What could be the origin of p-adic length scale hypothesis?
The argument would explain the existence of preferred p-adic primes. It does not yet explain p-adic length scale hypothesis stating that p-adic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3-adic time scale hierarchies in living matter (see this) and in music both 2-adicity and 3-adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this).
The weak form of NMP might come in rescue here.
- Entanglement negentropy for a negentropic entanglement characterized by n-dimensional projection operator is the log(Np(n) for some p whose power divides n. The maximum negentropy is obtained if the power of p is the largest power of prime divisor of p, and this can be taken as definition of number theoretic entanglement negentropy. If the largest divisor is pk, one has N= k× log(p). The entanglement negentropy per entangled state is N/n=klog(p)/n and is maximal for n=pk. Hence powers of prime are favoured which means that p-adic length scale hierarchies with scales coming as powers of p are negentropically favored and should be generated by NMP. Note that n=pk would define a hierarchy of heff/h=pk. During the first years of heff hypothesis I believe that the preferred values obey heff=rk, r integer not far from r= 211. It seems that this belief was not totally wrong.
- If one accepts this argument, the remaining challenge is to explain why primes near powers of two (or more generally p) are favoured. n=2k gives large entanglement negentropy for the final state. Why primes p=n2= 2k-r would be favored? The reason could be following. n=2k corresponds to p=2, which corresponds to the lowest level in p-adic evolution since it is the simplest p-adic topology and farthest from the real topology and therefore gives the poorest cognitive representation of real preferred extremal as p-adic preferred extermal (Note that p=1 makes formally sense but for it the topology is discrete).
- Weak form of NMP suggests a more convincing explanation. The density matrix of the state to be reduced is a direct sum over contributions proportional to projection operators. Suppose that the projection operator with largest dimension has dimension n. Strong form of NMP would say that final state is characterized by n-dimensional projection operator. Weak form of NMP allows free will so that all dimensions n-k, k=0,1,...n-1 for final state projection operator are possible. 1-dimensional case corresponds to vanishing entanglement negentropy and ordinary state function reduction isolating the measured system from external world.
- The negentropy of the final state per state depends on the value of k. It is maximal if n-k is power of prime. For n=2k=Mk+1, where Mk is Mersenne prime n-1 gives the maximum negentropy and also maximal p-adic prime available so that this reduction is favoured by NMP. Mersenne primes would be indeed special. Also the primes n=2k-r near 2k produce large entanglement negentropy and would be favored by NMP.
- This argument suggests a generalization of p-adic length scale hypothesis so that p=2 can be replaced by any prime.
For details see the article The Origin of Preferred p-Adic Primes?.
For a summary of earlier postings see Links to the latest progress in TGD.