General number-theoretical ideas about coupling constant evolution

The discrete coupling constant evolution would be associated with the scale hierarchy for CDs and the hierarchy of extensions of rationals.

1. Discrete p-adic coupling constant evolution would naturally correspond to the dependence of coupling constants on the size of CD. For instance, I have considered a concrete but rather ad hoc proposal for the evolution of Kähler couplings strength based on the zeros of Riemann zeta (see this). Number theoretical universality suggests that the size scale of CD identified as the temporal distance between the tips of CD using suitable multiple of CP₂ length scale as a length unit is integer, call it l. The prime factors of the integer could correspond to preferred p-adic primes for given CD.

2. I have also proposed that the so called ramified primes of the extension of rationals correspond to the physically preferred primes. Ramification is algebraically analogous to criticality in the sense that two roots understood in very general sense co-incide at criticality. Could the primes appearing as factors of l be ramified primes of extension? This would give strong correlation between the algebraic extension and the size scale of CD.

In quantum field theories coupling constants depend in good approximation logarithmically on mass scale, which would be in the case of p-adic coupling constant evolution replaced with an integer n characterizing the size scale of CD or perhaps the collection of prime factors of n (note that one cannot exclude rational numbers as size scales). Coupling constant evolution could also depend on the size of extension of rationals characterized by its order and Galois group.

In both cases one expects approximate logarithmic dependence and the challenge is to define “number theoretic logarithm” as a rational number valued function making thus sense also for p-adic number fields as required by the number theoretical universality.

Consider first the coupling constant as a function of the length scale l_CD(n)/l_CD(1)=n.

1. The number π(n) of primes p≤ n behaves approximately as π(n)= n/log(n). This suggests the definition of what might be called “number theoretic logarithm” as Log(n)== n/π(n). Also iterated logarithms such log(log(x)) appearing in coupling constant evolution would have number theoretic generalization.

2. If the p-adic variant of Log(n) is mapped to its real counterpart by canonical identification involving the replacement p→ 1/p, the behavior can very different from the ordinary logarithm. Log(n) increases however very slowly so that in the generic case one can expect Log(n)

   max, where p_max is the largest prime factor of n, so that there would be no dependence on p for p_max and the image under canonical identification would be number theoretically universal.

   For n=p^k, where p is small prime the situation changes since Log(n) can be larger than small prime p. Primes p near primes powers of 2 and perhaps also primes near powers of 3 and 5 – at least – seem to be physically special. For instance, for Mersenne prime M_k=2^k-1 there would be dramatic change in the step M_k→ M_k+1=2^k, which might relate to its special physical role.

3. One can consider also the analog of Log(n) as

Log(n)= ∑_p k_pLog(p) ,

where p^k_i is a factor of n. Log(n) would be sum of number theoretic analogs for primes factors and carry information about them.

For p∈ {2,3,5} one has Log(p)>log(p), where for larger primes one has Log(p) Log(2)=2>log(2)=.693…, Log(3)= 3k/2> log(3)= 1.099, Log(5)= 5/3=1.666..>log(5)= 1.609. For p=7
one has Log(7)= 7/4≈ 1.75this).


In particular, for Mersenne primes M_k=2^k-1 one would have Log(M_k) ≈ k log(2) for large enough k. For Log(2^k) one would have k × Log(2)=2k>log(2^k)=klog(2): there would be sudden increase in the value of Log(n) at n=M_k. This jump in p-adic length scale evolution might relate to the very special physical role of Mersenne primes strongly suggested by p-adic mass calculations (see this).

Consider next the dependence on the extension of rationals. The natural algebraization of the problem is to consider the Galois group of the extension.

1. Consider first the counterparts of primes and prime factorization for groups. The counterparts of primes are simple groups, which do not have normal subgroups H satisfying gH=Hg implying invariance under automorphisms of G. Simple groups have no decomposition to a product of sub-groups. If the group has normal subgroup H, it can be decomposed to a product H× G/H and any finite group can be decomposed to a product of simple groups.

All simple finite groups have been classified (see this). There are cyclic groups, alternating groups, 16 families of simple groups of Lie type, 26 sporadic groups. This includes 20 quotients G/H by a normal subgroup of monster group and 6 groups which for some reason are referred to as pariahs.

If its possible to define N(G) in a natural manner then for given G one can define the number π₁(N(G)) of simple groups (analogs of primes) not larger than G. The first guess is that that the number π₁(N(G)) varies slowly as a function of G. Since Z_i is simple group, one has π₁(N(G)) ≥ π(N(G)).

a) Log₁(N(G))= N(G)/π₁(N(G)) ,

b) Log₁(G)= ∑_i k_i Log₁(N(G_i)) ,
Log₁(N(G_i)) = N(G_i)/π₁(N(G_i)) .

Option a) does not provide information about the decomposition of G to a product of simple factors. For Option b) one decomposes G to a product of simple groups G_i: G= ∏_i G_i^k_i and defines the logarithm as Option b) so that it carries information about the simple factors of G.

a) Log₁(ord(G))= ord(G)/π₁(ord(G))

b) Log₁(ord(G))= ∑_i k_i Log(ord(G_i)) , Log₁(ord(G_i)) = ord(G_i)/π₁(ord(G_i)) .

Besides groups with prime orders there are non-Abelian groups with non-prime orders. The occurrence of same order for two non-isomorphic finite simple groups is very rare (see this). This would suggests that one has π₁(ord(G)) 1(ord(G))/ord(G)<1 would be true.

See the article The Recent View about Twistorialization in TGD Framework or the chapter chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

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