Wednesday, August 18, 2021

Could the notion of a polynomial of infinite degree make sense?

TGD provides motivations for the question whether the notion polynomial of infinite degree could make sense. In the following I consider this question from the point of view of physicist and start from the vision about physics as generalized number theory.

1. Background and motivations for the idea

M8-H-duality (H= M4× CP2) states that space-time surfaces defined as 4-D roots of complexified octonionic polynomials so that they have quaternionic normal space, can be mapped to 4-surfaces in H (see this, this, and this).

The octonionic polynomials are obtained by algebraic continuation of ordinary real polynomials with rational coefficients although one can also consider algebraic coefficients.

This construction makes sense also for analytic functions with rational (or algebraic) coefficients. For the twistor lift of TGD, cosmological constant Λ emerges via the coefficient of a volume term of the action containing also Kähler action (see this). This leads to an action consisting of Kähler action with both CP2 and M4 terms having very interesting and physically attractive properties, such as spin glass degeneracy. Λ=0 would correspond to an infinite volume limit making the QFT description possible as an approximate description. Also the thermodynamic limit could correspond to this limit (see this).

Irreducible polynomials of rational coefficients give rise to algebraic extensions characterized by the Galois group and these notions are central in adelic vision (see this).

I do not know of any deep reason preventing analytic functions with rational Taylor coefficients. These would make possible transcendental extensions. For instance, the product ∏p(ex-p) for some set of primes p would give as roots transcendental numbers log(p). The Galois group would be however trivial although the extension is infinite. Second example is provided by trigonometric functions sin(x) and cos(x) with roots coming as multiples of nπ and (2n+1)π/2. This might be necessary in order to have Fourier analysis. The translations by a multiple of π for x act permuted roots but do not leave rational numbers rational so that the interpretation as a Galois group is not possible so that also now Galois group would be trivial.

A long standing question has been whether these analytic functions could be regarded as polynomials of infinite order by posing some conditions to the Taylor coefficients. If so, one might hope that the notion of Galois group could make sense also now, and one would obtain a unified view about transcendental extensions of rationals.

  1. For polynomials as roots of octonionic polynomials space-time surfaces are finite and located inside finite-sized causal diamond (CD).

    In the TGD Universe cosmological constant Λ depends on the p-adic length scale and approaches zero at infinite length scale. At the Λ=0 limit, which corresponds also to QFT and thermodynamical limits, space-time surfaces would have infinite size. Only Kähler action with M4 and CP2 parts and having ground state degeneracy analogous to spin glass degeneracy would be present.

  2. The octonionic algebraic continuations of analytic functions with rational coefficients and subject to restrictions guaranteeing that the notion of prime function makes sense, would define space-time surfaces as their roots.
  3. Prime analytic functions defining space-time surfaces would in some sense be polynomials of infinite degree and could be even characterized by the Galois group. For real polynomials complex conjugations for the roots is certainly this kind of symmetry.

    These functions should have Taylor series at origin, which is a special point for octonionic polynomials with rational (or perhaps even algebraic) coefficients. The selection of origin as a preferred point relates directly to the condition eliminating possible problems due to the loss of associativity and commutativity.

    The prime property is possible only if the set of these polynomials fails to have a field property (so that the inverse of any element would be well-defined) since for fields one does not have the notion of prime. The field property is lost if the allowed functions vanish at origin so that one cannot have a Taylor series at origin and the inverse diverges at origin.

    The vanishing at origin guarantees that the functional composite f○g of f and g has the roots of g. Roots are inherited as algebraical complexity as a kind of evolution increases. In TGD inspired biology, the roots of polynomials are analogous to genes and the conservation of roots in the function composition would be analogous to the conservation of genes.

2. Intuitive view about the situation

Could one make anything concrete about this idea? What kind of functions f could serve as analogs of polynomials of infinite degree with transcendental roots. Could any analytic function with rational coefficients vanishing at origin have a possibly unique decomposition to prime analytic functions?

  1. Suppose that the analytic prime decomposes to a product over monomials x-xi with transcendental roots xi such that the Taylor series has rational coefficients. This requires an infinite Taylor series.
  2. One obtains an infinite number of conditions. Each power xn in f has a rational coefficient fn equal to the sum over all possible products ∏k=1n xik of n transcendental roots xik. This gives an infinite number of conditions and each condition involves an infinite number of roots. If the number N of transcendental roots is finite as it is for polynomials, each term involves a finite number of products and the conditions imply that the roots are algebraic. The number of transcendental roots must therefore be infinite. At least formally, these conditions make sense.
  3. The sums of products are generalized symmetric functions of transcendental roots and should have rational values equal to xn. This generalizes the corresponding condition for ordinary polynomials. Symmetric functions for Sn have Sn as a group of symmetries. For a Galois extension of a polynomial of order n, the Galois group is a subgroup of Sn. This suggests that the Galois group is a subgroup of S. S has the simple A of even permutations as a subgroup . The simple groups are analogs of primes for finite groups and one can hope that this is true for infinite and discrete groups (see this).
There are infinitely many ways to represent an algebraic extension in terms of a polynomial and the same is true for transcendental extensions with the rationality condition.
  1. Consider a general decomposition of the polynomial of an infinite order to a product of monomials with roots spanning the possibly transcendental extension. Could a suitable representation of extension as an infinite polynomial allow rational coefficients fn for the function ∑ fn xn defined by the infinite product?
  2. fn is the sum over all possible products of roots obtained by dropping n different roots from the product of all roots which should be finite and equal to one for the generalization of monic polynomials. Therefore there is an infinite sum of terms, which are inverses of finite products and therefore transcendental but one can hope that the infinite number of the summands allows the rationality condition to be satisfied.

3. Profinite groups and Galois extensions as inverse limits

Infinite groups indeed appear as Galois groups of infinite extensions. Absolute Galois groups, say Galois groups of algebraic numbers, provide the basic example.

  1. There exists a natural topology, known as Krull topology, which turns Galois group to a profinite group (totally disconnected, Hausdorff topological group) (see this), which is also Stone space (see this).
  2. Profinite groups are not countably infinite but are effectively finite just as hyper-finite factors of type II1 are finite-dimensional: they appear naturally in the TGD framework. Profinite groups have Haar measure giving them a finite volume. Profinite groups behave in many respects like finite groups (compact groups also behave in this manner as far representations are considered). Profiniteness is possessed by products, closed subgroups, and the coset groups associated with the closed normal subgroups.
  3. Every profinite infinite group is a Galois group for an infinite extension for some field K but one cannot control which field K is realized for a given profinite group. Additive p-adics groups and their products appear as Galois groups of an infinite extension for some field K. The Galois theory of infinite field extensions involves profinite groups obtained as Galois groups for the inverse limits of finite field extensions ..Fn→ Fn+1→ .
  4. This kind of iterated extensions are of special interest in the TGD framework and an infinite extension would be obtained at the limit (see this). The naive expectation is that the polynomial of infinite degree is a limit of a composite ...Pn○ Pn-1..○ P1 of rational polynomials. The number of infinite extensions obtained in this manner would be infinite.

    An interesting question is under what conditions the limiting infinite polynomial exists as an analytic function and whether the Taylor coefficients are rational or in some extension of rationals. The naive intuition is that the inverse limit preserves rationality.

  5. The identification as the iterate ...Pn○ Pn-1○ P1 is indeed suggestive. Infinite cyclic extension defined at the limit by the polynomial xN, N=∞, to be discussed below, has this kind of interpretation. The Galois group of this kind of extension is however not simple.

    Remark: The polynomials in question satisfying P(0)=0 are not irreducible: the composite of N polynomials has xN as a factor and has 0 as N-fold root. The origin of octonionic M8 appears as an isolated root.

  6. Is the infinite-D extension obtained as an inverse limit transcendental or algebraic? In the TGD framework the condition that the polynomial P1○ P2 has the roots of P1 as roots implies the loss of the field property of analytic functions making the notion of analytic prime possible. The roots of the infinite polynomial contain all roots of finite polynomials appearing in the sequences. This would suggest that the extension is not transcendental. Giving up the property Pi(0)=0 also leads to a loss of root inheritance.
For finite-dimensional Galois extensions, there exists an infinite number of polynomials generating the extension and one can consider families of extensions parametrized by a set of rational parameters. The Galois group does not change under small variations of parameters (see this). If the inverse limit based on an infinite composite of polynomials makes sense, the situation could be the same for possibly existing rational polynomials of infinite order? The study of infinite Galois groups could provide insights on the problem.

4. Could infinite extensions of rationals with a simple Galois group exist?

Simple Galois groups have no normal subgroups and are of special interest as the building bricks of extensions by functional composition of polynomials. The infinite Galois groups obtained as inverse limit have however an infinite hierarchy of normal subgroups and simple argument suggests that the extensions are algebraic. Could infinite-D transcendental extensions defined by an analytic function with rational coefficients and with a simple infinite Galois group, exist?

A is simple and could not be seen as an inverse limit. Also the groups PSLn(K) are infinite discrete groups for K=Z or Q. A further example is provided by Tarski monster groups (see this) having only cyclic groups Zp for a fixed p as subgroups and existing or p>1075. For these primes, there is a continuum of these monsters.

If the inverse limit is essential for profiniteness for infinite groups, then simple infinite groups are excluded as Galois groups. Indeed, the topology of an infinite simple group G cannot be profinite. The Krull topology has as a basis for open sets all cosets of normal subgroups H of finite index (the number of cosets gH is finite). Simple group has no normal subgroups except a trivial group consisting of a unit element and the group itself. The only open sets would be the empty set and G itself.

In fact, there is also a theorem stating that every Galois group is profinite (see this). All finite groups are profinite in discrete topology. This theorem however excludes infinite simple Galois groups. If one allows only polynomials with P(0)=0, the conservation of algebraic roots suggests that infinite polynomials with transcendental roots are not possible.

The condition for the failure of the field property however leaves the iterates of polynomials for which only the highest polynomial in the infinite sequence of functional compositions vanishes at origin. These infinite polynomials could have transcendental roots.

Two examples

In the following two examples are consider to test whether the notion of a polynomial of infinite order might work.

5.1 Cyclic extension as an example

The natural question is whether the transcendental roots be regarded as limits of roots for a polynomial with rational coefficients at the limit when the degree N approaches infinity. The above arguments suggest that the limits involve an infinite function composition.

Consider as an example cyclic extension defined by a polynomial XN, which can be regarded as a composite of polynomials xpi for ∏ pi=N. This is perhaps the simplest possible extension than one can imagine.

  1. The roots are now powers of roots of unity. The notion of the root of unity as ei2π/N does not make sense at the limit N→ ∞. One can however consider the roots ei2π M/N and its powers such that the limit M/N → α is irrational. The powers of ei nα give a dense subset of the circle S1 consisting of irrational points. Note that one obtains an infinite number of extensions labelled by irrational values of α.
  2. The polynomial should correspond to the limit PN(x)=xN-1, N→ ∞. For each finite value of N, one has PN(x)= ∏n=1N(x-Un)-1, U= ei2π/N . The reduction to P=xN-1 follows from the vanishing of all terms involving lower powers of x than xN.
  3. If these conditions hold true at the limit N→ ∞, one obtains the same result. The coefficient of xN equals to 1 trivially. The coefficient of xN-1 is the sum over all roots and should vanish. This is also assumed in Fourier analysis ∑n eiα n=0 for irrational α. For α=0 the sum equals to N=∞ identified as Dirac delta function. The lower terms give conditions expected to reduce to this condition. This can be explicitly checked for the coefficient f1.
  4. The Galois group is in this case the cyclic group U∞,α defined by the powers of Uα.
5.2 Infinite iteration yield contimuum or roots

The iterations of polynomials define an N→ ∞ limit, which can be handled mathematically whereas for an arbitrary sequence of polynomials in the functional composition it is difficult to say anything about the possible emergence of transcendental roots. Note however that the LimN→ ∞ (1+1/N)N=e shows that transcendentals can appear as limits of rationals. I have considered iterations of polynomials and approach to chaos from the point of view of M8-H duality in (see this).

Consider polynomials PN= QN○ R, where R with Q(0)=0 is fixed polynomial and QN=Q○N is the N:th iterate of some irreducible polynomial Q with Q(0)=0 and dQ/dz(0)=0. Origin is a fixed critical point of Q and the attractor towards which the points in the attractor basin of origin end up in the iteration and become roots of P and are roots at this limit. For the real points in the intersection of the positive real axis and attractor basin are roots at this limit so that one has a continuum of roots. The set of roots consists of a continuous segment [0,T) and a discrete set coming from the Julia set defining the boundary of the attractor basin.

Profiniteness suggests an interpretation of this set in terms of p-adic topology or a product of a subset p-adic topologies somehow determined by the number theoretic properties of Q. p-Adic number fields are indeed profinite and as additive groups can act as infinite Galois groups permuting the zeros. The action of p-adic translations could indeed leave the basin of attraction invariant.

In the TGD framework these roots correspond to values of M4 time (or actually energy!) in M8 mapped to actual time values in H by M8-H duality. I have referred to them as "very special moments in the life of self" with a motivation coming from TGD inspired theory of consciousness (see this, this). One might perhaps say that at this limit subjective time consisting of these moments becomes continuous in the interval [0,T].

See the article Does the notion of polynomial of infinite order make sense? or the chapter About the role of Galois groups in TGD framework.

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

Articles and other material related to TGD.

Friday, August 13, 2021

Empirical support for the Expanding Earth Model and TGD view about classical gauge fields

I learned about some new-to-me empirical facts providing further support for the Expanding Earth Model (EEM). The first strange finding is the large fluctuations of oxygen levels during the Cambrian Explosion. The general form of EEM applies to all astrophysical objects and could explain the strange lack of craters and volcanic activity in Venus suggesting a global resurfacing for 750 million years ago.

Contrary to expectations, the magnetic field of Venus vanishes. The TGD based view about gauge fields differs from the standard view in that it allows the notion of monopole flux. The monopole part field would be analogous to the external magnetic field H inducing magnetization M as the non-monopole part of B. Venus would be a perfect diamagnet and even a superconductor whereas Earth would be a paramagnet. In the TGD framework, superconductivity driven by the thermal energy feed from the interior of Venus would be possible. The interior of Venus could be a living system but in a very different sense than Earth.

See the article Empirical support for the Expanding Earth Model and TGD view about classical gauge fields.

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

Articles and other material related to TGD. 

Thursday, August 12, 2021

Expanding Venus model

News about unexpected findings relating to the physics of astrophysical objects emerge on an almost daily basis. The most recent news relates to Venus.

Has Venus turned itself inside-out?

The surface of Venus was expected to have craters, just like the surface of Earth, Moon, and Mars but the number of craters is very small. The surface of Venus also has weird features and many volcanoes. Also trace signs of erosion and tectonic shifts were found. The impression is that the surface of Venus had been turned inside out in a catastrophic event that occurred about 750 million years ago.

Since Venus is our sister planet with almost the same mass and radius, it is interesting to notice that the biology of Earth experienced Cambrian explosion 541 million years ago.

  1. The TGD explanation for Cambrian Explosion relies on Expanding Earth model (see this, this, this, and this).

    There was a relatively fast increase of the Earth's radius by factor, which led to the burst of underground oceans to the surface of the Earth and led to the formation of oceans. Standard cosmology predicts a continuous smooth expansion of astrophysical objects. Contrary to this prediction, astrophysical objects do not seem to expand smoothly. In the TGD Universe, the smooth expansion is replaced by rapid jerks and the Cambrian Explosion would be associated with this kind of phase transitions.

  2. In this expansion the multicellular photosynthesizing life burst to the surface. This explains the sudden emergence of highly evolved life forms during the Cambrian Explosion that Darwin realized to be a heavy objection against his theory.

  3. There are many objections to be circumvented. For instance, how photosynthesis could evolve in the underground ocean. Here TGD views dark matter as heff=nh0 phases of ordinary matter, which are relatively dark with respect to each other, come in rescue. Dark water blobs could leak into the interior of Earth and the solar light possessing dark portion could do the same so that photosynthesis became possible (see this).
  4. Did Venus experience a similar rapid expansion 200 million years earlier, about 750 million years ago (or maybe roughly at the same time). Venus does not have water at its surface. This can be understood in terms of heat from solar radiation forcing the evaporation of water and subsequent loss. This also prevented the leakage of the water to the interior of Venus. If there were no water reservoirs inside Venus, no oceans were formed. The cracks of the crust created expanding areas of magma, which were like the bottoms of the oceans at Earth. Also at Earth a fraction about 2/3 of the Earth's surface is sea bottom.

Why does Venus not possess a magnetic field?

Venus offers also a second puzzle. Venus does not have an appreciable magnetic field although it has been speculated that it has had it (see this). The solar dynamo mechanism would suggest its presence.

  1. TGD predicts that there are two kinds of flux tubes carrying Earth's magnetic field BE with a nominal value of .5 Gauss. This applies quite generally. The flux tubes have a closed cross section - this is possible only in TGD Universe, where the space-time is 4-surface in M4× CP2. The flux tubes can have a vanishing Kähler magnetic flux or non-vanishing quantized monopole flux: this has no counterpart in Maxwellian electrodynamics. For Earth, the monopole part would correspond to about .2 Gauss - 2/5 of the full strength of BE.
  2. Monopole part needs no currents to maintain it and this makes it possible to understand how the Earth's magnetic field has not disappeared a long time ago. This also explains the existence of magnetic fields in cosmological scales.

    The orientation of the Earth's magnetic field is varying. In the TGD based model the monopole part plays the role of master. When the non-monopole part becomes too weak, the magnetic body defined by the monopole part changes its orientation. This induced currents refresh the non-monopole part (this). The standard dynamo model is part of this model.

  3. There is an interesting (perhaps more than) analogy with the standard phenomenological description of magnetism in condensed matter. One has B= H+M. H field is analogous to the monopole part and the non-monopole part is analogous to the magnetization M induced by H. B= H+M would represent the total field. If this description corresponds to the presence of two kinds of flux tubes, the TGD view about magnetic fields would have been part of electromagnetism from the beginning!

    Flux tubes can also carry electric fields and also for them this kind of decomposition makes sense. Could also the fields D and H have a similar interpretation?

    In the linear model of magnetism, one has M= χH and B=μH= (1+χ)H. For diamagnets one has χ<0 and for paramagnets χ>0. Earth would be paramagnet with χ ≈ 3/2 if the linear model works. χ is a tensor in the general case so that B and H can have different directions.

  4. All stars and planets, also Venus, correspond to flux tube tangles formed from monopole flux tubes. This leaves only one possibility. Venus behaves like a super-conductor and is an ideal diamagnet with χ=-1 so that B vanishes. The monopole part would be present however.

    This could provide a totally new insight to the Meissner effect and loss of superconductivity. In TGD the based model (see this), monopole flux tubes carry supracurrent. The BSC model however requires the absence of a magnetic field. Could the induced non-monopole field cancel the monopole part. Venus would indeed be a superconductor!

  5. The tilt of the rotation axis relative to the plane of rotation around the Sun is very small for Venus, about 3 degrees and much smaller than for the Earth. This implies that the surface temperature of Venus is roughly constant. At Earth plate tectonics makes possible the heat transfer from the interior to the surface and its leakage to the outer space. For Venus this is not possible.

    Could this relate to the different magnetization properties of Earth and Venus? The TGD based model also predicts superconductivity driven by external energy feed. This would be possible also above critical temperature. The energy feed would increase the value of heff and below the critical temperature it would be provided by the energy liberated in the formation of Cooper pairs which need not actually be the current carriers since dark electrons can carry the current without dissipation. In TGD inspired biology and quite universally, the basic role of metabolic energy feed is to prevent the reductions of the values of heff.

    Could the superconductivity be forced by the thermal energy feed from the interior of Venus? Superconductivity means in the TGD framework large heff and therefore complexity, intelligence, and long quantum coherence length (see this). Could Venus be alive but in a very different sense than Earth? The same question can be of course made in the case of Sun.

    The possibility that life actually appears in cosmic scales and is associated with quantum coherent flux tube networks associated with active galactic nuclei usually identified as super-massive blackholes containing stellar and planetary systems as tangles is discussed here.

    Also Mars lacks the global magnetic field although it has auroras assigned with local fields. Could also Mars be alive in the same same sense as Venus? Note that the recent radius of Mars is about 1/2 of Earth's radius. If Venus expanded by factor 2, all these 3 planets would have had roughly the same radius for about 750 million years ago. Mars would be waiting for the moment of expansion.

See the article Updated version of Expanding Earth model or the chapter Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life.

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

Articles and other material related to TGD.

Tuesday, August 10, 2021

The conjectured duality between number theory and geometry from TGD point of view

There was a Quanta Magazine article about the link between geometry and numbers (see this). In the following I consider this idea from the TGD view point.

1. M8-M4× CP2 duality

What makes the proposed connection so interesting from the TGD point of view is that in TGD M8-M4× CP2 duality (see for instance this, this and this) states number theoretic and geometric descriptions of physics are dual and this duality is the generalization of wave-mechanical momentum-position duality having no generalization in quantum field theory since position is not an observable in quantum field theory but mere coordinate of space-time.

  1. M8 picture about space-time surface provides a number theoretic description of physics based on the identification of space-time surfaces as algebraic surfaces. Dynamics is coded by the condition that the normal space of the space-time surface is associative.
  2. H= M4× CP2 provides a geometric description of space-time surfaces based on differential geometry, partial differential equations, and action principle. The existence of twistor lift of TGD fixes the choice of H uniquely (this).

    The solutions of field equations reduce to minimal surfaces as counterparts for solutions of massless field equations and the simultaneous extremal of Kähler action implies a close connection with Maxwell's theory. Space-time surfaces would be analogous to soap films spanned by dynamically generated frames (this).

    Beltrami field property implies that dissipation is absent at the space-time level and gives support for the conjecture that the QFT limit gives Einstein-YM field equations in good approximation. The absence of dissipation is also a correlate for quantum coherence implying absence of dissipation (this).

It would be very nice if this duality between number theory and geometry would be present at the level of mathematics itself.

2. Adelic physics as unified description of sensory experience and cognition

Adelic physics involves both real and p-adic number fields (see for instance this). p-Adic variants of the space-time surface are an essential piece and give rise to mathematical correlates of cognition. Cognitive representations are discretizations, which consist of points of space-time surface a, whose imbedding space coordinates are in an extension of rationals characterizing a given adele are common to real and various p-adic variants of space-time, define the intersection of cognitive and sensory realities.

What is so nice from the physics point of view, is that these discretizations are unique for a given adele and adeles form an infinite evolutionary cognitive hierarchy . The p-adic geometries proposed by Scholze would be very interesting from this point of view and I wonder whether there might be something common between TGD and the work done by Scholze. Unfortunately, I do not have the needed knowledge about technicalities.

3. Langlands corresponds and TGD

Also Langlands correspondence, which I have tried to understand several times with my tiny physicist's brain, is involved.

  1. Global Langlands correspondence (GLC) states that there is a connection between representations of continuous groups and Galois groups of extensions of rationals.
  2. Local LC states (LLC) states this in the case of p-adics.
There is a nice interpretation for the two LCs in terms of sensory experience and cognition in the TGD inspired theory of consciousness.
  1. In adelic physics real numbers and p-adic number fields define the adele. Sensory experience corresponds to reals and cognition to p-adics. Cognitive representations are in their discrete intersection and for extensions of rationals belonging to the intersection.
  2. Sensory world, "real" world corresponds to representation of continuous groups/Galois groups of rationals: this would be GLC.
  3. "p-Adic" worlds correspond to cognition and representations of p-adic variants of continuous groups and Galois groups over p-adics: this would be LLC.
  4. One could perhaps talk also about Adelic LC (ALC) in the TGD framework. Adelic representations would combine real and p-adic representations for all primes and give as complete a view about reality as possible.

4. Galois groups, physics and cognition

TGD provides a geometrization for the action of Galois groups (see this and this).

  1. Galois groups are symmetry groups of TGD since space-time surfaces are determined by polynomials with rational (possibly also algebraic) coefficients continued to octonionic polynomials Galois groups relate to each other sheets of space-time time and a very nice physical picture emerges. Physical states correspond to the representations of Galois groups and are crucial in the dark matter sector, especially important in quantum biology. Space-time surface provides them and also the fermionic Fock states realize them.
  2. The order n of the Galois group over rationals corresponds to an effective Planck constant heff= nh0 so that there is a direct connection to a generalization of quantum physics (see for instance this). The phases of ordinary matter with heff=nh0 behave like dark matter. n measures the algebraic complexity of space-time surfaces and serves also as a kind of IQ. Evolution means an increase of n and therefore increase of IQ.

    The representation of real continuous groups assignable to the real numbers as a piece of adele would be related to the representations of Galois groups in GLC.

    Also p-adic representations of groups are needed to describe cognition and these p-adic group representations and representations of p-adic Galois groups would be related by LLC.

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

    Articles and other material related to TGD.

Saturday, August 07, 2021

Do we really understand how galaxies are formed?

The continual feed of unexpected observations is forcing a critical re-evaluation of what we really know about galaxies and their formation thought to be due a condensation of matter under gravitational attraction. Even the Milky Way yields one surprise after another. It is amusing to witness how empirical findings are gradually leaving TGD as the only viable option.

Today's surprise was from Science alert (see this). It tells at layman level about the findings reported in an article accepted to The Astrophysical Journal Letters (see this).

Cattail is a gigantic structure with a length which can be as much as 16,300 light-years, discovered in the Milky Way. It is a filament which does not seem to be analogous to a spiral arm since it does not follow the warping of the galactic disk which is thought to be an outcome of some ancient collision. In the TGD framework this structure would be associated with a cosmic string, which has in some places thickened to a flux tube and generated ordinary matter in this process.

Also the spiral arms might be accompanied by cosmic strings. In any case, there would be a long cosmic string orthogonal to the galactic plane (jets are parallel to it quite generally) having galaxies along it and generated by the thickening of the cosmic string generating blackhole-like entities as active galactic nuclei.

Just yesterday I learned that the Milky Way also offers other surprises (see this).

One of them is that the galactic disk contains old stars that should not be there but in the outskirts of the galaxy which is the place for oldies whereas younger stars live active life in the galactic disk. This if one assumes that the usual view about the formation of galaxies is correct. This applies also to the weird filaments mentioned above.

In the TGD Universe, galaxies are not formed by a condensation of gas but by a process replacing inflation with a process in which cosmic strings thicken and their string tension - energy density - is reduced. The liberated energy forms the ordinary matter giving rise to the galaxy. This solves the dark matter problem: strings define dark matter and energy and no halo is needed to produce a flat velocity spectrum of distant stars. The collisions of cosmic strings are unavoidable in 3-D space and could have induced the thickening process creating the active galactic nuclei (quasars).

This process would be opposite to what is believed to occur in the standard model. What comes to mind is that the oldies in the disk are formed from a cosmic string portion in the galactic plane. The tangle of the cosmic string can indeed extend in the galactic plane over long distances and there can also be cosmic strings (associated with galactic spiral arms?) in the galactic plane, which would have almost intersected a cosmic string orthogonal to the plane inducing the formation of the Milky Way.

For the TGD views see for instance this, this) and this). For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Fractons and TGD

In Quanta Magazine there was a highly interesting article about entities known as fractons (see this).

There seems to be two different views about fractons as one learns by going to Wikipedia. Fracton can be regarded as a as self-similar particle-like entity (see this or as "sub-dimensional" particle unable to move in isolation (see this). I do not understand the motivation for "sub-dimensional". It is also unclear whether the two notions are related. The popular article assigns to the fractons both the fractal character and the inability to move in isolation.

The basic idea is however that discrete translational symmetry is replaced with a discrete scaling invariance. The analog of lattice which is invariant under discrete translations is fractal invariant under discrete scalings.

One can also consider the possibility that the time evolution operator would act as scaling rather than translation. This is something totally new from quantum field theory (QFT) point of view. In QFTs energy corresponds to time translational symmetry and Hamiltonian generates infinitesimal translations. In string models the analog of stringy Hamiltonian is the infinitesimal scaling operator, Virasoro generator L0.

In TGD the extension of physics to adelic physics provides number theoretic and geometric descriptions as dual descriptions of physics (see for instance this, this, and this). This approach also provides insights about fractons as scale invariant entities and.

  1. In TGD the analog of time evolution between "small" state function reductions is the exponent of the infinitesimal scaling operator, Virasoro generator L0. One could imagine fractals as states invariant under discrete scalings defined by the exponential of L0. They would be counterparts of lattices but realized at the level of space-time surfaces having quite concrete fractal structure.
  2. In p-adic mass calculations the p-adic analog of thermodynamics for L0 proportional to mass squared operator M2 replaces energy. This approach is the counterpart of the Higgs mechanism which allows only to reproduce masses but does not predict them. I carried out the calculations already around 1995 and the predictions were amazingly successful and eventually led to what I call adelic physics fusing real and various p-adic physics (see this).
  3. Long range coherence and absence of thermal equilibrium are also mentioned as properties of fractons (at least those of the first kind). Long range coherence could be due to the predicted hierarchy of Planck constants heff=n×h0 assigned with dark matter and predicting quantum coherence in arbitrarily long scales and associated with what I called magnetic bodies.

    If translations are replaced by discrete scalings, the analogs of thermodynam equilibria would be possible for L0 rather than energy. Fractals would be the analogs of thermodynamic equilibria. In p-adic thermodynamic elementary particles are thermodynamic equilibria for L0 but it is not clear whether the analogy with fractal analog of a plane wave in lattice makes sense.

Fractons are also reported to be able to move only in combinations. This need not relate to the scaling invariance. What this actually means, remained unclear to me from the explanation. What comes to mind is color confinement: free quarks are not possible. Quarks are unable to exist as isolated entities, not only to move as in isolated entities.

In TGD number theoretical vision leads to the notion of Galois confinement analogous to color confinement. The Galois group of a given extension of rationals indeed acts as a symmetry at space-time level. In TGD inspired biology Galois groups would play a fundamental role. For instance, dark analogs of genetic codons, codon pairs, and genes would be singlets (invariant) under an appropriate Galois group and therefore behave as a single quantum coherent dynamical and informational unit. See (see this and this) .

Suppose that one has a system - say a fractal analog of a lattice consisting of Galois singlets. Could fracton be identified as a state which is analogous to quark or gluon and therefore not invariant under the Galois group. The physical states could be formed from these as Galois singlets and are like hadrons.

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

Articles and other material related to TGD.

Friday, August 06, 2021

Hyperon problem of stellar cores in TGD framework

Hyperon problem is a mystery related to the physics of neutron stars (see this). In neutron star neutrons temperature is zero in good approximation and Fermi statistics implies that the all states characterized by momentum and spin are filled up to maximum energy, known as Fermi energy EF identifiable as a chemical potential determined by the number density of fermions.

The increase of density inside a neutron star increases the total Fermi energy. Above a critical Fermi temperature possible in the core of the neutron star, the transformation of neutrons to hyperons which are baryons with some strange quarks becomes possible. λ hyperon with mass about 10 percent higher than neutron mass becomes possible. In a thermo-dynamical equilibrium the chemical potentials of hyperons and neutrons are identical. Note that chemical potentials are in a good approximation Fermi energies at zero temperature.

If part of neutrons transform to hyperons, the total energy decreases since the Fermi energy scales like 1/mass. One therefore expects the presence of hyperons in the cores of neutron stars, where the density and therefore also Fermi energy is high enough. The problem is that the maximal mass for known stars is above the maximal mass expected if hyperon fraction is present. Hyperon cores seem to be absent.

If further neutrons are added part of them transforms to hyperons and eventually all particles transform to neutrons and one can even think of the doomsday option that all matter transforms to hyperon stars.

Can one imagine any manner to prevent the formation of the hyperon core? Could the Fermi energy in the core remain below the needed critical Fermi energy by some new physics mechanism.

  1. Apart from numerical constants, the Fermi energy for effectively n-D system is given by EF= ℏ2 kF2/2, where kF is some power of number density (N/Vn)2/n, where Vn refers to volume, area, or length for n=3, 2, 1. Since zero temperature approximation is good, Fermi energy depends only on the density.
  2. Could one think that part of neutrons transforms to dark neutrons in the transformation heff→ kheff such that neither mass, energy, and Fermi energy are not affected but that wavelength is scaled up as also the volume. For an effectively 3-D system, dark neutrons would occupy a volume which is scaled up by factor k3.
  3. The Fermi energies as chemical potentials for both ordinary neutrons and their dark variants could remain the same in thermal equilibrium and remain below the critical value so that the transformation to hyperons would not take place? The condition that Fermi energies are the same implies that the numbers of ordinary and dark neutrons are the same. This would reduce individual Fermi energies by a factor 1/22/3 but is only a temporary solution.

    One can however introduce phases with k different values of heff and in this case the reduction of Fermi energies is 1/k2/3.

  4. Fermi statistics might however pose a problem. The second quantization of the induced spinor fields at the space-time surface is induced by the second quantization of free spinor fields in the embedding space M4×CP2. Could the CP2 degrees of freedom give additional degrees of freedom realized as many-sheeted structures allowing to avoid the problems with Fermi statistics?
See the article Solar Metallicity Problem from TGD Perspective.

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

Articles and other material related to TGD. 

Thursday, August 05, 2021

Time crystals in TGD framework

Google has reported about a realization of a time crystal as a spin system. A rather hypish layman article at here creates the impression that perpetuum mobile has been discovered. Also the Quanta Magazine article creates this impression. The original research article can be found in

It is interesting to look at the situation in the TGD framework. From the abstract of the article also from the Wikipedia article about time crystals one learns that the system has periodic energy feed and is therefore not closed so that the finding is not in conflict with the second law and perpetuum mobile is not in question.

1. What is time crystal?

The notion of time crystal (this) is a temporal analog of ordinary crystals in the sense that there is temporal periodicity, was proposed by Frank Wilczeck in 2012. Experimental realization was demonstrated in 2016-2017 but not in the way theorized by Wilczek. Soon also a no-go theorem against the original form of the time crystal emerged and motivated generalizations of Wilzeck's proposal.

The findings reported by Google are however extremely interesting. Very concisely, researchers study a spin system, which has two directions of magnetization and the external laser beam induces the system to oscillate between the two magnetization directions with a period, which is a multiple of the period of the laser beam. It is interesting to consider the system in TGD framework and I have actually discussed time crystals briefly in a recent article.

2. Space-time surfaces as periodic minimal surfaces as counterparts of time crystals

In TGD, classical physics is an exact part of quantum theory and quantum classical correspondence holds true. Hence it is interesting to consider first the situation at the classical space-time level. In TGD time crystals have as classical correlates space-time surfaces which are periodic minimal surfaces.

It is possible to have analogs of time-crystals and also more general structures built as piles of lego like basic pieces in time direction bringing in mind sentences of language and DNA, which is quasi-periodic structure and more general than crystal.

3. What about thermodynamics of time crystals?

Could the time crystal be possible also in thermodynamic sense and even for thermodynamically closed systems? In TGD Negentropy Maximization Principle (NMP) (see this) and zero energy ontology (ZEO) (see this and this) forces to generalize thermodynamics to allow both time arrows. ZEO is forced by TGD inspired theory of consciousness and solves the basic paradox of quantum measurement theory. The arrow of time would change in ordinary ("big") state function reduction (BSFR) and would remain unaffected in "small" SFR (SSFR). Second law holds true at the level of real physics but in the cognitive sector information increases and NMP holds true.

4. Is new quantum theory making possible quantum coherence in long scales needed?

Also new quantum theory might be needed to explain why the period is multiple of the driving period. The first possibly needed new element is hierarchy of effective Planck constants heff= n×h_0 having number theoretical interpretation. heff measures the scale of quantum coherence and has also interpretation as the order of Galois group for a polynomial defining the space-time surface in M8 mapped to M4×CP2 by M8-H duality (see this and this).

The replacement of h with heff scales the periods by n and keeps energies unchanged. In TGD inspired biology heff hierarchy is in a crucial role and its levels behave relative to each other like dark matter.

In the recent case, the magnetic body (MB)of the spin system controlling its behavior would have heff=nh. Each period would be initiated by BSFR at the level of MB and change the arrow of time and induce effective change of it also at the level of the ordinary matter.

In ZEO, time crystal-like entities, which live in cycle by extracting back part of the energy that they have dissipated in a time reversed mode, are in principle possible. System "breathes". Various bio-rhythms could correspond to time crystals. The biological analogy is obvious and we know that life requires a metabolic energy feed: in TGD Universe it prevents the decrease of heff (see this).

5. Perpetuum mobile?: almost but not quite!

For an thermodynamically open system, part of the dissipated energy leaks into the external world during each half cycle. Same happens in the time reversed mode and would mean that the system apparently receives positive energy also from the external world. Could this energy feed compensate for the energy loss to the external world by dissipation so that no external energy feed would be needed? Perhaps this might be the case in the ideal situation.

One would have almost a perpetuum mobile! Periodic driving feeding energy to the system would be needed to take care that heff is not reduced.

See the chapter Quantum Criticality and Dark Matter: part II.

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

Articles and other material related to TGD. 

Huge fluctuations in oxygen concentration during Cambrian Explosion and Expanding Earth model

I encountered two interesting articles related to the Great Oxidation Event that started long before the Cambrian Explosion (CE) and reached its climax during CE (about 541 million years ago) leading to the oxygen based multicellular life in a very rapid time scale.

The standard view is that oceans before CE had very low oxygen content. The emergence of photosynthesizing cyanobacteria producing oxygen as a side product led to the oxygenation of the atmosphere and to mysteriously rapid evolution of life. How this is possible at all is not understood.

The first article (see this) proposes that the slowing down of the spinning of Earth was somehow related to this.

The second article in Quanta Magazine (see this) tells about finding that during the Cambrian Explosion (see this") the oxygen content of the studied shallow ocean show fluctuations with with about 4-5 peaks. The reduction/increase of the oxygen content was even 40 per cent, which is a huge number. The reduction of oxygen content caused extinctions and its increase was accompanied by the emergence of new species. The mystery is how this could happen so fast and which caused the fluctuations.

1. Expanding Earth hypothesis

Expanding Earth theory hypothesis is not originally TGD based but TGD provides its realization. The proposal is that the Cambrian Explosion was caused by a rapid increase of the radius of Earth by factor 2 (see this and this).

This hypothesis also solves one of the basic mysteries of cosmology. Astrophysical objects participate in cosmological expansion by comoving with it but do not expand themselves. Why? The prediction that the expansion of the astrophysical objects did not occur smoothly but as rapid phase transitions and the expansion was very slow in the intermediate states. Cambrian Explosion would correspond to one particular jerk of this kind in which the radius of Earth grew by a factor 2 (p-adic length scale hypothesis). The length of the day increased by factor 4 from conservation of angular momentum. This might relate to the conjecture of the first article.

The rapid expansion led to the breakage of the Earth crust and to the birth of plate tectonics. It also led to the burst of underground oceans to the surface of the Earth. The photosynthesizing multicellular life had developed in these oceans and emerged almost instantaneously and led to a rapid oxygenation of the atmosphere. One can say that life evolved in the womb of Mother Gaia shielded from meteorites and cosmic rays. No superfast evolution was needed. Already Charles Darwin realized that the sudden appearance of trilobites was a heavy objection against the theory of natural selection.

Possible scenarios for the phase transition are discussed here. The thickening of magnetic flux tubes for water blobs at the surface of Earth led to the increase of the volume of water blob and induced the increase of heff a factor 2 for valence electrons but not for the inner electrons. Since valence electrons are responsible for chemistry, atoms became effectively dark and the water blobs could leak to the interior of Earth. By their darkness they could have much lower temperature and pressure than the matter around them and the life could evolve.

2. How photosynthesis was possible underground?

What made photosynthesis possible in the underground oceans? One possible explanation is that the photons from the Sun propagated along flux tubes of the "endogenous" part of the Earth's magnetic field as dark photons with heff=nh_0>h. Endogenous part would be the part of Earth's magnetic field with a strength about 2/5 of the Earth's magnetic field for which flux tubes carry monopole flux: this is possible in TGD but not in Maxwell's theory.

Since these photons behave like dark matter with respect to the ordinary matter, they were not absorbed considerably and reached the water blobs (or actually their magnetic bodies consisting of flux tubes) in underground oceans having a portion with the same value of heff>h. Of course, several values of heff were possible since this is the case in quantum critical system (large values of heff characterize the quantum scales of long range fluctuations). One can also consider other variants of the model. The ordinary matter in Earth's crust had heff =h/2 and photons with heff=h propagated to the interior and reached the water blobs with heff=h.

3. The sudden emergence of multicellulars and oxygen fluctuations

Before the expansion period was much like the surface of Mars now and contained no oceans, perhaps some ponds allowing primitive monocellular lifeforms. As the ground of Earth broke here and there during the rapid expansion period, lakes and oceans were formed at the surface of Earth. The multicellulars bursted to these oceans and oxygenation of the atmosphere started locally.

Since the oxygen rich water was mixed with the water in the shallow oceans, the local oxygen content of the burst water was reduced and this led to an eventual extinction of many multicellulars in the burst. Burgess Shale fauna contained entire classes, which suffered extinction. In the average sense the oxygen concentration increased and led to the apparent very rapid evolution of multicellulars, which had actually already occurred underground. Of course, also evolution at the surface of Earth took place.

See the article Updated version of Expanding Earth model or the chapter Expanding Earth Model and Pre-Cambrian Evolution of Continents, Climate, and Life .

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

Articles and other material related to TGD. 

Tuesday, August 03, 2021

Gentral engine of galactic nuclei as a time reversed blackhole-like object

The work to be summarize was inspired by a Quanta Magazine article "Physicists Identify the Engine Powering blackhole Energy Beams" (see this) telling about an empirical support for the model of Blandford and Znajek (BZ model in the sequel) for the central engine providing energy for jets from active galactic nuclei (AGNs). In the BZ model (see this) AGN is identified as a blackhole and the Penrose process would provide the energy of the jets emerging from the blackhole. The energy would come basically from the blackhole mass. The empirical support is found by studying the supermassive blackhole associated with a galaxy known as Messier 87 (M87).

The basic problem is the identification of the central engine of the active galactic nuclei (AGNs) (see this) providing the huge energy feed to the the jets.

1. Typical properties of active galactic nuclei

The power emitted by active galactic nuclei (AGNs) is typically of the order of 1038 W corresponding to a transformation of a mass of 1022 kg per second to energy. The typical radius of the AGN is R ≈ 2 AU for the active region.

One must distinguish between magnetic fields associated with the interior of the central objects, the region near its surface, and the jet region with the scale of visible jets about 105 ly. According to the estimate of (see this), the magnetic field is about 10 Gauss in the jet region. About 106-107 Gauss near horizon. Also near-horizon magnetic fields in the range 108-1011 Tesla have been proposed for some AGNs.

Quasars are examples of AGNs and also M87 central region idenfied as a blackhole is such. In this case the mass is 6.5× 109MSun and Scwartschild radius is about 2× 1010 km = 1.3× 103 AU. For M87 central object the magnetic field in the jet region is that of refrigerator magnet and about 100 Gauss. For Sagittarius A has a in the center of the Milky Way the radius of the central object .4 AU.

One can pose some general conditions on the central engine serving as the energy source for the jets. The time scale Δt for the luminosity fluctuations in the power should satisfy Δt< R. For M87 one has Δt ≤ 104 s. The gravitational force is assumed to be balanced by the radiation pressure of the outward radiation.

Consider now the observations about the M87 blackhole-like entity (see this).

  1. The mass of the M87 black-hole-like entity is about 6.5× 109MSun.
  2. There are two 5,000 ly long white hot plasma jets travelling in opposite time directions with emitted power of 3× 1036 W. They have blobs at their ends. Synchrotron radiation is emitted at radio wavelengths in the magnetic field, which according to the popular article has the strength of a refrigerator magnet, so that one would have B ≈ 100 Gauss. Both the intensity of B and the size of the emitting region contribute to the intensity of the energy flow.
  3. There are two alternatives for the BZ process that have been developed and explored in hundreds of computer simulations in recent decades. They have acronyms MAD and SANE.

    For the SANE option B is weak: charged matter dominates over B. For the MAD option B is strong, has a spiral structure and acts as a "boss" of the matter. The tight spiral structure forms a sleeve around the jet preventing charges from entering the central object. This inspires a critical question: doesn't the object look more like a whitehole.

    The strongly polarized light in the Event Horizon Telescope's new photo suggests strong magnetic fields, and supports the MAD version. B has a strength of about 100 Gauss, that is 200 times the strength of the Earth's magnetic field with the nominal value BE≈ .5 Gauss. The polarization pattern for the radio waves is found to be stripy and the polarization in a plane locally: this allows us to conclude that the magnetic field is indeed helical and non-random.

2. Central engine as a Penrose process?

In the BZ model, the central object is assumed to be a blackhole and Penrose process would provide the energy feed to the jets of length about 5000 ly. Note that the Milky Way is about 1,0000 ly thick.

  1. The blackhole is surrounded by an accretion disk from which the matter ends down to the BH.
  2. Kerr solution of Einstein-Maxwell field equations (this) involving magnetic field is the starting point. Matter falling into the Kerr blackhole rotates and the magnetic field lines are twisted to helical shape. By Faraday law, an electric field along field lines is generated by the rotation of the flux lines. Electrons and positrons created in the annihilation photons emitted as the particles fall to the region near the blachole, start to flow along the field lines of the electric field in opposite directions and generate the jets.
  3. The model assumes that the electromagnetic field is force free so that it does not dissipate and Lorentz force vanishes. At a single particle level this implies the condition E+qv× B=0. Vanishing dissipation requires v· E=0. This helical structure would be in the direction of the jet.
  4. The basic question has been whether it is accretion disk or magnetic field that controls the dynamics. The first option, known as SANE, corresponds to weak and incoherent magnetic fields. The second option, known as MAD, corresponds to strong and coherent magnetic fields.
  5. MAD is favoured by the recent observations. Magnetic field would form a sleeve around the jet and the synchrotron radiation pressure would prevent matter from falling into the blackhole. Matter can only occasionally leak to blackhole.

    One can however wonder whether it makes sense to talk about blackhole anymore! Doesn't this look more like white hole as a time reversal of blackhole feeding energy and matter to the environment?

3. TGD inspired view of the central engine

In the TGD framework the model of the central engine as a Penrose process is replaced by the following picture. The key concepts are following:

  1. Space-time is identified as a 4-D minimal surface in H=M4× CP2 or as an algebraic surface in complexified M8 having octonionic interpretation. These descriptions are related by M8-H duality analogous to momentum-position duality, which does not generalize from wave mechanics to quantum field theory (QFT). Therefore the points or M8 are 8-momenta.

    The classical dissipation is absent for the generalized Beltrami fields and the proposal is that minimal surfaces (apart form singularities defining dynamically generated frame for space-time surfaces as analog of a soap film) define locally generalized Beltrami fields.

  2. Zero energy ontology (ZEO) predicts that time reversal occurs in the TGD counterparts of ordinary state function reductions ("big" SFRs) but not in "small" SFRs (SSFRs).
  3. The hierarchy of effective Planck constants predicts a hierarchy of phases of ordinary matter labelled by the values of effective Planck constant heff= nh0. The phases with different values of heff behave in many respects like dark matter with respect to each other. The findings of Randell Mills suggest ℏ/ℏ0=6 but also larger values for this ratio can be considered. I have proposed that the ℏ0/ℏ is equal to the ratio lP/R of Planck length lP to CP2 radius R.

    As a special case, one obtains gravitational Planck constant satisfying heff= hgr= GMm/β0, where β0=v0/c and β0<c has dimensions of velocity, as a generalization of Nottale's hypothesis. The gravitational Compton length λgr=ℏgr/m=GM/β0 does not depend on m and is equal to Schwartschild radius rs for β0= 1/2. Also the cyclotron energy spectrum Ec=n GMqB/β0 is independent of the mass of the charged particle.

    The hierarchy of Planck constants, the notion of ℏgr, and coupling constant evolution are discussed in detail here.

Consider next the key elements of the model.
  1. TGD leads to a general model for the formation of galaxies, stars, planets,... in terms of cosmic strings thickening to flux tubes. The energy of the flux tube, which consists of a volume energy and Kähler magnetic energy, is transformed to ordinary matter as the string tension is reduced in a sequence of phase transitions reducing the length scale dependent cosmological constant λ.

    This process is analogous to the decay of an inflaton field to matter. The model (there are actually several basic variants of it) explains the flat velocity spectra associated with the spiral galaxies. For the first option, a long cosmic string normal to the galactic plane causes the gravitational field explaining the flat velocity spectrum of spiral galaxies. For galaxies formed around closed flux loops the velocity spectrum is not flat. There is no dark matter halo although it is possible that the galactic plane contains cosmic strings parallel to the plane.

  2. Zero energy ontology (ZEO), which predicts that the TGD counterparts of ordinary state function reductions (SFRs) involve time reversal, is involved in an essential manner. TGD predicts both blackhole-like objects (BH) and whitehole-like objects (WH) as the time reversals of BHs. The seed of the galaxy, active galactic nucleus (AGN), involves WH. Quasars are cases of AGNs as WHs.
  3. In the TGD framework, the Kerr blackhole is replaced with a whitehole-like object (WH). Kerr blackhole indeed has an opposite arrow of time reversal as the distant environment. The WH is time reversal of BH and feeds matter and energy to the environment. This serves as an analog of the Penrose process in the TGD based model.
  4. The TGD analog for the rotation of spacetime and the twisting of the magnetic field lines near the Kerr blackhole is very concrete. Space-time is a 4-surface and the flux tubes carrying monopole flux are pieces of 3-space as a 3-surface. They quite concretely rotate and get twisted in the process. Analogous process occurs in the Sun with a period of 11 years ending as reconnections untwist the flux tubes.
  5. WH would correspond to a tangle of a long cosmic string in the direction of the jet thickened to a flux tube but still carrying an extremely strong magnetic field. The helical magnetic field in the exterior of the jet would not represent return flux of this field as one might first think. There is a current ring associated with the equator of Earth, which carries a parallel magnetic field analogous to the helical magnetic field.

    The magnetic field in the exterior of WH is associated with a space-time surface, which is many-sheeted with respect to CP2 rather than M4 so that either CP2 or cosmic string world sheet M2× S2⊂ M4× CP2) would serve as the arena of physics rather than M4, which is quantum coherent flux tube bundle analogous to BE-condensate. M4 coordinates as functions of CP2 or M2× CP2 coordinates would be many-valued rather than vice versa. This picture is very natural if one accepts M8-H duality.

    Cosmic strings dominate during the primordial cosmology in TGD Universe, and the analog of the inflationary period corresponds to the transition to a phase in which the Einsteinian space-time with M4 as the arena of physics is a good approximation. Hence the M2× CP2 option looks more plausible.

  6. The force-free em fields appearing in the BZ model correspond to space-time surfaces as minimal surfaces realizing a 4-D generalization of 3-D Beltrami fields, which do not not dissipate classically. The interpretation of the non-dissipating Kähler currents is as classical correlates for supracurrents. The prediction is that charged particles flow without dissipation that is as supra currents: not only Cooper pairs but also charged fermions. Also the analogs of laser beams of dark photons are expected.
  7. The hierarchy of Planck constants is an important piece of the picture emerging from adelic physics. From ℏgr=GMm/β0 realizing Equivalence Principle, the gravitational Compton length λgr= rs/2β0 is universal and equals to rS for β0== β0=1/2.

    All astrophysical objects are predicted to be quantum coherent in the scale of λgr= rs/2β0 at least. The quantum coherence would be at the level of magnetic body (MB). WH/BH as a thickened flux tube tangle would not have large heff but would be accompanied by a large scale quantum object.

    The astroscopic quantum coherence would be associated with the helical magnetic field surrounding the long cosmic string having BH or WH as a tangle.

  8. Also the cyclotron energy spectrum is universal and does not depend on the mass of the charged particles so that all charged particles rather than only electrons are expected to form supracurrents. Dark matter would flow along flux tubes and form the dark core of the jet, perhaps extending over cosmic distances to other galaxies identified as tangles of the one and the same cosmic string.

    Stars and even planets would be parts of this fractal network. Dark cyclotron states have huge energies for heff=hgr serving also as a measure for algebraic complexity and, in the TGD inspired theory of consciousness, also for intelligence and scale of quantum coherence. The analogy with a cosmic nervous system is obvious.

  9. The decay of quantum coherent states to ordinary states takes place by the loss of quantum coherence in which heff= hgr is reduced. This would create the visible jets and blobs at their ends. For M87, which is elliptical for which the velocity spectrum is not flat, the flux tubes would be closed in a relatively short scale. Their length scale could be that of the jets in the case of ellipticals. The thickening of the cosmic string at the core leads to the reduction of mass of WH and gives rise to the flow of mass and energy to the environment. One could see this process as a time reversal for the generation of BH and perhaps also as an analogy for the evaporation of BH.
  10. M8-H duality and adelic physics help to understand the decoherence process geometrically. The reduction of heff and thus of the length scale of quantum coherence, allows a number theoretic description at the level of M8. An irreducible polynomial, which depends on parameters, reduces to a product of polynomials for some critical values of the parameters. This gives rise to a set of disjoint space-time surfaces, which are not correlated. This means decoherence. This includes as a special case the description of catastrophic changes in catastrophe theory of Thom. The maximal decoherence produces a product of first order polynomials with rational roots.

    At the level of H =M4×CP2 this corresponds to a decay of coherent flux tube bundle to disjoint uncorrelated flux tubes.

See the article TGD view of the engine powering jets from active galactic nuclei or the chapter with the same title.

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

Articles and other material related to TGD.

Evolution of Kähler coupling strength

Evolution of Kähler coupling strength

The evolution of Kähler coupling strength αK= gK2/2heff gives the evolution of αK as a function of dimension n of EQ: αK= gK2/2nh0. If gK2 corresponds to electroweak U(1) coupling, it is expected to evolve also with respect to PLS so that the evolutions would factorize.

Note that the original proposal that gK2 is renormalization group invariant was later replaced with a piecewise constancy: αK has indeed interpretation as piecewise constant critical temperature

  1. In the TGD framework, coupling constant as a continuous function of the continuous length scale is replaced with a function of PLS so that coupling constant is a piecewise constant function of the continuous length scale.

    PLSs correspond to p-adic primes p, and a hitherto unanswered question is whether the extension determines p and whether p-adic primes possible for a given extension could correspond to ramified primes of the extension appearing as factors of the moduli square for the differences of the roots defining the space-time surface.

    In the M8 picture the moduli squared for differences ri-rj of the roots of the real polynomial with rational coefficients associated with the space-time surfaces correspond to energy squared and mass squared. This is the case of p-adic prime corresponds to the size scale of the CD.

    The scaling of the roots by constant factor however leaves the number theoretic properties of the extension unaffected, which suggests that PLS evolution and dark evolution factorize in the sense that PLS reduces to the evolution of a power of a scaling factor multiplying all roots.

  2. If the exponent Δ K/log(p) appearing in pΔ K/log(p))=exp(Δ K) is an integer, exp(Δ K) reduces to an integer power of p and exists p-adically. If Δ K corresponds to a deviation from the Kähler function of WCW for a particular path in the tree inside CD, p is fixed and exp(Δ K) is integer. This would provide the long-sought-for identification of the preferred p-adic prime. Note that p must be same for all paths of the tree. p need not be a ramified prime so that the trouble-some correlation between n and ramified prime defining padic prime p is not required.

  3. This picture makes it possible to understand also PLS evolution if Δ K is identified as a deviation from the Kähler function. pΔ K/log(p))=exp(Δ K) implies that Δ K is proportional to log(p). Since Δ K as 6-D Kähler action is proportional to 1/αK, log(p)-proportionality of Δ K could be interpreted as a logarithmic renormalization factor of αK∝ 1/log(p).

  4. The universal CCE for αK inside CDs would induce other CCEs, perhaps according to the scenario based on M"obius transformations.
Dark and p-adic length scale evolutions of Kähler coupling strength

The original hypothesis for dark CCE was that heff=nh is satisfied. Here n would be the dimension of EQ defined by the polynomial defining the space-time surface X4subset M8c mapped to H by M8-H correspondence. n would also define the order of the Galois group and in general larger than the degree of the irreducible polynomial.

Remark: The number of roots of the extension is in general smaller and equal to n for cyclic extensions only. Therefore the number of sheets of the complexified space-time surface in M8c as the number of roots identifiable as the degree d of the irreducible polynomial would in general be smaller than n. n would be equal to the number of roots only for cyclic extensions (unfortunately, some former articles contain the obviously wrong statement d=n).

Later the findings of Randell Mills, suggesting that h is not a minimal value of heff, forced to consider the formula heff=nh0, h0=h/6, as the simplest formula consistent with the findings of Mills. h0 could however be a multiple of even smaller value of heff, call if h0 and the formula h0=h/6 could be replaced by an approximate formula.

The value of heff=nh0 can be understood by noticing that Galois symmetry permutes "fundamental regions" of the space-time surface so that action is n times the action for this kind of region. Effectively this means the replacement of αK with αK/n and implies the convergence of the perturbation theory. This was actually one of the basic physical motivations for the hierarchy of Planck constants. In the previous section, it was argued that h0 is given by the square of the ratio lP/R of Planck length and CP2 length scale identified as dark scale and equals to n0=(7!)2.

The basic challenge is to understand p-adic length scale evolutions of the basic gauge couplings. The coupling strengths should have a roughly logarithmic dependence on the p-adic length scale p≈ 2k/2 and this provides a strong number theoretic constraint in the adelic physics framework.

Since Kähler coupling strength αK induces the other CCEs it is enough to consider the evolution of αK.

p-Adic CCE of α from its value at atomic length scale?

If one combines the observation that fine structure constant is rather near to the inverse of the prime p=137 with PLS, one ends up with a number theoretic idea leading to a formula for αK as a function of p-adic length scale.

  1. The fine structure constant in atomic length scale L(k=137) is given α (k)=e2/2h ≈ 1/137. This finding has created a lot of speculative numerology.
  2. The PLS L(k)= 2k/2R(CP2) assignable to atomic length scale p≈ 2k corresponds to k=137 and in this scale α is rather near to 1/137. The notion of fine structure constant emerged in atomic physics. Is this just an accident, cosmic joke, or does this tell something very deep about CCE?

    Could the formula

    α(k)= e2(k)/2h= 1/k

    hold true?

There are obvious objections against the proposal.
  1. α is length scale dependent and the formula in the electron length scale is only approximate. In the weak boson scale one has α≈ 1/127 rather than α= 1/89.
  2. There are also other interactions and one can assign to them coupling constant strengths. Why electromagnetic interactions in electron Compton scale or atomic length scales would be so special?
The idea is however plausible since beta functions satisfy first order differential equation with respect to the scale parameter so that single value of coupling strength determines the entire evolution.

p-Adic CCE from the condition αK(k=137)= 1/137

In the TGD framework, Kähler coupling strength αK serves as the fundamental coupling strength. All other coupling strengths are expressible in terms of αK, and I have proposed that M"obius transformations relate other coupling strengths to αK. If αK is identified as electroweak U(1) coupling strength, its value in atomic scale L(k=137) cannot be far from 1/137.

The factorization of dark and p-adic CCEs means that the effective Planck constant heff(n,h,p) satisfies

heff(n,h,p)=heff(n,h) = nh .

and is independent of the p-adic length scale. Here n would be the dimension of the extension of rationals involved. heff(1,h,p) corresponding to trivial extension would correspond to the p-adic CCE as the TGD counterpart of the ordinary evolution.

The value of h need not be the minimal one as already the findings of Randel Mills suggest so that one would have h=n0h0.

heff= nn0h ,

αK,0= gK,max2/2h0 =n0 .

This would mean that the ordinary coupling constant would be associated with the non-trivial extension of rationals.

Consider now this picture in more detail.

  1. Since dark and p-adic length scale evolutions factorize, one has

    αK (n)= gK2(k)/2heff ,

    heff= nh0 .

    U(1) coupling indeed evolves with the p-adic length scale, and if one assumes that gK2(k,n0) (h=n0h0) is inversely proportional to the logarithm of p-adic length scale, one obtains

    gK2(k,n0) =gK2(max)/k ,

    αK = gK2(max)/2kheff .

  2. Since k=137 is prime (here number theoretical physics shows its power!), the condition αK (k=137,h0)=1/137 gives

    gK2(max)/2h0}= αK(max) =(7!)2 .

    The number theoretical miracle would fix the value of αK(max) to the ratio of Planck mass and CP2 mass n0= M2P/M2(CP2)= (7!)2 if one takes the argument of the previous section seriously.

    The convergence of perturbation theory could be possible also for heff=h0 if the p-adic length scale L(k) is long enough to make αK= n0/k small enough.

  3. The outcome is a very simple formula for αK

    αK(n,k) = n0/kn ,

    which is a testable prediction if one assumes that it corresponds to electroweak U(1) coupling strength at QFT limit of TGD. This formula would give a practically vanishing value of αK for very large values of n associated with hgr. Here one must have n>n0.

    For heff=nn0h characterizing extensions of extension with heff=h one can write

    αK(nn0,k) = 1/kn .

  4. The almost vanishing of αK for the very large values of n associated with ℏgr would practically eliminate the gauge interactions of the dark matter at gravitational flux tubes but leave gravitational interactions, whose coupling strength would be beta0/4pi. The dark matter at gravitational flux tubes would be highly analogous to ordinary dark matter.
See the article Questions about coupling constant evolution or the chapter with the same title.

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

Articles and other material related to TGD.

Minimal value of heff from the ratio of Planck mass and CP2 mass?

Minimal value of heff from the ratio of Planck mass and CP2 mass?

Could one understand and perhaps even predict the minimal value h0of heff? Here number theory and the notion of n-particle Planck constant heff(n) suggested by Yangian symmetry could serve as a guidelines.

  1. Hitherto I have found no convincing empirical argument fixing the value of r=ℏ/ℏ0: this is true for both single particle and 2-particle case.

    The value h0=h/6 as a maximal value of h0 is suggested by the findings of Randell Mills and by the idea that spin and color must be representable as Galois symmetries so that the Galois group must contain Z6=Z2× Z3. Smaller values of h0 cannot be however excluded.

  2. A possible manner to understand the value r geometrically would be following. It has been assumed that CP2 radius R defines a fundamental length scale in TGD and Planck length squared lP2= ℏ G =x-2 × 10-6R2 defines a secondary length scale. For Planck mass squared one has mPl2= m(CP2,ℏ)2× 106x2 , m(CP2,ℏ)2= ℏ/R2. The estimate for x from p-adic mass calculations gives x≈ 4.2. It is assumed that CP2 length is fundamental and Planck length is a derived quantity.

    But what if one assumes that Planck length identifiable as CP2 radius is fundamental and CP2 mass corresponds the minimal value h0 of heff(2)? That the mass formula is quadratic and mass is assignable to wormhole contact connecting two space-time sheets suggests in the Yangian framework that heff(2) is the correct Planck constant to consider.

One can indeed imagine an alternative interpretation. CP2 length scale is deduced indirectly from p-adic mass calculation for electron mass assuming heff=h and using Uncertainty Principle. This obviously leaves the possibility that R= lP apart from a numerical constant near unity, if the value of heff to be used in the mass calculations is actually h0= (lP/R)2ℏ. This would fix the value of ℏ0 uniquely.

The earlier interpretation makes sense if R(CP2) is interpreted as a dark length scale obtained scaling up lP by ℏ/ℏ0. Also the ordinary particles would be dark.

h0 would be very small and αK(ℏ0)= (ℏ/ℏ0K would be very large so that the perturbation theory for it would not converge. This would be the reason for why ℏ and in some cases some smaller values of heff such as ℏ/2 and ℏ/4 seem to be realized.

For R=lP Nottale formula remains unchanged for the identification M2P= ℏ/R2 (note that one could consider also ℏ0/R2 used in p-adic mass calculations).

Various options

Number theoretical arguments allow to deduce precise value for the ratio ℏ/ℏ0. Accepting the Yangian inspired picture, one can consider two options for what one means with ℏ.

  1. ℏ refers to the single particle Planck constant ℏeff(1) natural for point-like particles.
  2. ℏ refers to heff(2). This option is suggested by the proportionality M2∝ ℏ in string models due to the proportionality M2∝ℏ/G in string models. At a deeper level, one has M2 ∝ L0, where L0 is a scaling generator and its spectrum has scale given by ℏ.

    Since M2 is a p-adic thermal expectation of L0 in the TGD framework, the situation is the same. This also due the fact that one has In TGD framework, the basic building bricks of particles are indeed pairs of wormhole throats.

One can consider two options for what happens in the scaling heff→ kheff.

Option 1: Masses are scaled by k and Compton lengths are unaffected.

Option 2: Compton lengths are scaled by k and masses are unaffected.

The interpretation of MP2= (ℏ/ℏ0) M2(CP2) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.

The interpretation of MP2= (ℏ/ℏ0) M2(CP2) assumes Option 1 whereas the new proposal would correspond to Option 2 actually assumed in various applications.

For Option 1 mPl2= (ℏeff/ℏ) M2(CP2). The value of M2(CP2)= ℏ/R2 is deduced from the p-adic mass calculation for electron mass. One would have R2 ≈ (ℏeff/ℏ) lP2 with ℏeff/ℏ = 2.54× 107. One could say that the real Planck length corresponds to R.

Quantum-classical correspondence favours Option 2)

In an attempt to select between these two options, one can take space-time picture as a guideline. The study of the imbeddings of the space-time surfaces with spherically symmetric metric carried out for almost 4 decades ago suggested that CP2 radius R could naturally correspond to Planck length lP. The argument is described in detail in Appendix and shows that the lP=R option with heff=h used in the classical theory to determine αK appearing in the mass formula is the most natural.

Deduction of the value of ℏ/ℏ0

Assuming Option 2), the questions are following.

  1. Could lP=R be true apart from some numerical constant so that CP2 mass M(CP2) would be given by M(CP2)2= ℏ0/lP2, where ℏ0≈ 2.4× 10-7 ℏ (ℏ corresponds to ℏeff(2)) is the minimal value of ℏeff(2). The value of h0 would be fixed by the requirement that classical theory is consistent with quantum theory! It will be assumed that ℏ0 is also the minimal value of ℏeff(1) both ℏeff(2).
  2. Could ℏ(2)/ℏ0(2)=n0 correspond to the order of the product of identical Galois groups for two Minkowskian space-time sheets connected by the wormhole contact serving as a building brick of elementary particles and be therefore be given as n0=m2?
Assume that one has n0=m2.
  1. The natural assumption is that Galois symmetry of the ground state is maximal so that m corresponds to the order a maximal Galois group - that is permutation group Sk, where k is the degree of polynomial.

    This condition fixes the value k to k=7 and gives m=k!=7! = 5040 and gives n0= (k!)2= 25401600=2.5401600 × 107. The value of ℏ0(2)/ℏ(2)=m-2 would be rather small as also the value of ℏ0(1)ℏ(1). p-Adic mass calculations lead to the estimate mPl/m(CP2)= m1/2 m(CP2)=4.2× 103, which is not far from m=5040.

  2. The interpretation of the product structure S7 × S7 would be as a failure of irreducibility so that the polynomial decomposes into a product of polynomials - most naturally defined for causally isolated Minkowskian space-time sheets connected by a wormhole contact with Euclidian signature of metric representing a basic building brick of elementary particles.

    Each sheet would decompose to 7 sheets. ℏgr would be 2-particle Planck constant heff(2) to be distinguished from the ordinary Planck constant, which is single particle Planck constant and could be denoted by heff(1).

    The normal subgroups of S7 × S7 S7× A7 and A7× A7, S7, A7 and trivial group. A7 is simple group and therefore does not have any normal subgroups expect the trivial one. S7 and A7 could be regarded as the Galois group of a single space-time sheet assignable to elementary particles. One can consider the possibility that in the gravitational sector all EQs are extensions of this extension so that ℏ becomes effectively the unit of quantization and mPl the fundamental mass unit. Note however that for very small values of αK in long p-adic length scales also the values of heff<h, even h0, are in principle possible.

    The large value of αK ∝ 1/ℏeff for Galois groups with order not considerably smaller than m=(7!)2 suggests that very few values of heff(2)<h are realized. Perhaps only S7 × S7 S7× A7 and A7× A7 are allow by perturbation theory. Now however that in the "stringy phase" for which super-conformal invariance holds true, h0 might be realized as required by p-adic mass calculations. The alternative interpretation is that ordinary particles correspond to dark phase with R identified dark scale.

  3. A7 is the only normal subgroup of S7 and also a simple group and one has S7/A7= Z2. S7× S7 has S7× S7/A7× A7= Z2× Z2 with n=n0/4 and S7× S7/A7× S7= Z2 with n=n0/2. This would allow the values ℏ/2 and ℏ/4 as exotic values of Planck constant.

    The atomic energy levels scale like 1/ℏ2 and would be scaled up by factor 4 or 16 for these two options. It is not clear whether ℏ→ ℏ/2 option can explain all findings of Randel Mills in TGD framework, which effectively scale down the principal quantum number n from n to n/2.

  4. The product structure of the Nottale formula suggests

    n=n1× n2 = k1k2m2 .

    Equivalently, ni would be a multiple of m. One could say that MPl=(ℏ/ℏ0)1/2M(CP2) effectively replaces M(CP2) as a mass unit. At the level of polynomials this would mean that polynomials are composites P○ P0 where P0 is ground state polynomial and has a Galois group with degree n0. Perhaps S7 could be called the gravitational or ground state Galois group.

See the article Questions about coupling constant evolution or the chapter with the same title.

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

Articles and other material related to TGD.