Critical summary of the problems associated with the physical interpretation of the number theoretical vision

The physical interpretation of the number theoretical vision involves several ideas and assumptions which can be criticized.

p-Adic primes and p-adic length scale hypothesis

The basic notions are p-adic length scale hypothesis and hierarchy of Planck constants and they are motivated by the empirical input. p-Adic length scale hypothesis was originally motivated by p-adic mass calculations and p-adic length scale hypothesis and has now developed to to a rather nice picture (see this) solving the original interpretational problem due to needed tachyonic ground states.

1. The proposed interpretation of p-adic primes has been as ramified primes. The identification of ramified primes is however far from obvious since they are assignable to polynomials of a single complex variable: how this polynomial is determined. There are also huge number of polynomials that one must consider and it seems that the notion of p-adic prime should characterize large class of polynomials and be therefore rather universal.
2. A more promising interpretation is in terms of functional primes, which under some assumptions are mappable to ordinary primes by a morphism. The maps f are primes is they correspond to irreducible polynomials or ratios of such polynomials and if it is not possible to express f as f= g_º h. f is characterized by at most 3 primes corresponding to the 3 complex coordinates of H. Also for g primeness can be defined and if only f₁ is involved, ordinary prime characterizes it.

There would be morphism mapping these functional primes to ordinary primes perhaps identifiable as p-adic primes. This could also fit with the p-adic length scale hypothesis suggesting the pairing of a large prime pl and small prime ps: plarge∼ psmallk would be true. One expects that g=(g1,g2) with g2 fixed is the physically motivated option and one assign primes pl near powers of small prime ps to functional primes gpsº k/gr. The hierarchy of Planck constants heff=nh0

Consider first the evidence.

1. The quantal effects of ELF em fields on the brain provide support for very large values of h_eff of order 10¹⁴ scaling the Compton length and giving rise to long scale quantum coherence. There is also evidence for small values of h_eff.
2. There is also evidence for the gravitational resp. electric Planck constants ℏ_gr resp. ℏ_em, which are proportional to the product of large and small mass resp. charge) and therefore depend on the quantum numbers for the interacting particles. This distinguishes these parameters from ordinary Planck constant and its possible analog h_eff. The support for ℏ_gr and ℏ_em emerges from numerical coincidences and success in explaining features of certain astrophysical systems and bio-systems.

The proposal is that ℏ_gr and ℏ_em emerge in Yangian symmetries which replace single particle symmetries with multi-local symmetries acting at the local level on several particles simultaneously. One should be able to formulate this idea in a more precise manner. The basic mathematical ideas are following.

1. The proposal is that the scaling L_p→ (h_eff,2/h_eff,1)L_p(h_eff,1) takes place in the the transition h_eff,1→ h_eff,2 and increases the scale of quantum coherence. One cannot exclude L_p→ (h_eff,2/h_eff,1)^1/2L_p as an alternative.
2. The number theoretical vision motivates the proposal that h_eff corresponds to the order of the Galois group of a polynomial. It is however far from clear how one can assign to the space-time surface this kind of polynomial and I have made several proposals and the situation is unclear.

There are two sectors to be considered corresponding to the dynamical symmetries defined by g and the prime maps f_P. Consider first the g sector.

1. For g=(g₁,Id), the situation reduces to that for a single polynomial and h_eff could correspond the order of the Galois group of g₁ would define the dimension of the corresponding algebraic extension. The motivation is that the condition f₂=0 would define TGD counterpart of dynamical cosmological constant.
2. The first proposal was that h_eff/h₀ corresponds to the number of space-time sheets for the space-time surface, which can be connected and indeed is so for f_P. This number is the order of the polynomial involved in a single variable case and is in general much smaller than the order n of the Galois group which for polynomials with degree d has maximal value d_max=d!.

If the Galois group is cyclic, one has n=d. Could the proposal that for functional primes, the coefficients pk appearing in gkº gpº k commute with gp and each other, imply this? This condition might be seen as a theoretical counterpart for the assumption that the Abelian Cartan algebra of the symmetry group defines the set of mutually commuting observables. Consider next the f sector.

1. For a prime map f_P=(f₁,f₂), P could correspond to 3 ordinary primes assignable to the 3 complex coordinates of H: f₁ and f₂ could be prime polynomials with respect to all these coordinates. Does this mean that 3 p-adic length scales are involved or is there some criterion selecting one p-adic length scale, say assignable to the M⁴ complex coordinate or to the hypercomplex coordinate u?
2. For a prime map f_P, the space-time surface as a root is connected. The original hypothesis would state that h_eff/h₀ corresponds to the number space-time regions representing roots of f_P rather than to the order of the generalized Galois group associated to the surface f_P=0 and permuting the roots as space-time regions to each other. Again the cyclicity of the generalized Galois group would guarantee the consistency of the two views. Now however the polynomials are ordinary polynomials obeying ordinary commutative arithmetics. But is there any need to assign h_eff to f_P? As far as applications are considered, g seems to be enough.
3. g_p^k_º f has p^k disjoint roots of g_k. f=(g_p^k/g_r)_º h has p^k roots and r poles as roots of g_r. Also these are disjoint so that functional primeness for g does not imply connectedness. Functional primeness for f would be required.

Does Mother Nature love her theoreticians?

The hypothesis that Mother Nature is theoretician friendly (see this) and this) involves quantum field theoretic thinking, which can be motivated in TGD by the assumption that the long length scale limit of TGD is approximately described by quantum field theory. What this principle states is the following.

1. When the quantum states are such that perturbative quantum field theory ceases to converge, a phase transition h_eff→ nh_eff occurs and reduces the value of the coupling strength α_K ∝ 1/ℏ_eff by factor a 1/n so that the perturbation theory converges. This can take place when the coupling constant defined by the product of charges involved is so large that convergence is lost or at least that unitarity fails. The phase transition gives rise to quantum states, which are Galois singlets for the larger Galois group.
2. The classical interpretation would be that the number of space-time surfaces as roots of g_1º f₁ increases by factor n, where n is the order of polynomial g₁. The total classical action should be unchanged. This is the case if at the criticality for the transition the n space-time surfaces are identical.

Can the transition take place in BSFR or even SSFR? Can one associate a smooth classical time evolution with f→ gpº kº f producing p copies of the original surface at each step such that the replacement αk → αK/p occurs at each step?

1. The transition should correspond to quantum criticality, which should have classical criticality in algebraic sense as a correlate. Polynomials xⁿ have x=0 as an n-fold degenerate root. In mathematics degenerate roots are regarded as separate. Now they would correspond to identical space-time surfaces on top of each other such that even an infinitesimal deformation can separate them. If the copies are identical at quantum criticality, a smooth evolution leading to an approximate n-multiple of a single space-time surface is possible. The action would be preserved approximately and the proposed scaling down of α_K would guarantee this.
2. The catastrophe theoretic analogy is the vertex of a cusp catastrophe. At the vertex of the cusp 3 roots coincide and at the V-shaped boundary of the plane projection of the cusp 2 roots coincide. More generally, the initial state should be quantum critical with p^k degenerate roots. In the simplest one would have p degenerate roots and p=2 and p=3 and their powers are favored empirically and by the very special cognitive properties of these options (the roots can be solved analytically). Also this suggest that Mother Nature loves theoreticians.
3. g₁(f₁)=f₁^p would satisfy the condition. An arbitrary small deformation of f₁^p by replacing it with a_kº f₁^kp would remove the degeneracy. The functional counterpart of the p-adic number would be +_e sum of g_1,k= a_kº f₁^kp as product ∏_k g₁k. Each power would correspond to its own almost critical space-time surface and a_k=1 would correspond to maximal criticality. This would correspond to the number ∑ p^k and one would obtain Mersenne primes and general versions for p>2 naturally from maximal criticality giving rise to functional p-adicity. The classical non-determinism due to criticality would correspond naturally to p-adic non-determinism.

To sum up, the situation concerning the relationship between number theoretic and geometric views of TGD looks rather satisfactory but there are many questions to be asked and answered. The understanding of M⁸-H duality as one aspect of the duality between number theory and geometry as analog of momentum-position duality generalizes from point-like particles to 3-surfaces is far from complete: one can even ask whether the M⁸ view produces more problems than it solves.

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/critical-summary-of-problems-associated.html