Sunday, June 25, 2017

About McKay and Langlands correspondences in TGD framework

In adelic TGD Galois groups for extensions of rationals become discrete symmetry groups acting on dark matter, identified as heff/h=n phases of ordinary matter. n gives the number of sheet of covering assignable to space-time surface. Since Galois group acts on the cognitive representation defined by a discrete set of points of space-time surface with coordinates having values in extension of rationals, the action of Galois group defines n-sheeted covering, where n is the order of Galois group thus identifiable in terms of Planck constant.

Adelic TGD inspires the question whether the representations of Galois groups could correspond to representations of Lie groups defining the ground states of Kac-Moody representations emerging in TGD in two manners: as representations of Kac-Moody algebra assignable the Poincare-, color- and electroweak symmetries on one hand and with dynamical generated from supersymplectic symmetry assignable with the boundaries of causal diamond (CD) and extended Kac-Moody symmetres assignable to the light-like orbits of partonic 2-surfaces defining boundaries between space-time regions with Minkowskian and Euclidian signatures of the induced metric.

McKay correspondence states that the finite discrete subgroups of SU(2) can be characterized by McKay graphs characterizing the fusion rules for the tensor products for the representations of these groups. These graphs correspond to the Dynkin diagrams for Kac-Moody algebras of ADE type group (all roots have same unit length in Dynkin diagram). This inspires the conjecture that finite subgroups of SU(2) indeed correspond to Kac-Moody algebras. Could the representations of discrete subgroups appearing in the McKay graph define also representations for the ground states of corresponding ADE type Kac-Moodyt algebra? More generally, could the Mc-Kay graps of the Galois groups?

Number theoretic Langlands correspondence in turn states roughly that the representations of Galois group for extensions of rationals correspond to the so called automorphic representations of algebraic variants of reductive Lie groups. This is not totally surprising since the matrices defining algebraic matrix group has matrix elements in the extension of rationals. This raises the question how closely the number theoretic Langlands correspondence corresponds to the basic physical picture of TGD.

1. Could normal sub-groups of symplectic group and of Galois groups correspond to each other?

Measurement resolution realized in terms of various inclusion is the key principle of quantum TGD. There is an analogy between the hierarchies of Galois groups, of fractal sub-algebras of supersymplectic algebra (SSA), and of inclusions of hyperfinite factors of type II1 (HFFs). The inclusion hierarchies of isomorphic sub-algebras of SSA and of Galois groups for sequences of extensions of extensions should define hierarchies for measurement resolution. Also the inclusion hierarchies of HFFs are proposed to define hierarcies of measurement resolutions. How closely are these hierarchies related and could the notion of measurement resolution allow to gain new insights about these hierarchies and even about the mathematics needed to realize them?

  1. As noticed, SSA and its isomorphic sub-algebras are in a relation analogous to the between normal sub-group H of group Gal (analog of isomorphic sub-algebra) and the group G/H. One can assign to given Galois extension a hierarchy of intermediate extensions such that one proceeds from given number field (say rationals) to its extension step by step. The Galois groups H for given extension is normal sub-group of the Galois group of its extension. Hence Gal/H is a group. The physical interpretation is following. Finite measurement resolution defined by the condition that H acts trivially on the representations of Gal implies that they are representations of Gal/H. Thus Gal/H is completely analogous to the Kac-Moody type algebra conjecture to result from the analogous pair for SSA.

  2. How does this relate to McKay correspondence stating that inclusions of HFFs correspond to finite discrete sub-groups of SU(2) acting as isometries of regular n-polygons and Platonic solids correspond to Dynkin diagrams of ADE type Super Kac-Moody algebras (SKMAs) determined by ADE Lie group G. Could one identify the discrete groups as Galois groups represented geometrically as sub-groups of SU(2) and perhaps also those of corresponding Lie group? Could the representations of Galois group correspond to a sub-set of representations of G defining ground states of Kac-Moody representations. This might be possible. The sub-groups of SU(2) can however correspond only to a very small fraction of Galois groups.

Can one imagine a generalization of ADE correspondence? What would be required that the representations of Galois groups relate in some natural manner to the representations as Kac-Moody groups.

1.1 Some basic facts about Galois groups and finite groups

Some basic facts about Galois groups mus be listed before continuing. Any finite group can appear as a Galois group for an extension of some number field. It is known whether this is true for rationals (see this).

Simple groups appear as building bricks of finite groups and are rather well understood. One can even speak about periodic table for simple finite groups (see this). Finite groups can be regarded as a sub-group of permutation group Sn for some n. They can be classified to cyclic, alternating , and Lie type groups. Note that alternating group An is the subgroup of permutation group Sn that consists of even permutations. There are also 26 sporadic groups and Tits group.

Most simple finite groups are groups of Lie type that is rational sub-groups of Lie groups. Rational means ordinary rational numbers or their extension. The groups of Lie type (see this) can be characterized by the analogs of Dynkin diagrams characterizing Lie algebras. For finite groups of Lie type the McKay correspondence could generalize.

1.2 Representations of Lie groups defining Kac-Moody ground states as irreps of Galois group?

The goal is to generalize the McKay correspondence. Consider extension of rationals with Galois group Gal. The ground staes of KMA representations are irreps of the Lie group G defining KMA. Could the allow ground states for given Gal be irreps of also Gal?

This constraint would determine which group representations are possible as ground states of SKMA representations for a given Gal. The better the resolution the larger the dimensions of the allowed representations would be for given G. This would apply both to the representations of the SKMA associated with dynamical symmetries and maybe also those associated with the standard model symmetries. The idea would be quantum classical correspondence (QCC) space-time sheets as coverings would realize the ground states of SKMA representations assignable to the various SKMAs.

This option could also generalize the McKay correspondence since one can assign to finite groups of Lie type an analog of Dynkin diagram (see this). For Galois groups, which are discrete finite groups of SU(2) the hypothesis would state that the Kac-Moody algebra has same Dynkin diagram as the finite group in question.

To get some perspective one can ask what kind of algebraic extensions one can assign to ADE groups appearing in the McKay correspondence? One can get some idea about this by studying the geometry of Platonic solids (see this). Also the geometry of Dynkin diagrams telling about the geometry of root system gives some idea about the extension involved.

  1. Platonic solids have p vertices and q faces. One has [p,q]∈ { [3, 3], [4, 3], [3, 4], [5, 3], [3, 5]}. Tetrahedron is self-dual (see this) object whereas cube and octahedron and also dodecahedron and icosahedron are duals of each other. From the table of Wikipedia article one finds that the cosines and sines for the angles between the vectors for the vertices of tetrahedron, cube, and octahedron are rational numbers. For icosahedron and dodecahedron the coordinates of vertices and the angle between these vectors involve Golden Mean φ=(1+51/2)/2 so that algebraic extension must involve 51/2 at least.

    The dihedral angle θ between the faces of Platonic solid [p,q] is given by sin(θ/2)= cos(π/q)/sin(π/p). For tetrahedron, cube and octahedron sin(θ) and cos(θ) involve 31/2. For icosahedron dihedral angle is tan(θ/2)= φ. For instance, the geometry of tetrahedron involves both 21/2 and 31/2. For dodecahedron more complex algebraic numbers are involved.

  2. The rotation matrices for for the triangles of tetrahedron and icosahedron involve cos(2π/3) and sin(2π/3) associated with the quantum phase q= exp(i2π/3) associated with it. The rotation matrices performing rotation for a pentagonal face of dodecahedron involves cos(2π/5) and sin(2π/5) and thus q= exp(i2π/5) characterizing the extension. Both q= exp(i2π/3) and q= exp(i2π/5) are thus involved with icosahedral and dodecahedral rotation matrices. The rotation matrices for cube and for octahedron have rational matrix elements.

  3. The Dynkin diagrams characterize both the finite discrete groups of SU(2) and those of ADE groups. The Dynkin diagrams of Lie groups reflecting the structure of corresponding Weyl groups involve only the angles π/2, 2π/3, π-π/6, 2π- π/6 between the roots. They would naturally relate to quadratic extensions.

    For ADE Lie groups the diagram tells that the roots associated with the adjoint representation are either orthogonal or have mutual angle of 2π/3 and have same length so that length ratios are equal to 1. One has sin(2π/3)= 31/2/2. This suggests that 31/2 belongs to the algebraic extension associated with ADE group always. For the non-simply laced Lie groups of type B, C, F, G the ratios of some root lengths can be 21/2 or 31/2.

For ADE groups assignable to n-polygons (n>5) Galois group must involve the cyclic extension defined by exp(i2π/n). The simplest option is that the extension corresponds to the roots of the polynomial xn= 1.

2. A possible connection with number theoretic Langlands correspondence

I have discussed number theoretic version of Langlands correspondence in \citeallb/Langland,Langlandsnew trying to understand it using physical intuition provided by TGD (the only possible approach in my case). Concerning my unashamed intrusion to the territory of real mathematicians I have only one excuse: the number theoretic vision forces me to do this.

Number theoretic Langlands correspondence relates finite-dimensional representations of Galois groups and so called automorphic representations of reductive algebraic groups defined also for adeles, which are analogous to representations of Poincare group by fields. This is kind of relationship can exist follows from the fact that Galois group has natural action in algebraic reductive group defined by the extension in question.

The "Resiprocity conjecture" of Langlands states that so called Artin L-functions assignable to finite-dimensional representations of Galois group Gal are equal to L-functions arising from so called automorphic cuspidal representations of the algebraic reductive group G. One would have correspondence between finite number of representations of Galois group and finite number of cuspidal representations of G.

This is not far from what I am naively conjecturing on physical grounds: finite-D representations of Galois group are reductions of certain representations of G or of its subgroup defining the analog of spin for the automorphic forms in G (analogous to classical fields in Minkowski space). These representations could be seen as induced representations familiar for particle physicists dealing with Poincare invariance. McKay correspondence encourages the conjecture that the allowed spin representations are irreducible also with respect to Gal. For a childishly naive physicist knowing nothing about the complexities of the real mathematics this looks like an attractive starting point hypothesis.

In TGD framework Galois group could provide a geometric representation of "spin" (maybe even spin 1/2 property) as transformations permuting the sheets of the space-time surface identifiable as Galois covering. This geometrization of number theory in terms of cognitive representations analogous to the use of algebraic groups in Galois correspondence might provide a totally new geometric insights to Langlands correpondence. One could also think that Galois group represented in this manner could combine with the dynamical Kac-Moody group emerging from SSA to form its Langlands dual.

Skeptic physicist taking mathematics as high school arithmetics might argue that algebraic counterparts of reductive Lie groups are rather academic entities. In adelic physics the situation however changes completely. Evolution corresponds to a hierarchy of extensions of rationals reflected directly in the physics of dark matter in TGD sense: that is as phases of ordinary matter with heff/h=n identifiable as order of Galois group for extension of rationals. Algebraic groups and their representations get physical meaning and also the huge generalization of their representation to adelic representations makes sense if TGD view about consciousness and cognition is accepted.

In attempts to understand what Langlands conjecture says one should understand first the rough meaning of many concepts. Consider first the Artin L-functions appearing at the number theoretic side. Consider first the Artin L-functions appearing at the number theoretic side.

  1. L-functions (see this) are meromorphic functions on complex plane that can be assigned to number fields and are analogs of Riemann zeta function factorizing into products of contributions labelled by primes of the number field. The definition of L-function involves Direchlet characters: character is very general invariant of group representation defined as trace of the representation matrix invariant under conjugation of argument.

  2. In particular, there are Artin L-functions (see this) assignable to the representations of non-Abelian Galois groups. One considers finite extension L/K of fields with Galois group G. The factors of Artin L-function are labelled by primes p of K. There are two cases: p is un-ramified or ramified depending on whether the number of primes of L to which p decomposes is maximal or not. The number of ramified primes is finite and in TGD framework they are excellent candidates for physical preferred p-adic primes for given extension of rationals.

    These factors labelled by p analogous to the factors of Riemann zeta are identified as characteristic polynomials for a representation matrix associated with any element in a preferred conjugacy class of G. This preferred conjugacy class is known as Frobenius element Frob(p) for a given prime ideal p , whose action on given algebraic integer in OL is represented as its p:th power. For un-ramified p the characteristic polynomial is explicitly given as determinant det[I-tρ(Frob(p))]-1, where one has t= N(p)-s and N(p) is the field norm of p in the extension L (see this).

    In the ramified case one must restrict the representation space to a sub-space invariant under inertia subgroup, which by definition leaves invariant integers of OL/p that is the lowest part of integers in expansion of powers of p.

At the other side of the conjecture appear representations of algebraic counterparts of reductive Lie groups and their L-functions and the two number theoretic and automorphic L-functions would be identical.
  1. Automorphic form F generalizes the notion of plane wave invariant under discrete subgroup of the group of translations and satisfying Laplace equation defining Casimir operator for translation group. Automorphic representations can be seen as analogs for the modes of classical fields with given mass having spin characterized by a representation of subgroup of Lie group G (SO(3) in case of Poincare group).

    Automorphic functions as field modes are eigen modes of some Casimir operators assignable to G. Algebraic groups would in TGD framework relate to adeles defined by the hierarchy of extensions of rationals (also roots of e can be considered in extensions). Galois groups have natural action in algebraic groups.

  2. Automorphic form (see this) is a complex vector valued function F from topological group to some vector space V. F is an eigen function of certain Casimir operators of G. In the simplest situation these function are invariant under a discrete subgroup Γ⊂ G identifiable as the analog of the subgroup defining spin in the case of induced representations.

    In general situation the automorphic form F transforms by a factor j of automorphy under Γ. The factor can also act in a finite-dimensional representation of group Γ, which would suggest that it reduces to a subgroup of Γ obtained by dividing with a normal subgroup. j satisfies 1-cocycle condition j(g1,g2g3)= j(g1g2,g3) in group cohomology guaranteeing associativity (see this). Cuspidality relates to the conditions on the growth of F at infinity.

  3. Elliptic functions in complex plane characterized by two complex periods are meromorphic functions of this kind. A less trivial situation corresponds to non-compact group G=SL(2,R) and Γ ⊂ SL(2,Q).

There are more groups involved: Langlands group LF and Langlands dual group LG. A more technical formulation says that the automorphic representations of a reductive Lie group G correspond to homomorphisms from so called Langlands group LF (see this) at the number theoretic side to L-group LG or Langlands dual of algebraic G at group theory side (see this). It is important to notice that LG is a complex Lie group. Note also that homomorphism is a representation of Langlands group LF in L-group LG. In TGD this would be analogous to a homomorphism of Galois group defining it as subgroup of the group G defining Kac-Moody algebra.
  1. Langlands group LF of number field is a speculative notion conjectured to be a extension of the Weil group of extension, which in turn is a modification of the absolute Galois group. Unfortunately, I was not able to really understand the Wikipedia definition of Weil group (this). If E/F is finite extension as it is now, the Weil group would be WE/F= WF/WcE, WcE refers to the commutator subgroup WE defining a normal subgroup, and the factor group is expected to be finite. This is not Galois group but should be closely related to it.

    Only finite-D representations of Langlands group are allowed, which suggests that the representations are always trivial for some normal subgroup of LF For Archimedean local fields LF is Weil group, non-Archimedean local fields LF is the product of Weil group of L and of SU(2). The first guess is that SU(2) relates to quaternions. For global fields the existence of LF is still conjectural.

  2. I also failed to understand the formal Wikipedia definition of the L-group LG appearing at the group theory side. For a reductive Lie group one can construct its root datum (X*,Δ,X*, Δc), where X* is the lattice of characters of a maximal torus, X* its dual, Δ the roots, and Δc the co-roots. Dual root datum is obtained by switching X* and X* and Δ and Δc. The root datum for G and LG are related by this switch.

    For a reductive G the Dynkin diagram of LG is obtained from that of G by exchanging the components of type Bn with components of type Cn. For simple groups one has Bn↔ Cn. Note that for ADE groups the root data are same for G and its dual and it is the Kac-Moody counterparts of ADE groups, which appear in McKay correspondence. Could this mean that only these are allowed physically?

  3. Consider now a reductive group over some field with a separable closure K (say k for rationals and K for algebraic numbers). Over K G as root datum with an action of Galois group of K/k. The full group LG is the semi-direct product LG0⋊ Gal(K/k) of connected component as Galois group and Galois group. Gal(K/k) is infinite (absolute group for rationals). This looks hopelessly complicated but it turns it that one can use the Galois group of a finite extension over which G is split. This is what gives the action of Galois group of extension (l/k) in LG having now finitely many components. The Galois group permutes the components. The action is easy to understand as automorphism on Gal elements of G.

Could TGD picture provide additional insights to Langlands duality or vice versa?
  1. In TGD framework the action of Gal on algebraic group G is analogous to the action of Gal on cognitive representation at space-time level permuting the sheets of the Galois covering, whose number in the general case is the order of Gal identifiable as heff/h=n. The connected component LG0 would correspond to one sheet of the covering.

  2. What I do not understand is whether LG =G condition is actually forced by physical contraints for the dynamical Kac-Moody algebra and whether it relates to the notion of measurement resolution and inclusions of HFFs.

  3. The electric-magnetic duality in gauge theories suggests that gauge group action of G on electric charges corresponds in the dual phase to the action of LG on magnetic charges. In self-dual situation one would have G=LG. Intriguingly, CP2 geometry is self-dual (Kähler form is self-dual so that electric and magnetic fluxes are identical) but induced Kähler form is self-dual only at the orbits of partonic 2-surfaces if weak form of electric-magnetic duality holds true. Does this condition leads to LG=G for dynamical gauge groups? Or is it possible to distinguish between the two dynamical descriptions so that Langlands duality would correspond to electric-magnetic duality. Could this duality correspond to the proposed duality of two variants of SH: namely, the electric description provided by string world sheets and magnetic description provided by partonic 2-surfaces carrying monopole fluxes?

See the new chapter Are higher structures needed in the categorification of TGD? of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.

Saturday, June 24, 2017

Are Preferred Extremals Quaternion-Analytic in Some Sense?

A generalization of 2-D conformal invariance to its 4-D variant is strongly suggestive in TGD framework, and leads to the idea that for preferred extremals of action space-time regions have (co-)associative/(co-)quaternionic tangent space or normal space. The notion of M8-H correspondence allows to formulate this idea more precisely. The beauty of this notion is that it does not depend on the signature of Minkowski space M4 representable as sub-space of of complexified quaternions M4c, which in turn can be seen as sub-space of complexified octonions M8c.

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. This notion is however not so straightforward even in Euclidian signature, and the generalization to Minkowskian signature brings in further problems. The Cauchy-Riemann-Fuerter conditions make however sense also in Minkowskian quaternionic situation and the problem is whether they allow the physically expected solutions. One should also show that the possible generalization is consistent with (co)-associativity.

In this article these problems are considered. Also a comparison with Igor Frenkel's ideas about hierarchy of Lie algebras, loop, algebras and double look algebras and their quantum variants is made: it seems that TGD as a generalization of string models replacing string world sheets with space-time surfaces gives rise to the analogs of double loop algebras and they quantum variants and Yangians. The straightforward generalization of double loop algebras seems to make sense only at the light-like boundaries of causal diamonds and at light-like orbits of partonic 2-surfaces but that in the interior of space-time surface the simple form of the conformal generators is not preserved. The twistor lift of TGD in turn corresponds nicely to the heuristic proposal of Frenkel for the realization of double loop algebras.

See the article Are Preferred Extremals Quaternion-Analytic in Some Sense? or the chapter Unified Number Theoretical Vision
of "TGD as Generalized Number Theory".

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

Articles and other material related to TGD.

Philosophy of Adelic Physics

The p-adic aspects of Topological Geometrodynamics (TGD) will be discussed. Introduction gives a short summary about classical and quantum TGD. This is needed since the p-adic ideas are inspired by TGD based view about physics.

p-Adic mass calculations relying on p-adic generalization of thermodynamics and super-symplectic and super-conformal symmetries are summarized. Number theoretical existence constrains lead to highly non-trivial and successful physical predictions. The notion of canonical identification mapping p-adic mass squared to real mass squared emerges, and is expected to be a key player of adelic physics allowing to map various invariants from p-adics to reals and vice versa.

A view about p-adicization and adelization of real number based physics is proposed. The proposal is a fusion of real physics and various p-adic physics to single coherent whole achieved by a generalization of number concept by fusing reals and extensions of p-adic numbers induced by given extension of rationals to a larger structure and having the extension of rationals as their intersection.

The existence of p-adic variants of definite integral, Fourier analysis, Hilbert space, and Riemann geometry is far from obvious and various constraints lead to the idea of number theoretic universality (NTU) and finite measurement resolution realized in terms of number theory. An attractive manner to overcome the problems in case of symmetric spaces relies on the replacement of angle variables and their hyperbolic analogs with their exponentials identified as roots of unity and roots of e existing in finite-dimensional algebraic extension of p-adic numbers. Only group invariants - typically squares of distances and norms - are mapped by canonical identification from p-adic to real realm and various phases are mapped to themselves as number theoretically universal entities.

Also the understanding of the correspondence between real and p-adic physics at various levels - space-time level, imbedding space level, and level of "world of classical worlds" (WCW) - is a challenge. The gigantic isometry group of WCW and the maximal isometry group of imbedding space give hopes about a resolution of the problems. Strong form of holography (SH) allows a non-local correspondence between real and p-adic space-time surfaces induced by algebraic continuation from common string world sheets and partonic 2-surfaces. Also local correspondence seems intuitively plausible and is based on number theoretic discretization as intersection of real and p-adic surfaces providing automatically finite "cognitive" resolution. he existence p-adic variants of Kähler geometry of WCW is a challenge, and NTU might allow to realize it.

I will also sum up the role of p-adic physics in TGD inspired theory of consciousness. Negentropic entanglement (NE) characterized by number theoretical entanglement negentropy (NEN) plays a key role. Negentropy Maximization Principle (NMP) forces the generation of NE. The interpretation is in terms of evolution as increase of negentropy resources.

For details see the new chapter Philosophy of Adelic Physics of "Physics as Generalized Number Theory".

Wednesday, June 14, 2017

Why should stars be borne in pairs?

Stars seem to be born in pairs! For a popular article see this. The research article "Embedded Binaries and Their Dense Cores" is here.

For instance, our nearest neighbor, Alpha Centauri, is a triplet system. Explanation for this have been sought for for a long time. Does star capture occur leading to binaries or triplets. Or does its reverse process in which binary splits up to become single stars occur? There has been even a search for a companion of Sun christened Nemesis.

The new assertion is based on radio survey of a giant molecular cloud filled with recently formed sunlike stars (with age less than 4 million years) in constellation Perseus, a star nursery located 600 ly from us in Milky Way. All singles and twins with separations above 15 AUs were counted.

The proposed mathematical model was able to explain the observations only if all sunlike stars are born as wide binaries. "Wide" means that the mutual distance is more than 500 AU, where AU is the distance of Earth from Sun. After the birth the systems would shrink or split t within time about million years. It was found that wide binaries were not only very young but also tended to be aligned along the long axes of an egg-shaped dense core. Older systems did not have this tendency. For instance, triplets could form as binary captures a single star.

The theory says nothing about why the stars should born as binaries and what could be the birth mechanism. Could TGD say anything interesting about the how the binaries are formed?

  1. TGD based model for galaxies leads to the proposal that the region in which dark matter has constant density corresponds to a very knotted and possibly thickened cosmic string portion or closed very knotted string associated with long cosmic string. There would be an intersection of separate cosmic strings or self-intersection of single cosmic string giving rise to a galactic blackhole from which dark matter emerges and transforms to ordinary matter. Star formation would take place in this region 2-3 times larger than the optical region.

  2. Could an analogous mechanism be at work in star formation? Suppose that there is cosmic string in galactic plane and it has two nearby non-intersecting portions roughly parallel to each other. Deform the other one slightly locally so that it forms intersections with another one. The minimal number of stable intersections is 2 and even number in the general case. Single intersection corresponding to mere touching is a topologically unstable situation. If the intersections give rise to dark blackholes generating later the stars would have explanation for why stars are formed as twin pairs.

This would also explain why the blackholes possibly detected by LIGO are so massive (there is still debate about this going on): they would have not yet produced ordinary stars, a process in which part of dark matter and dark energy of cosmic strings transforms to ordinary matter.
  1. Suppose that these blackhole like objects are indeed intersections of two portions of cosmic string(s). The intersections have gravitational interaction and could move along the second cosmic string towards each other and eventually collide.

  2. More concretely, one can imagine a straight horizontal starionary string A (at x-axis with y=0 in (x,y)-coordinates) and a folded string B with a shape of an inverted vertical parabola (y=-ax2+y0(t), a>0, and moving downwards. In other words, y0(t) decreases with time. The strings A and B have two nearby intersections x+/-= +/- (y0(t)/a)1/2. Their distance decreases with time and eventually the intersection points fuse together at y0(t)=0 and give rise to the fusion of two black-hole like entities to single one.

See the the chapter TGD and astrophysics or the article TGD view about universal galactic rotation curves for spiral galaxies.

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

Articles and other material related to TGD.

Tuesday, June 13, 2017

Are higher structures needed in the categorification of TGD?

The notion of higher structures promoted by John Baez looks very promising notion in the attempts to understand various structures like quantum algebras and Yangians in TGD framework. The stimulus for this article came from the nice explanations of the notion of higher structure by Urs Screiber. The basic idea is simple: replace "=" as a blackbox with an operational definition with a proof for $A=B$. This proof is called homotopy generalizing homotopy in topological sense. n-structure emerges when one realizes that also the homotopy is defined only up to homotopy in turn defined only up...

In TGD framework the notion of measurement resolution defines in a natural manner various kinds of "="s and this gives rise to resolution hierarchies. Hierarchical structures are characteristic for TGD: hierarchy of space-time sheet, hierarchy of p-adic length scales, hierarchy of Planck constants and dark matters, hierarchy of inclusions of hyperfinite factors, hierarchy of extensions of rationals defining adeles in adelic TGD and corresponding hierarchy of Galois groups represented geometrically, hierarchy of infinite primes, self hierarchy, etc...

In this article the idea of n-structure is studied in more detail. A rather radical idea is a formulation of quantum TGD using only cognitive representations consisting of points of space-time surface with imbedding space coordinates in extension of rationals defining the level of adelic hierarchy. One would use only these discrete points sets and Galois groups. Everything would reduce to number theoretic discretization at space-time level perhaps reducing to that at partonic 2-surfaces with points of cognitive representation carrying fermion quantum numbers.

Even the"{world of classical worlds" (WCW) would discretize: cognitive representation would define the coordinates of WCW point. One would obtain cognitive representations of scattering amplitudes using a fusion category assignable to the representations of Galois groups: something diametrically opposite to the immense complexity of the WCW but perhaps consistent with it. Also a generalization of McKay's correspondence suggests itself: only those irreps of the Lie group associated with Kac-Moody algebra that remain irreps when reduced to a subgroup defined by a Galois group of Lie type are allowed as ground states.

See the new chapter Are higher structures needed in the categorification of TGD? of "Towards M-matrix" or the article with the same title.

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

Articles and other material related to TGD.

Friday, June 09, 2017

New view about galaxies and galactic blackholes

We had very interesting discussions with Gareth Lee Meredith in Beyond the Standard Models founded by Gareth. We talked about galaxy formation and various anomalies related to galactic dynamics popping up almost continually and challenging the halo model for dark matter. Unfortunately Gareth lost access to his FB and also Messenger account. It is extremely frustrating that this FB attack makes impossible to continue even discussions using Messenger.

There are good reasons to expect that some malevolent person has made an appeal to FB - maybe claiming that there is hate speech at his page. This is certainly not true: the page has a very friendly polite spirit. I have been also myself been a victim of this kind of FB attack: my posts to another FB page were not shown at all for months. I never learned what the reason was. FB should be better prepared for the possibility that some malevolent person, perhaps envious colleague, tries to make communications impossible.

One of the topics of discussion was results related to supermassive blackholes at the centers of galaxies. Gareth gave a link to an article telling about correlations between supermassive blackhole in galactic center and the evolution of galaxy itself.

  1. The size of the blackhole like object - that is its mass if blackhole in GRT sense is in question - correlates with the constant rotation velocity of distant stars for spiral galaxies.

  2. The relationship between the masses of black hole and galactic bulge are in constant relation: the mass ratio is about 700.

  3. A further finding is that galactic blackholes of very old stars are much more massive than the idea about galactic blackhole getting gradually bigger by "eating" surrounding stars would suggests. Unfortunately, I did not find link this article due to the strange FB episode.

This looks strange if one believes in the standard dogma that the galactic blackhole started to form relatively lately. What comes in mind is rather unorthodox idea. What if the large blackhole like entity was there from the beginning and gradually lost its mass? In TGD framework this could make sense!
  1. In TGD Universe galaxies are like pearls in a necklace defined by a long cosmic string. This explains the flat rotational spectrum and predicts essentially free motion along the string related perhaps to coherent motions in very long length scales. This explains also the old observation that galaxies form filament like structures and the correlations between spin directions of galaxies along the same filament since one expects that the spin is parallel to the filament locally. Filament can of course change its direction locally so that charge of direction of rotation gives information about the filament shape.

  2. The channelling of gravitational flux in the radial direction orthogonal to the string makes gravitational force very long ranged (1/transversal distance instead of 1/r2) and also stronger and predicts rotational spectrum. This model of dark matter differs dramatically from the fashionable halo model and involves only the string tension as a parameter unlike the halo model.

    The observed rigid body rotation within radius 2-3 times the optical radius (region inside which most stars are) can be understood if the long cosmic string is either strongly knotted or has closed galactic string around long cosmic string. The knotted portion would formed a highly knotted spaghetti like structure giving approximately constant mass density. Stars would be associated with the knotted structure as sub-knots. Light beams from supernovas could be along the string going through the star. Maybe even planets might be associated with thickened strings! One can also imagine intersections of long cosmic strings and Milky Way could contain such.

  3. Galactic black hole like object could correspond to a self intersection of the long cosmic string or of closed galactic cosmic string bound to it. There could be several intersections. They would contain both dark matter and energy in TGD sense and located inside the string. Matter antimatter asymmetry would mean that there is slightly more antimatter inside string and slightly more matter outside it. Twistor lift of TGD predicts the needed new kind of CP breaking. What is new that the galactic blackhole like objects would be present from the beginning and lose their dark mass gradually. Time evolution would be opposite to what it has been usually thought to be!

    Most of the energy of the cosmic string would be magnetic energy identifiable as dark energy. During the cosmic evolution various perturbations would force the cosmic string to gradually thicken so that in M4 projection ceases to be pointlike. Magnetic monopole flux is conserved (BS= constant, S the transversal area), which forces magnetic energy density per unit length - string tension - to be reduced like 1/S. The lost energy becomes ordinary matter: the energy of inflaton field would be replaced with dark magnetic energy and the TGD counterpart for inflationary period would be transition from cosmic string dominated period to radiation dominated cosmology and also the emergence of space-time in GRT sense.

    The primordial cosmic string dominated phase would consist of cosmic strings in M2×CP2. The explanation for the constancy of CMB temperature would suggest quantum coherence in even cosmic scales made possible by the hierarchy of dark matters labelled by the valued of Planck constant heff/h=n. Maybe characterization as a super-fluid rather than gas discussed with Garrett is more precise manner to say it. What would be fantastic that these primordial structures would be directly visible nowadays.

  4. The dark matter particles emanating from the dark supermassive blackhole would transform gradually to ordinary matter so that galaxy would be formed. This would explain the correlation of the bulge size with the mass (and size) of the blackhole correlating with the string tension. The rotational velocity of distant stars with string tension so that the strange correlation between velocity of distant stars and size of galactic blackhole is implied by a common cause.

    This also explains the appearance of Fermi bubbles. Fermi bubbles are formed when dark particles from the blackhole scatter with dark matter and partially transform to ordinary cosmic rays and produce dark photons transformed to visible photons partially. This occurs only within the region where the spaghetti like structure containing dark matter inside the cosmic string exists. Fermi bubbles indeed have the same size as this region.

  5. While writing this I realized that also the galactic bar (2/3 of spiral galaxies have it) should be understood. This is difficult if there is nothing breaking the rotational symmetry around the long cosmic string. The situation changes if one has a portion of cosmic string along the plane of galaxy.

    There is indeed evidence for the second straight string portion: in Milky Way there are mini-galaxies rotating in the plane forming roughly 60 degrees angle with respect to galactic plane and the presence of two cosmic strings portions roughly orthogonal to each other could explain this (see this). Galactic blackhole could be associated with the intersection of string portions. The horizontal string portion could be part of long cosmic string, a separate closed cosmic string, or even another long cosmic string. One can imagine two basic options for the formation of the bar.

    1. The first option is that galactic bar is formed around the straight portion of string. The gravitational force orthogonal to the string portion would create the bar. The ordinary matter in rigid body rotation would be accelerated while approaching the bar and then slow down and dissipate part of its energy in the process. The slowed down stars would after a further rotation of π tend to stuck around the string portion forming bound states with it and start to rotate around it: a kind of galactic traffic jam. Bars would be asymptotic outcomes of the galactic dynamics. Recent studies have confirmed the idea that bars are now are signs of full maturity as the "formative years" end (see this).

    2. Second option is that the bar is formed as dark matter inside bar is transformed to ordinary matter as the portion thickens and loses dark energy identified as Kähler magnetic energy by a process analogous to the decay of inflaton vacuum energy. Bars would be transients in the evolution of galaxies rather than final outcomes. This option is not consistent with the idea that that only the galactic blackhole serves as the source of dark matter transforming to ordinary matter.

  6. The pearls in string model explains also why elliptic galaxies have declining rotational velocity. They correspond to "free" closed strings which have not formed bound states with long cosmic strings transforming them to spiral galaxies. The recently found 10 billion old galaxies with declining rotational velocity could correspond to elliptical galaxies of this kind.

    One can also imagine the analog of ionization. The bound state of closed cosmic string and long cosmic string decays and spiral galaxy starts to decay under centrifugal force not anymore balanced by the gravitational force of the long cosmic strings and would transform to elliptic galaxy. Also the central bulge would start to increase in size.

    It would also lose its central blackhole if is associated with the long cosmic string. I am grateful for Garreth for giving a link to a popular article telling about this kind of elliptic galaxy with very large size of one million light years and without central blackhole and unusually large bulge region.

This view about galactic blackholes also suggests a profound revision of GRT based view for the formation of blackholes. Note that in TGD one must of course speak about blackhole like objects differing from their GRT counterparts inside Schwartschildt radius and also outside it in microscopic scales (gravitational flux is mediated by magnetic flux tubes carrying dark particles). Perhaps also ordinary blachholes were once intersections of dark cosmic strings containing dark matter which gradually produce the stellar matter! If so, old blackholes would be more massive than the young ones.
  1. This new thinking conform with the findings of LIGO. All the three stellar blackholes have been by more than order of magnitude massive than expected. There are also indications that the members of the second blackhole pair merging together did not have parallel spin directions. This does not fit with the idea that a twin pairs of stars was in question. It is very difficult to understand how two blackholes, which do not form bound system could find each other. Similar problem is encountered in bio-catalysis: who to biomolecules manage to find each other in the molecular crowd. The solution to the both problem is very similar.

  2. TGD suggests that the collision could have occurred when to blackholes travelling along strings or portions of the same knotted string arrived from different directions. The gravitational attraction between strings would have helped to generate the intersection and strings would have guided the blackholes together. In biological context even a phase transition reducing Planck constant to the flux tube connecting the molecules could occur and bring the molecules together.

See the article TGD view about universal galactic rotation curves for spiral galaxies.

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

Articles and other material related to TGD.

Friday, June 02, 2017

Neutron production from an arc current in gaseous hydrogen: 66 year old nuclear physics anomaly

I learned about nuclear physics anomaly new to me (actually the anomaly is 64 years old) from an article of Norman and Dunning-Davies in Research Gate (see this). Neutrons are produced from an arc current in hydrogen gas with a rate exceeding dramatically the rate predicted by the standard model of electroweak interactions, in which the production should occur through e-+p→ n+ν by weak boson exchange. The low electron energies make the process also kinematically impossible. Additional strange finding due to Borghi and Santilli is that the neutron production can in some cases be delayed by several hours. Furthermore, according to Santilli neutron production occurs only for hydrogen but not for heavier nuclei.

In the following I sum up the history of the anomaly following closely the representation of Norman and Dunning-Davies (see this): this article gives references and details and is strongly recommended. This includes the pioneering work of Sternglass in 1951, the experiments of Don Carlo Borghi in the late 1960s, and the rather recent experiments of Ruggiero Santilli (see this).

The pioneering experiment of Sternglass

The initial anomalously large production of neutrons using an current arc in hydrogen gas was performed by Earnest Sternglass in 1951 while completing his Ph.D. thesis at Cornell. He wrote to Einstein about his inexplicable results, which seemed to occur in conditions lacking sufficient energy to synthesize the neutrons that his experiments had indeed somehow apparently created. Although Einstein firmly advised that the results must be published even though they apparently contradicted standard theory, Sternglass refused due to the stultifying preponderance of contrary opinion and so his results were preemptively excluded under orthodox pressure within discipline leaving them unpublished. Edward Trounson, a physicist working at the Naval Ordnance Laboratory repeated the experiment and again gained successful results but they too, were not published.

One cannot avoid the question, what physics would look like today, if Sternglass had published or managed to publish his results. One must however remember that the first indications for cold fusion emerged also surprisingly early but did not receive any attention and that cold fusion researchers were for decades labelled as next to criminals. Maybe the extreme conservatism following the revolution in theoretical physics during the first decades of the previous century would have prevented his work to receive the attention that it would have deserved.

The experiments of Don Carlo Borghi

Italian priest-physicist Don Carlo Borghi in collaboration with experimentalists from the University of Recife, Brazil, claimed in the late 1960s to have achieved the laboratory synthesis of neutrons from protons and electrons. C. Borghi, C. Giori, and A. Dall'Olio published 1993 an article entitled "Experimental evidence of emission of neutrons from cold hydrogen plasma" in Yad. Fiz. 56 and Phys. At. Nucl. 56 (7).

Don Borghi's experiment was conducted via a cylindrical metallic chamber (called "klystron") filled up with a partially ionized hydrogen gas at a fraction of 1 bar pressure, traversed by an electric arc with about 500V and 10mA as well as by microwaves with 1010 Hz frequency. Note that the energies of electrons would be below .5 keV and non-relativistic. In the cylindrical exterior of the chamber the experimentalists placed various materials suitable to become radioactive when subjected to a neutron flux (such as gold, silver and others). Following exposures of the order of weeks, the experimentalists reported nuclear transmutations due to a claimed neutron flux of the order of 104 cps, apparently confirmed by beta emissions not present in the original material.

Don Borghi's claim remained un-noticed for decades due to its incompatibility with the prevailing view about weak interactions. The process e-+p→ n+ν is also forbidden by conservation of energy unless the total cm energy of proton and the electron have energy larger than Δ E= mn-mp-me=0.78 MeV. This requires highly relativistic electrons. Also the cross section for the reaction proceeding by exchange of W boson is extremely small at low energies (about 10-20 barn: barn=10-28 m2 represents the natural scale for cross section in nuclear physics). Some new physics must be involved if the effect is real. Situation is strongly reminiscent of cold fusion (or low energy nuclear reactions (LENR), which many main stream nuclear physicists still regard as a pseudoscience.

Santilli's experiments

Ruggero Santilli (see this) replicated the experiments of Don Borghi. Both in the experiments of Don Carlo Borghi and those of Santilli, delayed neutron synthesis was sometimes observed. Santilli analyzes several alternative proposals explaining the anomalyn and suggests that new spin zero bound state of electron and proton with rest mass below the sum of proton and electron masses and absorbed by nuclei decaying then radioactively could explain the anomaly. The energy needed to overcome the kinematic barrier could come from the energy liberated by electric arc. The problem of the model is that it has no connection with standard model.

Both in the experiments of Don Carlo Borghi and those of Santilli, delayed neutron synthesis was sometimes observed. According to Santilli: According to Santilli:

" A first series of measurements was initiated with Klystron I on July 28,2006, at 2 p.m. Following flushing of air, the klystron was filled up with commercial grale hydrogen at 25 psi pressure. We first used detector PM1703GN to verify that the background radiations were solely consisting of photon counts of 5-7 μR/h without any neutron count; we delivered a DC electric arc at 27 V and 30 A (namely with power much bigger than that of the arc used in Don Borghi's tests...), at about 0.125" gap for about 3 s; we waited for one hour until the electrodes had cooled down, and then placed detector PM1703GN against the PVC cylinder. This resulted in the detection of photons at the rate of 10 - 15 μR/hr expected from the residual excitation of the tips of the electrodes, but no
neutron count at all.

However, about three hours following the test, detector PM1703GN entered into sonic and vibration alarms, specifically, for neutron detections off the instrument maximum of 99 cps at about 5' distance from the klystron while no anomalous photon emission was measured. The detector was moved outside the laboratory and the neutron counts returned to zero. The detector was then returned to the laboratory and we were surprised to see it entering again into sonic and vibrational alarms at about 5' away from the arc chamber with the neutron count off scale without appreciable detection of photons, at which point the laboratory was evacuated for safety.

After waiting for 30 minutes (double neutron's lifetime), we were surprised to see detector PMl703GN go off scale again in neutron counts at a distance of 10' from the experimental set up, and the laboratory was closed for the day."

TGD based model

The basic problems to be solved are following.

  1. What is the role of current arc and other triggering impulses (such as microwave radiation or pressure surge mentioned by Santilli): do they provide energy or do they have some other role?

  2. Neutron production is kinematically impossible if weak interactions mediate it. Even if kinematically possible, weak interaction rates are quite too slow. The creation of intermediate states via other than weak interactions would solve both problems. If weak interactions are involved with the creation of the intermediate states, how there rates can be so high?

  3. What causes the strange delays in the production in some cases but now always? Why hydrogen gas is preferred?

The effect brings strongly in mind cold fusion for which TGD proposes a model (see this) in terms of generation of dark nuclei with non-standard value heff=n× h of Planck constant formed from dark proton sequences at flux tubes. The binding energy for these states is supposed to be much lower than for the ordinary nuclei and eventually these nuclei would decay to ordinary nuclei in collisions with metallic targets attracting positively charged magnetic flux tubes. The energy liberated would be of the essentially the ordinary nuclear binding energy. Note that the creation of dark proton sequences does not require weak interactions so that the basic objections are circumvented.

TGD explanation (see this) could be the same for Tesla's findings, for cold fusion (see this), Pollack effect (see this) and for the anomalous production of neutrons. Even electrolysis would involve in an essential manner Pollack effect and new physics.

Could this model explain the anomalous neuron production and its strange features?

  1. Why electric arc, pressure surge, or microwave radiation would be needed? Dark phases are formed at quantum criticality (see this) and give rise to the long range correlations via quantum entanglement made possible by large heff=n× h. The presence of electron arc occurring as di-electric breakdown is indeed a critical phenomenon.

    Already Tesla discovered strange phenomena in his studies of arc discharges but his discoveries were forgotten by mainstream. TGD explanation (see this) could be the same for Tesla's findings, for cold fusion (see this), Pollack effect (see this) and for the anomalous production of neutrons. Even electrolysis would involve in an essential manner Pollack effect and new physics.

    Also energy feed might be involved. Quite generally, in TGD inspired quantum biology generation of dark states requires energy feed and the role of metabolic energy is to excite dark states. For instance, dark atoms have smaller binding energy and the energies of cyclotron states increase with heff/h. For instance, part of microwave photons could be dark and have much higher energy than otherwise.

    Could the production of dark proton sequences at magnetic flux tubes be all that is needed so that the possible dark variant of the reaction e-+p→ n+ν would not be needed at all?

  2. If also weak bosons appear as dark variants, their Compton length is scaled up accordingly and in scales shorter than the Compton length, they behave effectively as massless particles and weak interactions would become as strong as electromagnetic interactions. This would make possible the decay of dark proton sequences at magnetic flux tubes to beta stable dark isotopes via p→ n+e++ν. Neutrons would be produced in the decays of the dark nuclei to ordinary nuclei liberating nuclear binding energy. Note however that TGD allows also to consider p-adically scaled variants of weak bosons with much smaller mass scale possible important in biology, and one cannot exclude them from consideration.

  3. The reaction e-+p→ n+ν is not necessary in the model. One can however ask, whether there could exist a mechanism making the dark reaction e-+p→ n+ν kinematically possible. If the scale of dark nuclear binding energy is strongly reduced, also p→ n+e++ν in dark nuclei would become kinematically impossible (in ordinary nuclei nuclear binding energy makes n effectively lighter than p).

    TGD based model for nuclei as strings of nucleons (see this and this) connected by neutral or charged (possibly colored) mesonlike bonds with quark and antiquark at its ends could resolve this problem. One could have exotic nuclei in which proton plus negatively charged bond could effectively behave like neutron. Dark weak interactions would take place for neutral bonds between protons and reduce the charge of the bond from q=0 to q= -1 and transform p to effective n. This was assumed also in the model of dark nuclei and also in the model of ordinary nuclei and predicts large number of exotic states. One can of course ask, whether the nuclear neutrons are actually pairs of proton and negatively charged bond.

  4. What about the delays in neutron production occurring in some cases? Why not always? In the situations, when there is a delay in neutron production, the dark nuclei could have rotated around magnetic flux tubes of the magnetic body (MB) of the system before entering to the metal target, one would have a delayed production.

  5. Why would hydrogen be preferred? Why for instance, deuteron and heavier isotopes containing neutrons would not form dark proton sequences at magnetic flux tubes. Why would be the probability for the transformation of say D=pn to its dark variant be very small?

    If the binding energy of dark nuclei per nucleon is several orders of magnitude smaller than for ordinary nuclei, the explanation is obvious. The ordinary nuclear binding energy is much higher than the dark binding energy so that only the sequences of dark protons can form dark nuclei. The first guess (see this) was that the binding energy is analogous to Coulomb energy and thus inversely proportional to the size scale of dark nucleus scaling like h/heff. One can however ask why D with ordinary size could not serve as sub-unit.

See the article Anomalous neutron production from an arc current in gaseous hydrogen or the chapter Cold Fusion Again of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy".

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

Articles and other material related to TGD.

Third gravitational wave detection by LIGO collaboration

The news about third gravitational wave detection managed to direct the attention of at least some of us from the doings of Donald J. Trump. Also New York Times told about the gravitational wave detection by LIGO, the Laser Interferometer Gravitational-Wave Observatory. Gravitational waves are estimated to be created by a black-hole merger at distance of 3 billion light years. The results are published in the article "Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2" in Phys Rev Lett.

Two black holes with masses 19× M(Sun) and 31× M(Sun) merged to single blackhole hole of with mass of 49× M(Sun) meaning that roughly one solar mass was transformed to gravitational radiation. During the the climax of the merger, they were emitting more energy in the form of gravitational waves than all the stars in the observable universe.

The colliding blackholes were very massive in all three events. There should be some explanation for this. An explanation considered in the article is that the stars giving rise to blackholes were rather primitive containing light elements and this would have allowed large masses. The transformation to blackholes could have occurred directly without the intervening supernova phase. There is indeed quite recent finding showing a disappearance of very heavy star with 25 solar masses suggesting that direct blackhole formation without super-nova explosion is possible for heavy stars.

It is interesting to take a fresh look to these blackhole like entities in TGD framework. This however requires brief summary about the formation of galaxies and stars in TGD Universe (see this and this).

  1. The simplest possibility allowed by TGD is that galaxies as pearls in necklace are knots (or spagettilike substructures) in long cosmic strings. This does not exclude the original identification as closed strings around long cosmic string. These loops must be however knotted. Galactic super-blackhole could correspond to a self-intersection of the long cosmic string. This view is forced by the experimental finding that for mini spirals, there is volume with radius containing essentially constant density of dark matter. The radius of this volume is 2-3 times larger than the volume containing most stars of the galaxy. This region would contain a galactic knot.

    The important conclusion is that stars would be subknots of these galactic knots as indeed proposed earlier. Part of the magnetic energy would decay to ordinary matter giving rise to visible part of start as the cosmic string thickens. This conforms with the finding that the region in which dark matter density seems to be constant has size few times larger than the region containing the stars (size scale is few kpc).

  2. The light beams from supernovas would most naturally arrive along the flux tubes being bound to helical orbits rotating around them. Primordial cosmic string as stars, galaxies, linear structures of galaxies, even elementary particles, hadrons, nuclei, and biomolecules: all these structures would be magnetic flux tubes possibly knotted and linked. The space-time of GRT as a small deformation of M4 would have emerged from cosmic string dominated phase via the TGD counterpart of inflationary period. The signatures of the primordial cosmic string dominated period would be directly visible in all scales! We would be seeing the incredibly simple truth but our theories would prevent us to become aware about what we are seeing!

The crucial question concerns the dark matter fraction of the star.
  1. The fraction depends on the thickness of the deformed cosmic string having originally 1-D projection E3⊂ M4. If Kähler magnetic energy dominates, the energy per length for a thickened flux tube is proportional to 1/S, S the area of M4 projection and thus decreases rapidly with thickening. The thickness of the flux tube would be in minimum about CP2 size scale of 104 Planck lengths. If S is large enough, the contribution of cosmic string to the mass of the star is smaller than that of visible matter created in the thickening.

  2. What about very primitive stars - say those associated with LIGO mergers. The proportion of visible matter in star should gradually increase as flux tube thickens. Could the detected blackhole fusion correspond to a fusion of dark matter stars rather than that of Einsteinian blackholes? If the radius of the objects satisfies rS=2GM, the blackhole like entities are in question also in TGD. The space-time sheet assigable to blachhole according to TGD has however two horizons. The first horizon would be a counterpart of the usual Schwartschild horizons. At second horizon the signature of the induced metric would become Euclidian - this is possible only in TGD. Cosmic string would topologically condense at this space-time sheet.

  3. Could most of matter be dark even in the case of Sun? What can we really say about the portion of the ordinary matter inside Sun? The total rate of nuclear fusion in the solar core depends on the density of ordinary matter and one can argue that existing model does not allow a considerable reduction of the portion of ordinary matter.

    There is however also another option - dark fusion - which would be at work in TGD based model of cold fusion (see this) (low energy nuclear reactions (LENR) is less misleading term) and also in TGD inspired biology (there is evidence for bio-fusion) as Pollack effect (see this), in which part of protons go to dark phase at magnetic flux tubes to form dark nuclear strings creating negatively charged exclusion zone). Dark fusion would give rise to dark proton sequences at magnetic flux tubes decaying by dark beta emission to beta stable nuclei and later to ordinary nuclei and releasing nuclear binding energy.

    Dark fusion could explain the generation of elements heavier than iron not possible in stellar cores (see this). Standard model assumes that they are formed in supernova explosions by so called r-process but empirical data do not support this hypothesis. In TGD Universe dark fusion could occur outside stellar interiors.

  4. But if heavier elements are formed via dark fusion, why the same could not be true for the lighter elements? The TGD based model of atomic nuclei represents nucleus as a string like object or several of them possibly linked and knotted. Thickened cosmic strings again! Nucleons would be connected by meson like bonds with quark and antiquark at their ends.

    This raises a heretic question: could also ordinary nuclear fusion rely on similar mechanism? Standard nuclear physics relies on potential models approximating nucleons with point like particles: this is of course the only thing that nuclear physicists of past could imagine as children of their time. Should the entire nuclear physics be formulated in terms of many-sheeted space-time concept and flux tubes? I have proposed this kind of formulation long time ago (see this). What would distinguish between ordinary and dark fusion would be the value of heff=n× h.

After this prelude it is possible to speculate about blackholes in the spirit of TGD .
  1. Also the interiors of blackholes would contain dark knots and have magnetic structure. This predicts unexpected features such as magnetic moments not possible for GRT blackholes. Also the matter inside blackhole would be dark (the TGD based explanation for Fermi bubbles assumes this (see this). Already the model for the first LIGO event explained the unexpected gamma ray bursts in terms of the twisting of rotating flux tubes as effect analogous to what causes sunspots: twisting and finally reconnection.

  2. One must also ask whether LIGO blackholes are actually dark stars with very small amount of ordinary matter. If the radius is indeed equal to Schwarschild radius rS= 2GM and mass is really what it is estimated to be rather than being systematically smaller, then the interpretation as TGD counterparts of blackholes makes sense. If mass is considerably smaller, the radius would be correspondingly large, and one would not have genuine blackhole. I do not however take this option too seriously.

  3. What about collisions of blackholes? Could they correspond to two knots moving along same string in opposite directions and colliding? Or two cosmic strings intersecting and forming a cosmic crossroad with second blackhole in the crossing? Or self-intersection of single cosmic string? In any case, cosmic traffic accident would be in question.

    The second LIGO event gave hints that the spin directions of the colliding blackholes were not the same. This does not conform with the assumption that binary blackhole system was in question. Since the spin direction would be naturally that of long cosmic string, this suggests that the traffic accident in cosmic cross road defined by intersection or self-intersection created the merger. Note that intersections tend to occur (think of moving strings in 3-D space) and could be stablized by gravitational attraction: two string world sheet at 4-D space-time surface have stable intersections just like strings in plane unless they reconnect.

See the article LIGO and TGD or the chapter Quantum astrophysics of "Physics in many-sheeted space-time".

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

Articles and other material related to TGD.