About p-adic length scale hypothesis and dark matter hierarchy

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 T_p must be inverse integer in order that the analogs of Boltzmann factors identified essentially as p^L₀/T_p.

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 2^k. 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 h_eff=nh₀

The identification of dark matter as phases of ordinary matter with effective Planck constant h_eff=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

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=r_n 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. M⁸_H 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 L_p would be proportional nL_p, 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. In the article 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 L_p 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 nL_p for the extension and that the sub-extensions of extensions could define hierarchy of sub-CDs.

See the article About p-adic length scale hypothesis and dark matter hierarchy.

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

Articles and other material related to TGD.

Rejuvenation and zero energy ontology

Biologist Harold Katcher (see this) claims that the epigenetic age (there are several measures for it such as methylation level of DNA) of rats has been reduced up to 50 percent. The theory goes that epigenetic age of molecules would be controllable by hormonal signalling globally.

I have been just working with the view about state function reduction in zero energy ontology of TGD providing a theory of quantum measurement free of its basic paradox and having profound implications in the understanding of mysteries of life and death.

For ordinary "big" state function reductions (BSFRs) the arrow of time changes. BSFR would mean death of conscious entity and its reincarnation with opposite arrow of time. The system would rejuvenate in the transition starting a new life in opposite time direction from childhood so to say- rejuvenation would be in question. Doing this twice would lead to life with original arrow of time but starting in rejuvenated state.

Returning cells to the stem cell state inducing de-differentiation as reversal of differentiation would be one example of rejuvenation. The claims of the group suggests that living matter is doing this systematically using hormonal control.

See the article When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?.

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

Articles and other material related to TGD.

When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?

In ZEO based view about quantum measurement theory as theory of consciousness one has two kinds of state function reductions (SFRs) ( see this and this). The ordinary "big" SFRs (BSFRs) and "small" SFRs (SSFRs) (see this). BSFR changes the arrow of geometric time and is identified as death of self identified as a sequence of SSFRs, which do not change arrow of time but increase the size of self by keeping passive boundary in place and states at it unaffected but increasing the size of CD by shifting the upper boundary towards future. Both boundaries increase in size. The 3-surfaces at the active boundary form a kind of log file about events in the life of self and - contrary to expectations - the memories are stored to geometric future.

Under what conditions does "big state function reduction (BSFR) changing the arrow of time take place? I have proposed several ad hoc guesses about this. One example is following. If the h_eff=n× h₀ assignable to the CD or its active boundary does not change in SSFRs, the entanglement can become such that the diagonalized density matrices does not have eigenvalues in the extension of rationals considered and one can argue that BSFR is forced to occur. The proposal for how the sequence of SSFR could in special case correspond to a sequence of iterations for a polynomial of degree n (see this) is however in conflict with the constancy of n.

The hypothesis is that BSFR corresponds to the death of self followed by re-incarnation with opposite arrow of geometric time in universal sense. This suggests that one should look what one can learn from what happens in the death and birth of biological organism, which should now take in opposite arrow of time.

1. Death certainly occurs if there is no metabolic energy feed to the system. Metabolic energy feed is guaranteed by nutrition using basic molecules as metabolites. Since the increase of h_eff quite generally requires energy if other parameters are kept constant and since the reduction of h_eff can take spontaneously, the metabolic energy is needed to keep the distribution of values of h_eff stationary or even increase it - at least during the growth of organism and perhaps also during the mature age when it would go to increase of h_eff at MB.
   

   If the size of CD for at least MB correlates with the maximum value of h_eff or its average, the size of CD cannot grow and can be even reduced if the metabolic energy feed is too low. The starving organism withers and its mental abilities are reduced. This could correspond to the reduction of maximum/average value of h_eff and also size of CD.
   

   One can argue that if the organism loses metabolic energy feed or is not able to utilize the metabolic energy death and therefore also BSFR must take place.
   
   

2. In ZEO self-organization reduces to the second law in reversed direction of geometric time at the level of MB inducing effective change of arrow of time at the level of biological body (see this)). The necessary energy feed correspond to dissipation of energy in opposite time direction. In biological matter energy feed means its extraction from the metabolites fed to the system. One could say that system sends negative energy to the systems able to receive it. A more precise statement is that time reversed subs-system dissipates and metabolites receive the energy but in reversed time direction.
   

   In living matter sub-systems with non-standard arrow of time are necessary since their dissipation is needed to extract metabolic energy. The highest level dissipates in standard time direction and there must be a transfer of energy between different levels. This hierarchy of levels with opposite arrows of geometric time would be realized at the level of MB.
   
   

These observations suggest that one should consider the reincarnation with opposite arrow of time with wisdom coming from the death of biological systems.

1. We know what happens in death and birth in biological systems. What happens in biological death should have analogy at general level. In particular, in death the decay of the system to components should occur. Also the opposite of this process with reversed arrow of time should take place and lead at molecular level to the replication of DNA and RNA and build-up of basic biomolecules and at the cell level to cell replications and development of organs. How these processes could correspond to each other?
   
   
2. The perceived time corresponds to the hyperplane t=T/2 of CD, where T is the distance between the tips of CD and therefore to maximal size of temporal slice of CD. The part of CD above it shifts towards future in SSFRs. In BSFR part of the boundary of space-time surfaces at the active boundary of CD becomes unchanging permanent part of re-incarnate - kind of log file about the previous life. One can say that the law of Karma is realized.
   

   If CD decreases in size in BSFR the former active boundary keeps its position but its size as distance between its tips is scaled down: T → T₁≤ T. The re-incarnate would start from childhood at T-T₁/2 and would get partially rid of the permanent part of self-hood so that new permanent part would be between T/2 and T-T₁/2 . Reincarnate would start almost from scratch, so to say. The part between T-T₁/2 and T would be preserved as analog of what was called BIOS in personal computers.
   
   

3. At the moment of birth CD possibly would thus decrease in size and the former passive boundary between t=T/2 hyperplane and lower tip of new CD at T-T₁ would becomes active and the seat of sensory experience. Where the analog of biological decay is located? The region of CD above T/2 and T-T₁/2 is the only possible candidate. This region is also the place, where the events related to birth in opposite time direction should take place.
   

   The decay of previous organism should correspond to the development and birth of re-incarnated organism. The decay of organism dissipates energy in standard time direction: this energy could used by the re-incarnate as metabolic energy.
   

   This vision might be tested. The replication of DNA and RNA and build-up of various bio-molecules should be time-reversals for their decays. The same applies to the replication of cells and generation of organs. Replication of DNA is self-organization process in which second DNA strand serves as a template for a new one. The decay of DNA should therefore involve two DNA strands such that the second DNA strand serves as a template for the time reversed replications. The double strand structure indeed makes possible for the other strand to decay first. One could even ask whether the opposite inherent chiralities of DNA strands correspond to opposite arrows of time. Maybe this could be seen as a kind of explanation for the double strand structure of DNA.
   

   In biology pairs of various structures often occur and maybe they could correspond in some sense to time reversals of each other. Also cell replication should use another cell as replicate and same would happen in the cell decay.
   
   

4. Eastern philosophies talk about the possibility of liberation from Karma's cycle. Can one imagine something like this? The above picture would suggest that in this kind of process the reduction of the size of CD does not occur at all and therefore there would be no decay process equivalent to the growth of time reversed organism. This would serve as an empirical signature for the liberation if possible at all. CD would continue to increase in size or perhaps keep its size. It would seem that a new kind of non-biological source of metabolic energy is needed.
   
   
5. Mental images should correspond to sub-selves and therefore sub-CDs of CD. The idea that the re-incarnations of mental images correspond to re-incarnations with a reversed arrow of time is very attractive. After images is the basic example. Only the after images with standard arrow of time would be experienced by us. Are the after images sensory memories of subjective past involving communication with re-incarnated visual mental image?
   

   The original, rather natural, proposal was that the after image is in the geometric past but according to the new view it would be shifting with the active boundary of CD towards geometric future at the active boundary of CD as a kind of log file. To remember it as sensory mental image requires communication with it along active boundary involving both future and past directed signals.
   

   One can imagine also more mundane explanation for after images in terms of propagation of dark photon signals along closed magnetic loops giving rise to periodically occurring mental images.
   
   

See either the article Some comments related to Zero Energy Ontology (ZEO), the article When does "big" state function reduction as universal death and re-incarnation with reversed arrow of time take place?, or the chapter Life and Death and Consciousness.

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

Articles and other material related to TGD.

Could Universe could have North-South direction: How?

Wes Johnson gave ) told about very interesting observations suggesting that cosmology has North-South (N-S) axis in the sense that fine structure constant has N-S variation with respect to this axis. See the popular article. Here is the abstract of the article of Webb et al.

Observations of the redshift z = 7.085 quasar J1120+0641 are used to search for variations of the fine structure constant, a, over the redshift range 5.5 to 7.1. Observations at z = 7.1 probe the physics of the universe at only 0.8 billion years old. These are the most distant direct measurements of a to date and the first measurements using a near-IR spectrograph. A new AI analysis method is employed. Four measurements from the X-SHOOTER spectrograph on the Very Large Telescope (VLT) constrain changes in a relative to the terrestrial value (α₀). The weighted mean electromagnetic force in this location in the universe deviates from the terrestrial value by Δ α/α = (α_z- α₀)/α₀= (-2.18 ± 7.27) × 10^-5, consistent with no temporal change. Combining these measurements with existing data, we find a spatial variation is preferred over a no-variation model at the 3.9 σ level.

To repeat: the difference from earthly value of α is small and consistent with no temporal change. If the measurements are combined with existing data, one finds that the model assuming spatial variation in north-south direction is preferred over no-variation model at 3.9 sigma level.

This kind of variation was reported years ago (see this). Thanks for Richard Ruquist for the link. I also wrote about the claim (see this).

The findings are very strange and counterintuitive and the effect probably disappears: there are many uncertainties involved since data from several experiments are combined. If the effect is real, there is challenge to understand it so that one cannot avoid the temptation for intellectual exercise.

In TGD framework many-sheeted space-time serves as a starting point.

1. The notion of space-time sheet requires that the M^4 projection of space-time surfaces is 4-D: I call these space-time sheets Einstenian. This was not true in primordial cosmology during which cosmic strings with 2-D M⁴ projection dominated (2-D in good approximation) - space-time was not Einsteinian yet. During the analog of inflationary period cosmic strings thickened to flux tubes and liberated energy giving rise to ordinary particles. Transition to radiation dominated cosmology took place during this period.
   
   
2. The fluctuations in the density of matter tell that this transition did not take at exactly the same value of cosmic time T but there are fluctuations of order ΔT/T ≈ 10^-5. This happens to be same order of magnitude as the reported value of Δα/α along North-South direction, which puts bells ringing. Could same cosmic parameter determine fluctuation amplitude ΔT/T and the relative change Δα/α along N-S direction?
   

   Could it be that the transition to radiation dominated cosmology took place in a wave propagating in North-South (N-S) direction so that there would be a gradient of T along N-S direction: ΔT/T - not fluctuation. This does not require gradient in fluctuations Δ T/T and Δ ρ/ρ. Could this gradient also explain the gradient in α along N-S direction?
   
   

How the N-S gradient in α could be understood?

1. At QFT limit particle experiences the sum of induced gauge fields assignable to the space-time sheets which it necessarily touches because it has same size of order CP₂size as the sheets on top of each other in CP₂directions. Standard model gauge fields can be indeed defined as sums of these induced gauge fields. Same applies to gravitational field identified in terms of metric of Einsteinian space-time having 4-D M⁴ projection.
   
   
2. The many-sheeted space-time was not quite the same thing in today and in ancient universe. The number of space-time sheets could have been different. Space-time sheets carried also induced classical fields with different strength.
   

   Monopole flux tubes created during the analog of inflationary period from cosmic strings indeed evolve during cosmic evolution. Their thickness increases in rapid jerks and in average sense this corresponds to a smooth cosmic expansion. This conforms with the fact that astrophysical objects do not seem to expand themselves in cosmic expansion although they co-move as particles in this expansion.
   

   The increase of the thickness of monopole magnetic flux tube reduces its magnetic field strength since monopole flux is conserved. This in turn reduces the contribution of this space-time sheet to the classical em field experienced by a charged particle. In particular, this would affect the binding energies of atoms slightly.
   
   

3. Could this together with the wave like progression of the transition to radiation dominated cosmology be responsible for the dependence α on N-S direction with the increase Δα/α ≈ 10^-5?
   
   

See the chapter More TGD inspired cosmology.

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

Articles and other material related to TGD.