Axioms of set theory from TGD viewpoint

Lubos made interesting comments related to the work of set theorist set theorist Paul Cohen. Cohen proved that axiom of choice (AC) cannot be proved nor disproved from Zermelo-Fraenkel axioms for set theory. He proved same also for continuum hypothesis (CH). Checking the definitions of various notions from Wikipedia led to an intellectual adventure of Sunday morning.

Axiom of choice from physics point of view

As a physicist I tend to share the skepticism of Lubos concerning AC. Stating it more formally: given a collection of sets indexed by some index set and not containing empty set, one can select from each set single element. What AC says that Cartesian product exists as a set. For finite sets there is no problem but already for a Cartesian power of reals one has a problem. Can one really choose from each factor R single element? Physicist might regard this kind of choice as too strong an idealization since there is always a finite accuracy involved. One can fix a given point of real axis only in finite accuracy since in the real world one must always perform a cutoff on the expansion of the real number in powers of 10, 2, or prime, or any natural number larger than one..

Physicist might prefer to weaken the axiom to a choice of a finite open ball around a point. This would lead from set theory to topology. One could return to set theory by postulating discrete topology in which each point is an open set. Physicist has however motivations to believe that physical world obeys less trivial topology and metric topology with the notion of distance defining the concept of nearness is natural for physicist. One can characterize the measurement resolution in terms of the radius of the ball replacing point. In p-adic context the situation is simpler than in real context since p-adic open balls are either disjoint or nested. Also the ordering, which is un-necessary luxury below measurement resolution, is lost in p-adic context. Since we want to minimize thinking, this is is a good reason for cognition to be p-adic;-).

TGD framework the notion of finite measurement resolution (cognitive resolution, numerical accuracy) plays a key role and has very powerful implications for the proposed mathematical framework of physics. For instance, the inclusions of hyper-finite factors relating closely to quantum groups often assigned to Planck scale represent finite measurement resolution at quantum level and at space-time and imbedding space level discretization become the counterpart for the finite measurement resolution at the level of geometry.

In p-adic context total disconnectedness implies difficulties with the notion of p-adic manifold and with integration and TGD suggests a notion of p-adic manifold based on coordinate charts mapping p-adic manifold to its real counterpart so one can induce real topology and well-orderedness of reals to p-adic context. Also here finite measurement resolution plays a key role.

Infinite primes and axioms of set theory

Cardinal numbers are defined as cardinalities of sets: two sets have the same cardinality if there is a bijection between them. Ordinals in turn can be seen as ordered sets of sets: set theoretically a given ordinal is the ordered sequence of ordinals defined by the cardinalities of the sets of smaller cardinality. The successor axiom of Peano arithmetics is essential here. Finite cardinals and ordinals can be identified but the notion of infinite ordinal is more refined notion than that of cardinal: this is obvious from the fact that infinite ordinals x and x+1 are not equivalent unlike corresponding cardinals.

Power set axiom states that the set of subsets of set exists. The cardinality of the power set is larger than the cardinality of set. In the case of natural numbers the cardinality of power set would be the cardinality of continuum and CH states that there are no cardinals between cardinality of natural numbers and cardinality of continuum postulated to that for the power set of natural numbers. CH can be extended to apply to any set and its power set. Cohen suggests that CH is not true although he only proves that both CH and not-CH are consistent with ZF axioms.

The notion of infinite prime provides number theoretic notion of infinity, which does not seem to reduce to the notions of infinite ordinals and cardinals. There is infinite hierarchy of infinite primes and infinite integers have a detailed number theoretic anatomy distinguishing them from cardinals and also "ordinary" ordinals. Infinity ceases to be some kind of limit and becomes something quite concrete expressed by an explicit formula having as a basic building brick the products of all primes at levels below a given level of hierarchy expressed as formal variables.

Physically this infinite hierarchy corresponds to a repeated second quantization of an arithmetic quantum field theory. The many-particle states of previous level become elementary particle states of the new level. Simple infinite primes correspond Fock states consisting of fermions and bosons labelled by prime valued "energy". There are also not so simple infinite primes analogous to bound states, which is rather interesting from the point of view of quantum field theories. Many-sheeted space-time could serve as a natural space-time correlate for this hierarchy in the sense that even galaxy sized objects could be seen in some aspects as elementary particles and proton could be also characterized as elementary fermion at particular level of the hierarchy.

The construction of infinite primes can be easily understood in terms of repeated second quantization.

1. The Dirac vacuum is what one starts from and is at the first level of the hierarchy identified as the product X₁=∏_i p_i of all ordinary primes. At the level n one has objects X_k, k=1,..n with X_k defined as the product of all primes belonging to levels m<k.
   
   

   
   

2. At the lowest level simple infinite primes are obtained by dividing X with a square free integer m and adding to the result m: X→ X/m+/- m. The physical analogy is kicking of the fermions corresponding to prime factors of m from Dirac sea to positive energy states. After this one can add bosons (p^m corresponds to m particles with "energy" p). One can also multiply X/m by an arbitrary integer n having no common factors with m: its decomposition to prime powers tells the numbers of bosons in corresponding modes. It is also possible to multiply m with integer r with any integer consisting of primes dividing m: also now the powers characterize the boson numbers in various modes. By construction nX/m and rm have no common prime factors so that the infinite integer is nmX/m+/- rm represents infinite prime. Depending on the sign factor one obtains two kinds of primes and X+/- 1 represent the simplest infinite primes and obviously differ by 2.
   
   

   
   Side remark: One can add to/subtract from X+/- 1 any integer and obtain an integer always divisible by some prime (any prime factor in the prime decomposition of X). One obtains therefore infinitely long range of infinite integers containing no primes. Analogous theorem for finite integers states that the range defined by the numbers n! +m, 0≤ m<n+1 contains no primes.
   

How do infinite primes and integers relate to the axioms of set theory? I am not a set theorist and can make only some observations of a dilettante. In the following I restrict the consideration to the infinite primes at the lowest level so that they have only ordinary primes below them.

The notion of infinity emerges in two different manners in the construction of infinite primes.

1. How the infinite size of a given infinite prime relates to the infinite cardinals and ordinals? To me the relationship is not obvious. The number theoretical anatomy of infinite primes and the hierarchy of Dirac seas X_n defined by products of primes at various levels seem to distinguish the notion of infinite natural number from infinite ordinals and cardinals. In particular, infinite prime at a given level (first level now) has infinite number predecessors at the same level. Hence bi-directional ordering seems to replace ordinary uni-directional ordering of natural numbers. Does this mean something genuinely new or can it be reduced to set theory? Certainly it is difficult to imagine a set having number of elements given by infinite prime: infinite prime is more like energy than a number of elements.
   
   
2. What is the number of infinite primes/integers/rationals at given level of hierarchy and how it relates to the notions of infinite cardinal and ordinal? The number of infinite primes at given level depends on what one allows. Consider the construction of infinite primes at the first level. The numbers n and m could be chosen to be finite. In this case the number of simple infinite primes would be the number of finite rationals n/rm. This implies denumerably of simple infinite primes (cardinality is same as for natural numbers).
   

   If one allows n or m to have infinite number of factors, the choices of m correspond to the set of subsets of finite primes and the resulting set is not denumerable and larger than the set of natural numbers (by CH it would be same as the cardinality of reals). One can argue that since the states with infinite fermion or boson number have infinite energy as physical states, they should be excluded so that the simple infinite primes would form a denumerable set. This result might generalize also to not so simple infinite primes, and might hold true at all levels of the hierarchy. Finite energy would correspond to denumerability. One can of course, ask "What about the entire Universe": should it correspond to the limit when the hierarchy level approaches infinite or to infinite particle numbers.
   

Infinite primes relate in an interesting manner to AC and so called well-ordering theorem often thought to be one of the holy truths of set theory.

1. Well-ordering theorem states that every set can be well-ordered. The existence of well-ordering of course depends on axiomatics. For real numbers well-ordering is natural and generalizes to Cartesian powers of reals in obvious manner. In p-adic number fields well-ordering is not possible if one stays with the restrictions of p-adic topology. As already mentioned, canonical identification map to reals can induce well-ordering of positive reals to p-adic context.
   
   
2. To my opinion the status of AC is not a mere academic question even for physicist. Banach-Tarski paradox illustrates this. If AC is true, one can decompose sphere into parts and re-arrange them by using only rotations and translations, which are volume preserving operations, so that the volume of the resulting sphere is two times larger than the original! The operation involves a construction of non-measurable sets having no well-defined volume and an infinite number of choices. This result would suggest that set theory alone is not enough for the needs of physics. The notion of measurable set is needed.
   
   
3. Well-ordering theorem is implied by AC. By definition set X is well-ordered by a strict total order if ever nonempty subset of X has a least element. This seems to be essential for the induction hypothesis generalized from naturals to infinite context: in induction the first step is to show the triviality of the theorem for n=0 and then show it for arbitrary n using induction hypothesis. For infinite positive integers at given level the least element does not however exist. Consider the infinite integers at the first level of the hierarchy. Infinite primes nmX/m -rm are always positive but there is no smallest infinite prime of this kind. In fact, also the integers X-n have no lower limit just as X+n do not have any upper limit. One can have a least element only by allowing a jump to a lower level, and this limit is of course 0. Unless one is ready to make this jump, well-ordering theorem is not true and therefore also AC cannot hold true. Maybe one could replace the uni-directional induction by bi-directional induction for infinite integers.
   
   

Planck and CMB

Several bloggers have already commented the new Planck data (28 articles!). The posting of Phil Gibbs explains in some detail what is measured and what are the basic results from the point of view of cosmology. Second blog posting discussing the results from particle physics point of view is by Resonaances.

The basic observable are the temperature fluctuations of cosmic microwave background coded by the correlation function for the local CMB temperature and deducible from the data. The correlation function is expressible in terms of spherical harmonics which are functions of the angular separation Ω for the points of celestial sphere.

The role of standard model is played Λ CDM model involving cosmological constant and cold dark matter. Inflationary scenario is behind this model. The basic prediction is that the spectrum is Gaussian just like density fluctuations are predicted to be Gaussian. This is the simplest option that one can assume. The amplitude of the density fluctuations (and thus temperature fluctuations) is one of the non-trivial predictions. Although I am not a specialist, I dare to guess that Gaussian character dictates the predictions to a high degree.

As Phil Giggs describes, the new data provide a much more accurate view about temperature fluctuations since higher harmonics l, which corresponds to higher scales of angular (and therefore also length scale -) resolution for a given distance/redshift, are included. The highest angular resolution is .1 degrees for Planck data.

The data allow to deduce fractions of of dark energy, dark matter, and visible matter as 69.2+/- 1 %, 25.8+/- .4 %, and 4.82+/- .05 %. Also the age of the Universe can be estimated. The age of the Universe comes as 13.82 +/- 0.05 Giga-years.

From particle physics point of view the interesting outcome is the estimate for the number of light neutrinos which is now N_eff=3.3+/- 0.5. This does not support the idea about inert neutrinos having been in thermal equilibrium with other particles. This results is interesting also from TGD point of view (see also the earlier posting Is there an inert neutrino there?) since right handed neutrino has an exceptional role in TGD and despited the fact that it has no spinor couplings to induced electroweak gauge fields, one must consider also the possibility that via its coupling to the space-time geometry via modified Dirac equation it has managed to achieve a thermal equilibrium with other particles during early cosmology.

Amplitude anomaly

There are two kinds of anomalies associated with small angular momenta: amplitude anomalies and directional anomalies. Anomalies at the low frequency end. 2.5 to 3 sigma significance. Anomalies appear at large angular separations and thus large length scales hinting about the existence of unexpectedly large structures. By Uncertainty Principle of harmonic analysis large angular distances correspond to small "angular momenta" and lowest spherical harmonics.

The correlation function for the temperature fluctuations at two points of celestial sphere can be developed in terms of spherical harmonics as a function of Ω=(θ,φ) defining the relative angular coordinates of the points.

⟨ T(Ω₁)T(Ω₂)⟩= ∑_{l,| m|=0,...,l}Q_l^mY^l_m(Ω) .

Q_l,m are multipole strengths. Inflationary cosmological predicts Gaussian distribution in for the fluctuations as function of the angular separation. This leads to predictions for the coefficients of Q_l,m.

Dipole anomaly is the lowest anomaly but can be identified in terms of the motion of Earth with respect to CMB and by a suitable choice of the rest frame it can be eliminated. There are anomalies for also higher harmonics in Planck data up to l=10 at least. Higher harmonics are not shown in the graph of the posting of Phil Gibbs. For these values of l, odd harmonics are stronger whereas even harmonics are weaker than predicted. Why this?

Axis of Evil

To a given l one can assign direction as direction Ω for which l^th contribution ∑_mQ_lm²Y_l,m has maximum of modulus squared of this contribution. These directions are expected to be random. For small even values of l they tend to be aligned with each other and with what is called Axis of Evil, which is almost in the orbital plane of Earth. The celestial direction angles of the Axis of Evil are (RA= 212 ^o, δ= 25 ^o).

A short digression: Right ascension RA and declination δ relate to spherical coordinates Ω=(θ,φ) via the relationship (δ= π/2-θ, RA=φ). δ is the counterpart of Norther latitude and RA the counterpart of eastern longitude. RA =0 holds true, when the center of Sun is in the plane as the Earth's equator (note that spin axes of Earth is slightly tilted with respect to the orbital plane (see this ). Axis of Evil points to a region in which the number of radio galaxies is lower than expected: the size scale of this low temperature region is 300 MPc (10⁹ ly). The alignment of galactic spin directions along this direction.

Possible explanations for the Axis of Evil has been discussed. The first article that web search gave was Goodness in the axis of evil by Schild and Gibson. The article relies on WMAP data but gives a glimpse about the problem for the interested reader. The abstract of the article mentions the unexpected alignment of 2-4-8-16 cosmic microwave background spherical harmonic directions with the direction of a surprisingly large WMAP temperature minimum ("Axis of Evil") defining a large low temperature structure and radio galaxy void.

This cold region is not in the same direction as the Cold Spot for which one has RA= 3h 15min 5s and δ = −19^o 35' 02". 3h makes roughly 3× 30=90 degrees as anyone familiar with clock can conclude). The authors do not state that odd harmonics would lie in the direction of Axis of Evil: what the situation is for the new data is not clear to me.

Article mentions also the unexpected alignment and handedness of galaxy spins in the same direction up to distances of order 1.5 Gpc (pc= 3.26 Iy). The size scale of the low temperature radio galaxy void is 10⁹ ly to be compared with the size scale 10⁸ ly of large voids, which seem to form a honeycomb structure (for which I have considered a model based on many-sheeted space-time). I tend to agree with the authors that these long range correlations can be seen as a serious challenge the Λ QCM model which assumes cosmological constant, inflation, and cold dark matter.

In TGD framework a natural explanation for the Axis of Evil would be in terms of long galactic cosmic strings. Note that these cosmic strings have only name in common with the ordinary GUT cosmic strings. As mentioned in Not Even Wrong, Planck data show now evidence for GUT cosmic strings.

1. Cosmic strings in TGD sense are key players of TGD inspired cosmology. During primordial period they corresponds to string like objects with 2-D CP₂ and M⁴ projections and later their M⁴ projection expands so that it becomes 4-D. The outcome is Kähler magnetic flux tubes with Kähler magnetic fields explaining the magnetic fields filling the Universe but having no natural explanation in standard cosmology. Since magnetic monopole flux having purely topological origin is in question, these fields need no current as source as ordinary magnetic fields.
   
   
2. Kähler magnetic energy is identifiable as dark energy and the magnetic tension gives rise to the "negative pressure" needed to achieve accelerated cosmic expansion. The flux tubes can decay to ordinary and dark matter in the same manner as inflaton type fields are assumed to decay the ordinary and dark matter in inflationary scenarios. In TGD framework the flatness of 3-space corresponds to quantum criticality (no scales) rather than inflation. The Kähler magnetic energy of galaxies are assumed to be located along these string like objects like pearls along necklace. The correlations between galactic spins could be inherited from primordial period when galaxies were very near to each other and their dynamics were correlated.
   
   
3. One should understand the amplitude anomalies in terms of this model. Why even harmonics have weaker amplitudes and odd harmonics stronger amplitudes than predicted? Could one say that some power has been transferred from even to odd harmonics? Why odd parity amplitudes would be stronger? How the selection of preferred axis by the presence of cosmic string could be responsible for this effect.
   

Sacred geometry and biology

Platonic solids and Golden ratio are in central role in what some people call sacred geometry. I know that the average colleagues cannot tolerate words like "sacred geometry". In my humble opinion names (nor even Names) do not matter much, and one should look for what is behind the names. These Platonic structures are extremely interesting since they relate to discrete and finite subgroups of 3-D rotation group characterizing the symmetries of also biomolecules. In particular, tey involve Golden ratio known to pop up again and again in biology. They also relate to the inclusions of hyperfinite factors and there exists a fascinating correspondence discovered by McKay between the discrete subgroups and laced Lie algebras. For instance, exceptional Lie groups correspond to tetrahedron, octahedron, and icosahedron in McKay correspondence.

The following considerations suggests that tetrahedral and icosahedral geometries might have deep connections to biology and genetic code in the sense that genetic code could have geometric representation in terms of tetrahedrons and icosahedrons. For more details see the article Could one find a geometric realization for genetic and memetic codes? and for background the article What are the counterparts of Einstein’s equations in TGD?. In the following I try to represent the ideas as "slides" (used originally as a synopsis of a Skype discussion) for an imagined audience (this is just a trial motivated by my laziness).

A brief summary

1. The problem of sacred geometrician (icosahedron consists of 20 tetrahedrons which are quite not regular) has a solution which requires deformation of Euclidian space to 3-sphere locally.
   

   This is possible in sub-manifold gravity. This is not however enough: this sub-manifold gravity must be strong. TGD predicts both sub-manifold gravity and a hierarchy of strong gravities.
   
   

2. Tetrahedron has 4 faces (number of code letters A, T, C, G of genetic code) and icosahedron 20 faces (number of amino-acids: accident or connection of genetic code with sacred geometry?)
   

   Could symmetry breaking for icosahedron assign to a given face of icosahedron a unique amino-acid? What could select this face so that one could say that one particular icosahedron corresponds to a given amino-acid? Does it involve gluing of amino-acid to particular face?
   
   

3. TGD predicts besides genetic code also memetic code.
   

   Combinatorial hierarchy defined recursively M₂= 3 ,M₃=7, M₇ =127, M₁27= 2¹²⁷-1,... The lowest Mersenne numbers are primes. The rest also if Hilbert was right.
   

   One can assign codes to these Mersenne primes. Number of codons: 2^3-1=4 for M₃, 2^7-1= 64 for M₇ (genetic code). 2^127-1= 2¹²⁶ for memetic code.
   

   Genetic code corresponds to M₇=127 and is followed by memetic code for which codewords correspond to 2¹²⁶ sequences of 21 DNA codons.
   

   !21=20+1. Is this an accident?
   

   Could icosahedral geometry provide a geometric realization of memetic code?
   
   

4. TGD suggests a dark realization of genetic code realized in terms of dark proton sequences.
   

   Genetic code realized at the level of dark nuclear physics!
   

   !Lowest magic numbers for nuclei correspond to the numbers of faces for tetrahedron (4), octahedron (8), and icosahedron (20). Could nuclear physics have a connection with dark geometry In nuclear string model the folding of nuclear strings completely analogous to protein folding to these shapes could indeed give rise to magic nuclei.
   

   !?Could memetic codons be realized as sequences of 21 dark protons folded to icosahedron with 20 faces +1 dark proton attached together with amino-acid to one face! Amino-acid sequences accompanied by sequences of dark memetic codons. Do proteins carry a huge amount of memetic information which we know nothing about!!??
   

   !?Could some parts of intronic portion of ordinary DNA realize memetic code chemically? Could cultural evolution as opposed to purely biological relate to the intronic portion of DNA? Could dark protons at cell membrane be connected by magnetic flux tubes to DNA codons could make possible topological quantum computations and evolution of software making possible cultural evolution? If so, cultural evolution would correspond also to the evolution of magnetic body leading to a hierarchy of levels of collective consciousness.
   

The problem of sacred geometrician and the first observation

Icosahedron (12 vertices, 20 faces ) is Platonic solid as are also tetrahedron, cube, octahedron, dodecahedron.

Observation: Icosahedron consists of tetrahedrons, which are howevernot quite regular. The ratio of "surface" edges to radial edges is 1.05 rather than 1. Scaling does not help. Sacred geometrician does not feel happy;-).

1. Is there any manner to obtain 20 regular tetrahedrons?
   

   !There is! If Euclidian 3-space E³ is replaced with 3-sphere S³, one obtains 20 regular tetrahedrons.
   

   Could gravitation modify distances by making space-time non-flat. Could gravitation allow to deform piece of E³ containing icosahedron to a piece of S³ so that one would have sacred geometry?
   

   !In TGD universe sub-manifold gravity allows to take a piece of E³ realized as 3-surface and deform it it CP₂ directions to make it look locally like 3-sphere! One obtains 20 *regular* tetrahedrons!
   

2. Is gravitational deformation physically possible?
   

   Gravitation according to General Relativity cannot produce local S³ geometry in scales of biology and and condensed matter physics. For a system with mass M the scale of piece of S³ would be Schwartschild radius r_s= 2GM. This length scale is extremely short scale if one works in the scales of condensed matter. Even for Sun r_s is only 3 km! Huge gravitational constant is require in order to have r_s ∼ hbar/M, the Compton length of particle with mass M. G=∼ hbar/2GM². Gravitation would be extremely strong.
   

   ! This need not be a problem! TGD predicts hierarchy of strong gravities assignable to p-adic length scale hierarchy. Gravitational constant has a spectrum of values. For instance, inside hadrons gravitation is strong and gravitational constant is about 10³⁸ times larger than ordinary gravitational constant. Only in very long length scales gravitation gets extremely weak. Size of piece of S³ of order G_strongM and can be of order of the size of say atomic nucleus or of icosahedral water molecule cluster.
   

   ! TGD predicts a fractal hierarchy of scaled copies of hadron physics and thus also strong gravities. In biology these scaled copies would be especially important in 10 nm- 5 micrometer length scale range.
   

   TGD inspired estimate for gravitational constant is given by
   

   G= (L_p²/hbar) × exp(-S_K(CP₂)) , L_p² = p × R²(CP₂) .
   

   Here R(CP₂) is CP₂ size of order 10⁴ Planck lengths.
   

   S_K(CP₂) is Kähler action for a deformation of CP₂ type vacuum extremal representing graviton line of a generalized Feynman diagram as a deformation of CP₂ type vacuume extremal. In long length scales, the magnitude for the value of Kähler action becomes maximal, and one obtains ordinary gravitational constant for p=M₁₂₇ and α_K equal to fine structure constant. M₁₂₇ is the p-adic prime characterizing electron and largest Mersenne prime which does not correspond to completely super-astrophysical length scale.
   

   In short scales (short graviton lines) the strong gravitation limit G∼ L_p² is approached since Kähler action becomes small. Hadron physics and its various fractal copies would correspond to strong gravities. Schwartschild radius r_s= 2GM = 2L_p²M/hbar ≈ 2 L_p would be of order Compton length!
   

Second observation: sub-manifold geometry allows to realize quasi-lattices consisting of pieces of S³

!In sub-manifold gravity one can construct from pieces of S³ realized as 3-D surfaces in M⁴× CP₂ large quasi-lattice like structures by gluing them together along boundaries. Icosahedrons decomposed of *regular* tetrahedrons could serve as a basic supercell.

Could icosahedral water clusters in nanometer length scale correspond to this kind of quasi-lattice like structures?

Could it be that strong gravitation is essential for living matter where ordered water plays a key role? Penrose and Hameroff speculated also that quantum gravitation important but the problem is that ordinary gravitation is so weak.

Third observation/question: sacred geometries ↔ genetic code

Two intriguing observations:

1. Tetrahedron has 4 faces. DNA has for different code letters: A, T, C, G.
   
   
2. Icosahedron has 20 faces. There are 20 different amino-acids.
   

Two questions:

1. Could A,T,C,G somehow correspond to faces of tetrahedron?
   
   
2. Could amino-acids somehow correspond to faces of icosahedron?
   

One must be able to achieve two things.

1. To distinguish between faces of tetrahedron/icosahedron in order to assign them code letter or amino-acid. Symmetry breaking is required.
   

   !One can look these objects in two manners. In S³ geometry of the space-time sheet. Or in E³ subset M⁴× CP₂ geometry of imbedding space! 4- and 8-dimension perspectives! The regular tetrahedrons of S³ geometry do not regular in E³ geometry! In the same manner the faces of icosahedron need not be identical in E³ geometry although they are so in S³ geometry. Could this be enough for the needed symmetry breaking or is more needed?! Need the breaking be purely geometric?
   

2. To select one of the 4 faces of tetrahedron and one of the twenty faces of icosahedron in order to say that this tetrahedron represents particular code letter or this faces represents particular amino-acids. How to achieve this?
   

Fourth observation: genetic code → hierarchy of codes. Memetic code

TGD leads to a generalization of genetic code.

1. Combinatorial Hierarchy defined recursively:
   

   M(m+1)= M_M(n)= 2^M(n)-1 .
   

   Mersenne numbers
   

   M₂= 3, M₃=7, M₇ =127, M₁27= 2¹²⁷-1,...
   

   The lifted Mersenne numbers are primes. The rest also if Hilbert was right.
   
   

2. One can assign to these Mersenne numbers codes.
   

   M_n=2ⁿ-1: 2ⁿ sequences of n bits. Boolean algebra. One half of Boolean statements can be simultaneously true and define "axioms". Codes are axioms systems with 2^n-1 axioms in the case of M_n.
   

   Number of codons is 2^3-1=4 for M₃, 2^7-1= 64 for M₇ (genetic code). 2^127-1= 2¹²⁶ for memetic code.
   

The interpretation of the hierarchy is in terms of statements about statements about statements.... Reflective hierarchy of codes would give Combinatorial Hierarchy. Hierarchy of codes labelled by certain Mersenne primes.

1. Genetic code corresponds to M₇=127. 64=2⁶ code letters.
   
   
2. The next code after genetic code - memetic code - corresponds to Mersenne prime 2¹²⁷-1, which is the p-adic prime characterizing electron and ordinary gravitation in long length scales. This code has 2¹²⁶ =2^{6× 21} = (2⁶)²¹ codewords, which can be represented as sequences of 21 DNA codons. Huge amount of information! Could this code be realized for introns and be assignable to topological quantum computation.
   

There are two further observations.

1. First observation: number of DNA codons in memetic codon is 21 =20+1= number of aminoacids +1 (stopping sign as "aminoacid")? What could this mean!? Is there a connection between memetic codon, the number of amino-acids, and icosahedral geometry?
   

   Could it be that one could take DNA strand and fold it to icosahedral shape?! This would be like protein folding. Could one have 20 DNA codons forming icosahedron and 21:st glued to one of the 20 faces of the icosahedron?
   

   Could this select icosahedral face ↔ amino-acid? Could this involve also gluing of real amino-acid to the face? Could amino-acid sequences be accompanied by sequences of icosahedrons so that one would have realization of memetic code? Could amino-acid sequences much more that mere building bricks: software instead of mere hardware? Could huge amounts of hidden/dark information not realized chemically!? Could memetic code perhaps responsible for the cultural evolution be associated with aminoacid sequences?
   
   

2. Second observation: DNA possesses pearls-in-necklace structure. Pearls identifiable as nucleosomes. Could they correspond to 21 DNA codons in icosahedral configuration?
   

   This need not be the case. There are 47 codons in single nucleosome. 2× 21+5. 2 memetic codons plus 5 surplus DNAs, which corresponds to 5 nm length: p-adic length scale L(149) defining thickness of lipid layer of cell membrane? Note that 5 DNAs correspond to a twist of π for DNA double strand. Why this? Could there be some other realization?
   

Further observations and further ideas

Further observations lead to further ideas.

1. Fifth observation:nuclear string model ↔ protein folding ↔ sacred geometries. In nuclear string model atomic nuclei correspond to folded strings. Protons are folded into string and neutrons too. The string structure orders the nucleons and is something new and in principle it should be possible to find tests for this.
   

   Folding of nuclear strings completely analogous to the folding of proteins!
   

   4, 8, 20 are nuclear magic numbers. These are numbers of faces for tetrahedron, octahedron, and icosahedron. Sacred geometries! Is this a mere accident? Note that usual explanation in terms of nuclear shell model with harmonic oscillator potential.
   
   

2. Sixth observation: in atto-second time scale water obeys chemical formula H_1.5O. 1/4:th or protons invisible in neutron diffraction and electron scattering (see Chaplin's homepage). Where these protons lurk?
   

   !Could they be dark in TGD sense? Could they reside at dark space-time sheets and have non-standard value of Planck constant. This leads to a model for dark protons as protons lost by water molecules to dark space-time sheets.
   
   

3. Seventh observation/idea: dark realization of DNA, mRNA, tRNA, amino-acids and vertebrate genetic code
   

   !The model for dark proton predicts that its states are in 1-1 correspondence with DNA, RNA, tRNA, and amino-acids and also predicts with natural assumptions that the numbers of dark "DNA type" protons which correspond to dark "amino-acid type" protons are same as the number of DNA codons mapped to a given amino-acid in vertebrate genetic code! Sequences of dark protons define dark nuclei as analogs of DNA codon sequences. Dark realization of genetic and possibly also memetic code .
   

   !This could make possible R keno& D department in which various genetic modifications could be tested and successful variants could be taken in use by transcribing dark genetic codons to ordinary ones. The very rapid evolution of genes related to immune system could involve active experimentation in the virtual world of dark genes and dark amino-acids. Evolution would not be a completely random process but an outcome of active experimentation just as the technological evolution is.
   

   Non-reductionistic side remark: No one considers seriously of throwing copper and silicon to pool and waiting the computers evolve from this soup! But this is how standard biologist would do.
   

   !?Could it be that dark DNA 21-plets folded to icosahedral configuration accompany amino-acid sequences! Each amino-acid would correspond to dark protonic string of 20 dark protons folded to icosahedron + 1 additional dark proton to which amino-acid is glued. Completion to memetic codon would select unique amino-acid!! Two flies killed with single blow!
   

   ! The folded dark proton string of 20 nucleons would correspond to especially stable magic dark nucleus having an icosahedral shape.
   

Eighth observation: water molecule clusters and geometric realization of memetic and genetic codes

Tetrahedral and icosahedral geometries assignable to water molecules to which one can assign dark protons.

!?Could water molecule clusters accompany dark proton sequences? One can imagine two options.

1. Water molecule has tetrahedral structure. H⁺ nuclei and two pairs of lonely electrons are in vertices of tetrahedron. Could water molecule be associated with a piece of S³ instead of E³? Could water molecules combine to form icosahedrons? One dark proton for each water molecule.
   

   ?Could this structure correspond to single dark DNA codon? The size might be too small. Ordinary DNA codons corresponds to 1 nm. Water molecules is roughly 10 time smaller.
   
   

2. 14 water molecules form a regular tetrahedron and these in turn combine to form icosahedral water clusters (see the homepage of Chaplin about anomalies of water). This structure has nanometer scale. Could this structure define one dark genetic codon containing one dark proton? In this case one would have 1 dark proton per 14 water molecules.
   

Background: progress in understanding of TGD

The consideration above were preceded by a development of ideas about preferred extremals of K&ayml;hler action. For preferred extremals of Kähler action energy momentum tensor of Kähler action must have vanishing divergence.This corresponds to local conservation of energy momentum. If Einstein-Maxwell equations with cosmological constant assumed, this is true. Λ and G emerge as predictions. This distinguishes TGD from general relativity and allows a hierarchy of strong gravitations assignable to scaled variants of strong interactions. These would be important even in condensed matter scales and living matter.

?Problem: Einstein-Maxwell equations with cosmological constant might allow too limited set of solutions. Does not make sense for vacuum extremals for which energy momentum tensor vanishes but Einstein tensor is non-vanishing. Would require infinite gravitational constant and vanishing cosmological constant.

The interpretation of vacuum extremals suggests something more general.

1. Einstein-Maxwell equations cannot be true for them. The solution ansatz guaranteing the vanishing of divergence of energy momentum tensor must be generalized. This is possible: two or three cosmological "constants", which are actually not constants! This was discussed in previous posting.
   

2. Interpretational problem: Vacuum extremals are vacua of Kähler action but their Einstein tensor non-vanishing. Einstein equations say that there must be energy momentum associated with these vacuum extremals. What does this mean?
   

   !As I have suggested earlier, Einstein tensor characterizes the topologically condensed matter, smaller space-time sheets present and glued to the vacuum extremal space-time sheet. The sheet carries information about the energy momentum tensor of topologically condensed matter in its Einstein tensor.
   

What are the counterparts of Einstein's equations in TGD?

Whether TGD is consistent with Einstein's equations or not has been a continual source of worry and my interpretations have fluctuated. The working with a quite different problem forced however to ask the basic questions again and led to challenge the assumption inspired by the construction of preferred extremals that the local conservation of energy momentum tensor of Kähler action always forces Einstein-Maxwell action with cosmological term. As a matter, fact much milder conditions are enough to achieve this and imply the existence of several cosmological "constants" (not constants actually) multiplying mutually orthogonal projectors, whose sum replaces the ordinary cosmological constant term. This view unifies the two quite different interpretations of cosmological constant.

The first - "Einsteinian" - interpretation is as a fundamental constant not characterizing real energy density. According to the second interpretation cosmological constant characterizes genuine contributions to energy assigned to inflaton type fields. In the recent case these contributions would correspond to topologically condensed matter such as K\"ahler magnetic flux tubes and particles. As a matter fact, this was the original interpretation, which I challenged after the discovery of preferred extremals. I glue below the abstract of the new chapter What are the counterparts of Einstein's equations in TGD?.

The original motivation of this work was related to Platonic solids. The playing with Einstein's equations and the attempts to interpret them physically forced the return to an old interpretational problem of TGD. TGD allows enormous vacuum degeneracy for Kähler action but the vacuum extremals are not gravitational vacua. Could this mean that TGD forces to modify Einstein's equations? Could space-time surfaces carrying energy and momentum in GRT frameword be vacua in TGD context? Of course, also in GRT context cosmological constant means just this and an experimental fact, is that cosmological constant is non-vanishing albeit extremely small.

Trying to understand what is involved led to the realization that the hypothesis that preferred extremals correspond to the solutions of Einstein-Maxwell equations with cosmological constant is too restricted in the case of vacuum extremals and also in the case of standard cosmologies imbedded as vacuum extremals. What one must achieve is the vanishing of the divergence of energy momentum tensor of Kähler action expressing the local conservation of energy momentum currents. The most general analog of Einstein's equations and Equivalence Principle would be just this condition giving in GRT framework rise to the Einstein-Maxwell equations with cosmological constant.

One can however wonder whether it could be possible to find some general ansätze allowing to satisfy this condition. This kind of ansätze can be indeed found and can be written as kG+∑Λ_iP_i=T, where Λ_i are cosmological "constants" and P_i are mutually orthogonal projectors such that each projector contribution has a vanishing divergence. One can interpret the projector contribution in terms of topologically condensed matter, whose energy momentum tensor the projectors code in the representation kG=-∑Λ_iP_i+T. Therefore Einstein's equations with cosmological constant are generalized. This generalization is not possible in General Relativity, where Einstein's equations follow from a variational principle.

The suggested quaternionic preferred extremals and the preferred extremals involving Hamilton-Jacobi structure might allow identification as different families characterized by the little group of particles involved and assignable to time-like/light-like local direction. One should prove that this ansatz works also for all vacuum extremals. If not, the local conservation of K¨hler energy momentum tensor would be the general formulation for the counterpart of Equivalence Principle in TGD framework. This progress - if it really is progress - provides a more refined view about how TGD Universe differs from the Universe according to General Relativity and leads also to a model for how the cosmic honeycomb structure with basic unit cells having size scale 108 ly could be modelled in TGD framework.

For details see the article What are the counterparts of Einstein's equations in TGD? or the new chapter of "Physics in many-Sheeted space-time".

Nothing new about Higgs but 3 sigma indications of M89 kaon

The news about Moriond conference (for details see for the posting of Phil Gibbs) did not bring anything really new concerning the situation with Higgs. The two-photon discrepancy is still there although the production rate is now about 1.6 times higher than predicted. The error bars are however getting narrower so that there are excellent reasons to hope/fear that unexpected kind of new physics is trying to tell about itself. Also the masses deduced from gamma pair and Z pair decay widths are slightly different.

The TGD-based explanation would be in terms of M₈₉ hadron physics, a fractal copy of ordinary hadron physics with 512 times higher overall mass scale. If the pion of this new physics has mass not too far from 125 GeV its decays to gamma and Z pairs would affect the observed decay rates of Higgs to gamma and Z pairs if one assumes just standard model. Fermi anomaly suggests mass of about 135 GeV for the pion of M₈₉ hadron physics. The observations of RHIC and those from proton-heavy nucleus collisions - correlated pairs of charged particles moving in same or opposite directions- could be understood in terms of decays of M₈₉ mesons behaving like hadronic strings in low energies in the relevant energy scale.

Lubos tells in his recent posting about 3 sigma excess for new charged and neutral particles with mass around 420 GeV. They would be produced as pairs of charged and neutral particle. M₈₉ physics based explanation would be in terms of kaons of M₈₉ hadron physics. The naive scaling by the ratio r=m(π⁺₁₀₇)/m(K⁺₁₀₇) of masses of ordinary pion and kaon predicts that the M₈₉ pion should have mass m(π⁺₈₉)= r× 420 GeV. This would give m(π⁺₈₉)=119 GeV not too far from 125 GeV to affect the apparent decay rates of Higgs to gamma and Z pairs since its width as strongly interacting particle decaying to ordinary quarks and gluons is expected to be large. This mass however deviates from the 135 GeV mass suggested by Fermi data by 18 per cent.

Update: The CMS data from Higgs came from Moriond (see for instance this and this) and tell that photon pair production rate is .78+/- .27 from the predicted rate. The mean value would be less than half of that found by ATLAS! The groups use different detectors so that the large difference could be due to statistical fluctuations. Unless it is due to the different assumptions in the data analysis! I believe that Higgs like state is here and might well behave just as the standard model tells it must behave. As an innocent outsider I however cannot avoid making innocent questions. Could the anomalies be there too? What these anomalies could tell about the new physics that was expected to emerge at TeV energies? And how much pre-existing beliefs affect the analysis in which one must know precisely what one is searching for ("Standard model explains everything!", "Maybe there is new physics of expected kind, say SUSY, can be searched for", "Maybe even new physics outside the mainstream is worth searching for")? It is a pity that we have the next opportunity to answer these questions only after 2015.

Seeing without brains?

Science News tells about a finding that transplanted eyes located far outside the head of vertebrate can see without a direct connection to brain. The connection to spine is however present.

The experimenters surgically removed donor embryo eye primordia, marked them with fuorescent proteins, and grafted them into the posterior region of recipient embryos. This induced the growth of the ectopic eyes. The natural eyes of recipients were removed. Fluorescent spectroscopy revealed the natural innervation patterns but none of the animals developed connections to brain.

To determine whether the animals having only ectopic eyes could see the training system was divided to quadrants of water illuminated by either red or blue LED light, and experimenters gave slight electric shocks in a particular quadrant. What was found that about 19 per cent of animals with optic nerves connected to the spine learned to avoid the quadrant in which they received electric shocks.

What experiments show that it is possible to see without neural connections to brain. The question is whether only the spinal cord or also the brain was involved with the learning. Probably neuroscientist could immediately answer this question but for an innocent layman like me the answer is far from obvious. Experimenters seem to think that brain is involved. As Douglas J. Blackinston, the first author of the paper "Ectopic Eyes Outside the Head in Xenopus Tadpoles Provide Sensory Data For Light-Mediated Learning," in the February 27 issue of the Journal of Experimental Biology, states "Here, our research reveals the brain's remarkable ability, or plasticity, to process visual data coming from misplaced eyes, even when they are located far from the head."

If brain is involved and the learned response is not a mere reflex involving only the spine, there must be information transfer to brain - perhaps along spine - but not as nerve pulses.

In TGD framework these findings inspire several questions.

1. Does the ability to see colors mean that visual colors are perceived at the level of retina rather than brain? The phenomenon of phantom limb supports strongly the standard view that various qualia emerge at the level of brain. On the other hand, the almost-prediction of TGD inspired theory of consciousness is that the primary sensory percept - and therefore also color qualia - can be assigned with the sensory organs. In TGD framework brain and body are 4-dimensional so that the pain in non-existing limb would be pain in the real limb of the geometric past.
   

   Brain would build cognitive representations - standardized mental images - about the sensory input by decomposing the perceptive field to objects. Brain would of course induce also motor response by associating to these standardized mental images motor actions.
   
   

2. In order to build standardized mental images brain would generate feedback as a virtual sensory input to the sensory receptors. Virtual sensory input would be realized using what I have called dark photons having "topological light rays" as space-time correlates and assignable to the magnetic flux tubes connecting body parts together. Two new notions are involved: magnetic body - the primary intentional agent - and the signalling using photons, which are dark in the sense that they are characterized by a large effective value hbar_eff of Planck constant coming as an integer multiple of hbar so that for say energy of visible photon the wavelength can be much longer than micrometer.
   
   
3. It has of course known for a long time that EEG carries precise information about the state of brain, and the natural question is why so? Magnetic body must receive data from biological body and the hypothesis is that EEG and its variants and possible scaled variants of EEG involving dark photons with large enough value of Planck constant to make their energies higher than thermal energy make this communications possible. Dark photons would be assigned to what I have used to call "topological light rays" assignable to magnetic flux tubes. The basic functions of EEG would therefore be communication to and control by magnetic body.
   

   For instance, quite recent experiment involved two rats as model animals. The first rat learned to press one of the two levers in response to a light signal over the correct level to get the reward. Second rat received the EEG response of the first rat and learned to respond in the same manner on basis of this response only so that this sensory response served as a virtual sensory or cognitive input for it.
   

   Magnetic body would generate also motor response using brain as a control instrument. Is the motor response in the recent case a kind of reflex action using only spine? Or are brain and magnetic body involved? Certainly the magnetic body could use brain as an intermediate control instrument. How much of the plasticity usually assigned with brain is actually flexibility of the magnetic body? And who is learning: is it brain or the magnetic body?
   
   

4. The communication using dark photons and the presence of magnetic body would make possible the participation of also brain to the learning process. For instance, the communication from the ectopic eye to brain could utilize quantum coherent dark photons travelling along the route ectopic eye → appropriate layer of magnetic body → brain. One can imagine also a dark photon communication along magnetic flux tubes parallel to spine.
   

Still about non-planar twistor diagrams

A question about how non-planar Feynman diagrams could be represented in twistor Grassmannian approach inspired a re-reading of the recent article by recent article by Nima Arkani-Hamed et al.

This inspired the conjecture that non-planar twistor diagrams correspond to non-planar Feynman diagrams and a concrete proposal for realizing the earlier proposal that the contribution of non-planar diagrams could be calculated by transforming them to planar ones by using the procedure applied in knot theories to eliminate crossings by reducing the knot diagram with crossing to a combination of two diagrams for which the crossing is replaced with reconnection. The Wikipedia article about magnetic reconnection explains what reconnection means. More explicitly, the two reconnections for crossing line pair (AB,CD) correspond to the non-crossing line pairs (AD,BC) and (AC,BD).

I do not bother to type the 5 pages of text here. Instead I give a link to the article Still about non-planar twistor diagrams at my homepage. For background see the chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-matrix".

Comments on the recent experiments by the group of Michael Persinger

Michael Persinger's group reports three very interesting experimental findings related to EEG, magnetic fields, photon emissions from brain, and macroscopic quantum coherence. The findings provide also support for the proposal of Hu and Wu that nerve pulse activity could induce spin flips of spin networks assignable to cell membrane.

In this article (see also the new chapter of "TGD based view about living matter and remote mental interactions") I analyze the experiments from TGD point of view. It turns out that the experiments provide support for several TGD inspired ideas about living matter. Magnetic flux quanta as generators of macroscopic quantum entanglement, dark matter as a hierarchy of macroscopic quantum phases with large effective Planck constant, DNA-cell membrane system as a topological quantum computer with nucleotides and lipids connected by magnetic flux tubes with ends assignable to phosphate containing molecules, and the proposal that "dark" nuclei consisting of dark proton strings could provide a representation of the genetic code. The proposal of Hu and Wu translates to the assumption that lipids of the two layers of the cell membrane are accompanied by dark protons which arrange themselves to dark protonic strings defining a dark analog of DNA double strand (see this ).

Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD

The generation of matter-antimatter asymmetry is still poorly understood. There exists a multitude of models but no convincing one. In TGD framework the generation of matter-antimatter asymmetry can be explained in terms of cosmic strings carrying dark energy identified as Kähler magnetic energy (see this). Their decay to ordinary and dark matter would be the analog for the decay of the inflaton field to matter and the asymmetry would be generated in this process. The details of the process have not been considered hitherto.

The stimulus for constructing a general model for this process came from attempt to understand the notion of sphaleron claimed to allow a non-perturbative description for a separate non-conservation of baryon and lepton numbers in standard model. The separate non-conservation of B and L would make possible models of baryo-genesis and even lepto-genesis assuming that in the primordial situation only right-handed inert neutrinos are present. To my opinion these models however fail mathematically because they equate the non-conservation of axial fermion numbers - which is on a mathematically sound basis - with the non-conservation of fermion numbers. This kind of assumption is unjustified and to my opinion is misuse of the attribute "non-perturbative".

The basic vision about lepto-genesis followed by baryo-genesis is however very attractive. This even more so because right-handed neutrino is in a completely unique role in TGD Universe. The obvious question therefore is whether this vision could make sense also in TGD framework. It would be wonderful if cosmic strings - infinitely thin Kähler magnetic flux tubes carrying magnetic monopole field, which later develop finite sized and expanding M⁴ projection - carrying only right-handed neutrinos were the fundamental objects from which matter would have emerged in a manner analogous to the decay of vacuum expectations of instanton fields (see this). Even better, Kähler magnetic energy has interpretation as dark energy and magnetic tension gives rise to the negative "pressure" inducing accelerated expansion of the Universe.

The basic question is whether B and L are conserved separately or not. In TGD Universe one can consider two options depending on the answer to this question. For option I - the "official" version of TGD - quarks and leptons correspond to opposite 8-D chiralities of the induced spinor fields and B and L are conserved separately. For option II (see this) only leptonic spinor fields would be fundamental, and the idea is that quarks could be fractionally charged leptons. This option could lead to genuine baryo-genesis, and in the simplest model baryons would be generated from 3-leptons as 3-sheeted structures for which fractionization of color hyper-charge occurs. Leptonic imbedding space spinors moving in triality zero color partial waves would be replaced with triality +/- 1 partial waves assigned with quarks. Whether this replacement is on a mathematically sound basis, is far from obvious since induced spinor fields at space-time level would couple to induced spinor fields with leptonic couplings.

In any case, one can check whether leptogenesis, baryogenesis, and matter antimatter asymmetry are possible for either of both of these options. It turns out that for both option I and II one can construct simple model in terms for the generation of quarks from leptons via emission of lepto-quarks analogous to gauge bosons but differing from their counterparts in GUTs. Option II allows also genuine baryogenesis from leptons. The conclusion is that the "official" version of TGD predicting separate conservation of B and L allows an elegant vision about the generation of matter from cosmic strings containing only right-handed neutrinos in the initial states.

For details see the article Matter-antimatter asymmetry, baryo-genesis, lepto-genesis, and TGD of the chapter TGD and Astrophysics of "Physics in Many-Sheeted Space-time".

p-Adic symmetries

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the third one and devoted to p-adic symmetries.

A further objection relates to symmetries. It has become already clear that discrete subgroups of Lie-groups of symmetries cannot be realized p-adically without introducing algebraic extensions of p-adics making it possible to represent the p-adic counterparts of real group elements. Therefore symmetry breaking is unavoidable in p-adic context: one can speak only about realization of discrete sub-groups for the direct generalizations of real symmetry groups. The interpretation for the symmetry breaking is in terms of discretization serving as a correlate for finite measurement resolution reflecting itself also at the level of symmetries.

This observation has led to TGD inspired proposal for the realization of the p-adic counterparts symmetric spaces resembling the construction of P¹(K) in many respects but also differing from it.

1. For TGD option one considers a discrete subgroup G₀ of the isometry group G making sense both in real context and for extension of p-adic numbers. One combines G₀ with a p-adic counterpart of Lie group G_p obtained by exponentiating the Lie algebra by using p-adic parameters t_i in the exponentiation exp(t_iT_i).
   
   
2. One obtains actually an inclusion hierarchy of p-adic Lie groups. The levels of the hierarchy are labelled by the maximum p-adic norms |t_i|_p= p^-n_i, n_i ≥ 1 and in the special case n_i=n - strongly suggested by group invariance - one can write G_p,1 ⊃ G_p,2 ⊃ ...G_p,n .... G_p,i defines the p-adic counterpart of the continuous group which gets the smaller the larger the value of n is. The discrete group cannot be obtained as a p-adic exponential (although it can be obtained as real exponential), and one can say that group decomposes to a union of disconnected parts corresponding to the products of discrete group elements with G_p,n.
   

   This decomposition to totally uncorrelated disjoint parts is of course worrying from the point of view of algebraic continuation. The construction of p-adic manifolds by using canonical identification to define coordinate charts as real ones allows a correspondence between p-adic and real groups and also allows to glue together the images of the disjoint regions at real side: this induces gluing at p-adic side. The procedure will be discussed later in more detail.
   

3. There is a little technicality is needed. The usual Lie-algebra exponential in the matrix representation contains an imaginary unit. For p mod 4 =3 this imaginary unit can be introduced as a unit in the algebraic extension. For p mod 4 =1 it can be realized as an algebraic number. It however seems that imaginary unit or its p-adic analog should belong to an algebraic extension of p-adic numbers. The group parameters for algebraic extension of p-adic numbers belong to the algebraic extension. If the algebraic extension contains non-trivial roots of unity U_m,n= exp(i2 π m/n), the differences U_m,n-U^*_m,n are proportional to imaginary unit as real numbers and one can replace imaginary unit in the exponential with U_m,n-U^*_m,n. In real context this means only a rescaling of the Lie algebra generator and Planck constant by a factor (2sin(2 π m/n))^-1. A natural imaginary unit is defined in terms of U_1,pⁿ.
   
   
4. This construction is expected to generalize to the case of coset spaces and give rise to a coset space G/H identified as the union of discrete coset spaces associated with the elements of the coset G₀/H₀ making sense also in the real context. These are obtained by multiplying the element of G₀/H₀ by the p-adic factor space G_p,n/H_p,n.
   

One has two hierarchies corresponding to the hierarchy of discrete subgroups of G₀ requiring each some minimal algebraic extension of p-adic numbers and to the hierarchy of G_p:s defined by the powers of p. These two hierarchies can be assigned to angles (actually phases coming as roots of unity) and p-adic length scales in the space of group parameters.

The Lie algebra of the rotation group spanned by the generators L_x,L_y,L_z provides a good example of the situation and leads to the question whether the hierarchy of Planck constants kenociteallb/Planck could be understood p-adically.

1. Ordinary commutation relations are [L_x,L_y]= i hbar L_z. For the hierarchy of Lie groups it is convenient to extend the algebra by introducing the generators L_iⁿ⁾= pⁿL_i and one obtains [L_x^m),L_yⁿ⁾]= i hbar L_z^m+n). This resembles the commutation relations of Kac-Moody algebra structurally.
   
   
2. For the generators of Lie-algebra generated by L_i^m) one has [L_x^m),L_y^m)]= ip^m hbar L_z^m). One can say that Planck constant is scaled from hbar to p^m hbar. Could the effective hierarchy of Planck constants assigned to the multi-furcations of space-time sheets correspond in p-adic context to this hierarchy of Lie-algebras?
   
   
3. The values of the Planck constants would come as powers of primes: the hypothesis has been that they comes as positive integers. The integer n defining the number of sheets for n-furcation would come as powers n=p^m. The connection between p-adic length scale hierarchy and hierarchy of Planck constants has been conjectured already earlier but the recent conjecture is the most natural one found hitherto. Of course, the question whether the number sheets of furcation correlates with the power of p characterizing "small" continuous symmetries remains an open question. Note that also n-adic and even q=m/n-adic topology is possible with norms given by powers of integer or rational. Number field is however obtained only for primes. This suggests that if also integer - and perhaps even rational valued scales are allowed for causal diamonds, they correspond to effective n-adic or q-adic topologies and that powers of p are favored.
   

The difficult questions concern again integration. The integrals reduce to sums over the discrete subgroup of G multiplied with an integral over the p-adic variant G_p,n of the continuous Lie group. The first integral - that is summation - is number theoretically universal. The latter integral is the problematic one.

1. The easy way to solve the problem is to interpret the hierarchy of continuous p-adic Lie groups G_p,n as analogs of gauge groups. But if the wave functions are invariant under G_p,n, what is the situation with respect to G_p,m for m<n? Infinitesimally one obtains that the commutator algebras [G_p,k,G_p,l] ⊂ G_p,k+l must annihilate the functions for k+l ≥ n. Does also G_p,m, m<n annihilate the functions for as a direct calculation demonstrates in the real case. If this is the case also p-adically the hierarchy of groups G_p,n would have no physical implications. This would be disappointing.
   
   
2. One must however be very cautious here. Lie algebra consists of first order differential operators and in p-adic context the functions annihilated by these operators are pseudo-constants. It could be that the wave functions annihilated by G_p,n are pseudo-constants depending on finite number of pinary digits only so that one can imagine of defining an integral as a sum. In the recent case the digits would naturally correspond to powers p^m, m<n. The presence of these
   functions could be purely p-adic phenomenon having no real counterpart and emerge when one
   leaves the intersections of real and p-adic worlds. This would be just the non-determinism of imagination assigned to p-adic physics in TGD inspired theory of consciousness.
   

Is there any hope that one could define harmonic analysis in G_p,n in a number theoretically universal manner? Could one think of identifying discrete subgroups of G_p,n allowing also an interpretation as real groups?

1. Exponentiation implies that in matrix representation the elements of G_p,n are of form g= Id+ pⁿg₁: here Id represents real unit matrix. For compact groups like SU(2) or CP₂ the group elements in real context are bounded above by unity so that this kind of sub-groups do not exist as real groups. For non-compact groups like SL(2,C) and T⁴ this kind of subgroups make sense also in real context.
   
   
2. Zero energy ontology suggests that discrete but infinite sub-groups Γ of SL(2,C) satisfying certain additional conditions define hyperbolic spaces as factor spaces H³/ Γ (H³ is hyperboloid of M⁴ lightcone). These spaces have constant sectional curvature and very many 3-manifolds allow a hyperbolic metric with hyperbolic volume defining a topological invariant. The moduli space of CDs contains the groups Γ defining lattices of H³ replacing it in finite measurement resolution. One could imagine hierarchies of wave functions restricted to these subgroups or H³ lattices associated with them. These wave functions would have the same form in both real and p-adic context so that number theoretical universality would make sense and one could perhaps define the inner products in terms of "integrals" reducing to sums.
   
   
3. The inclusion hierarchy G_p,n ⊃ G_p,n+1 would in the case of SL(2,C) have interpretation in terms of finite measurement resolution for four-momentum. If G_p,n annihilate the physical states or creates zero norm states, this inclusion hierarchy corresponds to increasing IR cutoff (note that short length scale in p-adic sense corresponds to long scale in real sense!). The hierarchy of groups G_p,n makes sense also in the case of translation group T⁴ and also now the interpretation in terms of increasing IR cutoff makes sense. This picture would provide a group theoretic realization for with the vision that p-adic length scale hierarchies correspond to hierarchies of length scale measurement resolutions in M⁴ degrees of freedom.
   

Canonical identification and the definition of p-adic counterparts of Lie groups

For Lie groups for which matrix elements satisfy algebraic equations, algebraic subgroups with rational matrix elements could regarded as belonging to the intersection of real and p-adic worlds, and algebraic continuation by replacing rationals by reals or p-adics defines the real and p-adic counterparts of these algebraic groups. The challenge is to construct the canonical identification map between these groups: this map would identify the common rationals and possible common algebraic points on both sides and could be seen also a projection induced by finite measurement resolution.

A proposal for a construction of the p-adic variants of Lie groups was discussed in previous section. It was found that the p-adic variant of Lie group decomposes to a union of disjoint sets defined by a discrete subgroup G₀ multiplied by the p-adic counterpart G_p,n of the continuous Lie group G. The representability of the discrete group requires an algebraic extension of p-adic numbers. The disturbing feature of the construction is that the p-adic cosets are disjoint. Canonical identification I_k,l suggests a natural solution to the problem. The following is a rough sketch leaving a lot of details open.

1. Discrete p-adic subgroup G₀ corresponds as such to its real counterpart represented by matrices in algebraic extension of rationals. G_p,n can be coordinatized separately by Lie algebra parameters for each element of G₀ and canonical identification maps each G_p,n to a subset of real G. These subsets intersect and the chart-to-chart identification maps between Lie algebra coordinates associated with different elements of G₀ are defined by these intersections. This correspondence induces the correspondence in p-adic context by the inverse of canonical identification.
   
   
2. One should map the p-adic exponentials of Lie-group elements of G_p,n to their real counterparts by some form of canonical identification.
   
   1. Consider first the basic form I=I_{0, ∞} of canonical identification mapping all p-adics to their real counterparts and maps only the p-adic integers 0 ≤ k<p to themselves.
      

      The gluing maps between groups G_p,n associated with elements g_m and g_n of G₀ would be defined by the condition g_m I(exp(it_aT^a)= g_n I(exp(iv_aT^a). Here t_a and v_a are Lie-algebra coordinates for the groups at g_m and g_n. The delicacies related to the identification of p-adic analog of imaginary unit have been discussed in the previous section. It is important that Lie-algebra coordinates belong to the algebraic extension of p-adic numbers containing also the roots of unity needed to represent g_n. This condition allows to solve v_a in terms of t_a and v_a= v_a(t_b) defines the chart map relating the two coordinate patches on the real side. The inverse of the canonical identification in turn defines the p-adic variant of the chart map in p-adic context. For I this map is not p-adically analytic as one might have guessed.
      
      

   2. The use of I_k,n instead of I gives hopes about analytic chart-to chart maps on both sides. One must however restrict I_k,n to a subset of rational points (or generalized points in algebraic extension with generalized rational defined as ratio of generalized integers in the extension). Canonical identification respects group multiplication only if the integers defining the rationals m/n appearing in the matrix elements of group representation are below the cutoff p^k. The points satisfying this condition do not in general form a rational subgroup. The real images of rational points however generate a rational sub-group of the full Lie-group having a manifold completion to the real Lie-group.
      

      One can define the real chart-to chart maps between the real images of G_p,k at different points of G₀ using I_k,l(exp(iv_aT^a)= g_n^-1g_m × I_k,l(exp(it_aT^a). When real charts intersect, this correspondence should allow solutions v_a,t_b belonging to the algebraic extension and satisfying the cutoff condition. If the rational point at the other side does not correspond to a rational point it might be possible to perform pinary cutoff at the other side.
      

      Real chart-to-chart maps induce via common rational points discrete p-adic chart-to-chart maps between G_p,k. This discrete correspondence should allow extension to a unique chart-to-chart map the p-adic side. The idea about algebraic continuation suggests that an analytic form for real chart-to-chart maps using rational functions makes sense also in the p-adic context.
      

3. p-Adic Lie-groups G_p,k for an inclusion hierarchy with size characterized by p^-k. For large values of k the canonical image of G_p,k for given point of G₀ can therefore intersect its copies only for a small number of neighboring points in G₀, whose size correlates with the size of the algebraic extension. If the algebraic extension has small dimension or if k becomes large for a given algebraic extension, the number of intersection points can vanish. Therefore it seems that in the situations, where chart-to-chart maps are possible, the power p^k and the dimension of algebraic extension must correlate. Very roughly, the order of magnitude for the minimum distance between elements of G₀ cannot be larger than p^-k+1. The interesting outcome is that the dimension of algebraic extension would correlate with the pinary cutoff analogous to the IR cutoff defining measurement resolution for four-momenta.
   

For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.

Could canonical identification make possible definition of integrals in p-adic context?

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the second one and devoted to p-adic integration.

The notion of p-adic manifold using using real chart maps instead of p-adic ones allows an attractive approach also to p-adic integration and to the problem of defining p-adic version of differential forms and their integrals.

1. If one accepts the simplest form of canonical identification I(x): ∑_n x_npⁿ → ∑ x_np^-n, the image of the p-adic surface is continuous but not differentiable and only integers n<p are mapped to themselvs. One can define integrals of real functions along images of the p-adically analytic curves and define the values of their p-adic counterparts as their algebraic continuation when it exists.
   

   In TGD framework this does not however work. If one wants to define induced quantities - such as metric and K ähler form - on the real side one encounters a problem since the image surface is not smooth and the presence of edges implies that these quantities containing derivatives of imbedding space coordinates possess delta function singularities. These singularities could be even dense in the integration region so that one would have no-where differentiable continuous functions and the real integrals would reduce to a sum which do not make sense.
   
   

2. In TGD framework finite measurement resolution realized in terms of pinary cutoff however saves the situation. The canonical identification I_k,l(m/n) = I_k,l(m)/I_kl(n) maps rationals to themselves for m<p^k,n<p^k. The second pinary cutoff m<p^l,n<p^l, l>k implies that the chart map takes a discrete subset of p-adic rationals to a discrete set of real rationals. The completion of the discrete image of p-adic preferred extremal under I_k,l to a real preferred extremal is very natural. This preferred extremal can be said to be unique apart from a finite measurement resolution represented by the pinary cutoffs k and l. All induced quantities are well defined on both sides.
   

   p-Adic integrals can be defined as pullbacks of real integrals by algebraic continuation when this is possible. The inverse image of the real integration region in canonical identification defines the p-adic integration region.
   
   

3. The integrals of p-adic differential forms can be defined as pullbacks of the real integrals. The integrals of closed forms, which are typically integers, would be the same integers but interpreted as p-adic integers.
   

It is interesting to study the algebraic continuation of K ähler action from real sector to p-adic sectors.

1. K ähler action for both Euclidian and Minkowskian regions reduces to the algebraic continuation of the integral of Chern-Simons-K ähler form over preferred 3-surfaces. The contributions from Euclidian and Minkowskian regions reduce to integrals of Chern-Simons form over 3-surfaces. I have somewhere considered the possibility that the 3-surfaces for Minkowskian and Euclidian contribution might be identical: this cannot be the case since the space-like 3-surfaces at the boundaries of CD for Minkowskian and Euclidian regions are disjoint.
   

   The contribution from Euclidian regions defines K ähler function of WCW and the contribution from Minkowskian regions giving imaginary exponential of K ähler action has interpretation as Morse function whose stationary points are expected to select special preferred extremals. One would expect that both functions have a continuous spectrum of values. In the case of K ähler function this is necessary since K ähler function defines the K ähler metric of WCW via its second derivatives in complex coordinates by the well-known formula. Note that by the above observation K ähler and Morse functions are not in general proportional to each other.
   
   

2. The algebraic continuation of the exponent of K ähler function for a given p-adic prime is expected to require the proportionality to pⁿ so that not all preferred extremals are expected to allow a continuation to a given p-adic number field. This kind of assumption has been indeed made in the case of deformations of CP₂ type extremals in order to derive formula for the gravitational constant in terms of basic parameters of TGD but without real justification (see this).
   
   
3. The condition that the action exponential in the Minkowskian regions is a genuine phase factor implies that it reduces to a root of unity (one must have an algebraic extension of p-adic numbers). Therefore the contribution to the imaginary exponent K ähler action from these regions for the p-adicizable preferred extremals should be of form 2 π (k+m/n).
   

   If all preferred real extremals allow p-adic counterpart, the value spectrum of the Morse function on the real side is discrete and could be forced by the preferred extremal property. If this were the case the stationary phase approximation around extrema of K ähler function on the real side would be replaced by sum with varying phase factors weighted by K ähler function.
   

   An alternative conclusion is that the algebraic continuation of K ähler action to any p-adic field is possible only for a subset of preferred extremals with a quantized spectrum of Morse function. One the real side stationary phase approximation would make sense. It however seems that the stationary phases must obey the above discussed quantization rule.
   

Also holomorphic forms allow algebraic continuation and one can require that also their integrals over cycles do so. An important example is provided by the holomorphic one-forms integrals over cycles of partonic 2-surface defining the Teichmueller parameters characterizing the conformal equivalence class of the partonic 2-surfaces as Riemann surface. The p-adic variants exist of these parameters exist if they allow an algebraic continuation to a p-adic number. The algebraic continuation from the real side to the p-adic side would be possible on for certain p-adic primes p if any: this would allow to assign p-adic prime or primes to a given real preferred extremal. This justifies the assumptions of p-adic mass calculations concerning the contribution of conformal modular degrees of freedom to mass squared (see this).

For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.

Could canonical identification allow construction of path connected topologies for p-adic manifolds?

The recent progress in the formulation of the notion of p-adic manifold is so important for the program of defining quantum TGD in mathematically rigorous manner that it deserves a series of more detailed postings devoted to the notion of p-adic manifold, p-adic integration, and p-adic symmetries. This posting is the first one and devoted to the notion of p-adic manifold.

Total disconnectedness of p-adic numbers as the basic problem

The total dis-connectedness of p-adic topology and lacking correspondence with real manifolds could be seen as genuine problem in the purely formal construction of p-adic manifolds. Physical intuition suggests that path connected should be realized in some natural manner and that one should have a close connection with real topology which after all is the "lab topology".

In TGD framework one of the basic physical problems has been the connection between p-adic numbers and reals. Algebraic and topological approaches have been competing also here.

1. Algebraic approach suggests the identification of reals and various p-adic numbers along common rationals but this correspondence is non-continuous. Above some resolution defined by power of p it must be replaced with a correspondence is continuous unless one uses pinary cutoff. Below this cutoff the pseudo-constants of p-adic differential equations would naturally relate to the identification of p-adics and reals along common rationals (plus common algebraics in the case of algebraic extensions).
   
   
2. Topological approach relies on canonical identification and its variants mapping p-adic numbers to reals in a continuous manner. This correspondence is however problematic in the sense that does not commute with the basic symmetries as correspondence along common rationals would do for subgroups of the symmetries represented in terms of rational matrices. A further problematic aspect of canonical identification is that it does not commute with the field equations.
   
   
3. The notion of finite measurement resolution allows to find a compromise between the symmetries and continuity (that is, algebra and topology). Canonical identification can be modified so that it maps rationals to themselves only up to some pinary digits but is still continuous in p-adic sense. Canonical identification could map only a skeleton formed by discrete point set - analogous to Bruhat-Tits building - from real to p-adic context and the preferred extremals on both sides would contain this skeleton.
   

Canonical identification combined with the identification of common rationals in finite pinary resolution suggests also a manner of replacing p-adic topology with a path connected one. This topology would be essentially real topology induced to p-adic context by canonical identification used to build real chart leafs.

1. Canonical identification maps p-adic numbers ∑ x_npⁿ to reals and is defined by the formula I(x) = ∑ x_np^-n. I is a continuous map from p-adic numbers to reals. Its inverse is also continuous but two-valued for a finite number of pinary digits since the pinary expansion of real number is not unique (1=.999999.. is example of this in 10-adic case). For a real number with a finite number of pinary digits one can always choose the p-adic representative with a finite number of pinary digits.
   
   
2. Canonical identification is used to map the predictions of p-adic mass calculations to map the p-adic value of the mass squared to its real counterpart. It makes also sense to map p-adic probabilities to their real counterparts by canonical identification. In TGD inspired theory of consciousness canonical identification is a good candidate for defining cognitive representations as representations mapping real preferred extremals to p-adic preferred extremals as also for the realization of intentional action as a quantum jump replacing p-adic preferred extremal representing intention with a real preferred extremal representing action. Could these cognitive representations and their inverses actually define real coordinate charts for the p-adic "mind stuff" and vice versa?
   
   
3. Canonical identification has several variants. For instance, one can map p-adic rational number m/n regarded as a p-adic number to a real number I(m)/I(n). In this case canonical identification respects rationality but is ill-defined for p-adic irrationals. This is not a catastrophe if one has finite measurement resolution meaning that only rationals for which m<p^l,n<p^l are mapped to the reals (real rationals actually).
   

   One can also express p-adic number as expansion of powers fo p^k and define canonical identification I_k as ∑ x_np^kn → ∑ x_np^-kn. Also the variant I_k,l(m/n)=I_k,l(m)/I_k,l(n) with l defining pinary cutoff for m and l makes sense. One can say that I_k,l(m/n) identifies p-adic and real numbers along common rationals for p-adic numbers with a pinary cutoff defined by k and maps them to rationals for pinary cutoff defined by l. Discrete subset of rational points on p-adic side is mapped to a discrete subset of rational points on real side by this hybrid of canonical identification and identification along common rationals. This form of canonical identification is the one needed in TGD framework.
   
   

4. Canonical identification does not commute with rational symmetries unless one uses the map I_k,l(m/n)=I_k,l(m)/I_k,l(n) and also now only in finite resolutions defined by k. For the large p-adic primes associated with elementary particles this is not a practical problem (electron corresponds to M₁₂₇=2¹²⁷-1!) The generalization to algebraic extensions makes also sense. Canonical identification breaks general coordinate invariance unless one uses group theoretically preferred coordinates for M⁴ and CP₂ and subset of these for the space-time region considered.
   

What is very remarkable is that canonical identification can be seen as a continuous generalization of the p-adic norm defined as N_p(x) == I_k,l(x) having the highly desired Archimedean property. I_k,l is the most natural variant of canonical identification.

1. Canonical identification for the various coordinates defines a chart map mapping regions of p-adic manifold to Rⁿ₊. That each coordinate is mapped to a norm N_p(x) means that the real coordinates are always non-negative. If real spaces Rⁿ₊ would provide only chart maps, it is not necessary to require approximate commutativity with symmetries. Also Berkovich considers norms but for a space of formal power series assigned with the p-adic disk: in this case however the norms have extremely low information content.
   
   
2. I_k,l(x) indeed defines the analog of Archimedean norm in the sense that one has N_p(x+y) ≤ N_p(x)+N_p(y). This follows immediately from the fact that the sum of pinary digits can vanish modulo p. The triangle inequality holds true also for the rational variant of I. N_p(x) is however not multiplicative: only a milder condition N_p(pⁿx)=N(pⁿ)N(x)=p^-n N(x) holds true.
   
   
3. Archimedean property gives excellent hopes that p-adic space provided with chart maps for the coordinates defined by canonical identification inherits real topology and its path connectedness. A hierarchy of topologies would be obtained as induced real topologies and characterized by various norms defined by I_k,l labelled by a finite measurement resolution. This would give a very close connection with physics.
   
   
4. The mapping of p-adic manifolds to real manifolds would make the construction of p-adic topologies very concrete. For instance, one can map real preferred subset of rationalp oints of a real extremal to a p-adic one by the inverse of canonical identification by mapping the real points with finite number of pinary digits to p-adic points with a finite number of pinary digits. This does not of course guarantee that the p-adic preferred extremal is unique. One could however hope that p-adic preferred extrremals can be said to possess the invariants of corresponding real topologies in finite measurement resolution.
   
   
5. The maps between different real charts would be induced by the p-adically analytic maps between the inverse images of these charts. At the real side the maps would be consistent with the p-adic maps only in the discretization below pinary cutoff.
   
   
6. As already mentioned, one must restrict the p-adic points mapped to reals to rationals since I_k,l(m/n) is not well-defined for p-adic irrationals (having non-periodic pinary expansion: note however that one can consider also p-adic integers). For the restriction to finite rationals the chart image on real side would consist of rational points. The cutoff would mean that these rationals are not dense in the set of reals. Preferred extremal property could however allow to identify the chart leaf as a piece of preferred extremal containing the rational points in the measurement resolution use. This would realize the dream of mapping p-adic p-adic preferred extremals to real ones playing a key role in number theoretical universality.
   

To sum up, chart maps are constructed in two steps and works in both directions. For p-adic-to-real case a subset of rational points of the p-adic preferred extremal would be mapped using I_k,l to rational points of the real preferred extremal. Field equations for the preferred extremal would be then used to complete the resulting discrete skeleton to a full map leaf. Of course also algebraic extensions can and must be considered. This kind of completion performed in iterative manner has been also proposed assuming that space-time surfaces are quaternionic surfaces (tangent spaces are in well-defined sense quaterionic sub-space of octonionic space containing complex octonions as a preferred sub-space this).

What about p-adic coordinate charts for a real preferred extremal?

What is remarkable that one can also build p-adic coordinate charts about real preferred extremal using the inverse of the canonical identification assuming that finite rationals are mapped to finite rationals. There are actually good reasons to expect that coordinate charts make sense in both directions.

Algebraic continuation from real to p-adic context is one such reason. At the real side one can calculate the values of various integrals like K ähler action. This would favor p-adic regions as map leafs. One can require that K ähler action for Minkowskian and Euclidian regions (or their appropriate exponents) make sense p-adically and define the values of these functions for the p-adic preferred extremals by algebraic continuation. This could be very powerful criterion allowing to assign only very few p-adic primes to a given real space-time surface. This would also allow to define p-adic boundaries as images of real boundaries in finite measurement resolution. p-Adic path connectedness would be induced from real path-connectedness.

p-Adic rationals include also the ratios of integers, which are infinite as real integers so that the pinary expansion of the rational is not periodic asymptotically. In principle one could imagine of mapping also these to real numbers but the resulting skeleton might be too dense and might not allow to satisfy the preferred extremal property. Furthermore, the representation of a p-adic number as a ratio of this kind of integers is not unique and can be always tranformed to an infinite p-adic integer multiplied by a power of p . In the same manner real points which can be regarded as images of ratios of p-adic integers infinite as real integers could be mapped to p-adic ones but same problem is encountered also now.

In the intersection of real and p-adic worlds the correspondence is certainly unique and means that one interprets the equations defining the p-adic space-time surface as real equations. The number of rational points (with cutoff) for the p-adic preferred extremal becomes a measure for how unique the chart map in the general case can be. For instance, for 2-D surfaces the surfaces xⁿ+yⁿ=zⁿ allow no nontrivial rational solutions for n>2 for finite real integers. This criterion does not distinguish between different p-adic primes and algebraic continuation is needed to make this distinction.

Chart maps for p-adic manifolds

The real map leafs must be mutually consistent so that there must be maps relating coordinates used in the overlapping regions of coordinate charts on both real and p-adic side. On p-adic side chart maps between real map leafs are naturally induced by identifying the canonical image points of identified p-adic points on the real side. For discrete chart maps I_k,l with finite pinary cutoffs one one must complete the real chart map to - say diffeomorphism. That this completion is not unique reflects the finite measurement resolution.

In TGD framework the situation is dramatically simpler. For sub-manifolds the manifold structure is induced from that of imbedding space and it is enough to construct the manifold structure M⁴ × CP₂ in a given measurement resolution (k,l). Due to the isometries of the factors of the imbedding space, the chart maps in both real and p-adic case are known in preferred imbedding space coordinates. As already discussed, this allows to achieve an almost complete general coordinate invariance by using subset of imbedding space coordinates for the space-time surface. The breaking of GCI has interpretation in terms of presence of cognition and selection of quantization axes.

For instance, in the case of Riemann sphere S² the holomorphism relating the complex coordinates in which rotations act as M öbius tranformations and rotations around -call it z-axis- act as phase multiplications - the coordinates z and w at Norther and Southern hemispheres are identified as w=1/z restricted to rational points at both side. For CP₂ one has three poles instead of two but the situation is otherwise essentially the same.

For details and background see the article the article What p-adic icosahedron could mean? And what about p-adic manifold? at my homepage.