If the polynomials satisfy P(0)=0 requiring P(x)= xP1(x), the roots of P are inherited. In this case fixed points correspond to the points with P(x)=1. Assume also that the coefficients are rational. Monic polynomials are an especially interesting option.
For a k:th iterate of P, the mass squared spectrum is obtained as a union of spectra obtained as images of the spectrum of P under iterates P-r, r< k, for the inverse of P, which is an n-valued algebraic function if P has degree n. This set is a subset of Fatou set (see this) and for polynomials a subset of filled Julia set.
At the limit of large k, the limiting contributions to the spectrum approach a subset of Julia set defined as a P-invariant set which for polynomials is the boundary of the set for which the iteration divergences. The iteration of all roots except x=0 (massless particles) leads to the Julia set asymptotically.
All inverse iterates of the roots of P are algebraic numbers. The Julia set itself is expected to contain transcendental complex numbers. It is not clear whether the inverse iterates at the limit are algebraic numbers or transcendentals. For instance, one can ask whether they could consist of n-cycles for various values of n consisting of algebraic points and forming a dense subset of the Julia set. The fact that the number of roots is infinite at this limit, suggests that a dense subset is in question.
The basis properties of Julia set
The basic properties of Julia set deserve to be listed.
- At the real axis , the fixed points satisfying P(x)=x with |dP/dx|>1 are repellers and belong to the Julia set. In the complex plane, the definition of points of the Julia set is |P(w)-P(z)|> |w-z| for point w near to z.
- Julia set is the complement of the Fatou set consisting of domains. Each Fatou domain contains at least one critical point with dP/dz=0. At the real axis, this means that P has maximum or minimum. The iteration of P inside Fatou domain leads to a fixed point inside the Fatou set and inverse iteration to its boundary. The boundaries of Fatou domains combine to form the Julia set. In the case of polynomials, Fatou domains are labeled by the n-1 solutions of dP/dz= P1 +zdP1/dz=0.
- Julia set is a closure of infinitely many periodic repelling orbits. The limit of inverse iteration leads towards these orbits. These points are fixed points for powers Pn of P.
- For rational functions Julia set is the boundary of a set consisting of points whose iteration diverges to infinity. For polynomials Julia set is the boundary of the so-called filled Julia set consisting of points for which the iterate remains finite.
The critical points of P with dP/dz=0 for z= zcr located inside Fatou domains are analogous to point z=0 for Q(z) associated with Fatou domains and quadratic polynomial a+b(z-zcr)2, b>0, would serve as an approximation. The variation of a is determined by the variation of the coefficients of P required to leave zcr invariant.
Feigenbaum studied iteration of a polynomial a-x2 for which origin is unstable critical point and found that the variation of a leads to a period doubling sequence in which a sequence of 2n-cycles is generated (see this). Origin would correspond to an unstable critical point dP(z)/dz=0 belonging to a Julia set.
About physical implications
The physical implications of this picture are highly interesting.
- For a large number of interacting quarks, the mass squared spectrum of quarks as roots of the iterate of P in the interaction region would approach the Julia set as infinite inverse iterates of the roots of P. This conforms with the idea that the complexity increases with the particle number.
Galois confinement forces the mass squared spectrum to be integer valued when one uses as a unit the p-adic mass scale defined by the larger ramified prime for the iterate. The complexity manifests itself only as the increase of the microscopic states in interaction regions.
- Julia set contains a dense set consisting of repulsive n-cycles, which are fixed points of P and the natural expectation is that the mass spectrum decomposes into n-multiplets. Whether all values of n are allowed, is not clear to me. The limit of a large quark number would also mean an approach to (quantum) criticality.
There is a useful objection against this picture. M8-H duality requires the same momentum and mass spectrum for quarks in M8 and H. However, the mass spectrum of color partial waves for quark spinors in H is very simple and characterized by 2 integers labeling triality t=1 representations of SU(3) (see this). How can these pictures be consistent with each other? Do the quarks in M4 and H differ from each other and what does this mean?
To answer this question, one must ask what one means with quark in H and in M8.
- There are good reasons to assume that the quark spinor modes in H are annihilated only by the H d'Alembertian but not by the H Dirac operator (see this). This allows different M4 chiralities to propagate separately and solves problems related to the notion of right-handed neutrino νR and also conforms with the right and left-handed character of standard model couplings.
- Apart from νR all quark partial harmonics have CP2 mass scale and also the correlation between color and electroweak quantum numbers is wrong (see this). Therefore the physical quarks cannot correspond to the solutions of H spinor d'Alembertian.
The M4 Kähler structure forced by the twistor lift of TGD (see this) is part of the solution. It predicts that νR, if modeled as a mode of H spinor with Kähler coupling giving correct leptonic charges, has a tachyonic mass. The first guess is that the physical states contain an appropriate number of right handed neutrinos to build a tachyonic ground state from which one can construct a massless state. A more general approach allows roots of P with a negative real part as tachyonic virtual quarks. The virtual particles of standard QFT would correspond to quarks with masses coming as roots of P and they can also be tachyonic. Galois singlets would be analogous to on-mass shell particles.
- How to construct quark states which are physical in the sense that they are massless and color-electroweak correlation is correct? The reduction to quark masses to zero requires a tachyonic ground state in p-adic mass calculations (see this) . Also colored operators are required to make all quarks state color triplets.
The solution of the problem is provided by the identification of physical quarks as states of super-symplectic representations. Also the generalized Kac-Moody algebra assignable to the light-like partonic orbits or both of these representations can be considered. These representations could correspond to inertial and gravitational representations realized at "objective" embedding space level and "subjective" space-time level.
Supersymplectic generators are characterized by a conformal weight h completely analogous to mass squared. The conformal weights naturally correspond to algebraic integers associated with P. The mass squared values for the "physical" quarks are algebraic integers and Galois confinement forces integer-valued conformal weights for the physical states consisting of quarks. This conforms with the earlier picture about conformal confinement.
- These "physical" quarks constructed as states of super-symplectic representation, as opposed to modes of H spinor field, would correspond to quarks in M8. The complex momenta and mass squared values would be generated by supersymplectic generators with conformal weights h coming as algebraic integers associated with P. Most importantly, the modes of H-spinors would have integer-valued momenta and mass spectrum of the spinor d'Alembertian.
- Here several new concepts lend a hand. Galois confinement could solve the problems if one considers only Galois singlets as physical particles. ZEO replaces quantum states with entangled pairs of positive and negative energy states at the boundaries of CD and entanglement coefficients define transition amplitudes.
The notion of the unitary time evolution is replaced with the Kähler metric in quark degrees of freedom and its components correspond to transition amplitudes. The analog of the time evolution operator assignable to SSFRs corresponds naturally to a scaling rather than time translation and mass squared operator corresponds to an infinitesimal scaling.
- The complex eigenvalues of mass squared as roots of P be allowed when unitarity at quark level is not required to achieve probability conservation. For complex mass squared values, the entanglement coefficients for quarks would be proportional to mass squared exponents exp(im2λ), λ the scaling parameter analogous to the duration of time evolution. For Galois singlets these exponentials would sum up to imaginary ones so that probability conservation would hold true.
For a summary of earlier postings see Latest progress in TGD.