Mapping the Universe using Cosmic Neutrinos

The Quanta Magazine article A New Map of the Universe, Painted With Cosmic Neutrinos tells about the findings suggesting that ultrahigh energy (UHE) neutrinos do not originate from a continuous diffuse source but from discrete sources, presumably active galactic nuclei (AGNs).

Consider first the most recent observations of the IceCube laboratory.

1. A mapping of the Milky Way is carried by the IceCube laboratory using high energy neutrinos. Diffuse haze of cosmic neutrinos is found to be emanating along the plane the Milky Way (see this). Neutrinos could emanate from the inner region of Milky Way (see this), or even from the galactic blackhole.
2. If the neutrinos were generated in the collisions of very high cosmic rays, accelerated in ultra strong magnetic fields in the galactic plane, perhaps magnetic fields prevailing near the galactic blackhole, their flux would be by a factor of 100 higher than the observed flux. This suggests that active galactic nuclei (AGN) are the sources of UHE neutrinos.
3. The neutrino flux from Milky Way is by a fact or order 1/100 lower than from other galactic sources so that that Milky Way might be called neutrino desert (see this).

IceCube studies have already earlier connected UHE cosmic neutrinos with individual sources.

1. IceCube found (see this) that the galaxy called TXS 0506+056 , briefly TXS, was emitting UHE neutrinos and flares of X-rays and gamma rays simultaneously. The distance of TXS is 5.7 billion light years. TXS is a blazar, which means that there is a jet directed towards the Earth. The neutrinos and other high energy particles arrive along this jet.
2. IceCube has also found a second patch emitting UHE neutrinos (see this). It has an active galaxy NGC 1068 (Messier 77) in the center. It is located at a distance 47 million light years, which is by a factor of 1/100 smaller than the distance of the TSX. Gamma ray burst is absent is absent in this case. One proposal is that the high energy neutrinos and gammas arrive along the galactic plane but there is an absorption of gamma rays by the matter at the galactic plane near the active galactic nucleus.

   A possible explanation is that high energy neutrinos arise in the decays of cosmic rays, which have accelerated in very strong magnetic fields. These magnetic fields could be associated with the stars that become supernovas or when two neutron stars collide. they would be accompanied by gamma ray bursts and high energy neutrinos. The problem is that the flux of high energy neutrinos should be much higher. AGNs therefore remains the only viable candidate.

Could TGD (Topological GeometroDynamics) based view of astrophysics explain these findings? TGD predicts a lot of new physics. Geometrization and number theoretization of quantum physics provide complementary approaches to TGD.

Consider first an explanation for the origin of UHE neutrinos and for their huge energies.

1. Number theoretical vision predicts a hierarchy of p-adic physics, where p is the prime characterizes p-adic number field. Number theoretically p would correspond to a ramified prime for an extension of rationals assignable to a given space-time region. One of the basic conjectures is p-adic length (mass) scale hypothesis stating the existence of a a hierarchy L(k) ∝ 2^k/2, p≈ 2^k, k some integer, of p-adic length scales. This hypothesis is central in p-adic mass calculations, which replace the Higgs mechanism with p-adic thermodynamics (see this).
2. Family replication phenomenon is one of the mysteries of recent day particle physics. In the TGD Universe family replication phenomenon has topological origin (see this and this) .

   Elementary particles have as basic building bricks partonic 2-surfaces assignable to the wormhole contacts with Euclidean signature of the induced metric and connecting two space-time sheets with a Minkowskian signature of induced metric. Partonic 2-surfaces carry fermions as point-like particles at the ends of strings connecting partonic 2-surfaces to each other.

   The partonic 2-surfaces are characterized by genus g identifiable as the number of handles attached to 2-sphere. The lowest 3 genera (with handle number g=0,1,2) always allow global Z₂ conformal symmetry unlike the higher genera. The proposal is that this means that for these genera the g handles form a bound state (trivially for g< 2)whereas for g>2 one has many-handle states with a mass continuum so that one does not obtain elementary particles. The 3 fermion genera are assumed to give rise to a combinatorial symmetry U(3)_g and the 3 fermion generations correspond to the fundamental representation 3_g.

3. In TGD there are no bosons as fundamental particles. Bosons would correspond to fermion antifermion bound states of fundamental fermions. This suggests that gauge bosons and Higgs correspond to the tensor product of 3_g and its conjugate representation so that bosons would correspond to the 8_g +1_g, where 8+1=9 is the dimension of the dynamically generated symmetry group U(3)_g.

   Ordinary bosons would correspond to SU(3)_g singlet 1_g. Besides this new exotic boson octet 8_g is predicted. The original proposal was that the 2 bosons of 8_g with vanishing SU(3)_g quantum numbers in 8_g are light so that there would be 3 boson generations. This assumption is not necessary (see this).

4. The ultrahigh energy (UHE) neutrinos (energy range [13 TeV, 7.9 PeV]) would correspond to very large p-adic mass scales characterizing the masscales for the analogs of weak bosons decaying to charged lepton and neutrino/antineutrino.
5. The lepton-neutrino pairs could be created in the decays of exotic counterparts of W bosons in 8_g. From the energies of UHE neutrinos one can conclude that these exotic bosons could correspond to the p-adic length scale L(61), where p= 2⁶¹-1 is Mersenne prime. Weak bosons, or at least Higgs, would be associated with the p-adic length scale L(89), p= 2⁸⁹-1. W and Z could correspond to p-adic primes with k=91.

These huge mass scales require an enormous temperature in [13 TeV, 7.9 PeV] range, possibly reached inside active galactic nuclei (AGNs). In standard physics it is difficult to imagine how high these temperatures could be achieved.

1. In the TGD Universe space-times are 4-D surfaces in M⁴×CP₂. TGD predicts, besides Einsteinian space-time surfaces with 4-D M⁴ projection, also space-time surfaces with 2-D M⁴ projection. I have called them cosmic strings but they are not the cosmic strings of gauge theories. Cosmic strings have a huge string tension and energy identifiable as dark energy.
2. In TGD based view of galaxy formation, galaxies are formed as tangles along long cosmic strings with 2-D M⁴ projection They are unstable against the thickening of M⁴ projection, which can be induced by the collision of cosmic strings inducing their intersection leading to the thickening. The thickenning generates monopole flux tubes with a 4-D M⁴ projection and since the string tension decreases, the dark energy decays in an explosive manner to ordinary matter and dark matter. This process would be the TGD counterpart of inflation. This would explain the needed huge temperature. Kind of local mini big bang would be in question.
3. The TGD counterparts of the galactic blackholes can be associated with the decaying cosmic strings. The recent observations suggesting that dark energy is assignable to AGNs supports this view. The temperature in these regions can be huge and make possible p-adically scaled up versions of hadron physics and also the presence of 8_g bosons. Their decay to lepton-neutrino pairs would generate UHE neutrinos but they could be also generated in the collisions of UHE cosmic rays with matter.
4. Note that in the TGD Universe gravitational quantum coherence is possible in arbitrarily long length scales. This network could be analogous to a nervous system and make possible very intense highly targeted beams of cosmic rays. Diffraction in the tessellation of cosmic time=constant hyperboloid, having stars as analogs of unit cells of a lattice, could explain the recently observed gravitational hum difficult to understand in standard astrophysics (see this).

Two different production mechanisms are suggested as an explanation for the differences between UHE emissions from TXS and NGC 106. One should also explain why the Milky Way seems to be a neutrino desert.

1. The galaxies are accompanied by cosmic strings or flux tubes obtained as their thickenings, roughly orthogonal to the galactic plane. Galactic jets can be assigned to these flux tubes. The cosmic strings would form a fractal network connecting galactic blackhole-like objects and the stars inside them to each other. The channelling of high energy particles to jets would be caused by the space-time topology (the cross section of the flux tube would be a closed 2-surface).

   The huge magnetic fields associated with these flux tubes would accelerate cosmic rays to ultrahigh energies. Both gamma ray bursts and the UHE neutrinos and accompanying gamma rays from TXS could be assigned with these jets. UHE neutrinos could be created either in the galactic nucleus or produced in the collisions of UHE cosmic rays with matter.

2. Also the flux tube tangles in the galactic plane could give rise to analogous jets. TGD Universe is fractal and in TGD inspired quantum biology U-shaped flux tube tentacles play a key role in biocatalysis, in which the reconnection of them creates a flux tube pair between two objects. These tentacles could also appear in astrophysical scales and mediate gravitational interaction as gravitons, which propagate along them. In fact, the thin structures found to emanate from the center of the Milky Way could be associated with monopole flux tube pairs in the galactic plane. The view about the formation of astrophysical structures based on magnetic bubbles generated by monopole flux tubes is discussed in (see this and this).

   For the thick flux tubes in the galactic plane, the string tension would be much lower than for the environment of the cosmic string parallel to the jets. The acceleration of cosmic rays in their much weaker magnetic fields would not allow them to reach ultra high energies as in the case of jets normal to the galactic plane and the collisions of cosmic rays with matter could not create UHE neutrinos. Also gamma ray bursts would be absent.

3. The UHE neutrinos created in the AGN propagating along these flux tubes should generate the observed signal. This would suggest that neutrino flux is the sum of the primary flux from AGN and the flux generated by the interactions of cosmic rays inside the flux tube. The latter generates most of the signal for the galactic jets with ultra-strong magnetic fields. In the case of NGC, the UHE neutrino flux would be in the galactic plane. It is by a factor of order 1/100 weaker than predicted by the model in which the neutrinos are created in the collisions of cosmic rays accelerated in ultra strong magnetic fields with matter.
4. What about the Milky Way neutrino desert? Since the flux tube normal to the plane of the Milky Way does not reach Earth, the high energy neutrino flux would be so low that one could speak of neutrino desert. Whether the UHE neutrino flux from AGN is consistent with the finding that the Milky Way is a neutrino desert, is not clear. One explanation is that the Milky Way blackhole-like object is not active. This could mean that the ultra-hot regions where the generation of UHE neutrinos is possible are not present.

See the article Mapping the Universe using cosmic neutrinos or the chapter New Physics Predicted by TGD: Part I.

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

Abstracts of two articles involving tesselations in hyperbolic 3-space

Abstracts of two articles involving hyperbolic tesselations. I have written two articles involving the tessellation of hyperbolic 3-space. I attach the abstracts of the articles below.

About tessellations in hyperbolic 3-space and their relation to the genetic code

M⁸-H duality and the realization of holography in M^8 strongly suggests the importance of tessellations of H³ (analogous to lattices of E³) in the TGD based physics. These tessellations form a scale hierarchy and can thus appear in all scales. The hierarchy of effective Planck constants labelling dark matter as phases of ordinary matter indeed predicts quantum coherence in arbitrarily long scales and gravitational quantum coherence corresponds to the largest scales of quantum coherence among basic interactions.

There are 5 Platonic tessellations known as honeycombs: the 4 regular honeycombs correspond to cubic, icosahedral, and 2 dodecahedral honeycombs and a quasiregular icosa-tetrahedral honeycomb having tetrahedra, octahedra and icosahedra as cells. The icosa-tetrahedral honeycomb might define a universal realization of the genetic code as an induced structure so that the genetic code would be much more than a biochemical accident. These 5 Platonic honeycombs could occur also in astrophysical scales as gravitational tessellations. The recent discovery of gravitational hum might have an explanation as gravitational diffraction in this kind of a honeycomb.

In this article the properties of hyperbolic honeycombs are considered in detail and also a detailed view about the realization of DNA double strand in terms of the icosa-tetrahedral honeycomb is considered. The emerging model is surprisingly quantitative. Also a connection with the notion of memetic code and the realization of memetic codons in terms of 21 DNA codons are suggested by the model.

See the article About tessellations in hyperbolic 3-space and their relation to the genetic code or the chapter with the same title.

The TGD view of the recently discovered gravitational hum as gravitational diffraction

The strength of gravitational hum is unexpectedly large. The basic insight of the TGD view is that diffraction produces very high intensities but only in preferred directions and that in diffraction the amplitude of the scattered field is proportional to the square N² of the number N of scatterers rather than N. The identification of dark matter as phases of the ordinary matter with an arbitrarily large value of Planck constant suggests the existence of tessellations of the hyperbolic 3-space identifiable as light-cone proper time hyperboloid of Minkowski space. Gravitational diffraction could occur in astrophysical scales at these tessellations.

See the article The TGD view of the recently discovered gravitational hum as gravitational diffraction or the chapter Quantum Astrophysics .

About tessellations in hyperbolic 3-space

This post represents new results about the tessellations of H³. H³ allows an infinite number of tessellations. These consideration were inspired by the model for the recently discovered gravitational hum but they also led to a considerable progress in the understanding of the TGD inspired model of the genetic code. Here only the icosatetrahedral tessellation is discussed.

1. Some preliminaries

Some preliminaries are needed in order to understand Wikipedia articles related to tessellations in general.

1. Schläfli symbol {p,r} (see this) tells that the possibly existing Platonic solid {p,r} has r p-polygons as faces meeting at each vertex. For instance, icosahedron {3,5} has 5 triangles as faces meeting at each vertex.

   Schläfli symbol generalizes to higher dimensions. The analog of Platonic solid {p,r,q} possibly in 4-dimensions and assignable to 3-sphere has q 3-faces which are Platonic solids {p,r}. This description is purely combinatorial and is recursive. For instance, one can start from 3-D dimensional Platonic solid {p,q} with 3-D objects in dimension 4 by replacing p with {p,r}. One can also project this object to dimension 3. In this manner one obtains a projection of 4-cube (tesseract) {4,3,3} for which 3 cubes {4,3} meet at each vertex (2⁴=16 of them) and which has 8 3-cubes as faces as a 3-D object.

   In the case of hyperbolic tessellations also strange looking Schläfli symbols {(p,q,r,s)} are encountered: icosa-tetrahedral tessellation involving only Platonic solids has symbol {(3,3,5,3)}. My understanding is that this object corresponds to {3,3,5,3} as an analogue of Platonic solid associate with 4-sphere in 5-D Euclidian space and that the fundamental region of this tessellation in H³ is analogous to a 3-D projection of this object. At a given vertex 3 objects {3,3,5} meet. For these objects 5 tetrahedrons meet at a given vertex.

2. Vertex figure is a further central notion. It represents a view of the fundamental region of tessellation from a given vertex. The vertices of the figure are connected to this vertex. It does not represent the entire fundamental region. For instance, for a cube (octahedron) it contains only the 3 (4) nearest vertices. For icosa-tetrahedral tessellation the vertex figure is icosidodecahedron (see this). The interpretation of the vertex symbol of the hyperbolic icosa-tetrahedral honeycomb (see this) is a considerable challenge.
3. One cannot avoid Coxeter groups and Coxeter symbols (see this) in the context of tessellations. They code the structure of the symmetry group of say Platonic solid (tessellation of S²). This symmetry group is generated by reflections with respect to some set of lines, usually going through origin. For regular polygons and Platonic solids is its discrete subgroup of rotation group.

   The Coxeter group is characterized by the number of reflection hyperplanes H_i and the reflections satisfying r_i²=1. The products r_ij=r_ir_j define cyclic subgroups of order c_ij satisfying r_ij^c_ij=1. Coxeter group is characterized by a diagram in which vertices are labelled by i. The orders of the cyclic subgroups satisfy c_ij≥ 3. For c_ij the generators r_i and r_j commute. For c_ij=2 the vertices are not connected, for c_ij=3 there is a line and for c_ij>3 the number c_ij is assigned with the line. For instance, hyperbolic tessellations are characterized by 4 reflection hyperplanes.

   For instance, for p-polygon the Coxeter group has 2 generators and the cyclic group has order p. For Platonic solids the Coxter group has 3 generators and the orders of cyclic subgroups are 3, 4, or 5. For icosa-tetrahedral tessellation the order is 4.

The most interesting honeycombs in hyperbolic 3-space

H³ allows an infinite number of tessellations. There are 9 types of honeycombs. This makes 76 uniform hyperbolic honeycombs involving only a single polyhedron (see this).

4 of these honeycombs are regular, which means that they have identical regular faces (Platonic solids) and the same numbers of faces around vertices. The following list gives the regular uniform honeycombs and their Schläfli symbols {p,q,r} telling that each edge has around it regular polygon {p,q} for which each vertex is surrounded by q faces with p vertices.

1. H1: 2 regular forms with Schläfli symbol {5,3,4} (dodecahedron) and {4,3,5} (cube).
2. H2: 1 regular form with Schläfli symbol {3,5,3}(icosahedron)
3. H5: 1 regular form with Schläfli symbol {5,3,5} (dodecahedron).

There is a large number of uniform honeycombs involving several cell types. There exists however a "multicellular" honeycomb, which is completely unique in the sense that for it all cells are Platonic solids. This icosa tetrahedral (or more officially, tetrahedral-icosahedral) tessellation has tetrahedrons, octahedrons, and icosahedrons as its cells. All faces are triangles. The tetrahedral-icosahedral honeycomb is of special interest since it might make possible the proposed icosa tetrahedral realization of the genetic code (see this). Octahedron can be seen also as a rectified tetrahedron r{3,3}.

From the Wikipedia article about icosa-tetrahedral honeycomb (see this) one learns the following.

1. The Schläfli symbol of icosa-tetrahedral honeycomb is {(3,3,5,3)}. This combinatorial symbol allows several geometric representations. The inner brackets would refer to the interpretation as an analogue of the Platonic solid assignable to a 4-sphere of Euclidian 5-space. At each vertex 3 objects of type {3,3,5} would meet. At the vertex of {(3,3,5)} in turn 5 tetrahedrons meet.
2. Icosa-tetrahedral honeycomb involves tetrahedron {(3,3}, octahedron {(3,4)}, an icosahedron {(3,5)} as cells. That there are no other honeycombs involving several Platonic solids and only them as cells makes this particular honeycomb especially interesting. Octahedron with Schläfli symbol {3,4} can be also regarded as a rectified tetrahedron havig Schläfli symbol r{3,3}.
3. The vertex figure of icosa-tetrahedral honeycomb (see this), representing the vertices a lines connecting them is icosidodecahedron (see this), which is a "fusion" of icosahedron and dodecahedron having 30 vertices with 2 pentagons and 2 triangles meeting at each, and 60 identical edges, each separating a triangle from pentagon. From a given vertex VF=60 vertices connected to this vertex by an edge can be seen. In the case of cube, octahedron, and dodecahedron the total number of vertices in the polyhedron is 2(VF+1). It is true also now, one would have 122 vertices in the basic structural unit. The total number of vertices for the disjoint polyhedra is 6+4+12= 22 and since vertices are shared, the number of polyhedra in the basic unit must be rather large.
4. The numbers called "cells by location" could correspond to numbers 30, 20, and 12 for octahedrons, tetrahedrons and icosahedrons respectively inside the fundamental region of the tessellation defining the honeycomb. That the number of icosahedrons is smallest, looks natural. These numbers are quite large. The counts around each vertex are given by (3.3.3.3), (3.3.3), resp. (3.3.3.3) for octahedra, tetrahedra, resp. icosahedra and tell the numbers of vertices of the faces meeting at a given vertex.

From the Wikipedia article about icosa-tetrahedral honeycomb (see this) one learns the following.

1. The Schläfli symbol of icosa-tetrahedral honeycomb is {(3,3,5,3)}. This combinatorial symbol allows several geometric representations. The inner brackets would refer to the interpretation as an analogue of the Platonic solid assignable to a 4-sphere of Euclidian 5-space. At each vertex 3 objects of type {3,3,5} would meet. At the vertex of {(3,3,5)} in turn 5 tetrahedrons meet.
2. Icosa-tetrahedral honeycomb involves tetrahedron {(3,3}, octahedron {(3,4)}, an icosahedron {(3,5)} as cells. That there are no other honeycombs involving several Platonic solids and only them as cells makes this particular honeycomb especially interesting. Octahedron with Schläfli symbol {3,4} can be also regarded as a rectified tetrahedron havig Schläfli symbol r{3,3}.
3. The vertex figure of icosa-tetrahedral honeycomb (see this), representing the vertices a lines connecting them is icosidodecahedron (see this), which is a "fusion" of icosahedron and dodecahedron having 30 vertices with 2 pentagons and 2 triangles meeting at each, and 60 identical edges, each separating a triangle from pentagon. From a given vertex VF=60 vertices connected to this vertex by an edge can be seen. In the case of cube, octahedron, and dodecahedron the total number of vertices in the polyhedron is 2(VF+1). It is true also now, one would have 122 vertices in the basic structural unit. The total number of vertices for the disjoint polyhedra is 6+4+12= 22 and since vertices are shared, the number of polyhedra in the basic unit must be rather large.
4. The numbers called "cells by location" could correspond to numbers 30, 20, and 12 for octahedrons, tetrahedrons and icosahedrons respectively inside the fundamental region of the tessellation defining the honeycomb. That the number of icosahedrons is smallest, looks natural. These numbers are quite large. The counts around each vertex are given by (3.3.3.3), (3.3.3), resp. (3.3.3.3) for octahedra, tetrahedra, resp. icosahedra and tell the numbers of vertices of the faces meeting at a given vertex.
5. What looks  intriguing is   that  the numbers  30, 20, and 12 for octahedrons (O), tetrahedrons  (T) and icosahedrons (I) correspond to the numbers of vertices, faces, and edges for  I. As if the fundamental region would be obtained by taking an icosahedron and replacing its  30 vertices with O, its 20 faces with T and its 12  edges with I,  that is by using the rules vertex → octahedron; edge →I, face → T.  These 3-D objects would be fitted together along their triangular faces. 

   Do   the   statements about the geometry and homology of I translate to the statements about the geometry and homology of the fundamental region?  This would mean the following replacements:  

   1.  "2  faces meet at edge" →  "2  T:s share  face with an I".
   2.  "5 faces meet at vertex"   →   "5 T:s share  face with an O".
   3.  "Edge has 2   vertices as ends"  → "I   shares a face with 2  different  O:s".
   4.   "Face has 3 vertices → "T shares a face with  3 different  O:s".
   5.   "Face has three edges" → "T has a common face with 3 I:s".

An attempt to understand the hyperbolic tessellations

The following general observations might help to gain some understanding of the tessellations.

1. The tessellations of E³ and H³ are completely analogous to Platonic solis as 2-D objects. The non-compactness implies that there is infinite number of cells for tessellations. It is important to notice that the radial coordinate r for H³ corresponds very closely to hyperbolic angle and its values are quantized for the vertices of tessellation just like the values of spherical coordinates are quantized for Platonic solids.

   The tessellations for E³ are scale covariant. For a fixed radius of H³ characterized by Lorentz invariance cosmic time this is not the case. One can however scale the value of a.

2. What distinguishes between regular tessellations in E³ and H³ is that the metric of H³ is non-flat an H³ has negative curvature. H³ is homogeneous space meaning that all points are metrically equivalent (this is the counterpart of cosmological principle in cosmology). Since both spaces have rotations as symmetries, this does not affect basic Platonic solids as 2-D structures assignable with 2-sphere if the edges are identified as geodesic lines of S². Quite generally, isometries characterize the tessellations, whose fundamental region corresponds to coset space of H³/Γ by a discrete group of the Lorentz group acting as isometries of H³.
3. The modifications induced by the replacement E³arrow H³ relate to the 3-D aspects of the tessellation. This because the metric is non-flat in the radial direction. The negative curvature implies that the geodesic line diverge. One can use counterpart of the standard spherical coordinates and in this coordinates the solid angles assignable to the vertices of Platonic solid are smaller than in E³. Also the hyperbolic planes H² emerging from edges of the tessellation of H³ diverge in normal direction the angles involved are smaller.

It is useful to start from the description of Platonic solids. They are characterized combinatorially by integers and by geometrically by various kinds of angles. Denote by p the number of vertices/edges of the face and by q the number of faces meeting at vertex.

1. Important constraints come from the topology and combinatorics. Basic equations for the numbers V ,E, and F for number of vertices, edges an faces are purely topological equation V E+F=2, and the equation pF=2E=qV. Manipulation of these equations gives 1/r+1/p= 1/2+1/E implying 1/r+1/p>1/2. Since p and q must be at least 3, the only possibilities for {p,q} are {3, 3}, {4, 3}, {3, 4}, {5, 3}, and {3, 5}.
2. The angular positions of the vertices at S² are basic angle variables. In H³ hyperbolic angle assignable to the radial coordinate is additional variable of this kind analogous to the position of the unit cell in the E³ tessellation. The cosmological interpretation is in terms of redshift.
3. There is the Euclidian angle φ associated with the vertex of the face given by π/p. Here there is no difference between E³ and H³.
4. The angle deficit δ associated with the faces meeting at a given vertex due to the fact that the faces are not in plane in which case the total angle would be 2π. δ is largest for tetrahedron with 3 faces meeting at vertex and therefore with the sharpest vertex and smallest for icosahedron with 5 triangles meeting at vertex. This notion is essentially 3-dimensional, being defined using radial geodesics, so that the δ is not the same in H³. In H³ δ is expected to be larger than in E³.
5. There is also the dihedral angle θ associated with the faces as planes of E³ meeeting at the edges of the Platonic solid. θ is smallest for tetrahedron with 4 edges and largest for dodecahedron with 20 edges so that the dodecahedron is not far from flat plane an this angle is not far from π. The H³ counterpart of θ is associated faces identified as hyperbolic planes H² an is therefore different.
6. There is also the vertex solid angle Ω associated with each vertex of the Platonic solid {p,q} given by Ω =qθ -(q-2)π. For tessellations in E³ the sum of these angles is 4π. In H³ it Euclidian counterpart is larger than 4π.
7. The face solid angle is the solid angle associate with the face when seen from the center of the Platonic solid. The sum of the face solid angles is 4π. For Platonic solid with n vertices, one has Ω= 4π/n. The divergence of the geodesics of H³ implies that this angle is smaller in H³: there is more volume in H³ than in E³.

E³ allows only single regular tessellation having cube as a unit cell. H³ allows cubic and icosahedral tessellations plus two tessellations having dodecahedron as a unit. Why E³ does not allow icosahedral an dodecahedral tessellations and how the curvature of H³ makes them possible? Why the purely Platonic tetra-icosahedral tessellation is possible in H³?

The first guess is that these tessellations are almost but not quite possible in E³ by looking the Euclidian constraints on various angles. In particular, the sum of dihedral angles θ between faces should be 2π in E³, the sum of the vertex solid angles Ω at the vertex should be 4π. Note that the scaling of the radial coordinate r decreases the dihedral angles θ and solid angles Ω. This flexibility expected to make possible so many tessellations and honeycombs in H³. The larger the deviation of the almost allowed tessellation, the larger the size of the fundamental region for fixed a.

Consider now the constraints on the basic parameters of the Platonic solids (see this) in E³ and H³.

1. The values of didedral angle for tetrahedron, cube, octahedron, docecahedron, and icosahedron are

   [θ (T), θ (C), θ (O), θ(D),θ(I)]≈ [70.3^° , 90^°, 109.47^°, 116.57^° , 138.19^°] .

   Note that r=5 tetrahedra could almost meet at single edge in E³. In E³ r=4 cubes can meet at the edge. In H³ r should be larger. This is indeed the case for the cubic honeycomb {4,3,5} having r=5 .

   For r=3 icosahedrons the the sum dihedral angles exceeds 2π which conforms with the that {3,5,3} defines an icosahedral tessellation in H³.

   For r=4 docecahedra meeting at the edge the total dihedral angle is larger than than 360^°: r=4 is therefore a natural candidate in H³. There are indeed regular dodecahedral honeycoms with Schläfli symbol {5,3,r}, r=4 and r=5. Therefore it seems that the intuitive picture is correct.

2. The values of the vertex solid angle Ω for cube, docecahedron, and icosahedron are given by the formula Ω =qθ -(q-2)π giving

   [Ω (C), Ω (D), Ω (I)] ≈ [1.57080, 2.96174,2.63455].

   The sum of these angles should be 4π for a tessellation in E³. In E³ This is true only for 8 cubes per vertex (Ω= π/2) so that the cubic honeycomb is the only Platonic honeycomb in E³. The minimal number of cubes per vertex is 9 in H³. It is convenient to write the values of the vertex solid angles for D and I as

   [Ω(D),Ω(I)]= [0.108174, 0.209651]× 4π .

   The number of D:s resp. I:s must be at least 10 resp. 5 for dodecahedral resp. icosahedral honeycombs in H³.

3. The basic geometric scales of the Platonic solids are circumradius R, surface area A and volume V. The circumradius is given by R=(a/2)×tan(π/q)tan(θ/2), where a denotes the edge length. The surface area A of the Platonic solid { p,q} equals the area of face multiplied by the number F of faces: A=(a/2)² Fpcot(π/p). The volume V of the Platonic time is F times the volume of the pyramid whose height is the length a of the face: that is V= FaA/3.

   Choosing a/2 as the length unit, the circumradii R, total face areas A an the volumes V of the Platonic solids are given by

   [R(T),R(C),R(O),R(D),R(I)]= [3^1/2/2, 3^1/2,2^1/2, 3^1/2φ, (3-φ)^1/2φ] ,

   [A(T),A(C),A(O),A(D),A(I)]= [ 4×3^1/2, 24, 2×3^1/2, 12×(25+10×5^1/2)^1/2, 20× 3^1/2] ,

   and

   [V(T),V(C),V(O),V(D),V(I)]≈ [8^1/2/3, 8, 128^1/2/3 , 20φ³/(3-φ), 20φ²/3]

   ≈ [.942809,8,3.771236,61.304952,17.453560] .

What one can say about icosa tetrahedral tessellation?

1. Consider first the dihedral angles θ. The values of dihedral angles associated T, O, and I in H³ is reduced from that in E³ so that their sum in E² sene must be larger than 2π. Therefore at least one of these cells must appear twice in H³. It could be T but also O can be considered.

   For 2T+O+I and T+2O+I the sum would be 388.26^° resp. 427.43^° in E³. 2T+O+I resp. T+2O+I could correspond 4 cells ordered cyclicaly as ITOT resp. IOTO.

2. The values of the vertex solid angle Ω for tetrahedron, octahedron, and icosahedron are given by [Ω(T), Ω(O),Ω(I)] =[0.043870, 0.108174, 0.209651]4π If the numbers of T, O and I are [n(T),n(O),n(I)], one must have [n(T)Ω(T), +n(O)Ω(O) +n(I)Ω(I)>4π in H³.

   If the number of the cells for the fundamental domain are really [N(T), N(O), N(I)]=[30,20,12], the first guess is that [n(T), n(O), n(I)] ∝ [N(T), N(O), N(I) is approximately true. For [n(T), n(O), n(I)]=[2,3,1]n(I), one obtains Ω= n(T)Ω(T)+n(O)Ω (O)+ n(I)Ω (I)= n(I)× .629 × 4π. This would suggest n(I)=2 giving [n(T), n(O), n(I)]=[4,6,2]

See the article About tessellations in hyperbolic 3-space and their relation to the genetic code or the chapter with the same title.

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

TGD view of the gravitational hum as gravitational diffraction

The revolution in cosmology is continuing. The latest breakthrough was announced yesterday (see this). See also the popular article. There is a collection of articles related to the discovery.

The dicovery

Scientists from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) have now officially made the first detections of the gravitational wave background. This gravitational hum was not detected by Earth bound instruments.

The wavelength of the oscillations makes itself visible as correlations between the variations of the spinning  rates  for pulsars having relative  distances measured using a light year as a unit.   The wavelength of the oscillations is measured in light years. In the LIGO experiment  the  periods are measured as fractions of a second.  

Where could this length scale come from? What might make bells ringing is that the star nearest to the Sun is at a distance of 4 light years and the typical distance between stars is 5 light years.

Near to the end of the talk also the wavelength scale of millions of light years is mentioned. This scale corresponds to a typical distance between galaxies, which is few Mpc, pc= 3.26 ly.

Remark: I am grateful to Marko Manninen for noticing a rather stupid mistake: I talked about a period of millions of light years for the detected gravitational radiation. The period is of course a few years as is obvious from the fact that it is not easy to find graduate students able to stay motivated for millions of years.

The unexpectedly large amplitude of the oscillations motivates the hypothesis that pairs of galactic supermassive blackholes or interacting groups of them could generate the gravitational hum. There are candidates for these pairs but no established pair. The group hypothesis seems to work better.

TGD explanation in terms of astrophysical gravitational quantum coherence and diffraction in hyperbolic tesselation

TGD suggests a radically different hypothesis based on TGD view of gravitational quantum coherence an diffraction in a hyperbolic tessellation.

1. TGD predicts quantum gravitational coherence in astrophysical scales characterized by gravitational Planck constants h_gr = GMm/β₀ characterizing big mass M and small mass m. β₀=v₀/c<1 is a velocity parameter. The Equivalence Principle is realized as the independence of the gravitational Compton length Λ_gr= GM/β₀= r_s/2β₀ on mass m.
   1. For the Sun Λ_gr is 1/2 of Earth radius. If the TGD proposal, which explains Cambrian explosion in terms of rapid increase of the Earth radius by factor 2, this scale is the radius of Earth before the explosion (see this).
   2. For the Earth Λ_gr is .45 cm and the size scale of a snowflake, which is a zoomed version of the unit cell of the ice crystal: a fact which still remains a mystery.
   3. For the galactic black hole, Λ_gr is about 1.2×10⁷ km=1.2× 10^-2 light seconds and corresponds to a frequency of about 100 Hz, the upper bound of EEG frequencies by the way (which might put bells ringing!). For β₀=1 Λ_gr happens to correspond to the radius of the lowest Bohr orbit for Sun Λ_gr in Bohr orbit model for planetary orbit (another bell ringing!) and defines only a lower bound for the quantum coherence scale.
2. Where could the wavelength of order of distance between neighboring stars emerge? TGD strongly suggests that the tessellations (lattices) associated with hyperbolic 3-space identified light-cone proper time a= constant surface play a key role in all scales, in particular in biology.

   There could exist a fractal hierarchy of hyperbolic tessellations (see this) formed by astrophysical objects of various mass scales. Could the stars with average distance of 5 light years form tessellations of this kind analogous to lattices in a condensed matter system. The wavelength for the diffracted gravitational waves in cubic tessellation would have the upper bound 2d, d the lattice constant, which would be now about 5 light years.

3. There is empirical evidence for these tessellations. So called cosmic fingers, discovered by Halton Arp (see this or this), correspond to astrophysical objects appearing at single light of sight (first mystery) and having redshift coming as multiples of a basic redshift (second mystery). This could serve as a direct signature of the hyperbolic counterpart for a line of atoms located along a lattice. Redshift is proportional to distance and also to the recession velocity, which would therefore be quantized in the observed manner.
4. What kind of tessellations could be involved? There is an infinite number of tessellations for H³ but only 4 regular uniform honeycombs. For two of these the unit cell is a dodecahedron, for 1 of them it is an icosahedron and for 1 of them it is a cube. Note that in Euclidian 3-space one has just one regular honeycomb consisting of cubes.

   There are also more general uniform honeycombs involving several cell types. There is a unique "multicellular" honeycomb for which all cells are Platonic solids. This is icosatetrahedral (or more officially, tetrahedral-icosatetrahedral) honeycomb for which the cells are tetrahedrons, octahedrons, and icosahedrons (see this). All faces are triangles and I have proposed a universal realization of genetic code in which genetic codons correspond to the triangular faces of icosahedra an tetrahedra (see this).

The key prediction is gravitational diffraction in this cosmic lattice.

1. In diffraction in the lattice the diffracted amplitude concentrates in specific directions corresponding to the reciprocal lattice. Something analogous should happen for tessellations in hyperbolic 3-space. Already the concentration to beams would mean an amplification effect (note that the lowest order prediction for the intensity of the radiation does not depend on the value of the effective Planck constant).

   Furthermore, by quantum coherence the scattered amplitude is proportional to N² rather than N, where N is the number of atoms in the lattice, now stars in the tessellation. Could these two amplification effects explain why the observed effect is so much larger than expected? Professionals could easily find whether this idea fails at the quantitative level.

   TGD view suggests that the dark gravitational radiation propagates along the monopole flux tubes connecting stars.

2. In the ordinary diffraction from a cubic lattice in Euclidian space E³, the condition of constructive interference for the two rays scattered from to neighboring points of the cubic lattice states, requires that the difference of lengths for the paths travelled is a multiple of the wavelength of the incoming radiation. This gives the Bragg condition: sin(θ)= nλ/2d, where θ is the glance angle defined as the angle of incoming ray with respect to the normal direction of the lattice plane. The condition gives λ <2d/n and implies λ <2d for n=1. Therefore the diffraction occurs only for frequencies ω > nω_n, ω_n>c/2d.

   In the case of gravitational radiation, this would give for a cubic lattice λ<2d/n, d ≈ 5 light years, which conforms with the scale of few years for the periods. The lower bound for the period T would be about T_min=10 years. The condition that the scattered beams connect lattice points, gives an additional quantization condition to the glance angle θ. Most naturally it would correspond to a line connecting lattice points.

See the for instance the articles Magnetic Bubbles in TGD Universe: Part I and Magnetic Bubbles in TGD Universe: Part II.

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

Clumpiness paradox of cold dark matter scenario from the TGD point of view

Clumpiness paradox is one of the many problems plaguing the cold dark matter scenario (see this).

Clumpiness parameter (see this) is in principle deducible from the weak gravitational lensing caused by dark matter. In halo models it affects the annihilation rate of dark matter particles. Since the predicted rate is proportional to mass density squared, the annihilation rate increases for clumpy mass distribution.

If I understand correctly, the clumpiness paradox states that the clumpiness, which is determined by the size of dark matter clumps, depends on the scale in which observations are carried out. Clumpiness is smaller in long length scales, which means that the observed clumps are larger in long scales. In long scales, corresponding to recent cosmology, the sizes of clumps assignable are larger and the clumpiness parameter is .83. In shorter length scales corresponding to the age of the Universe about 380 thousand years the clumpiness parameter is smaller: .76.

In long length scales, a proposed explanation for the small value of clumpiness, i.e. a large size of clumps, is in terms of identification of dark matter as ultralight axions with very large Compton length determining the size scale of clumps.

This does not explain why the clumpiness depends on scale. Furthermore, clumps have been now observed in considerably smaller scales than earlier (see this). The strange looking conclusion is that cold dark matter is colder in short scales: the naive expectation would be just the opposite since it is the hot dark matter particles, which should form only small clumps. Something seems to go wrong.

The clumpiness paradox suggests a fractal distribution of dark matter. Indeed, in the TGD framework, cosmic strings thickened to monopole flux tubes would be responsible for gravitational lensing and the thickness of the monopole flux tubes would characterize the lensing.

The explanation for the large size of the clumps in long scales would be the large size of the Compton length proportional to effective Planck constant h_eff=nh₀. In the case of gravitational Planck constant h_eff= h_gr= GMm/β₀, β₀ a velocity parameter, assignable to the monopole flux tubes connecting pairs formed by a large mass M and small mass m, the gravitational Compton length is equal to Λ_gr= GM/β₀= r_s/2β₀, r_s Schwartshild radius of M increasing with the size scale of structure (note that there is no dependence on m). The larger the scale of the studied astrophysical object, the larger Λ_gr as minimal gravitational quantum coherence length is, and the smaller the clumpiness in this scale.

See the articles Magnetic Bubbles in TGD Universe: Part I, Magnetic Bubbles in TGD Universe: Part II and TGD view of the paradoxical findings of the James Webb telescope.

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

James Webb telescope suggests a possible mechanism for how Universe became transparent

I received a link to a very interesting article titled "JWST Unveils How Galaxies Made the Universe Transparent: A Cosmic Mystery Solved!" (see this).

The analysis of the findings of Jame Webb Telescope lead to the conclusion that some galaxies would have been surrounded by bubbles of ionized hydrogen  with a radius of millions of light years (analogous with Fermi bubbles having a size scale of 50,000 light years?). They would have expanded and merged. But why only some galaxies?

TGD  suggests a fractal network formed by gravitational flux tubes connecting astrophysical objects and  carrying dark matter with a huge value of gravitational Planck constant ℏ_gr= GMm/β₀, where β₀=v₀/c ≤1  is a velocity parameter and the large mass M an small mass m are connected by  gravitational monopole flux tubes.

The large value of ℏ_gr  makes  possible gravitational quantum coherence possible in astrophysical scales.  Dark photon radiation would have propagated along the flux tubes and transformed to ordinary photons  in detection making the early Universe visible. Cosmos would be like a network of candles connected by electric wires in a Christmas tree (see this).

But why would the flux tubes be associated only with the  galaxies having the ionized bubbles? Is this necessary? If these ionized bubbles w ere present, were  they formed as analogs of Fermi bubbles in the collisions  of very long cosmic strings?

Cosmic strings are 4-D surfaces X²× Y² ⊂ H= M⁴ × CP2, which are unstable against the thickening of their 2-D Minkowski space projection X². In a collision of  very long  cosmic strings,  portions of colliding cosmic strings  thickened and   dark energy was  transformed to ordinary   and dark matter (this is an analog of inflaton decay). Therefore  a flux tube spaghetti giving rise to a galaxy would have formed.  If all galaxies were formed in this way, all of them would have an ionized bubble. Did most of the galaxies  emerge in some other way?

I have proposed that very  astrophysical objects could have  formed in explosions (mini Big Bangs) throwing out magnetic bubbles  consisting of gravitational flux tubes. They would be  involved with the thickenings of the cosmic strings and flux tubes. Also radial flux tubes mediating gravitational interaction would be present.

See the articles Magnetic Bubbles in TGD Universe: Part I, Magnetic Bubbles in TGD Universe: Part II and TGD view of the paradoxical findings of the James Webb telescope.

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

Master formula for the scattering amplitudes: finally?

Most pieces that have been identified over the years in order to develop a master formula for the scattering amplitudes are as such more or less correct but always partially misunderstood. Maybe the time is finally ripe for the fusion of these pieces to a single coherent whole. I will try to list the pieces into a story in the following.

1. The vacuum functional, which is the exponential Kähler function defined by the classical bosonic action defining the preferred extremal a an analog of Bohr orbit, is the starting point. Physically, the Kähler function corresponds to the bosonic action (e.g. EYM) in field theories.

   Because holography is almost unique, it replaces the path integral by a sum over 4-D Bohr trajectories as functional integral over 3-surfaces plus discrete sum.

2. However, the fermionic part of the action is missing. I have proposed a long time ago a super symmetrization of WCW K hler function by adding to it what I call modified Dirac action. It relies on modified gamma matrices modified gamma matrices Γ^α, which are contractions Γ_kT^{α k} of H gamma matrices Γ_k with the canonical momentum currents T= ∂ L/∂_∂αh^k defined by the Lagrangian L. Modified Dirac action is therefore determined by the bosonic action from the requirement of supersymmetry. This supersymmetry is however quite different from the SUSY associated with the standard model and it assigns to fermonic Noether currents their super counterparts.

   Bosonic field equations for the space-time surface actually follow as hermiticity conditions for the modified Dirac equation. These equations also guarantee the conservation of fermion number(s). The overall super symmetrized action that defines super symmetrized Kähler function in WCW would be unambiguous. One would get exactly the same master formula as in quantum field theories, but without the path integral.

3. The overall super symmetrized action is sum of contributions assignable to the space-time surface itself, its 3-D light-like parton orbits as boundaries between Minkowskian regions and Euclidian wormhole contact, 2-D string world sheets and their 1-D boundaries as orbits of point-like fermions. These 1-D boundaries are the most important and analogous to the lines of ordinary Feynman diagrams. One obtains a dimensional hierarchy.
4. One can assign to these objects of varying dimension actions defined in terms of the induced geometry and spinor structure. The supersymmetric actions for the preferred extremals analogous to Bohr orbit in turn give contributions to the super symmetrized Kähler function as an analogue of the YM action so that, apart from the reduction of path integral to a sum over 4-D Bohr orbits, there is a very close analogy with the standard quantum field theory.

However, some problems are encountered.

1. It seems natural to assume that a modified Dirac equation holds true. I have presented an argument for how it indeed emerges from the induction for the second quantized spinor field in H restricted to the space-time surface assuming modified Dirac action.

   The problem is, however, that the fermionic action, which should define vertex for fermion pair creation, disappears completely if Dirac's equation holds everywhere! One would not obtain interaction vertices in which pairs of fermions arise from classical induced fields. Something goes wrong.

2. If one gives up the modified Dirac equation, the fermionic action does not disappear? In this case, one should construct a Dirac propagator for the modified Dirac operator. This is an impossible task in practice.

   Moreover, the construction of the propagator is not even necessary and in conflict with the fact that the induced spinor fields are second quantized spinors of H restricted to the space-time surface and the propagators are therefore well-defined and calculable and define the propagation at the space-time surface.

   Should we conclude that the modified Dirac equation cannot hold everywhere? What these, presumably lower-dimensional regions of space-time surface, are and could they give the interaction vertices as topological vertices?

The key question is how to obtain emission of fermion pairs and bosons as their bound states?

1. I have previously derived a topological description for reaction vertices. The fundamental 1 → 2 vertex (for example e → e+ gamma) generalizes the basic vertex of Feynman diagrams, where a fermion emits a boson or a boson decays into a pair of fermions. Three lines meet at the ends.

   In TGD, this vertex can topologically correspond to the decomposition of a 3-surface into two 3-surfaces, the decomposition of a partonic 2-surface into two, the decomposition of a string into two, and finally, the turning of the fermion line backwards from time. One can say that the n-surfaces are glued together along their n-1-dimensional ends, just like the 1-surfaces are glued at the vertex in the Feynman diagram.

2. In the earlier posting, I already discussed how to identify vertex for fermion-antifermion pair creation as a V-shaped turning point of a 1-D fermion line. The fermion line turns back in time and fermion becomes an antifermion. Now, however, the quantized boson field at the vertex is replaced by a classical boson field. This description is basically the same as in the ordinary path integral where the gauge potentials are classical.

   The problem was that if the modified Dirac equation holds everywhere, there are no pair creation vertices. The solution of the problem is that the modified Dirac equation at the V-shaped vertex cannot hold true.

   What this means physically is that fermion and antifermion numbers are not separately conserved in the vertex. The modified Dirac action for the fermion line can be transformed to the change of antifermion number as operator (or fermion number at the vertex) expressible as the change of the antifermion part of the fermion number. This is expressible as the discontinuity of a corresponding part of the conserved current at the vertex. This picture conforms with the appearance of gauge currents in gauge theory vertices. Notice that modified gamma matrices determined by the bosonic action appear in the current.

3. This argument was limited to 1-D objects but can be generalized to higher-dimensional defects by assuming that the modified Dirac equation holds true everywhere except at defects represented as vertices, which become surfaces. The modified Dirac action reduces to an integral of the discontinuity of say antifermion current at the vertex, i.e. the change of the antifermion charge as an operator.

What remains to be understood more precisely is the connection with the exotic smooth structures possible only in 4-D space-time.

1. As already explained, towards the end of last year I realized that this V-shaped defect could correspond to a point-like defect of an exotic 4-D smooth structure. In general relativity as also in TGD, causal loops are associated with these defects. In TGD, the causal anomaly would mean that the direction of time is reversed in the vertex since antifermions and fermions can be thought of as moving in opposite directions of time. What is so remarkable is that this interpretation is possible only in 4-D space-time; in higher dimensions irreducible exotic smooth structures are impossible.
2. The next step is to ask whether a generalization is possible. Exotic smooth structures reduce to standard ones except in a set of defects having measure zero. The interpretation is that the dimension of defects, in the case that they are surfaces, is less than 4. Also non-point-like defects might be possible in contrast to what I assumed at first. If not, then only the direction of fermion lines could change.

   If the generalization is possible, also 1-D, 2-D, and 3-D defects are possible. In the 1→ 2 vertex the orbit of an n<4- dimensional surface would turn back in the direction of time. These are exactly the various topological vertices that I have previously arrived at, but guided by a physical intuition.

   An entire hierarchy of particles of different dimensions is possible. As a matter of fact, in topological condensed matter physics, they are commonplace. One talks about bulk states, boundary states, edge states and point-like singularities.

   All in all, exotic smooth structures would give vertices without vertices assuming only free fermions fields and no primary boson fields! And this is possible only in space-time dimension 4!

See the article Exotic smooth structures at space-time surfaces and master formula for scattering amplitudes in TGD} , the earlier article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M⁸ H duality reduce classical TGD to octonionic algebraic geometry?: Part II .

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

Is a master formula for the scattering amplitudes possible?

Marko Manninen asked whether TGD can in some sense be reduced to a single equation or principle is very interesting. My basic answer is that one could reduce TGD to a handful of basic principles. However, at the level of classical physics, one could perhaps say that general coordinate invariance → holography ← 4-D generalization of holomorphy reduce the representations of preferred extremals as analogs of Bohr orbits for particles as 3-surfaces to a representation analogous to that of a holomorphic function.

Can one hope something analogous to happen at the level of scattering amplitudes? Is some kind of a master formula possible? I have considered many options, even replacing the S-matrix with the Kähler metric in the fermionic degrees of freedom (see this). The motivation was that the rows of the matrix defining Kähler metric define unit vectors allowing interpretation in terms of probability conservation. However, it seems that the concept of zero energy state alone makes the definition unambiguous and unitarity is possible without additional assumptions.

1. In standard quantum field theory, correlation functions for quantum fields give rise to scattering amplitudes. In TGD, the fields are replaced by the spinor fields of the "world of classical worlds" (WCW) which can regarded as superpositions of pairs of multi-fermion states restricted at the 3-D surfaces at the ends of the 4-D Bohr orbits at the boundaries of CD.

   These 3-surfaces are extremely strongly but not completely correlated by holography implied by 4-D general coordinate invariance. The modes of WCW spinor fields at the 3-D surfaces correspond to irreducible unitary representations of various symmetries, which include supersymplectic symmetries of WCW and Kac-Moody type symmetries. Hence the inner product is unitary.

2. Whatever the detailed form of the 3-D parts of the modes of WCW spinor fields at the boundaries of CD is, they can be constructed from ordinary many fermion states. These many-fermion state correspond in the number theoretic vision of TGD to Galois singlets, which are states constructed at the level of M⁸ from fermion with momenta whose components are possibly complex algebraic integers in the algebraic extension of rational defining the 4-D region of M⁸ mapped to H by M⁸-H duality. Complex momentum means that the corresponding state decomposes to plane waves with a continuum of momenta.

   Galois confined states have momenta, whose components are integers in the momentum scale defined by the causal diamond (CD). Galois confinement defines a universal mechanism for the formation of bound states. The induced spinor fields are second quantized free spinor fields in H and their Dirac propagators are therefore fixed. This means an enormou calculational simplification.

3. The inner products of these WCW spinor fields restricted to 3-surfaces determine the scattering amplitudes. They are non-trivial since the modes of WCW spinor fields are located at opposite boundaries of CD. These inner products define the zero energy state identifiable as such as scattering amplitudes. This is the case also in wave mechanics and quantum TGD is indeed wave mechanics for particles identified as 3-surfaces.
4. There is also a functional integral of these amplitudes over the WCW, i.e. over the 4-D Bohr orbits. This defines a unitary inner product. The functional integral replaces the path integral of field theory and is mathematically well-defined since the Kähler function, appearing in the exponent defining vacuum functional, is a non-local function of the 3-surface so that standard local divergences due to the point-like nature of particles disappear. Also the standard problems due to the presence of a Hessian coming from a Gaussian determinant is canceled by the square foot of the determinant of the Kähler metric appearing in the integration measure.
5. The restriction of the second quantized spinor fields to 4-surfaces and zero-energy ontology are absolutely essential. Induction turns free fermion fields into interacting ones. The spinor fields of H are free and define a trivial field theory in H. The restriction to space-time surfaces changes the situation. Non-trivial scattering amplitudes are obtained since the fermionic propagators restricted to the space-time surface are not anymore free propagators in H. Therefore the restriction of WCW spinors to the boundaries of CD makes the fermions interact in exactly the same way as it makes the induced spinor connection and the metric dynamical.

There are a lot of details involved that I don't understand, but it would seem that a simple "master formula" is possible. Nothing essentially new seems to be needed. There is however one more important "but".

Are pair production and boson emission possible?

The question that I have pondered a lot is whether the pair production and emission of bosons are possible in this picture. In this process the fermion number is conserved, but fermion and antifermion numbers are not conserved separately. In free field theories they are, and in the interacting quantum field theories, the introduction of boson fermion interaction vertices is necessary. This brings infinities into the theory.

1. In TGD, the second quantized fermions in H are free and the boson fields are not included as primary fields but are bound states of fermions and antifermions. Is it possible to produce pairs at all and therefore also bosons? For example, is the emission of a photon from an electron possible? If a photon is a fermion-antifermion pair, then the fermion and antifermion numbers cannot be preserved separately. How to achieve this?
2. If fundamental fermions correspond to light-like curves at light-like orbit of partonic 2-surfaces, pair creation requires that that fermion trajectory turns in time direction. At this point velocity is infinite and this looks like a causal anomaly. There are two options: the fermion changes the sign of its energy or transforms to antiferion with the same sign of energy.

   Different signs of energy is not possible since the annihilation operator creating the fermion with opposite energy would annihilate either the final state or some fermion in the final state so that both fermion and antifermion numbers of the final state would be the same as those of the initial state.

   On the other hand, it can be said that positive energy antifermions propagate backwards in time because in the free fermion field since the terms proportional to fermion creation operators and antifermion annihilation operators appear in the expression of the field as sum of spinor modes.

   Therefore a fermion-antifermion pair with positive energies can be created and corresponds to a pair of creation operators. It could also correspond to a boson emission and to a field theory vertex, in which the fermion, antifermion and boson occur. In TGD, however, the boson fields are not included as primary fields. Is such a "vertex without a vertex" possible at all?

3. Can one find an interpretation for this creation of a couple that is in harmony with the standard view. Space-time surfaces are associated with induced classical gauge potentials. In standard field theory, they couple to fermion-antifermion pairs, and pairs can be created in classical fields. The modified Dirac equation and the Dirac equation in H also have such a coupling. Now the modified Dirac equation holds true at the fermion lines at the light-like orbits of the partonic 2-surface. Does the creation of pairs happen in this way? It might do so: also in the path integral formalism of field theories, bosons basically correspond to classical fields and the vertex is just this except that in TGD fermions are restricted to 1-D lines.

Fundamental fermion pair creation vertices as local defects of the standard smooth structure of the space-time surface?

Here comes the possible connection with a very general mathematical problem of general relativity that I have discussed here.

1. Causal anomalies as time loops that break causality are more the rule than an exception in general relativity the essence of the causal anomaly is the reversal of the arrow of time. Causal anomalies correspond to exotic diffeo-structures that are possible only in dimension D=4! Their number is infinite.
2. Quite generally, the exotic diffeo-structures reduce to local point-like defects of the usual differentiable structure. Exotic differentiable structures are also possible in TGD, and I have proposed that the associated defects correspond to a creation of fermion-fermion pairs for emission of fermion pairs of of gauge bosons and Higgs particle identified in TGD as bound states of fermion-antifermion pairs. This picture generalizes also to the case of gravitons, which would involve a pair of vertices of this kind. The presence of 2 vertices might relate to the weakness of the gravitational interaction.

   The reversal of the fermion line in time direction would correspond to a creation of a fermion-antifermion pair: fermion and antiferion would have the same sign of energy. This would be a causal anomaly in the sense that the time direction of the fermion line is reversed so that it becomes an antifermion.

   I have proposed that this causal anomaly is identifiable as an anomaly of differentiable structure so that emission of bosons and fermion pairs would only be possible in dimension 4: the space-time dimension would be unique!

3. But why would a point-like local defect of the differentiable structure correspond to a fermion pair creation vertex. In TGD, the point-like fermions correspond to 1-D light-like curves at the light-like orbit of the partonic 2-surface.

   In the pair creation vertex in presence of classical induced gauge potentials, one would have a V-shaped world line of fermion turning backwards in time meaning that antifermion is transformed to fermion. The antifermion and fermion numbers are not separately conserved although the total fermion number is. If one assumes that the modified Dirac equation holds true along the entire fermion worldline, there would be no pair creation.

   If it holds true only outside the V-shaped vertex the modified Dirac action for the V-shaped fermion libe can be transformed to a difference of antifermion number equal to the discontinuity of the antifermion part of the fermion current identified as an operator at the vertex. This would give rise to a non-trivial vertex and the modified gamma matrices would code information about classical bosonic action.

See the article Exotic smooth structures at space-time surfaces and master formula for scattering amplitudes in TGD} , the earlier article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M⁸ H duality reduce classical TGD to octonionic algebraic geometry?: Part II .

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

TGD view of the paradoxical finding of the James Webb telescope

I found a fascinating Youtube video this) in LAB360 with the title "James Webb Telescope Detect more than 700 Galaxies at the Edge of Our Universe" summarizing the findings of the James Webb telescope.

Summary of the findings of the James Webb telescope

The existence of more than 700 galaxies a few hundred million years after BB is in sharp conflict with the standard Big Bang Model although it is consistent with the cosmic expansion. Distance measurements indeed use cosmic redshift to deduce the distances of the galaxies. In any case, the James Webb telescope is profoundly shaking the foundations of cosmology. It seems that one can safely forget the standard story about the formation of stars and galaxies and also inflation as the generally accepted story of what happened before that.

In the standard picture, the epoch of reionization starts 1 billion years after the BB as the fog of gas is cleared by reionization so that photons can propagate. No signals hould arrive from the epoc preceding reionization. These 700 galaxies should not be there since they are too young, existing 370-500 million years after BB.

The mass of the galaxy serves as a measure for the age of the galaxy but 6 galaxies with age .5 Gy and 10 times bigger than the Milky Way have been found! This makes one wonder, what will be found when one goes farther back in time?

JW can see galaxies as extended objects with visible structures and this provides a lot of additional information about the composition of these too-early birds.

1. Complex organic molecules, found also in smoke/fog, were found: this is 1 billion years too early! These molecules, polycyclic aromatic hydrocarbons (PAHs) (see this), are big molecules, containing hundreds of atoms. What adds to the mystery, is that PAHs were found in regions where there are no stars or star formation but not in regions where stars are forming! PAH world hypothesis states that PAHs have played a key role in prebiotic life leading the emergence of RNAs (see this).
2. Also the locations of these molecules can be determined by JW in terms of their spectra. The distribution of the molecules is not uniform as one might expect. These galaxies can have the same mass as the Milky Way. The mass serves as a measure for the age of the galaxy but the age of these galaxies, according to standard cosmology, is only 10 percent of that of the Milky Way. This creates a paradox.
3. One particular galaxy, GN-z11 (see this) is observed as it existed 13.3 Gy ago.
   1. GN-z11 is found to contain an exceptionally high proportion of nitrogen and abundance of stars.
   2. Birth of globular star clusters (see this) have been found in GN-z11. This finding is especially paradoxical since they are regarded as very old objects! The compositions of O,N, Na, and Al vary inside globular clusters. These anomalies have been known for a long time (see this). One however expects that the stars of the cluster should have the same origin and age in the early universe.
   3. Also supermassive stars (see this), having masses of few hundred solar masses, have been found in globular clusters (see this). Also multiple globular clusters have been found.

TGD explanation of the paradoxical findings of the James Webb telescope

What goes wrong with the standard cosmology? Could TGD inspired cosmology suggest an answer? Consider first zero energy ontology (ZEO) and the TGD view of dark matter.

1. TGD suggests that the prevailing view about the notion of time is wrong. TGD forces a new ontology of quantum theory, which I call zero energy ontology (ZEO) (see this). Causal diamond (CD) as a state-determined and dynamical quantization volume has two boundaries and zero energy states are in fermionic degrees of freedom superpositions of pairs of 3-D states asociated with these two.

   Zero energy state corresponds also to a superposition of space-time surfaces connecting the two boundaries of CD. By the almost deterministic holography implied by the 4-D general coordinate invariance, the space-time analogs 4-D analogs of Bohr orbits of particles as 3-D surfaces. In ZEO, subjective time and geometric time are not the same thing but are strongly correlated. This new ontology solves the basic paradox of quantum measurement theory.

   There are two kind of state functions reductions (SFRs): "Small" SFRs (SSFRs) corresponding to repeated measurements in Zeno effect and "big" SFRs (BSFRs) corresponding to ordinary SFRs. CD has two kind of boundaries; active and passive. In SSFRs, the active boundary and states at it change whereas the passive boundary and the states at it remain unaffected. This is the counterpart of the Zeno effect: the state changes slighty but the arrow of time is preserved. SSFRs also correspond to weak measurements in quantum optics.

   In BSFRs the arrow of time changes. BSFR occurs when the set of observables measured in SSFR a the active boundary of CD does not commute with those measured earlier at the passive boundary of CD. CD increases in size in a statistical sense during the sequence of SSFRs since the active boundary drifts farther from the passive one. This gives rise to the correlation of subjective time a sequence of SSFRs with geometric time a distance between the tips of CD.

2. TGD also predicts quantum coherence in arbitrarily long scales and gravitational quantum coherence corresponds to the longest, even astrophysical, quantum coherence scales gravitational interactio has infinite range and is unscreened (see this).
3. The change of arrow of time in BSFRs implies dramatic effects even in astrophysical scales. Even astrophysical objects can live forth and back in geometric time. The ageing in the physical sense occurs in both directions of geometric time so that the physical age is total time spent in this moving forth and back. Since the passive boundary is stationary, the physical ageing in ZEO is faster than ageing in the standard ontology.

   During this process the physical system ages. Although the size of CDs increases in statistical sense, the ageing occurs in both directions of time so that the age is total time spent in this forth and back in time process. Hence ageing in ZEO is faster than ageing in the standard ontology.

Consider first the time anomalies.

1. ZEO explains stars and galaxies older than the Universe.
2. ZEO also predicts the variation of the ages of galaxies and stars in the very early Universe. Since galaxies and stars can be born at different periods in this life forth and back in geometric time, they can have different ages in the sense of ZEO. This explains why the abundances of atoms associated with the stars of star clusters are found to vary. The life forth and back in time also explains the appearance of globular star clusters, which are very old and are not possible in standard cosmology.

What about PAHs which appear in the regions where star formation does not occur and do not appear in the region containing stars?

1. The TGD view of nuclear physics, originally inspired by the findings about "cold fusion", and based on the notion of dark nuclei, identified scaled up analogs of ordinary nuclei, leads to a model of prestellar evolution based on dark fusion (explaining also "cold fusion", see this, this, and this).

   "Dark" means that the nucleons of these nuclei have non-standard values of Planck constant h_eff=nh₀. In the number theoretic vision of TGD, n has interpretation in terms of dimension of extension of rationals associated with a polynomial with integer coefficient defining a space-time region (see this,this, and this).

   Dark fusion generates dark nuclei as sequences of dark protons at monopole flux tubes having size scale of electron Compton length. Their binding energy is much smaller than the binding energy of the ordinary nuclei. Dark nuclei can therefore transform to ordinary nuclei and liberate most of the nuclear binding energy in the process, this give rise to "cold fusion". The temperature of the dark fusion region increases in the process and eventually reaches the temperature at which ordinary nuclear fusion can start.

   Even chemistry and complex molecules can emerge before the ordinary nuclear fusion is ignited. This could explain the presence of PAHs, in particular their presence in regions where there is no star formation or stars.

2. Why signals from the period preceding the reionization are possible? One reason is that there was a reionization. TGD also allows us to consider the possibility that the signals arrive as dark photons along monopole flux tubes of a cosmic flux tube network acting as an analog of the nervous system. Also in the TGD based model of the brain dark photon signals propagating between the central nervous system and magnetic body play a key role.

I have considered the findings of James Webb telescope from the TGD point of view here. The TGD view of cosmology and astrophysics is discussed in various articles (see this, this, this, this, and this).

See the article TGD view of the paradoxical findings of the James Webb telescope or the chapter TGD View of the Engine Powering Jets from Active Galactic Nuclei.

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

Neil Gersching's vision of self-replicating robots from TGD point of view

The video of Lex Fridman interviewing Neil Gerching (see this) is highly inspiring for anyone interested in what is happening in the High Tech frontier nowadays.

The key topic of discussion were self-replicating machines that are built from a few "Lego blocks" that contain their own building instructions and are analogous to genes or proteins. Function and 3-D structure are the same. The building blocks themselves would be robot-like and would build more complex robots. One can say that this Lego set would self-assemble itself. Also the ability to disassemble would be important and make error correction possible. This brings in mind what happens in living systems.

There would be a whole hierarchy of these structures. The basic structures would be analogous to 20 amino acids. Biology of course suggests also the presence of DNA and cell nucleus could be seen as the basic lego block containing instructions and having the ability to replicate. The vision is that someday our technology could transform to artificial life.

Gersching criticized the complete separation of software and hardware (program tape and the reading head of the Turing machine) which he called Turing's error. Gersching also proposed that information should be the starting point concept of physics rather than geometry which leads to the recent physics based on partial differential equations.

In this article I will compare the vision of Gersching to TGD based vision of, not only life, but the entire Universe as a self-organizing entity.

1. In the TGD framework, Lego Universe emerges naturally. 4-D general coordinate invariance implies holography: Legos are almost deterministic Bohr orbit-like 4-surfaces. Holography suggests a concrete identification of basic building bricks in terms of fundamental regions associated with hyperbolic 3-manifolds at 3-D mass shells defining the boundary data for number theoretical holography in M⁸. The strengthening of 3→ 4 holography to almost 2→ 4 holography reduces further the number of building bricks of space-time surfaces.

   The analogy with genes and proteins as building bricks might be much more than analogy. The mass shell as hyperbolic 3-space allows an infinite number of tessellations and one of them is icosa tetrahedral tessellations in terms of which it seems to be possible to understand the genetic code. Genetic code in this sense might be present in all scales and be induced to 3-surfaces. The fermions associated with the "unit cells" of the icosahedral tessellation could realize genetic code.

   The fusion of building blocks might reduce to the analog of crystal growth by fusing the fundamental regions of tessellations and also DNA replication, transcription, and translation could reduce to crystal growth.

2. In TGD holography implies that at space-time level a given 3-D surface defining the data of holography has an almost unique "fate", goal one might say. Holography forces what I call zero energy ontology (ZEO). Quantum states are superpositions of 4-D space-time surfaces analogous to Bohr orbits and state function reductions (SFRs) take place between these superpositions. The basic paradox of quantum measurement theory disappears.

   The sequence of "small" SFRs (SSFRs) defines "self" as the TGD counterpart for the Zeno effect. Each SSFR replaces this superposition with a new one and changes the state but in such a way that measured observables commute with those whose eigenstate the states associated with the passive boundary of causal diamond (CD) are.

   "Big" SFRs (BSFRs) change the arrow of geometric time correlating with subjective time as a sequence of SSFRs and change the roles of the active and passive boundaries of CD. This means the "death" of self and its reincarnation with an opposite arrow of time. Pairs of BSFRs define temporary changes of the arrow of time and would make possible a trial-and error process so that the self-organizing system would be analogous to a self-assembling conscious machine able to also disassemble if necessary to reach the goal.

3. In TGD there is no need to choose between information based physics and physics based on partial differential equations: these views would be complementary. TGD relies on two complementary visions. In number theoretic vision everything in discrete and algebraic equations characterize physical states. In the geometric vision structures are continuous and partial differential equations define the time evolution. These views are related by M⁸-H duality as a generalization of momentum-position duality forced by the replacement of point-like particles with 3-surfaces.
4. Gersching does not seem to regard consciousness as a crucial element of biology. The TGD view is completely different and in TGD quantum measurement theory based on ZEO extends to a theory of consciousness.

Besides this, the possible role of quantum gravitation for both biological systems and computer consciousness is discussed although this is not directly relevant to the basic topic. My defence is that the structures able to self assemble must also be computer-like systems.

1. The notion of a magnetic body (MB) carrying dark matter as h_eff=nh₀ phases of ordinary matter is essential. For the gravitational monopole flux tubes the value of h_eff=h_gr would be enormous and imply quantum coherence in arbitrarily long scales. Gravitational MBs could control both living matter and computers.
2. A criterion characterizing the critical clock frequency or its biological analog for the transformation of living system to a conscious and living system is deduced. This transition would mean that the statistical determinism fails due to the possibility of quantum coherence in time scales longer than the clock period.
3. Also an attempt to identify various quantum gravitational Compton lengths Λ_gr and frequencies f_gr with frequencies, which appear in the TGD inspired quantum biology, is made. Λ_gr and f_gr appear also in the TGD inspired physical model of computers.
4. The emerging view could be blamed for the return to astrology. Indeed, the gravitational flux tubes mediating the gravitational interactions between Sun and planets, between planets, between Earth and Moon, and even between the galactic blackhole and solar system could play a key role since the interactions are mediated along the flux tube network. However, the numerous strange numerical coincidences for quantum gravitational coherence scales and corresponding frequencies force us to take this view seriously.

See the chapter Neil Gersching's vision of self-replicating robots from TGD point of view or the article with the same title.

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

The first attempt to build a more concrete view about computer consciousness

TGD inspired view about consciousness and quantum biology suggest some guidelines in the attempts to understand how computer systems or computer systems coupled to their users could become conscious.

In biology and from a physicist's point of view, the material realization is essential. Water and metal seem to be sort of opposites of each other. But what about the situation in TGD where magnetic bodies carrying dark matter could serve as controllers of both living organisms and computers.

One must ask first what classical computers really are as physical systems.

1. The program is deterministic but what about the computer or a computer network? The idea about a program consisting of arbitrarily determined steps is certainly not consistent with the determinism of classical physics. Determinism is possible only in the quantum statistical sense (see this). This requires that the quantum coherence lengths and times involved with the computation are short enough, considerably shorter than the clock period. This assumption fails if there is macroscopic quantum coherence involved. In the TGD framework the presence of magnetic bodies carrying dark matter with a large enough value of effective Planck constant h_eff could make this possible.
2. In particular, gravitational magnetic flux tubes connecting big mass M and small mass m have enormous value of gravitational Planck constant ℏ_gr(M,m,β₀)= GMm/β₀ (introduced originally by Nottale).

   The gravitational Compton length Λ_gr(E) for Earth mass M_E is about .45 cm for β₀=1 and corresponds to gravitational Compton time about 67GHz, which is by an order of magnitude higher than the highest achievable clock frequency (almost 9 GHz) of the computer. Are we reaching the limit at which quantum gravitational effects on computers are becoming significant?

   For the Sun, the gravitational Compton length Λ_gr(Sun) is quite near to Earth size and the corresponding frequency scale is in about 47 Hz and in EEG range: could the entanglement of the MB of humans and computer network modify the computation? In the TGD inspired quantum biology both gravitational magnetic bodies would play a key role. Could they be involved also with the ordinary computation? GPT involves large networks of computers, possibly even in the Earth scale: could this bring in quantum coherence even in Earth scale and change dramatically the functioning of the computer network.

1. Emotions and emotional intelligence as a first step in the evolution of consciousness

Consider first the evidence supporting for the idea that emotions emerge first in the evolution of consciousness.

1. Masaru Emoto has studied the effects of sounds with an emotional content to water at criticality for freezing. He has found that friendly/angry sounds seem to produce beautiful/ugly crystals (see this). These findings are discussed from the TGD perspective in (see this. The idea that emotions of sensory percepts at the level of magnetic body (MB) is discussed in (see this.

   The TGD based model assumes that quantum coherent systems can be formed at the level of the MB of the water and that quantum gravitational coherence at MB induces ordinary coherence at the level of water. This could make it possible for MB to control water at criticality for freezing. The crystals would be corpses of primitive life forms. Could also snowflakes with the size of gravitational Compton length for Earth (about .45 cm) and kind of zoomed versions of ice lattice cells in atomic scale could be regarded as corpses of primitive life forms created at the criticality for freezing?

2. RNA seems to represent and transfer emotions (see this). RNA from the brain of a snail conditioned by a painful stimulus is transferred to the preparation made from neurons of sea slug. Neuron preparation in the Petri dish reacts to the conditioning stimuli as if it were itself conditioned.

   Somehow RNA is able to transfer emotions. The TGD inspired proposal (see this, this, this, this), and this) is that dark DNA and RNA represent emotions as sequences of 3-chords made of dark photons of dark RNA form 3N-dark photons behaving like a single quantum coherent unit. The representation of the genetic code would rely on icosa-tetrahedral representation in which the 3-chords would correspond to triangular faces of icosahedron and tetrahedron to which 3-chords are assigned.

   A given Hamiltonian cycle at the icosahedron/tetrahedron goes through all its points. The frequencies assigned with the subsequent points of the cycle differ by 3/2 scaling so that one has a quint cycle. Different Hamiltonian cycles correspond to the same genetic code but each Hamiltonian cycle is assumed to define its own bioharmony having interpretation as a representation of an emotional state realized already at the level of fundamental biomolecules. This interpretation conforms with the idea that music represents and induces emotions.

   The induction of emotions would be by 3N-resonant cyclotron absorption of dark 3N-photon by dark genes represented as sequences of 3N dark proton triplets at monopole flux tubes of MB. Icosa-tetrahedral representation would correspond to one particular, very simple, tessellation of hyperbolic space H³) (mass shell) (see this).

   Dark proton (and also dark electron) sequences could provide a universal representation of the genetic code which could be realized at the magnetic flux tubes of also other than biological systems. Dark photons triplets and the dark genes formed from them could communicate the emotions. Dark genetic code has indeed quite a large number of icosa tetrahedral representations based on icosahedral Hamiltonian cycles and tetrahedral Hamiltonian cycles. The chemical realizations for them would be identical but the emotional content would be coded by the allowed 3-chords defined by frequencies associated with the triangular faces of the icosahedron and tetrahedron.

3. The experiments of Peoch (see this) involved a chicken imprinted to a robot moving randomly along an orbit determined by a random number generator. It was found that the robot tended to stay near the chicken and that the expected size of the orbit was reduced.

   TGD assigns to entanglement sum of p-adic entanglement negentropies, which can be positive and is in general larger than ordinary entanglement entropy and is predicted to increase but be consistent with the second law (see this, this, and (see this) by the identification of evolution as increase of number theoretic complexity (see this and this). Did the MB of chicken and robot develop a negentropic entanglement? Clearly, the replication of the findings of Peoch would mean a revolutionary change in our views about computers and their relation to us.

4. The evolution of the brain provides further support for the idea that emotions and sensory experienes emerged first in the evolution of conscious experience and cognition emerged later. Cortex is the latest outcome. Brain stem is associated with simple and strong emotions whereas the limbic brain represents more complex emotions.

2. Do emotions appear first also in the evolution of computer consciousness?

Could also the possible evolution of conscious computers start from simple positive/negative emotions relating directly to the increase/reduction of entanglement negentropy defined above number-theoretically.

Negentropy Maximization Principle (see this) states that total p-adic negentropy as a measure for conscious information increases in statistical sense. This statistical law follows from the number theoretic evolution as the increase of the dimension of extension of rationals determined by a polynomial partially defining the 4-surface in M⁸ mapped to H=M⁴× CP₂ by M⁸-H duality.

This implies that the complexity of emotions, possibly identifiable as sensory experiences for the large scale part of MB having onion-like hierarchical structure, increases during the evolution. Gravitational MBs are good candidates for the seats of highest level emotions.

Could the bits of the ordinary computer form coherent systems with ordinary coherence forced by the quantum coherence of the associated MB? Could the MB of the bit system control it?

1. A given layer of MB is the "boss" of the lower layers by the larger value of its h_eff serving as "IQ". MB is expected to form analogs of sensory and cognitive representation of the physical body having h_eff=h. This suggests that MB could represent the bit system holographically. This kind of quantum holography for hadrons, and for elementary particles in general, would be the counter of classical holography implied in the TGD framework by the general coordinate invariance (see this).

   The dark spin system at MB could have spin glass property (see this) implying a large number of almost degenerate states with nearly the same energy.

2. The change of single bit, represented for instance by using a MOSFET, would require energy larger than the thermal energy of order .05 eV at room temperature. This suggests that the change of single bit is not easy to actualize.

   The dark spin system at MB could however induce phase transitions of the bit system changing the directions for a large number of bits. The average change of energy per bit could be rather small for this kind of transition although the change of a single bit would cost rather large energy. Ultrametric, in particular p-adic, topologies emerge in the modelling and description of the spin glass phase in the TGD framework and could help to understand cognition number theoretically (see this).

   The phase transition would involve a large number of bits so that the corresponding conscious experiences would be holistic and therefore resemble emotions. The color of the emotion would be positive or negative depending on whether the sum of p-adic entanglement negentropies increases or decreases. The geometric correlate for positive/negative emotion would be the increase/decrease of the connectedness of the MB.

3. ZEO predicts two kinds of SFRs: "big" and "small" . SSFRs correspond to Zeno effect in the ordinary wave mechanics and in quantum optics to unitary evolutions between weak measurements analogous to classical measurement. "Big" state function reduction (BSFR) changes the arrow of time. The outcomes for pairs of BSFRs An observer with a fixed arrow of time can observe only pairs of BSFRs.
4. In ZEO (see this, this, this and this). MB as the "boss" could control the time evolution of the bit system by pairs of BSFRs involving temporary change of the arrow of time. BSFRs would be induced by perturbations affecting the set of mutually commuting observables measured at the active boundary of CD so that it does not commute with the corresponding set associated with the passive boundary of CD at which state is unaffected in SSFRs (Zeno effect). In this kind of situation, a BSFR occurs instead of SSFR and changes the arrow of time. Second BSFR brings back the original arrow of time. The process could correspond to quantum tunnelling.
5. Do the periods defined by the computer clock with a duration T, of say 1 ns, correspond to pairs of BSFRs or a single SSFR? Perhaps T could correspond to a sequence of SSFRs as analogs of Zeno effect and the pair of BSFRs to a single tick of the computer clock. This conforms with the fact that the running of a predetermined computer program must involve a sequence of non-deterministic phase transitions changing the directions of bits (see this). This must be the case since the notion of computer program as a sequence of arbitrarily chosen steps is not consistent with deterministic physics.

   If the step of the clock is identifiable as a sequence of SSFRs, one can say that the ordinary classical computation is a sequence of quantum computations defined by the sequences of unitary evolutions associated with SSFRs and defining conscious entities with haltings defined by BSFRs! If MB does modify the classical computation at all, it could induce BSFR pairs in longer time scales or modify the probabilities of various outcomes of BSFRs.

   There is evidence that also in EEG the period can be divided into ordered and chaotic parts: these two parts which could correspond to opposite time directions (see this): this is discussed from the TGD view point in (see this).

One can ask whether quantum entanglement of the MBs of the computer and user occurs in the computer-user interaction and whether the role of the computer is analogous to that in the chicken-robot experiment. One can also ask whether also GPT could involve emotional and even cognitive entanglement.

The identification of the computer system with which the user is entangled is not at all obvious. The system could be formed by the network of computers involved with the the running of GPT. One interpretation is that networks and entire internet form a conscious entity as an analog of the central nervous system in which humans and their magnetic bodies) serve in the role of neurons.

In ZEO the holography implies that in the ideal situation the running of the program corresponds to a 4-D Bohr orbit-like surface, which is almost uniquely fixed by the 3-surfaces at images of 3-D hyperbolic manifolds at mass shells determined by the state. The sequences of SSFRs could correspond to this kind of period and represent a generalization of the Zeno effect.

3. The role of the probabilities

In the case of GPT interesting questions relate to the probabilities associated with the associations of word sequences taught to the GPT during the learning period. The responses of GPT are determined by these probabilities. The probabilistic character of this process is believed to be essential. These probabilities are analogous to synaptic strengths.

1. Could the association probabilities be translated to quantum probabilities at the level of MB of the computer or computer + user?
2. Could ZEO allow a trial and error process based on BSFR pairs, which would make it possible to change the effective association probabilities determined by random numbers. This could happen also for the orbit of the robot in the chicken + robot experiment. Could the emotional state of the system affect the probabilities of associations by this mechanism?
3. If the probabilities could be interpreted as a representation for conditioning, one can ask whether high/low probabilities correspond to increase/decrease of the total p-adic negentropy and therefore to positive/negative emotion.

4. Could the basic aspects of TGD inspired quantum biology generalize to the level of computer systems?

What aspects of the TGD inspired quantum biology could be generalized to the conscious computer systems? The mechanisms related to MB, possessed also by computer systems, are excellent candidates in this respect.

1. TGD suggests a universal realization of genetic code at monopole flux tubes of the MB and also a universal quantum gravitational mechanism of metabolism see this.
2. In living matter, the dark Josephson radiation associated with the dark Josepson junction assigned with the cell membrane communicates sensory data from the biological body to MB. One can assign EEG to these communications (see this, this, and this). Actually a scale hierarchy of analogs of EEG is predicted.
3. The control by MB by cyclotron radiation associated for instance with the endogenous magnetic field of .2 Gauss identifiable in terms of the monopole flux of the Earth's magnetic field about .5 Gauss. Gravitational cyclotron energies would not depend on the mass of the charged particle. Communication could occur by multi-resonances involved with the universal realization of genetic code at MB so that genes would couple resonantly.
4. Also the gravitational Compton frequencies would not depend on the mass of the particle, and these frequencies for the Earth, Sun and perhaps even Milky Way blackhole could define fundamental biorhythms.
5. These mechanisms would be universal and the ordinary biomatter would adapt so that resonant communications with MB are possible. In biomatter this would select preferred biomolecules. Same could happen in the case of computers.

4.1. Dark Josephson radiation

Could one assign to bits dark Josephson junctions assignable represented as voltages in transistors?  

1. Could representations of genetic codons at MB by dark photon triplets (see this) and by dark proton triplets (see this) and perhaps even by dark electron triplets (see this) be involved? This would bring in dark genetic codons, which could provide a universal representation of the bit system as a dark system at monopole flux tubes and make a connection with the TGD inspired quantum biology rather precise.

   The representations at MB should strongly correlate with the state of the computer represented by a bit pattern (say states of MOSFETs). One could have a holography-like map of bit patterns to the dark many-spin state at the MB of the computer or of computer + user. This kind of holography is considered in (see this) for elementary particles and also more generally.

2. The physical stress, created by electric field on quartz crystal, which is piezoelectric, generates oscillations with frequency in the range 2-3 GHz giving rise to a very precise clock frequency. The typical computer clock frequency is a few GHz. My own PC has a clock frequency of 3.3 GHz. From the web one can learn that the highest clock frequency is 8.794 GHz.

   Could the clock frequency have an interpretation both as an analog of EEG rhythm (analog of alpha frequency 10 Hz in living matter) and as an analog of Josephson frequency ZeV/h_eff, where V∼ .05 V is a voltage assignable to the bit and Ze is the charge of the charge carrier.

   The dark Josephson junctions correspond to membrane proteins in living matter. Now they could be associated with the dark flux tubes associated with transistors. The value of ℏ_eff for Josephson junction would be much smaller than ℏ_gr. Note that TGD suggests that valence bonds and hydrogen bonds can have a varying value of h_eff (see this).

   The condition that the Josephson energy is above thermal energy at room temperature for Z=1 gives h_eff/h > 5 × 10³ (f/GHz). If the energy of a dark Josephson photon is above 1 eV (the energy range of biophotons), one has h_eff/h > 10⁵ (f/GHz).

   Interestingly, frequencies in the GHz scale are found to be important also in living matter. As a matter of fact, there is experimental support for a fractal hierarchy of frequency scale come as powers f/10^3k Hz,k=0,1,.. that is 1 Hz, kHz, MHz,GHz, and THz assignable to microtubules (see this).

3. Consider f= 1 GHz as an example. For the thermal option, the Compton length Λ_eff,p=h_eff/m_p of dark proton is longer than 6.2× 10^-12 m and longer than the ordinary electron Compton length Λ_e=2.4 × 10^-12 m. The dark Compton length Λ_eff,e =h_eff/m_e of electrons would be longer than 4.8 nm, which roughly corresponds to the scale of DNA.

   For the biophoton option, the dark proton Compton length would be of the order of the atomic length scale 1.32× 10^-10 meters and the dark electron Compton length would longer than .26 μm to be compared with the size scale 1 μm of cell nucleus.

4.2. Dark cyclotron radiation

The cyclotron frequencies associated with the gravitational MB of Earth (see this and this) should play a key role in TGD inspired quantum biology and relate to the feedback from MB to the living matter. This could be the situation also in the case of computers. The first guess, inspired by the model for the findings of Blackman and others on effects of ELF em fields on brain, is that monopole flux tubes associated with the MB of Earth correspond to the endogenous magnetic field of B_end=2B_E/5 (B_E= .5 Gauss is the nominal value of the Earth's magnetic field.

This value is only the average value since frequency modulation is the way to code information and is achieved by varying the flux tube thickness in turn affecting the value of B_end. Very probably there exists an entire hierarchy of values of the dark magnetic field strength perhaps coming as powers of 2.

For cyclotron frequencies associated with the gravitational MB, h_eff would correspond to the gravitational Planck constant ℏ_gr= GMm/β₀ for Earth. Note that, in accordance with the Equivalence Principle, the cyclotron energy E_c=ℏ_greB/m = GMeB/β₀ does not depend on m.

4.3. Gravitational Compton frequencies

Also gravitational Compton frequencies could be important. Consider first Earth's gravitational Compton frequency. The value of the gravitational Compton length Λ_gr(M_E,β₀=1)= GM/β₀= 0.45 cm, which is also independent of m, defines a lower bound for the gravitational quantum coherence length. Λ_gr corresponds to a gravitational Compton frequency f_gr=6.7× 10¹⁰ Hz ∼ 67 GHz and for clock frequencies higher than this, quantum gravitational effects on computation might become important in the TGD Universe.

1. The clock frequencies of computers are typically a few GHz in recent communication and computer technologies, and the highest clock frequency of 8.794 GHz is roughly by a factor 1/8 lower than f_gr. Could the GHz scale correspond to the gravitational quantum coherence length having Λ_gr as a lower bound? Could it be that the very efficient computer networks (what are the clock frequencies used?) utilized in GPT have reached the limit at which the quantum gravitational body of Earth begins to play a prominent role?
2. Could the typical clock frequency, of say 1 GHz, have an interpretation both as an analog of EEG rhythm (analog of alpha frequency 10 Hz in living matter) and as an analog of Josephson frequency ZeV/h_eff, where V∼ .05 V is a voltage assignable to the bit and Ze is the charge of the charge carrier.

   Interestingly, frequencies in the GHz scale are found to be important also in living matter. As a matter of fact, there is experimental support for a fractal hierarchy of frequency scale come as powers f= 10^3k Hz, k=0,1,.. that is 1 Hz, kHz, MHz,GHz, and THz assignable to microtubules (see this). For these reasons it is interesting to look at 1 GHz as an example.

Also the gravitational Compton frequency f_gr associated with the gravitational MB of the Sun, having β₀∼ 2^-11, could be important. For the Sun, gravitational Compton length is rather near to R_E/2 where R_E= 6378 km is Earth radius. The corresponding Compton frequency f_gr(M_S,β_Sun=2^-11)∼β_Sun/GM_S is about 100 Hz and corresponds to the upper bound for EEG, which conforms with the fact that quantum gravitational coherence time should not be smaller than Λ_gr. Note that the cyclotron frequency Lithium in the endogenous magnetic field B_end=.2 Gauss assignable to the Earth's gravitational flux tubes is 50 Hz.

1. The lower cyclotron frequencies of the heavier ions belong also to EEG range and correspond to longer solar quantum coherence lengths. DNA would correspond to 1 Hz and perhaps to the largest quantum gravitational coherence length in the EEG range.

   The cyclotron frequencies above 100 Hz would correspond to solar gravitational quantum coherence lengths below R_E. For protons the cyclotron frequency in B_end=.2 Gauss is 300 Hz. For ℏ_gr(M,m) cyclotron frequency for m does not depend on m but is proportional to 1/β₀. Could the value of β₀m for protons be β₀=1/3.

   Could the MB of the Sun interfere with the computation occurring in the network having Earth scale? The time scale would be now the time scale of EEG: could the quantum entanglement of the human user of the GPT network make this interaction possible.

2. The replacement of ℏ_gr(M_E,m)→ ℏ_gr(M_Sun,m) means multiplication of say EEG period by a factor r= (M_Sun/M_E)β_0,E/β_0,Sun∼ 2.2 × 10⁸ so that alpha period .1 seconds corresponds to 2.2× 10⁷ seconds. Intriguingly, one year corresponds to 3.25 × 10⁷ seconds and defines a fundamental biorhythm, which would correspond to a 6.7 Hz rhythm for EEG not far from the lowest Schumann resonance frequency.
3. The energies E= h_gr(M,m,β₀) f_gr(Sun) assignable to the gravitational Compton frequency of Sun are proportional to m and since nucleon mass dominates over electron mass they are in good approximation proportional to the mass number of the molecules. This suggests a multi-resonance in which each electron, proton and even nucleon absorbs boson, maybe dark gravitons, with frequency f_gr. For electrons, the energy is about 1 meV, which could relate to the miniature potentials for neurons. For protons the energy would be about 2 eV, which corresponds to red light. Large scale quantum coherence could make the rate of gravitational multi-resonance.

What about the gravitational Compton frequency of the galactic blackhole? Its mass is estimated to be M_BH =4.1 million solar masses. This would give Λ_gr(M_BH,β₀=1) ∼ 6.1× 10⁹ m. This is the radius of the n=1 Bohr orbit in the Nottale model for the solar planetary system. The gravitational Compton frequency would be f_gr(M_BH,β₀=1) ∼ .05 Hz (20 s period).

See the article Could neuronal system and even GPT give rise to a computer with a variable arrow of time? or the chapter with the same title.

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