Does the universality of the holography-holomorphy principle make the notion of action un-necessary in the TGD framework?

It is gradually becoming clear that in the TGD framework the holography-holomorphy principle could make the notion of action defining the space-time surfaces un-necessary at the fundamental level. Only the Dirac action for the second quantized free spinors of H and the induced Dirac action would be needed. Geometrization of physics would reduce to its algebraic geometrization and number theoretical universality would allow to describe correlates of cognition. The four-dimensionality of space-time surfaces would be essential in making the theory non-trivial by allowing to identify vertices for fermion pair creation in terms of defects of the standard smooth structure of the space-time surface making it an exotic smooth structure.

Holography=holomorphy as the basic principle

Holography=holomorphy principle allows to solve the field equations for the space-time surfaces exactly by reducing them to algebraic equations.

1. Two functions f₁ and f₂ that depend on the generalized complex coordinates of H=M⁴xCP₂ are needed to solve the field equations. These functions depend on the two complex coordinates ξ₁ and ξ₂ of CP₂ and the complex coordinate w of M⁴ and the hypercomplex coordinate u for which the coordinate curves are light-like. If the functions are polynomials, denote them f₁==P₁ and f₂ ==P₂.

   Assume that the Taylor coefficients of these functions are rational or in the expansion of rational numbers, although this is not necessary either.

2. f₁=0 defines a 6-D surface in H and so does f₂=0. This is because the condition gives two conditions (both real and imaginary parts for f_i vanish). These 6-D surfaces are interpreted as analogs of the twistor bundles corresponding to M⁴ and CP₂. They have fiber which is 2-sphere. This is the physically motivated assumption, which might require an additional condition stating that ξ₁ and ξ₂ are functions of w) as analogs of the twistor bundles corresponding to M⁴ and CP₂. This would define the map mapping the twistor sphere of the twistor space of M⁴ to the twistor sphere of the twistor space of CP₂ or vice versa. The map need not be a bijection but would be single valued.

   The conditions f₁=0 and f₂=0 s give a 4-D spacetime surface as the intersection of these surfaces, identifiable as the base space of both twistor bundle analogies.

3. The obtained equations are algebraic equations. So they are not partial differential equations. Solving them numerically is child's play because they are completely local. TGD is solvable both analytically and numerically. The importance of this property cannot be overstated.
4. However, a discretization is needed, which can be number-theoretic and defined by the expansion of rationals. This is however not necessary if one is interested only in geometry and forgets the aspects related to algebraic geometry and number theory.
5. Once these algebraic equations have been solved at the discretization points, a discretization for the spacetime surface has been obtained.

   The task is to assign a spacetime surface to this discretization as a differentiable surface. Standard methods can be found here. A method that produces a surface for which the second partial derivatives exist because they appear in the curvature tensor.

   An analogy is the graph of a function for which the (y,x) pairs are known in a discrete set. One can connect these points, for example, with straight line segments to obtain a continuous curve.Polynomial fit gives rise to a smooth curve.

6. It is good to start with, for example, second-degree polynomials P₁ and P₂ of the generalized complex coordinates of H.

How could the solution be constructed in practice?

For simplicity, let's assume that f₁==P₁ and f₂==P₂ are polynomials.

1. First, one can solve for instance the equation P₂(u,w,ξ₁,ξ₂)=0 giving for example ξ₂(u,w,ξ₁) as its root. Any complex coordinates w, ξ₁ or ξ₂ is a possible choice and these choices can correspond to different roots as space-time regions and all must be considered to get the full picture. A completely local ordinary algebraic equation is in question so that the situation is infinitely simpler than for second order partial differential equations. This miracle is a consequence of holomorphy.
2. Substitute ξ₂(u,w,ξ₁) in P₁ to obtain the algebraic function P₁(u,w,ξ₁,ξ₂(u,w,ξ₁))= Q₁(u,w,ξ₁)
3. Solve ξ₁ from the condition Q₁=0. Now we are dealing with the root of the algebraic function, but the standard numerical solution is still infinitely easier than for partial differential equations.

   After this, the discretization must be completed to get a space-time surface using some method that produces a surface for which the second partial derivatives are continuous.

Algebraic universality

What is so remarkable is that the solutions of (f₁,f₂)=(0,0) to the variation of any action if the action is general coordinate invariant and depends only on the induced geometry. Metric and the tensors like curvature tensor associated with it and induced gauge fields and tensors associated with them. The reason is that complex analyticity implies that in the equations of motion there appears only contractions of complex tensors of different types. The second fundamental form (external curvature) defined by the trace of the tensor with respect to the induced metric defined by the covariant derivatives of the tangent vectors of the space-time surfaces is as a complex tensor of type (2,0)+(0,2) and the tensors contracted with it are of type (1,1). The result is identically zero. The holography-holomorphy principle provides a nonlinear analogy of massless field equations and the four surfaces can be interpreted as trajectories for particles that are 3-surfaces instead of point particles, i.e. as generalizations of geodesics. Geodesics are indeed 1-D minimal surfaces. We obtain a geometric version of the field-particle duality.

Number-theoretical universality

If the coefficients of the function f₁ and f₂ are in an extension of rationals, number-theoretical universality is obtained. The solution in the real case can also be interpreted as a solution in the p-adic cases p=2,3,5,7,... when we allow the expansion of the p-adic number system as induced by the rational expansions.

p-adic variants of space-time surfaces are cognitive representations for the real surfaces. The so-called ramified primes are selected for a special position, which can be associated with the discriminant as its prime factors. A prime number is now a prime number of an algebraic expansion. This makes possible adelic physics as a geometric correlate of cognition. Cognition itself is assignable to quantum jumps.

Is the notion of action needed at all at the fundamental level?

The universality of the space-time surfaces solving the field equations determined by holography=holomorphy principle forces us to ask whether the notion of action is completely unnecessary. Does restricting geometry to algebraic geometry and number theory replace the principle of action completely? This could be the case.

1. The vacuum functional exp(K), where the K hler function corresponds to the classical action , could be identified as the discriminant D associated with a polynomial. It would therefore be determined entirely by number theory as a product of differences of the roots of a polynomial P or in fact, of any analytic function. The problem is that the space-time surfaces are determined as roots of two analytic functions f₁ and f₂, rather than only one.
2. Could one define the 2-surfaces by allowing a third analytic function f_3 so that the roots of (f₁,f₂,f_3)=(0,0,0) would be 2-D surfaces. One can solve 3 complex coordinates CP₂ for each value of u as functions of the hypercomplex coordinate u whereas its dual remains free. One would have a string world sheet with a discrete set of roots for the 3 complex coordinates whose values depend on time. By adding a fourth function f_4 and substituting the 3 complex coordinates, f_4=0 would allow as roots values of the coordinate u. Only real roots would be allowed. A possible interpretation of these points of the space-time surface would be as loci of singularities at which the minimal surface property, i.e. holomorphy, fails.

   Note that for quadratic equations ax²+bx+c=0, the discriminant is D= b²-4ac and more generally the product of the differences of the roots. This formula also holds when f₁ and f₂ are not polynomials. The assumptions that some power of D corresponds to exp(K) and that K corresponds to the action imply additional conditions for the coupling constants appearing in the action , i.e. the coupling constant evolution.

3. This is not yet quite enough. The basic question concerns the construction of the interaction vertices for fermions. These vertices reduce to the analogs of gauge theory vertices in which induced fermion current assignable to the volume action is contracted with the induced gauge boson.

   The volume action is a unique choice in the sense that in this case the modified gamma matrices defined as contractions of the canonical momentum currents of the action with the gamma matrices of H reduce to induced gamma matrices, which anticommute to the induced metric. For a general action this is not the cae.

   The vertex for fermion pair creation corresponds to a defect of the standard smooth structure for the space-time surface and means that it becomes exotic smooth structure. These defects emerge in dimension D=4 and make it unique. In TGD, bosons are bound states of fermions and antifermions so that this also gives the vertices for the emission of bosons.

   For graviton emission one obtains an analogous vertex involving second fundamental form at the partonic orbit. The second fundamental form would have delta function singularity at the vertex and vanish elsewhere. If field equations are true also in the vertex, the action must contain an additional term, say Kähler action. Could the singularity of the second fundamental form correspond to the defect of the standard smooth structure?

4. If this view is correct, number theory and algebraic geometry combined with the geometric vision would make the notion of action un-necessary at the fundamental level. Geometrization of physics would be replaced by its algebraic geometrization. Action would however be a useful tool at the QFT limit of TGD.

See the article TGD as it is towards end of 2024: part II 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.