Infinite primes, the notion of rational prime, and holography= holomorphy principle

The notion of infinite prime cite{allb/visionc,infpc,infmotives} emerged a repeated quantization of a supersymmetric arithmetic quantum field theory in which the many-fermion states and many-boson formed from the single particle states at a given level give rise to free many-particle states at the next level. Also bound states of these states are included at the new level. There is a correspondence with rational functions as ratios R=P/Q of polynomials and infinite prime can be interpreted as prime rational function in the sense that P and Q have no common factors. The construction is possible for any coefficient field of polynomials identified as rationals or extension of rationals, call it E.

At a given level implest polynomials P and Q are products of monomials with roots in E, say rationals. Irreducible polynomials correspond to products of monomials with algebraic roots in the corresponding extension of rationals and define the counterparts of bound states so that the notion of bound state would be purely number theoretic. The level of the hierarchy would be characterized by the number of variables of the rational functions.

Holography= holomorphy principle suggests that the hierarchy of infinite primes could be used to construct the functions f₁: H→ C and f₂:H→ C defining space-time surfaces as roots f=(f₁,f₂). There is one hypercomplex coordinate and 3 complex coordinates so that the hierarchy for f_i would have 4 levels. The functions g:C²→ C² define a hierarchy of maps with respect to the functional composition _º. One can identify the counterparts of primes with respect to _º and it turns out that the notion of infinite prime generalizes.

The construction of infinite primes

Consider first the construction of infinite primes.

1. Two integers with no common prime factors define a rational r=m/n uniquely. Introduce the analog of Fermi sea as the product X = ∏_p p of all rational primes. Infinite primes is obtain as P= nX/r+ mr such that m=∏p_k is a product for finite number of primes p_k, n is not divisible by any p_k, and m has as factors powers of some of primes p_k. The finite and infinite parts of infinite prime correspond to the numerator and denominator of a rational n/m so that rationals and infinite primes can be identified. One can say that the rational for which n and m have no common factors is prime in this sense.

One can interpret the primes p_k dividing r as labels of fermions and r as fermions kicked out from the Fermi sea defined by X. The integers n and m as analogs of many-boson states. This construction generalizes also to athe algebraic extensions E of rationals.

One can generalize the construction to the second level of the hierarchy. At the second level one introduces fermionic vacuum Y as a product of all finite and infinite primes at the first level. One can repeat the construction and now integers r,m and n are products of the monomials P(m/n,X)= nX/r+mr represented as infinite integers and . The analog of r from the new fermionic vacuum away some fermions represented by infinite primes P(m/n,X)= nX/r+mr by kicking them out of the vacuum. The infinite integers at the second level are analogous to rational functions P/Q with the polynomials P and Q defined as the products of ratio of the monomials p(m/n,X)= X/r+mr taking the role of n and m. These polynomials are not irreducible.

One can however generalize and assume that they factor to monomials associated with the roots of some irreducible polynomial P (no rational roots) in some extension E of rationals. Hence also rational functions R(X)= P(X)/Q(X) with no common monomial factors as analogs of primes defining the analogs of primes for rational functions emerge. The lowest level with rational roots would correspond to free many-fermion states and the irreducible polynomials to a hierarchy of fermionic bound states.

The construction can be continued and one obtains an infinite hierarchy of infinite primes represented as rational functions R(X₁,X₂,..X_n)= P(X₁,X₂,..X_n)/Q(X₁,X₂,..X_n) which have no common prime factors of level n-1. At the second level the polynomials are P(X,Y)= ∑_k P_nk(X)Y^k. The roots Y_k of P(X,Y) are obtained as ordinary roots of a polynomials with coefficients P_nk(X) depending on X and they define the factorization of P to monomials. At the third level the coefficients are irreducible polynomials depending on X and Y and the roots of Z are algebraic functions of X and Y.

Physically this construction is analogous to a repeated second quantization of a number theoretic quantum field theory with bosons and fermions labelled/represented by primes. The simplest states at a given level of free many-particle states and bound states correspond to irreducible polynomials. The notion of free state depends on the extension E of rationals used.

Infinite primes and holography= holomorphy principle

How does this relate to holography= holomorphy principle? One can consider two options for what the hierarchy of infinite prime could correspond to.

1. One considers functions f=(f₁,f₂): H→ C², with f_i expressed in terms of rational functions of 3 complex coordinates and one hyperbolic coordinate. The general hypothesis is that the function pairs (f₁,f₂) defining the space-time surfaces as their roots (f₁,f₂)=(0,0) are analytic functions of generalized complex coordinates of H with coefficients in some extension E of rationals.
2. Now one has a pair of functions: (f₁,f₂) or (g₁,g₂) but infinite primes involve only a single function. One can solve the problem by using element-wise sum and product so that both factors would correspond to a hierarchy of infinite primes.
3. One can also assign space-time surfaces to polynomial pairs (P₁,P₂) and also to pairs rational functions (R₁,R₂). One can therefore restrict the consideration to f₁equiv f. f₂ can be treated in the same way but there are some physical motivations to ask whether f₂ could define the counterpart of cosmological constant and therefore could be more or less fixed in a given scale.

The allowance of rational functions forces us to ask whether zeros are enough or whether also poles needed?

1. Hitherto it has been assumed that only the roots f=0 matter. If one allows rational functions P/Q then also the poles, identifiable as roots of Q are important. The compactification of the complex plane to Riemann-sphere CP₁ is carried out in complex analysis so that the poles have a geometric interpretation: zeros correspond to say North Pole and poles to the South pole for the map of C→ C interpreted as map CP₁→ CP₁. Compactication would mean now to the compactification C²→ CP₁².

For instance, the Riemann-Roch theorem (see this) is a statement about the properties of zeros and poles of meromorphic functions defined at Riemann surfaces. The so called divisor is a representation for the poles and zeros as a formal sum over them. For instance, for meromorphic functions at a sphere the numbers of zeros and poles, with multiplicity taken into account, are the same.

The notion of the divisor would generalize to the level of space-time surfaces so that a divisor would be a union of space-time surfaces representing zero and poles of P and Q? Note that the iversion f_i→ 1/f_i maps zeros and poles to each other. It can be performed for f₁ and f₂ separately and the obvious question concerns the physical interpretation.

Infinite primes would thus correspond to rational functions R= P/Q of several variables. In the recent case, one has one hypercomplex coordinate u, one complex coordinate w of M⁴, and 2 complex coordinates ξ₁,ξ₂ of CP₂. They would correspond to the coordinates X_i and the hierarchy of infinite primes would have 4 levels. The order of the coordinates does not affect the rational function R(u,w,ξ₂,ξ₂) but the hypercomplex coordinate is naturally the first one. It seems that the order of complex coordinates depends on the space-time region since not all complex coordinates can be solved in terms of the remaining coordinates. It can even happen that the coordinate does not appear in P or Q.

The hypercomplex coordinate u is in a special position and one can ask whether rational functions for it are sensical. Trigonometric functions and Fourier analysis look more natural.

What could be the physical relationship between the space-time surfaces representing poles and zeros?

Could zeros and poles relate to ZEO and the time reversal occurring in “big” state function reduction (BSFR)? Could the time reversal change zero to poles and vice versa and correspond to f_i→ 1/f_i inducing P/Q → Q/P? Are both zeros and poles present for a given arrow of time or only for one arrow of time? One can also ask whether complex conjugation could be involved with the time reversal occurring in BSFR (it would not be the same as time reflection T).

For a meromorphic function, the numbers of poles and zeros are in a well-defined sense so that the numbers of corresponding space-time surfaces are the samel. What could this mean physically? Could this relate to the conservation of fermion numbers? There would be two conserved fermion numbers corresponding to f₁ and f₂. Could they correspond to baryon and lepton number.

P and Q would have no common polynomial (prime) factors. The zeros and poles of R as zeros of P and Q are represented as space-time surfaces. Could the zeros and poles correspond to matter and antimatter so that memomorphy would state that the numbers of particles and antiparticles are the same? Or do they correspond to the two fermionic vacuums assigned to the boundaries of CD such that the vacuum associated with the passive boundary is what corresponds to quantum states in 3-D sense.

Could infinite primes could have two representations. A representation as space-time surfaces in terms of holography= holomorphy principle and as fermion states involving a 4-levelled hierarchy of second quantizations for both quarks and leptons. What these 4 quantizations could mean physically?

Can the space-time surfaces defined by zeros and poles intersect each other? If BSFR permutes the two kinds of space-time surfaces, they should intersect at 3-surfaces defining holographic data. The failure of the exact classical determinism implies that the 4-surfaces are not identical.

Hierarchies of functional composites of g: C²→ C²

One can consider also rational functions g=(g₁,g₂) with g_i=R=P_i/Q_i: C²→ C² defining abstraction hierarchies. Also in this case elementwise product is possible but functional composition _º and the interpretation in terms of formation of abstractions looks more natural. Fractals are obtained as a special case. _º is not commutative and it is not clear whether the analogs of primes, prime decomposition, and the definition of rational functions exist.

1. Prime decompositions for g with respect to _º make sense and can identify polynomials f=(f₁,f₂) which are primes in the sense that they do not allow composition with g. These primal spacetime surfaces define the analogs of ground states.
2. The notion of generalized rational makes sense. For ordinary infinite primes represented as P/Q, the polynomials P and Q do not have common prime polynomial factors. Now / is replaced with a functional division (f,g)→ f_º g^-1 instead of (f,g)→ f/g. In general, g^-1 is a many-valued algebraic function. In the one-variable case for polynomials the inverse involves algebraic functions appearing in the expressions of the roots of the polynomial. This means a considerable generalization of the notion of infinite prime.
3. One obtains the counterpart for the hierarchy of infinite primes. The analog for the product of infinite primes at a given level is the composite of prime g:s. The irreducible polynomials as realization of bound states for ordinary infinite primes replaces the coefficient field E with its extension. The replacement of the rationals as a coefficient field with its extensions E does the same for the composes of g:s. This gives a hierarchy similar to that of irreducible polynomials: now the hierarchy formed by rational functions with increasing number of variables corresponds to the hierarchy of extensions of rationals.
4. The conditions for zeros and poles are not affected since they reduce to corresponding conditions for g_º f.

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.

Source: https://matpitka.blogspot.com/2025/04/infinite-primes-notion-of-rational.html