higgs field, diverges.

   These discontinuities give rise to defects of smooth structure and in 4-D case an exotic smooth structure emerges and makes possible description of fermion pair creation (boson emission) although the fermions are free particles. Fermions and also 3-surfaces turn backwards in time. This is possible only in dimension D=4.

One can criticize this picture as too heuristic and of the lack of explicit examples. I am grateful for Marko Manninen, a member of our Zoom group, who raised this question. In the following I try to make it clear that the outcome is extremely general and depends only on the very general aspects of what generalized holomorphy means. I hope that colleagues would realize that the TGD approach to theoretical physics is based on general mathematical principles and refined conceptualization: this approach is the diametric opposite of, say, the attempt to understand physics by performing massive QCD lattice calculations. Philosophical and mathematical thinking, taking empirical findings seriously, dominates rather than pragmatic model building and heavy numerics.

H-H principle and the solution of field equations

Consider first how H-H leads to an exact solution of the field equations in Minkowskian regions of the space-time surface (the solution can be found also in Euclidean regions).

1. The partial differential equations, which are extremely non-linear, reduce by generalized H-H to algebraic equations in which one has contractions of holomorphic tensors of different type vanishing identically if one has roots of f=(f₁,f₂)=(0,0). f₁ and f₂ and generalized analytic functions of generalized complex coordinates of H.

   This means a huge simplification since the Riemannian geometry reduces to algebraic geometry and partial differential equations reduce to local algebraic equations.

2. There are two kinds of induced gauge fields: induced metric and induced gauge potentials, Kähler gauge potential for the Kähler action. The variation with respect to induced metric gives a contraction of two holomorphic 2-tensors to the field equations. The variation with respect to gauge potential gives contraction of two holomorphic vector fields. The contractions are between tensors/vectors of different types and vanish identically.
   1. Consider the metric first. The contraction is between the energy momentum tensor of type (1,-1)+(-1,1) and the second fundamental form of type (1,1)+(-1,-1). Here 1 refers to a complex coordinate and -1 to its conjugate as tensor index. These contractions vanish identically.

      The vanishing of the trace of the second fundamental form occurs independently of the action and gives minimal surface except at singularities.

   2. Consider next the induced gauge potentials. In this case one has contraction of vector fields of different type (of type (1)and (-1) and also now the outcome is vanishing. In the case of more general action, such as volume + Kähler action, one also has a contraction of light-like Kähler current with a light-like vector field which vanishes too. The light-like Kähler current is non-vanishing for what I call "massless extremals". This miracle reflects the enormous power of generalized conformal invariance. \end{enumerate}
   3. For more general actions these results are probably true too but there I have no formal proof. If higher derivatives are involved one obtains higher derivatives of the second fundamental form which are of type (1,1,...,1) contracted with tensors which have mixed indices.

      Actions containing higher derivatives might be excluded by the requirement that only delta function singularities for the trace of the second fundamental form defining the analog of the Higgs field are possible.

   4. The result has analog already in ordinary electrodynamics in 2-D systems. The real and imaginary parts of an analytic function satisfy the field equations except at poles and cuts define the point charges and line charges. Also in string models the same occurs.
   Concerning explicit examples, I used 8 years after my thesis to study exact solutions of field equations of TGD \cite{all/class,prext}. The solutions that I found were essentially action independendent and had interpretation as minimal surfaces.

   Singularities as analogs of poles of analytic functions

   Consider now the singularities.

   1. The singularities 3-surfaces at which the generalized analyticity fails for (f₁,f₂): they are analogs of poles and zeros for analytic functions. At 3-D singularities the derivatives of H coordinates are discontinuous and the trace of the second fundamental form has a delta function singularity. This gives rise to edge.

      Singularities are analogous to poles of analytic functions and correspond to vertices and also to loci of non-determinism serving as seats of conscious memories.

   2. At singularities the entire action contributes to the field equations which express conservation laws of classical isometry charges. Note that the trace of the second fundamental form defines a generalized acceleration and behaves like a generalization of the Higgs field with respect to symmetries.

      Outside singularities the analog of massless geodesic motion with a vanishing acceleration occurs and the induced fields are formally massless. At singularities there is an infinite acceleration so that particles perform 8-D Brownian motion.

   3. Singularities as edges correspond to defects of the standard smooth structure as edges of space-time surface analogous to the frames of a soap film. The dependence of the loci of singularities on the classical action is expected from the condition that the field equations stating conservation laws are true for the entire action.

      It is possible that exotic smooth structure is at least partially characterized by the classical action having interpretation as effective action. For a mere volume action singularities might not be possible: if this is true it would correspond to the analog of massless free theory without fermion pair creation. In this case, the trace of the second fundamental form should vanish although its components should have delta function divergences.

      This makes it possible to interpret fermionic Feynman diagrams geometrically as Brownian motion of 3-D particles in H (see this, this and this). In particular, fermion pair creation (and also boson emission) corresponds to 3-surface and fermion lines turning backwards in time.

   4. The physical interpretation generalizes the interpretation in classical field theories, where charges are point-like. In massless field theories, charges as singularities serve as sources of fields. The trace of the second fundamental form vanishes almost everywhere (minimal surface property) stating that the analog of the charge density, serving as a source of massless field defined for H coordinates, vanishes except at the singularities. The generalized Higgs field defines the source concentrated to 3-D singularities.
   5. Classical non-determinism is an essential assumption. Already 2-D minimal surfaces allow non-determinism and soap films spanned by a given frame provide a basic example. The geomeric conditions under which non-determinism is expected, are known and can be generalized to 4-D context. Google LLM gives detailed information about the non-determinism in 2-D case and I have discussed the generalization to 4-D case in (see this and this).

   See the article What could 2-D minimal surfaces teach about TGD? or the chapter with the same title.

   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.