Separation of dynamics for cognitive and matter degrees of freedom and classical action as generalized discriminant

Holography= holomorphy hypothesis allows to solve field equations to purely algebraic equations. Space-time surfaces correspond to roots f=(f₁,f₂) for two analytic functions f_i: H=M^4× CP₂→ C². Analyticity means that the functions are analytic functions of one hypercomplex coordinate and complex coordinate of M⁴ and two complex coordinates of CP₂. The maps g:C₂→ C² are dynamical symmetries of the equations.

Interestingly, the dynamics associated with g and f separate in a well-defined sense. The roots of g_º f=0 are roots of g independently of f. This has an analogy in computer science. f is analogous to the substrate and g to the program. The assignment of correlates of cognition to the hierarchies of functional compositions of g is analogous to this principle but does not mean that conscious experience is substrate independent.

This suggests that the exponential of the classical action exponential is expressible as a product of action exponentials associated with f and g degrees of freedom. The proposal that the classical action exponential corresponds to a power of discriminant as the product of ramified primes for a suitably identified polynomial carries the essential information about polynomials and is therefore very attractive and could be kept. The action exponentials would in turn be expressible as suitable powers of discriminants defined by the roots of f=(f₁,f₂) resp. g=(g₁,g₂): D(f,g)= D(f)D(g).

How to define the discriminants D(f) an D(g)?

1. The starting point formula is the definition of D for the case of an ordinary polynomial of a single variable as a product of root differences D=∏_{i≠ j} (r_i-r_j).
2. How to generalize this? Restrict the consideration to the case f=(f₁,f₂). Now the roots are replaced with pairs (r_1,i|r2,j, r_2,j), where r_1,i|r2,j are the roots of f₁, when the root r_2,j of f₂ is fixed. For a fixed r_2,j, one can define discriminant D_1|r2,j using the usual product formula. The formula should be consistent with the strong correlation between the roots: the product of discriminants for f₁ and f₂ does not manifestly satisfy this condition. The discriminant should also vanish when two roots for f₁ or f₂ coincide.

3. The first guess for the discriminant for f₁,f₂ is as the product D_{1| 2}= ∏_{j≠ k} (r_2,jD_1|r2,j- r_2,kD_1|r2,k) . This formula is bilinear in the roots and has the required antisymmetries under the exchange of f₁ and f₂. The product differences of r_2,i-r_2,j do not appear explicitly in the expression. However, this expression vanishes when two roots of f₂ coincide, which is consistent with the symmetry under the exchange of f₁ and f₂. If this is not the case, the symmetry could be achieved by defining the discriminant as the product D_1,2 == D_{1| 2}D_2|1.

The action exponential should also carry information about the internal properties of the roots f=(r_1,j,r_2,j).

1. The assignment of action exponential, perhaps as a discriminant-like quantity, to each root f=(r_1,j,r_2,j) is non-trivial since the roots are now algebraic functions representing space-time regions the regions analogous to those associated with cusp catastrophe. The probably too naive guess is that the contribution to the action exponential is just 1: it would mean that this contribution to the action vanishes.
2. An alternative approach would require an identification of some special points in these regions of a natural coordinate as the dependent variable, say the hypercompex coordinate, as analog to the behavior variable of cusp. The problem is that this option is not a general coordinate invariant.
3. It would be nice if the proposed picture would generalize. The physical picture suggests that there is a dimensional hierarchy of surfaces with dimensions 4, 2, 0. The introduction of f₃ would allow us to identify 2-D string world sheets as roots of (f₁,f₂,f₃). The introduction of f₄ would make it possible to identify points of string world sheets as roots of (f₁,f₂,f₃,f₄) having interpretation as fermionic vertices. One could assign to these sets of these 2-surfaces and points discriminants in the way already described. The action exponential would involve the product of all these 3 discriminants. This would correspond to the assignment of action exponentials to these surfaces and also this would conform with the physical picture.
4. Locally, the analogs for the maps g for f₃ would be analytic general coordinate transformations mapping space-time surfaces to themselves locally.
   1. If they are 1-1, they give rise to a generalization of conformal invariance. If they are many-to-one or vice versa, they have a physical effect. The roots of g would be 2-surfaces. 2-D analogs of functional p-adics, of quantum criticality, etc... that I have assigned to elementary particles would be well defined notions and this would mean a justification of the physical picture behind the p-adic mass calculations involving string world sheets and partonic 2-surfaces.
   2. The conformal algebras in TGD have non-negative conformal weights and have an infinite fractal hierarchy of half -Lie algebras isomorphic to the entire algebra (see this). These algebras contain a finite-dimensional subalgebra transformed from a gauge algebra to a dynamical symmetry algebra. The interpretation in terms of the many-to-1 property of polynomial transformations g is natural. The action of symmetries on the pre-image of 2-surfaces as roots of g would affect all images simultaneously and would therefore be poly-local. Could the origin of the speculated Yangian symmetry (see this) be here? Could this relate to the gravitational resp. electric Planck constants which depend on the masses resp. charges of the interacting pair of systems.

See the article A more detailed view about the TGD counterpart of Langlands correspondence 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.