- 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.
- 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 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.
Coupling constant evolution associated with the size scale of CD
Consider first the coupling constant as a function of the length scale lCD(n)/lCD(1)=n.
- 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.
- 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)<pmax, where pmax is the largest prime factor of n, so that there would be no dependence on p for pmax and the image under canonical identification would be number theoretically universal.
For n=pk, 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 Mk=2k-1 there would be dramatic change in the step Mk→ Mk+1=2k, which might relate to its special physical role.
- One can consider also the analog of Log(n) as
Log(n)= ∑p kpLog(p) ,
where pki is a factor of n. Log(n) would be sum of number theoretic analogs for primes factors and carry information about them.
One can extend the definition of Log(x) to the rational values x=m/n of the argument. The logarithm Logb(n) in base b=r/s can be defined as Logb(x)= Log(x)/Log(b).
- For p∈ {2,3,5} one has Log(p)>log(p), where for larger primes one has Log(p)<log(p). One has
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≈ 1.75<log(7)≈ 1.946. Hence these primes and CD size scales n involving large powers of p∈ {2,3,5} ought to be physically special as indeed conjectured on basis of p-adic calculations and some observations related to music and biological evolution (see this).
In particular, for Mersenne primes Mk=2k-1 one would have Log(Mk) ≈ k log(2) for large enough k. For Log(2k) one would have k × Log(2)=2k>log(2k)=klog(2): there would be sudden increase in the value of Log(n) at n=Mk. 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).
- One can wonder whether one could replace the log(p) appearing as a unit in p-adic negentropy with a rational unit Log(p)= p/π(p) to gain number theoretical universality? One could therefore interpret the p-adic negentropy as real or p-adic number for some prime. Interestingly, |Log(p)|p=1/p approaches zero for large primes p (eye cannot see itself!) whereas |Log(p)|q=1/|π(p)|q has large values for the prime power factors qr of π(p).
Consider next the dependence on the extension of rationals. The natural algebraization of the problem is to consider the Galois group of the extension.
- 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.
- Suppose that finite groups can be ordered so that one can assign number N(G) to group G. The roughest ordering criterion is based on ord(G). For given order ord(G)=n one has all groups, which are products of cyclic groups associated with prime factors of n plus products involving non-Abelian groups for which the order is not prime. N(G)>ord(G) thus holds true. For groups with the same order one should have additional ordering criteria, which could relate to the complexity of the group. The number of simple factors would serve as an additional ordering criterion.
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 Zi is simple group, one has π1(N(G)) ≥ π(N(G)).
- One can consider two definitions of number theoretic logarithm, call it Log1.
a) Log1(N(G))= N(G)/π1(N(G)) ,
b) Log1(G)= ∑i ki Log1(N(Gi)) ,
Log1(N(Gi)) = N(Gi)/π1(N(Gi)) .
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 Gi: G= ∏i Giki and defines the logarithm as Option b) so that it carries information about the simple factors of G.
- One could organize the groups with the same order to same equivalence class. In this case the above definitions would give
a) Log1(ord(G))= ord(G)/π1(ord(G)) < Log(ord(G)) ,
b) Log1(ord(G))= ∑i ki Log(ord(Gi)) , Log1(ord(Gi)) = ord(Gi)/π1(ord(Gi)) .
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)) <ord(G) so that Log1(ord(G))/ord(G)<1 would be true.
- For orders n(G)∈ {2,3,5} one has Log1(n(G))=Log(n(G))>log(n(G)) so that the ordes n(G) involving large factors of p∈ {2,3,5} would be special also for the extensions of rationals. S3 with order 6 is the first non-abelian simple group. One has π(S3)=4 giving Log(6)= 6/4=1.5<log(6)=1.79 so that S3 is different from the simple groups below it.
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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