The following represents an introduction to an article summarizing my recent understanding of p-adic length scale hypothesis and dark matter hierarchy. These considerations lead to more detailed proposals. In particular, a proposal for explicit form of dark scale is made.
p-Adic length scale hypothesis
In p-adic mass calculations real mass squared is obtained by so called canonical identification from p-adic valued mass squared identified as analog of thermodynamical mass squared using p-adic generelization of thermodynamics assuming super-conformal invariance and Kac-Moody algebras assignable to isometries ad holonomies of H=M4× CP2. This implies that the mass squared is essentially the expectation value of sum of scaling generators associated with various tensor factors of the representations for the direct sum of super-conformal algebras and if the number of factors is 5 one obtains rather predictive scenario since the p-adic temperature Tp must be inverse integer in order that the analogs of Boltzmann factors identified essentially as pL0/Tp.
The p-adic mass squared is of form Xp+O(p2) and mapped to X/p+ O(1/p2). For the p-adic primes assignable to elementary particles (M127=2127-1 for electron) the higher order corrections are in general extremely small unless the coefficient of second order contribution is larger integer of order p so that calculations are practically exact.
Elementary particles seem to correspond to p-adic primes near powers 2k. Corresponding p-adic length - and time scales would come as half-octaves of basic scale if all integers k are allowed. For odd values of k one would have octaves as analog for period doubling. In chaotic systems also the generalization of period doubling in which prime p=2 is replaced by some other small prime appear and there is indeed evidence for powers of p=3 (period tripling as approach to chaos). Many elementary particles and also hadron physics and electroweak physics seem to correspond to Mersenne primes and Gaussian Mersennes which are maximally near to powers of 2.
For given prime p also higher powers of p define p-adic length scales: for instance, for electron the secondary p-adic time scale is .1 seconds characterizing fundamental bio-rhythm. Quite generally, elementary particles would be accompanied by macroscopic length and time scales perhaps assignable to their magnetic bodies or causal diamonds (CDs) accompanying them.
This inspired p-adic length scale hypothesis stating the size scales of space-time surface correspond to primes near half-octaves of 2. The predictions of p-adic are exponentially sensitive to the value of k and their success gives strong support for p-adic length scale hypothesis. This hypothesis applied not only to elementary particle physics but also to biology and even astrophysics and cosmology. TGD Universe could be p-adic fractal.
Dark matter as phases of ordinary matter with heff=nh0
The identification of dark matter as phases of ordinary matter with effective Planck constant heff=nh0 is second key hypothesis of TGD. To be precise, these phases behave like dark matter and galactic dark matter could correspond to dark energy in TGD sense assignable to cosmic strings thickened to magnetic flux tubes.
There are good arguments in favor of the identification h=6h0. "Effective" means that the actual value of Planck constant is h0 but in many-sheeted space-time n counts the number of symmetry related space-time sheets defining space-time surface as a covering. Each sheet gives identical contribution to action and this implies that effective value of Planck constant is nh0.
M8-H duality (H=M4× CP2) has taken a central role in TGD framework. M8-H duality allows to identify space-time regions as "roots" of octonionic polynomials in complexified M8. The polynomial is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. One obtains brane-like 6-surfaces as 6-spheres as universal solutions. They have M4 projection which is piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=rn of P. For monic polynomials these time values are algebraic integers and Galois group permutes them.
M8-H duality allows to map space-time surfaces in M8 to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M8 and as minimal surfaces with 2-D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for super-symplectic algebra actings as isometries for the "world of classical worlds" (WCW). Twistor lift allows variants of this duality. M8H duality predicts that space-time surfaces form a hierarchy induced by the hierarchy of extensions of rationals defining an evolutionary hierarchy. This forms the basis for the number theoretical vision about TGD.
During the writing of this article I realized that M8-H duality has very nice interpretation in terms of symmetries. For H=M4× CP2 the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP2. For octonionic M8 the subgroup SU(3) ⊂ G2 is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M8-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M8= M2× E6. The subgroup of the holonomy group of SO(4) for E4 factor of M8= M4× E4 is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M8 one has symmetry breaking from SO(6) to SU(3) and from SO(4)= SU(2)× SO(3) to U(2).
This interpretation gives a justification for the earlier proposal that the descriptions provided by the old-fashioned low energy hadron physics assuming SU(2)L× SU(2)R and acting acting as covering group for isometries SO(4) of E4 and by high energy hadron physics relying on color group SU(3) are dual to each other.
Number theoretic origin of p-adic primes and dark matter
There are several questions to be answered. How to fuse real number based physics with various p-adic physics? How p-adic length scale hypothesis and dark matter hypothesis emerge from TGD?
The properties of p-adic number fields and the strange failure of complete non-determinism for p-adic differential equations led to the proposal that p-adic physics might serve as a correlate for cognition, imagination, and intention. This led to a development of number theoretic vision which I call adelic physics. A given adele corresponds to a fusion of reals and extensions of various p-adic number fields induced by a given extension of rationals.
The notion of space-time generalizes to a book like structure having real space-time surfaces and their p-adic counterparts as pages. The common points of pages defining is back correspond to points with coordinates in the extension of rationals considered. This discretization of space-time surface is in general finite and unique and is identified as what I call cognitive representation. The Galois group of extension becomes symmetry group in cognitive degrees of freedom. The ramified primes of extension are exceptionally interesting and are identified as preferred p-adic primes for the extension considered.
The basic challenge is to identify dark scale. There are some reasons to expect correlation between p-adic and dark scales which would mean that the dark scale would depend on ramified primes, which characterize roots of the polynomial defining the extensions and are thus not defined completely by extension alone. Same extension can be defined by many polynomials. The naive guess is that the scale is proportional to the dimension n of extension serving as a measure for algebraic complexity (there are also other measures). Dark p-adic length scales Lp would be proportional nLp, p ramified prime of extension? The motivation would be that quantum scales are typically proportional to Planck constant. It turns out that the identification of CD scale as dark scale is rather natural. p-Adic length scale hypothesis and dark matter hierarchy are discussed from number theoretic perspective. The new result is that M8 duality allows to relate p-adic length scales Lp to differences for the roots of the polynomial defining the extension defining "special moments in the life of self" assignable causal diamond (CD) central in zero energy ontology (ZEO). Hence p-adic length scale hypothesis emerges both from p-adic mass calculations and M8-H duality. It is proposed that the size scale of CD correspond to the largest dark scale nLp for the extension and that the sub-extensions of extensions could define hierarchy of sub-CDs.
See the chapter Zero Energy Ontology and Matrices or the article About p-adic length scale hypothesis and dark matter hierarchy.