A possible generalization of number theoretic approach to analytic functions

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. 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.

1. 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.

2. Galois confinement provides a universal mechanism for the formation of bound states. Could evolution be a development of real states for which cognitive representations in terms of quarks become increasingly precise.

   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.

1. Analytic functions could be defined in terms of Taylor or Laurent series. In fact, any representation can be considered. Also now one can consider representation involving only integers, rationals, algebraic numbers, and even reals as parameters playing a role of Taylor coefficients.

2. Does the notion of algebraic integers generalize? The roots of the holomorphic functions defining the meromorphic functions as their ratios define an extension of rationals, which is in the general transcendental. It is plausible that the notion of algebraic integers generalizes and one can assume that quarks have momentum components, which are transcendental integers. One can also define the transcendental analog of Galois confinement.

3. One can form functional composites to construct scattering amplitudes and this would make possible particle reactions between particles characterized by analytic functions. Iteration of analytic functions and approach to chaos would emerge as the functions involved appear very many times as one expects in case of IR photons and gravitons.

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?

1. One can start from rational functions. In the case of rational functions R, one can generalize the notion of discriminant and define it as a ratio D= D₁/D₂ of discriminants D₁ and D₂ for the polynomials appearing as a numerator and denominator of R. The value of D is finite irrespective of the values of the degrees of polynomials.

2. Analytic functions define function fields. Could a generalization of discriminant exist. If the analytic function is holomorphic, it has no poles so that D could be defined as the product of squares of root differences.

   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.

3. Meromorphic functions are ratios of two holomorphic functions. If the numbers of zeros are finite, the ratio of the discriminants associated with the numerator and denominator is finite and rational under the same assumptions as for holomorphic functions.

4. M⁸-H duality leads to the proposal that the discriminant interpreted as a p-adic number for p-adic prime defined by the largest ramified prime, is equal to the exponent of exp(-K) of Kähler function for the space-time surface in H.

   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.

1. 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).

2. Could the basic restriction to cognition come from the Galois confinement: momenta of composite states must be integers using p-adic mass scale as a unit.

   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.

3. During the last centuries the evolution of mathematical consciousness has made huge steps due to the discovery of various mathematical concepts. Essentially a transformation of rational arithmetics with real analysis and calculus has taken place since the times of Newton. Could these evolutionary explosions correspond to the emergence of space-time surfaces defined by analytic functions or is it that only a conscious awareness about their existence has emerged?

3. Space-time surfaces defined by zeta functions and elliptic functions

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.

1. The proposal is that the zeros of zeta correspond to complex conformal weights s_n=1/2+iy_n. 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.
2. 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= r_n, where r_n 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.

3. Galois confinement states that physical states as many-quark states with quark momenta as algebraic integers in the extension defined by the polynomial have integer valued momentum components in the scale defined by the causal diamond also fixed by the p-adic prime identified as the largest ramified prime associated with the discriminant D of P.

   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.

1. 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?
2. 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 z_n= 1/2+iy_n 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.
3. Riemann zeta has only a single pole at s=1. Discriminant would be the product ∏_{m≠ n} (y_m-y_n²) ∏_{m≠ n} 4(m-n)² ∏_m,n(4m²+y_n²) since the pole gives D=1. D would be infinite.
4. Fermionic zeta ζ_F(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 z_n resp. z_n/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 M_n=2ⁿ-1, 2ⁿ would be equal to 1+ M_n=1 mod M_n and one would obtain an infinite power 1+M_n, which is equal to 1 mod M_n. Could this relate to the special role of Mersenne primes?

5. What about Riemann Hypothesis? By ζ(s^*)= ζ^*(s), the zeros of zeta appear in complex conjugate pairs. By functional equation, also s and 1-s are zeros. Suppose that there is a zero s₊= s₀+iy_n with s₀≠ 1/2 in the interval (0,1). This is accompanied by zeros s^*₊, 1-s₊, s_-= 1-s^*₊. The sum of these four zeros is equal to s=2. Therefore Galois singlet property does not allow us to say anything about the Riemann hypothesis.

3.3 Conformal confinement for elliptic functions

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.

1. 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.
2. 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.
3. 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.
4. 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+2×6^1/2)/2π) plus second part, which depends on τ (see formula 6) of the article.

5. Conformally confined states with conformal weights h= 1+(m₁+m₂)+(n₁+n₂)τ can be realized as pairs with conformal weights (z₊(m₁,n₁),z_-(m₂,n₂). The condition n₁=-n₂ guarantees integer-valued conformal weights and conformal confinement for a general value of τ.
6. A possible problem is that the total conformal weights can be also negative, which means tachyonicity. This is not a problem also in the case of Riemann zeta if trivial zeros are included.

   As a matter of fact, already at the level of M⁸, M⁴ Kähler structure implies that right-handed neutrino ν_R is a tachyon (see this). However, ν_R 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.

1. 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.
2. 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 Δ_m,n)² (Δ_m,n +2z₁) (Δ_m,n -2z₁), Δ_m,n =Δ m+Δ nτ, which is a complex integer valued if τ has integer components. D₁ would be a product of Gaussian integers.

3. The number of poles and zeros for the basic cell is the same so that D₂ as a product of the pole differences would have an identical general form. For large values of m,n, the factors in the product approach Δ_m,n for both zeros and poles so that the corresponding factors combine to a factor approaching unity.

   The double poles of P(z)= ∑_λ 1/(z-λ)² are at points of the lattice. One has D₂=∏_m,n Δ_m,n)⁴. This gives

   D= D₁/D₂=∏_m,n(1 +2z₀/Δ_m,n) (1 -2z₀/Δ_m,n)= ∏_m,n(1-8z₀/Δ_m,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.