A possible generalization of number theoretic approach to analytic functions
M8-H duality also allows the possibility that space-time surfaces in M8 are defined as roots of real analytic functions. This option will be considered in this section.
1. Are polynomials 4-surfaces only an approximation?
One of the open problems of the number-theoretic vision is whether the space-time surfaces associated with rational or even monic polynomials are an approximation or not.
- One could argue that the cognitive representations are only a universal discretization obtained by approximating the 4-surface in M8 by a polynomial. This discretization relies on an extension of rationals and more general than rational discretizations, which however appear via Galois confinement for the momenta of Galois singlets.
One objection against space-time surfaces as being determined by polynomials in M8 was that the resulting 4-surfaces in M8 would bre algebraic surfaces. There seems to be no hope about Fourier analysis. The problem disappeared with the realization that polynomials determine only the 3-surfaces as mass-shells of M4 and that M8-H duality is realized by an explicit formula subject to I(D)= exp(-K) condition.
That the quarks defining the active points of the representation are at 3-D mass shells would correspond to holography at the level of M8. At the level of H they would be at the boundaries of CD. This would explain why we experience the world as 3-dimensional. Also the 4-surfaces containing quark mass shells defined in terms of roots of arbitrary real analytic functions are possible.
What about p-adicization requiring the definition discriminant D and identification of the ramified primes and maximal ramified prime? Under what conditions do these notions generalize?
If the roots appear as complex conjugate pairs, D is real. This is guaranteed if one has f(z*) = f(z)*. The real analyticity of f guarantees this and is necessary in the case of polynomials. A stronger condition would be that the parameters such as Taylor coefficients are rational.
If the roots are rationals, the discriminant is a rational number and one can identify ramified primes and p-adic prime if the number of zeros is finite.
If one can assign ramified primes to D, it is possible to interpret D as a p-adic number having a finite real counterpart in canonical identification. For instance, if the roots of zeta are rationals, this could make sense. 2. Questions related to the emergence of mathematical consciousness
These considerations inspire further questions about the emergence of mathematical consciousness.
- Could some mathematical entities such as analytic functions have a direct realization in terms of space-time surfaces? Could cognitive processes be identified as a formation of functional composites of analytic functions? They would be analogs of particle reactions in which the incoming particles consist of quarks, which are associated with mass-shells defined by the roots of analytic function.
These composites would decay to products of polynomials in cognitive measurements involving a cascade of SSFRs reducing the entanglement between a relative Galois group and corresponding normal group acting as Galois group of rationals (see this).
Or could one think that the normal sub-group hierarchies formed by Galois groups actually give rise to hierarchies of states, which are Galois confined for an extension of the Galois group.
Could these higher levels relate to the emergence of consciousness about algebraic numbers. Could one extend computationalism allow also extensions of rationals and algebraic integers as discussed here).
Galois confinement for an extension of rationals would be analogous to the replacement of a description in terms of hadrons with that in terms of quarks and mean increase of cognitive resolution. Also Galois confinement could be generalized to its quantum version. One could have many quark states for which wave function in the space of total momenta is Galois singlet whereas total momenta are algebraic integers. S-wave states of a hydrogen atom define an obvious analog.
Several physical interpretations of Riemann zeta have been proposed. Zeta has been associated with chaotic systems, and the interpretation of the imaginary parts of the roots of zeta as energies has been considered. Also an interpretation as a formal analog of a partition function has been considered. The interpretation as a scattering amplitude was considered by Grant Remmen (see this).
3.1. Conformal confinement as Galois confinement for polynomials?
TGD suggests a totally different kind of approach in the attempts to understand Riemann Zeta. The basic notion is conformal confinement.
- The proposal is that the zeros of zeta correspond to complex conformal weights sn=1/2+iyn. Physical states should be conformally confined meaning that the total conformal weight as the sum of conformal weights for a composite particle is real so that the state would have integer value conformal weight n, which is indeed natural. Also the trivial roots of zeta with s=-2n, n>0, could be considered.
- In M8-H duality, the 4-surfaces X4⊂ M8 correspond to roots of polynomials P. M8 has an interpretation as an analog of momentum space. The 4-surface involves mass shells m2= rn, where rn is the root of the polynomial P, algebraic complex number in general.
The 4-surface goes through these 3-D mass-shells having M4 ⊂ M8 as a common real projection. The 4-surface is fixed from the condition that it defines M8-H duality mapping it to M4× CP2. One can think X4 as a deformation of M4 by a local SU(3) element such that the image points are U(2) invariant and therefore define a point of CP2. SU(3) has an interpretation as octonionic automorphism.
Mass squared in the stringy picture corresponds to conformal weight so that the mass squared values for quarks are analogous to conformal weights and the total conformal weight is integer by Galois confinement. 3.2. Conformal confinement for zeta functions
At least formally, TGD also allows a generalization of real polynomials to analytic functions. For a generic analytic function it is not possible to find superpositions of roots that would be integers and this could select Riemann Zeta and possible other analytic functions are those with infinite number of roots since they might allow a large number of bound states and be therefore winners in the number theoretic selection.
Riemann zeta is a highly interesting analytic function in this respect.
- Actually an infinite hierarchy of zeta functions, one for any extension of rationals and conjectured to have zeros at the critical line, can be considered. Could one regard these zetas as analogous to polynomials with an infinite degree so that the allowed mass squared values for quarks would correspond to the roots of zeta?
- Conformal confinement requires integer valued momentum components and total conformal weights as mass squared values. The fact that the roots of zetas appear as complex conjugates allows for a very large number of states with real conformal weights. This is however not enough. The fact that the roots are of the form zn= 1/2+iyn or z=-2n implies that the conformal weights of Galois/conformal singlets are integer-valued and the spectrum is the same as in conformal field theories.
- Riemann zeta has only a single pole at s=1. Discriminant would be the product ∏m≠ n (ym-yn2) ∏m≠ n 4(m-n)2 ∏m,n(4m2+yn2) since the pole gives D=1. D would be infinite.
- Fermionic zeta ζF(s)= ζ(s)/ζ(2s) is analogous to the partition function for fermionic statistics and looks more appropriate in the case of quarks. In this case, the zeros are zn resp. zn/2 and the ratio of determinants would reduce to an infinite power of 2. The ramified prime would be the smallest possible: p=2! D= D1/D2 would be infinite power of 2 and 2-adically zero so that exp(-K) should vanish and Kähler function would diverge. 3-adically it would be infinite power of -1. If one can say that the number of roots is even, one has D=1 3-adically. Kähler function would be equal to zero, which is in principle possible.
For Mersenne primes Mn=2n-1, 2n would be equal to 1+ Mn=1 mod Mn and one would obtain an infinite power 1+Mn, which is equal to 1 mod Mn. Could this relate to the special role of Mersenne primes?
Elliptic functions (see this) provide examples of analytic functions with infinite number of roots forming a doubly periodic lattice and are therefore cood candidates for analogs of polynomials with infinite degree.
- Elliptic functions are doubly periodic and characterized by the ratio τ of complex periods ω1 and ω2. One can assume the convention ω1=1 giving ω2=τ. The roots of the elliptic function for an infinite lattice and complex rational roots are of obvious interest concerning the generalization of Galois/conformal confinement.
- The fundamental set of zeros is associated with a cell of this lattice. The finite number of zeros (with zero with multiplicity m counted as m zeros) in the cell is the same as the number poles and characterizes partially the elliptic function besides τ.
- Weierstrass P-function and its derivative dP/dz are the building blocks of elliptic functions. A general elliptic function is a rational function of P and dP/dz. In even elliptic functions only the even funktion P appears.
- The roots of Weierstrass P-function P(z)= ∑λ 1/(z-λ)2 appear in pairs +/- z whereas the double poles at at the points of the modular lattice (see the article “The zeros of the Weierstrass P-function and hypergeometric series” of Duke and Imamoglu (see this).
The roots are given by Eichler-Zagier formula z+/-(m,n) =1/2+ m+nτ +/- z1, where z1 contains an imaginary transcendental part log(5+2×61/2)/2π) plus second part, which depends on τ (see formula 6) of the article.
As a matter of fact, already at the level of M8, M4 Kähler structure implies that right-handed neutrino νR is a tachyon (see this). However, νR provides the tachyon needed to construct massless super-symplectic ground states and also allows us to understand why neutrinos can be massive although right-handed neutrinos are not detected. The point is that only the square of Dirac equation in H is satisfied so that different M4 chiralities can propagate independently.
In M8-H duality, non-tachyonicity makes it possible to map the momenta at mass shell to the boundary of CD in H. Hence the natural condition would be that the total conformal weight of a physical state is non-negative. What about the notion of discriminant and ramified prime? One can assign to the algebraic extensions primes as prime ideals for algebraic integers and this suggests that the generalization of p-adicity and p-adic prime is possible. If this is the case also for transcendental extensions, it would be possible to define transcendental p-adicity.
One can however ask whether the discriminant is rational under some conditions. D could also allow factorization to the primes of the transcendental extension.
- Elliptic functions are meromorphic and have the same number of poles and zeros in the basic cell so that there are some hopes that the ratio of discriminants is finite and even rational or integer for a suitable choice of the modular parameter τ as the ratio of the periods and the other parameters. Discriminant D as the ratio D1/D2 of the discriminants defined by the products of differences of roots and poles could be finite although they diverge separately.
- For the Weierstrass P-function, the zeros appear as pairs +/- z0 and also as complex conjugate pairs. Complex pairs are required by real analyticity essential for the number theoretical vision. It might be possible to define the notion of ramified prime under some assumptions.
For z+(m,n) or z-(m,n), D1 in D1/D2 would reduce to a product ∏m,n Δm,n)2 (Δm,n +2z1) (Δm,n -2z1), Δm,n =Δ m+Δ nτ, which is a complex integer valued if τ has integer components. D1 would be a product of Gaussian integers.
The double poles of P(z)= ∑λ 1/(z-λ)2 are at points of the lattice. One has D2=∏m,n Δm,n)4. This gives
D= D1/D2=∏m,n(1 +2z0/Δm,n) (1 -2z0/Δm,n)= ∏m,n(1-8z0/Δm,n2) .
Therefore D is finite. z0 contains a transcendental constant term plus a term depending on τ (see this). The existence of values of τ for which D is rational, seems plausible. In the number theoretic vision, the construction of many-particle states corresponds to the formation of functional composites of polynomials P. If the condition P(0)=0 is satisfied, the n-fold composite inherits the roots of n-1-fold composites and the roots are like conserved genes. If one multiplies zeta functions and elliptic functions by z, one obtains similar families and the formation of composites gives rise to iteration sequences and approach to chaos (see this).
See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
Articles and other material related to TGD.
Source: http://matpitka.blogspot.com/2022/01/a-possible-generalization-of-number.html
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