In what sense the early Universe could contain dark matter or energy as primordial blackholes?

Are blackholes possible in the early Universe? Is the energy density in the early Universe so high that the gravitation collapses matter to blackholes? Could primordial blackholes explain the dark matter (see this)?

Before trying to cook up answers to these questions one should ask whether these questions are physically meaningful?  I believe that a more meaningful question concerns the reality of blackholes. They represent singularities,  at which general relativity fails. How should one modify general relativity to get rid of a system carrying the entire mass of the star in a single point? Perhaps this is the correct question.

TGD provides this modification. It solves the basic problem of GRT due the loss of the classical conservation laws and which also predicts the standard model symmetries and classical fields. In TGD, blackholes are replaced by blackhole-like objects, which can be regarded as tangles of monopole flux tubes filling the entire volume below the Schwartschild radius.

This leads to a new view of the  very early Universe. Cosmic strings with 2-D M^4 projection and 2-D CP_2 projection dominate in the very early Universe. Cosmic strings are unstable against the thickening of M^4 projection and this gives to quasars as blackhole-like objects, or rather, to  white-hole-like objects feeding energy into environment as the dark energy of the cosmic string transforms to ordinary matter as it thickens to monopole flux tube.

One might   say that these primordial blackhole-like objects evaporate and produces ordinary matter. One can also say that this  process is the TGD counterpart of inflation. The exponential expansion is not needed in TGD since quantum coherence in the scales  made possible by the presence of arbitrarily long cosmic strings with monopole flux making them stable against splitting implies the approximate constancy of  CMB temperature.

See the article About the recent TGD based view concerning cosmology and astrophysics or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Why the electric currents that should accompany magnetic substorms in the magnetotail are missing?

I found an interesting popular article (see this) telling about the surprising findings associated with the sub-storms of magnetic storms accompanying sunspots in the magnetotail of the Earth. The reconnections were observed and Maxwell's electrodynamics also predicts electric currents associated with them. However, there was no evidence for them.

TGD based view of electromagnetic fields predicts deviations from the Maxwellian view. In TGD, the magnetic field decomposes into two parts. The TGD counterpart of Maxwellian magnetic fields and the monopole flux part is not present in the Maxwellian theory.

1. The Maxwellian part consists of flux tubes with a cross section which has a boundary, say disk. The flux tubes correspond to space-time regions, or space-time sheets as I call them. The Maxwellian part requires currents to create it. At the quantum field theory (QFT) limit of TGD this gives rise to the Maxwellian magnetic fields.
2. The monopole part consists of closed monopole flux tubes, which have a closed 2-surface as cross section and the Maxwellian flux tubes with, say, disk-like cross section. These are not possible in field theories in Minkowski space. Monopole flux part would contribute roughly 2/5 to the total magnetic field strength of Earth at the QFT limit.

   What is important is that the monopole part does not require currents to create it. The monopole part is topologically stable and explains the puzzling existence of the magnetic fields in even cosmic scales and also the maintenance of the Earth's magnetic field. The Maxwellian part decays since the currents creating it dissipate (see this) .

Monopole flux tubes carry h_eff>h phases of ordinary matter behaving like dark matter.

1. These phases solve the missing baryon problem and the increasing fraction of missing baryons during cosmic evolution. The loss of baryons would be due to the gradual generation of effectively dark phases of nucleons (and other particles) with increasing values of h_eff. h_eff has an interpretation as a measure for an algebraic complexity of the space-time region measured by the dimension of the algebraic extension defined by the two polynomials associated with the region of space-time surface considered. A given polynomial with integer or rational coefficients defines an extension of rationals and the extensions associated with two polynomials define an extension containing both extensions. Mathematically, this increase is completely analogous to the unavoidable increase of entropy. This increase of complexity would give to evolution, also biological evolution. Dark matter in this sense plays a key role in the TGD inspired quantum biology.
2. Notice that in TGD, the galactic dark matter is actually dark energy of cosmic strings (extremely thin monopole flux tubes) and of the monopole flux tubes to which they thicken as extremely thin flux tubes. Therefore one should speak of galactic dark energy. The recent discovery of what looks like MOND type gravitational anomaly for distant stars of binaries gives strong support for this view (see this).

Consider now the mystery of the missing currents. No electric currents associated with storm were observed also the signatures of reconnections were observed. Could the magnetopause be dominated by the monopole flux tubes carrying the h_eff>h phases of ordinary mater behaving like dark matter. The existence of the associated electric currents is not needed to create the monopole magnetic fields. Are electric currents very weak or are they only apparently absent since they are dark? How does magnetotail relate to this? Is it only because the reconnections occur here.

See the article at Magnetic Bubbles in TGD Universe: part II or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Could the TGD view of galactic dark matter make same predictions as MOND?

I learned about a very interesting findingso of Kyu-Hyun Chae  related to the dynamics of binaries of widely separated stars (this) . The dynamics seems to  violate  Newtonian gravitation for low accelerations,   which naturally emerge at large separations and the violations are roughly consistent with the MOND hypothesis. This raises the question whether the TGD based explanation of flat velocity spectra associated with galaxies could be consistent with the MOND hypothesis).  

The TGD based model for the binary system involving the monopole flux tubes associated with the stars of the binary leads to a prediction for the critical acceleration which is  of the same magnitude as the galactic critical accelerations. This result generalizes if the scaling law T²(m)/m= constant for the  system with mass m associated with a long monopole flux tube with string tension T(m) holds true.

See Could the TGD view of galactic dark matter make same predictions as MOND? and the chapter A HREF="https://tgdtheory.fi/pdfpool/3pieces.pdf">About the recent TGD based view concerning cosmology and astrophysics

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Intersection forms, exotic smooth structures, and particle vertices in the TGD framework

Gary Ehlenberg sent an email telling about his discussion with GPT  related to exotic smooth structures.     The timing was perfect.  I was just yesterday evening pondering what the exotic smooth structures as ordinary smooth  structures with defects could really mean and how to get a concrete grasp of them.    

Here are some facts about the exotic smooth structures. I do not count myself as a real mathematician but the results give a very useful perspective.

Summary of the basic findings about exotic smooth structures

The study of exotic R⁴'s   has led to numerous significant mathematical developments, particularly in the fields of differential topology, gauge theory, and 4-manifold theory. Here are some key developments:

1. Donaldson's Theorems

   Simon Donaldson's groundbreaking work in the early 1980s revolutionized the study of smooth 4-manifolds. His theorems provided new invariants, known as Donaldson polynomials, which distinguish between different smooth structures on 4-manifolds.

   Donaldson's Diagonalization Theorem: This theorem states that the intersection form of a smooth, simply connected 4-manifold must be diagonalizable over the integers, provided the manifold admits a smooth structure. This result was crucial in showing that some topological 4-manifolds cannot have a smooth structure.

   Donaldson s Polynomial Invariants: These invariants help classify and distinguish different smooth structures on 4-manifolds, particularly those with definite intersection forms.

2. Freedman's Classification of Topological 4-Manifolds

   Michael Freedman's work, which earned him a Fields Medal in 1986, provided a complete classification of simply connected topological 4-manifolds. His results showed that every such manifold is determined by its intersection form up to homeomorphism.

   h-Cobordism and the Disk Embedding Theorem: Freedman's proof of the h-cobordism theorem in dimension 4 and the disk embedding theorem were instrumental in his classification scheme.

3.  Seiberg-Witten Theory

   The development of Seiberg-Witten invariants provided a new set of tools for studying smooth structures on 4-manifolds, complementing and sometimes simplifying the methods introduced by Donaldson.

   Seiberg-Witten Invariants: These invariants are simpler to compute than Donaldson invariants and have been used to prove the existence of exotic smooth structures on 4-manifolds.

4.  Gauge Theory and 4-Manifolds

   Gauge theory, particularly through the study of solutions to the Yang-Mills equations, has provided deep insights into the structure of 4-manifolds.

   Instantons: The study of instantons (solutions to the self-dual Yang-Mills equations) has been crucial in understanding the differential topology of 4-manifolds. Instantons and their moduli spaces have been used to define Donaldson and Seiberg-Witten invariants.

5.  Symplectic and Complex Geometry

   The interaction between symplectic and complex geometry with 4-manifold theory has led to new discoveries and techniques.

   Gompf's Construction of Symplectic 4-Manifolds: Robert Gompf's work on constructing symplectic 4-manifolds provided new examples of exotic smooth structures. His techniques often involve surgeries and handle decompositions that preserve symplectic structures.

   Symplectic Surgeries: Techniques such as symplectic sum and Luttinger surgery have been used to construct new examples of 4-manifolds with exotic smooth structures.

6.  Floer Homology

   Floer homology, originally developed in the context of 3-manifolds, has been extended to 4-manifolds and provides a powerful tool for studying their smooth structures.

   Instanton Floer Homology: This theory associates a homology group to a 3-manifold, which can be used to study the 4-manifolds that bound them. It has applications in understanding the exotic smooth structures on 4-manifolds.

7.  Exotic Structures and Topological Quantum Field Theory (TQFT)

   The study of exotic R⁴'s  has also influenced developments in TQFT, where the smooth structure of 4-manifolds plays a crucial role.  TQFTs are sensitive to the smooth structures of the underlying manifolds, and exotic R⁴'s provide interesting examples for testing and developing these theories.

To sum up, the exploration of exotic R⁴'s has led to significant advances across various areas of mathematics, particularly in the understanding of smooth structures on 4-manifolds. Key developments include Donaldson and Seiberg-Witten invariants, Freedman s topological classification, advancements in gauge theory, symplectic and complex geometry, Floer homology, and topological quantum field theory. These contributions have profoundly deepened our understanding of the unique and complex nature of 4-dimensional manifolds.

How exotic smooth structures appear in TGD

The recent TGD view of particle vertices    relies  on exotic smooth structures emerging in D=4.  For a  background see this, this , and this .

1. In TGD string world sheets are replaced with 4-surfaces in H=M⁴xCP₂ which allow generalized complex structure as also M⁴  and H.

2. The notion of generalized complex structure.

   The generalized complex structure is introduced  for M⁴, for  H=M⁴× CP₂  and for the space-time surface X⁴ ⊂ H.

   1. The generalized complex structure of M⁴  is a fusion of hypercomplex structure and complex structure involving slicing of M⁴ by string world sheets and partonic 2-surfaces transversal to each other.  String world sheets allow  hypercomplex structure and  partonic 2-surface complex structure. Hypercomplex coordinates  of M⁴ consist of a pair of  light-like coordinates as a generalization of a light-coordinate of M² and complex coordinate as a generalization of a complex coordinate for E².  
   2. One obtains a generalized complex structure for H=M⁴×CP₂ with 1 hypercomplex coordinate and 3 complex coordinates.
   3. One  can use a  suitably selected  hypercomplex coordinate and a complex coordinate of H as generalized complex coordinates for X⁴ in regions where the induced metric is Minkowskian. In regions where it is Euclidean one has two complex coordinates for X⁴.    

3. Holography= generalized holomorphy  

   This conjecture gives a general solution of classical field equations.  Space-time surface X⁴ is defined as a zero locus for  two functions of generalized complex coordinates of H,  which are generalized-holomorphic and thus depend on 3 complex coordinates and one light-like coordinate. X⁴ is a minimal surface  apart from singularities at which the minimal surface property fails.  This irrespective of action assuming that it is constructed in terms of the induced geometry.   X⁴   generalizes the complex submanifold of algebraic geometry.  

   At  X⁴ the trace of the second fundamental form, H^k, vanishes. Physically this means that the generalized acceleration for a 3-D particle vanishes i.e one has free massless particle. Equivalently, one has a geometrization of a massless field.  This means particle-field duality.

4. What happens at the interfaces between Euclidean and Minkowskian regions of X⁴ are light-like 3-surfaces X³?  

   The light-like surface  X³ is topologically 3-D but metrically 2-D  and corresponds  to a  light-like orbit of a partonic  2-surface  at which the  induced metric of X⁴ changes its signature from Minkowskian to Euclidean.  At X³  a  generalized complex structure of X⁴ changes from Minkowskian to  its Euclidean variant.

   If the embedding is generalized-holomorphic, the induced  metric  of X⁴ degenerates to an  effective 2-D metric at at X³ so that   the topologically  4-D tangent space is effectively  2-D metrically.

5. Identification of the 2-D singularities (vertices) as regions at which the minimal surface property fails.

   At 2-D  singularities X², which I propose to be counterparts of 4-D smooth structure,  the minimal surface property fails. X² is  a hypercomplex analog  of a pole of complex functions and 2-D.   It is analogous to a source of a massless field.

   At X²  the generalized complex structure fails such that the trace of the second fundamental form generalizing acceleration for a point-like particle develops a delta function like singularity.  This singularity develops for the hypercomplex part  of the generalized complex structure and one has  as an  analog   a pole of analytic function at  which  analyticity fails. At X² the tangent space is 4-D rather than 2-D as elsewhere at the partonic orbit.

   At X²   there is an  infinite generalized acceleration. This  generalizes Brownian motion  of a point-like particle as a piecewise free motion. The partonic orbits could  perform Brownian motion and the 2-D singularities correspond to vertices for particle reactions.

   At least the creation of a   fermion-antifermion pair   occurs at this kind of singularity.    Fermion turns backwards in time. Without  these singularities fermion and antifermion number would be separately conserved and TGD  would be trivial as a physical theory.

6. One can identify the singularity X²  as a defect of the ordinary smooth structure.  

   This is the conjecture that I would like to understand better and here my limitations as a mathematician are the problem.

   I can  only ask questions inspired by the result that the intersection form I (X⁴)  for 2-D homologically non-trivial surfaces of X⁴  detects  the defects of the ordinary smooth structure,  which should correspond to surfaces X², i.e.  vertices for a pair creation.

   1. CP₂ has an intersection form corresponding to the homologically non-trivial 2-surfaces for which minimal intersection corresponds to a single point. The value of intersection form for 2 2-surfaces is essentially the product of integers characterizing their homology equivalence classes. If each wormhole contact contributes a single CP₂ summand to the total intersection form, there would be two summands per elementary particle as monopole flux tube.
   2. 2-D singularity gives rise to a creation of an elementary particle and would therefore add two CP₂ summands to the intersection form. The creation of a fermion-antifermion pair has an interpretation in terms of a closed monopole flux tube. A closed monopole flux tube having wormhole contacts at its "ends" splits into two by reconnection.
   See the article What gravitons are and could one detect them in TGD Universe? or the chapter with the same title.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

How to handle the interfaces between Minkowskian and Euclidean regions of space-time surface?

The treatment of  the dynamics at the   interfaces X³ between Minkowskian and Euclidean regions X³  of the  space-time surface identified as light-like partonic orbits has turned out to be a difficult technical problem. By holomorphy as a realization of generalized holography, the 4-metric at X³  degenerates to 2-D effective Euclidean metric apart from 2-D delta function singularities  X² at which the holomorphy fails but    the metric is  4-D.  

One must treat both the  bosonic and fermionic situations. There are two options for the treatment of the interface dynamics.

1.  The  interface X³ is  regarded as an independent dynamic unit.  The earlier approaches rely on this assumption.  By the light-likeness of X³, C-S-K action is the only possible option. The problem with U(1) gauge  invariance disappears if C-S-K action is identified as a total divergence emerging from  the instanton term for Kähler action.

   One can assign to  the instanton term a corresponding contribution to the modified Dirac action at X³.  It however seems that the instanton term associated with the 4-D modified Dirac action does not reduce to a total divergence  allowing to  localize  it a X³.

   In this approach, conservation laws require that the normal components of the canonical momentum currents from the Minkowskian and Euclidean sides  add up to the divergence of the canonical momentum currents associated with the C-S-K action.

2. Since the interface is not a genuine boundary, one  can argue that  one should treat the situation as 4-dimensional. This approach is adopted in this article.     In the bosonic degrees of freedom, the C-S-K term  is  present also for this option  could determine the bosonic  dynamics of the boundary apart from a 2-D delta function type singularities coming from the violation of the minimal surface property and of the generalized holomorphy. At vertices involving fermion pair creation this violation would occur.

In the 4-dimensional treatment  there are no analogs of the boundary conditions at the interface.

1. It is essential that the 3-D light-like orbit X³ is a 2-sided surface  between Minkowskian and Euclidean domains.  The variation of the C-S-K term emerging from a total divergence  could determine  the dynamics of the interface except possibly  at the singularities X³,  where the  interior contributions from the 2 sides  give rise to  a 2-D  delta function term.
2. The contravariant metric diverges at X³  since  by holography one has g_uv=0 at X³ outside X². The condition   J_uv= 0 could guarantee that  the contribution of the Kähler action remains finite.  The contribution from Kähler action  to field equations could   be even reduced to the divergence of the instanton term at X³  by what I have called  electric-magnetic duality   proposed years ago (see this).  At X³, the dynamics would be effectively reduced to 2-D Euclidean degrees of freedom outside X². Everything would be finite as far as Kähler action is considered.
3. Since the metric at X³ is effectively 2-D, the induced gamma matrices  are proportional to  2-D delta function and  by J_uv=0 condition the contribution of the volume term to the modified gamma matrices dominates over the finite contribution of the Kähler action. This holds true outside the 2-D singularities X². In this sense the idea that only induced gamma matrices matter at the interfaces, makes sense.

   In order to obtain the counterpart of Einstein's equations  the metric must be effectively 2-D also at X² so that det(g₂)=0 is true although holomorphy fails. It seems that   one must assume induced, rather than modified, gamma matrices (effectively reducing to the induced ones at X³  outside X²) since for the latter option the gravitational vertex would vanish by the field equations.

   The situation is very delicate and I cannot claim  that I understand it sufficiently. It seems that the edge of the partonic orbit due to the turning of the fermion line and involving hypercomplex conjugation is essential.

4. For the modified Dirac equation to make sense,  the vanishing of the covariant derivatives with respect to light-like coordinates   seems necessary. One would   have D_uΨ=0 and D_vΨ=0 in X³ except at the 2-D singularities X², where  the induced  metric would have diagonal components g_uu and g_vv. This would give rise to the gauge boson vertices involving emission of fermion-antifermion pairs.

5.  By the generalized holomorphy, the second fundamental form H^k  vanishes   outside X². At X²,  H^k   is proportional to a 2-D delta function and also the Kähler contribution can be of comparable size  This should give the TGD counterpart of Einstein's equations and Newtonian equations of motion and to the graviton vertex.

   The orientations of the tangent spaces at the two sides are different. The induced metric at   the Minkowskian side  would become 4-D.  At the Euclidean side it could be Euclidean and   even metrically 2-D.

The following overview of the symmetry breaking through the generation of 2-D  singularities is suggestive.  Masslessess and holomorphy are violated via the  generation of the analog of Higgs expectation at the vertices. The use of the  induced gamma matrices violates supersymmetry  guaranteed by the use of the modified gamma matrices   but only at the vertices.

There is however an objection. The use of the induced gammas in the modified Dirac equation seems necessary although the non-vanishing of H^k seems to violate  the hermiticity at the  vertices. Can the turning of the fermion line and the exotic smooth structure allow to get rid of this problem?

See the article What gravitons are and could one detect them in TGD Universe? or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Direct evidence for a mesoscale quantum coherence in living matter

This posting was inspired by Sabine Hossenfelder's video (see this) telling about the recently observed evidence for quantum coherence in mesoscales by Babcock et al (see this).

Experimental evidence for the mesoscale quantum coherence of living matter

The abstract to the article of Babcock et al summarizes the findings.

Networks of tryptophan (Trp) an aromatic amino acid with strong fluorescence response are ubiquitous in biological systems, forming diverse architectures in transmembrane proteins, cytoskeletal filaments, sub-neuronal elements, photoreceptor complexes, virion capsids, and other cellular structures.
We analyze the cooperative effects induced by ultraviolet (UV) excitation of several biologically relevant Trp mega-networks, thus giving insights into novel mechanisms for cellular signaling and control.
Our theoretical analysis in the single-excitation manifold predicts the formation of strongly superradiant states due to collective interactions among organized arrangements of up to > 10⁵ Trp UV-excited transition dipoles in microtubule (MT) architectures, which leads to an enhancement of the fluorescence quantum yield (QY) that is confirmed by our experiments.
We demonstrate the observed consequences of this superradiant behavior in the fluorescence QY for hierarchically organized tubulin structures, which increases in different geometric regimes at thermal equilibrium before saturation, highlighting the effect s persistence in the presence of disorder.
Our work thus showcases the many orders of magnitude across which the brightest (hundreds of femtoseconds) and darkest (tens of seconds) states can coexist in these Trp lattices.

From the article it is clear that the observed phenomenon is expected to be very common and not only related to MTs. From Wikipedia one learns that tryptophan is an amino acid needed for normal growth in infants and for the production and maintenance of the body's proteins, muscles, enzymes, and neurotransmitters. Trp is an essential amino acid, which means that the body cannot produce it, so one must get it from the diet.

Tryptophan (Trp) is important throughout biology and forms lattice-like structures. From the article I learned that Trp plays an essential role in terms of communications. There is a connection between Trp and biophotons as well. Trp's response to UV radiation is particularly strong and also to radiation up to red wavelengths.

What is studied is the UV excitation of the Trp network in the case of MTs. The total number of Trp molecules involved varies up to 10⁵. The scales studied are mesoscales: from the scale of a cell down to the scales of molecular machines. The wavelengths at which the response has been studied start at about 300 nm (4.1 eV, UV) and extend to 800 nm (1.55 eV, red light) and are significantly longer than tubulin's scale of 10 nm. This indicates that a network of this size scale is being activated. The range of time scales for the radiant states spans an enormous range.

UV excitation generates a superradiance meaning that the fluorescence is much more intense than it would be if the Trps were not a quantum-coherent system. The naive view is that the response is proportional to N² rather than N, where N is the number N of Trp molecules. Super-radiance is possible even in thermal equilibrium, which does not fit the assumptions of standard quantum theory and suggests that quantum coherence does not take place at the level of the ordinary biomatter.

In standard quantum physics, the origin of the mesoscale coherence is difficult to understand. Quantum coherence would be the natural explanation but the value of Planck constant is far too small and so are the quantum coherence lengths. The authors predict superradiance, but it is not clear what assumptions are involved. Is quantum coherence postulated or derived (very likely not).

TGD based interpretation

I have considered MTs in several articles (see for instance this, this and this).

In TGD, the obvious interpretation would be that the UV stimulus induces a sensory input communicated to the magnetic body of the Trp network, analogous to the EEG, which in turn produces superradiance as a "motor" reaction. The idea about MT as a quantum antenna is one of the oldest ideas of TGD inspired quantum biology (see this). The communication would be based on dark photons involved also with the communications of cell membrane to the MB of the brain and with DNA to its MB.

The Trp network could correspond to some kind of lattice structure or be associated with such a structure at the magnetic body of the system. The notion of bioharmony (see this and this) leads to a model of these communications based on the universal realization of the genetic code in terms of icosa tetrahedral tessellation of hyperbolic space H³.

The icosa tetrahedral tessellation (see this and this) is completely unique in that it has tetrahedrons, octahedrons, and icosahedrons as basic objects: usually only one platonic solid is possible. This tessellation predicts correctly the basic numbers of the genetic code and I have proposed that it could provide a realization of a universal genetic code not limited to mere biosystems. Could the cells of the Trp lattice correspond to the basic units of such a tessellation?

The work of Bandyopadhyay et al (see for instance this) gives support for the hypothesis that there is hierarchy of frequency scales coming as powers of 10³ (10 octaves for hearing in the case of humans) ranging from 1 Hz (cyclotron frequency of DNA) and extending to UV.

This hierarchy could correspond to a hierarchy of magnetic bodies. Gravitational magnetic bodies assignable to astrophysical objects (see this and this) and electric field bodies to systems with large scale electric fields (see this see this) can be considered. They possess a very large value of the gravitational/electric Planck constant giving rise to a long length scale quantum coherence.

Gravitational magnetic bodies have a cyclotron energy spectrum, which by Equivalence principle is independent of the mass of the charged particle. The discrete spectrum for the strengths of the endogenous magnetic field postulated by Blackman and identified as the non-Maxwellian monopole flux tube part of the magnetic field having minimal value of 2B_E/5=.2 Gauss would realize 12-note spectrum for the bioharmony. The spectrum of Josephson energies assignable to cell membrane is independent of h_eff (see this).

Both frequency spectra are inversely proportional to the mass of the charged particle, which makes them ideal for communication between ordinary biomatter and dark matter. Frequency modulated signals from say cell membrane to the magnetic body and coding the sensory input would propagate as dark Josephson photons to the magnetic body and generate a sequence of resonance pulses as a reaction, which in turn can induce nerve pulses or something analogous to them in the ordinary biomatter. In a rough sense, this would be a transformation of analog to digital.

Authors also propose that superradiance could involve a shielding effect, analogous to what happens in the Earth's magnetic field and might be based on a similar mechanism.

1. In the standard description, the Earth's magnetic field catches the incoming cosmic rays, such as UV photons, to the field lines, and thus prevents the arrival of the radiation to the surface of Earth. Van del Allen radiation belts are of special importance.
2. In the TGD description, a considerable fraction of incoming high energy photons and maybe also other higher energy particles would be transformed to their dark variants at the magnetic monopole flux tubes of the MB of the Earth with a field strength estimated to be B_end=2B_E/5, where B_E=.5 Gauss is the nominal value of the Earth's magnetic field. This mechanism would transform the high energy photons to low energy dark photons with much longer wavelengths which have very weak interactions with the ordinary biomatter. These in turn would be radiated away as ordinary photons and in this way become neutralized. The scaling factor for the wavelength would be ℏ_gr/h if the gravitational MB of the Earth is involved.

   Something similar would take place in biological systems at cellular level. The UV photons would be transformed to dark photons with much longer wavelengths and radiated away as ordinary photons.

Can one identify a range of biological scales perhaps labelled by the values of ℏ_eff/ℏ coming as powers of 10³.

1. The findings of Cyril Smith related to the phenomenon of water memory suggest that in living matter a scaling of photon frequency can take place with a scaling factor 2× 10¹¹ or is inverse. In the TGD framework, I christened this mechanism as "scaling law of homeopathy" (sounds suicidal in the ears of a mainstream colleague, see this). For a UV radiation with λ=300 nm frequency f=1.24× 10¹⁵ Hz this would mean scaling down of frequency to 6.8 kHz and scaling up of wavelength to .4× 10⁵.
2. The kHz scale is one of the preferred scales suggested by the work of Bandyopadhyay, suggesting also a hierarchy of the scaling factors 2× 10^11-3x, x=-1,0,2,... Could there exist a hierarchy of biological scales differing by powers of 10^{3? Could these scaling factors correspond to various values of h_eff/h₀?}
3. In the TGD inspired quantum biology, the Earth's gravitational magnetic body plays a key role. Could one assign the length scale with x=-1 with the Earth's gravitational magnetic body having gravitational Planck constant equal to ℏ_gr= GM_Em/β₀, β₀=v₀/c≈ 1, where M_E is the mass of Earth? By the Equivalence Principle, the gravitational Compton length is independent of mass m of the particle and for Earth is about .5 cm, the size scale of a snowflake.
4. The scaling hierarchy in powers of 10³ would predict besides .5 cm, the length scale 5 μm of cell nucleus, the length scale 5 nm characterizing the thickness of the lipid layer of cell membrane and of the DNA double strand, and the scale 5 × 10^-12 m to be compared with the Compton length 2.4 × 10^-12 m of electron. The scaling hierarchy would be naturally associated with the electron naturally. The wavelength scale corresponding to x=-2 is λ=.4× 10⁸ m, which is equal to the circumference of Earth 2π R_E≈ .4× 10⁸ m defining the lowest Schumann resonance frequency!
5. If β₀=v₀/c ≤ 1 is true, the scales with x=0,1,... cannot correspond to the values ℏ_gr for β₀ coming as positive powers of 10³ and its difficult to imagine hierarchy of masses as powers of 10³.

   Could the electric Planck constants as counterparts of gravitational Planck constants (see this, and this) defined as ℏ_em= Qe²/β₀, where Q is the charge of a system analogous to the electrode of a capacitor, give these scales as electric Compton length for electron? This would conform with the fact that cell interior and DNA are negatively charged.

There are good reasons to believe that these findings will be noticed by the people fighting with the problems related to quantum computers caused by the extreme fragility of quantum coherence in standard quantum theory. One might even hope that the basic assumptions of quantum theory could be questioned. The TGD based generalization of quantum theory could pave the way for building quantum computers and also raises the question whether ordinary computers could become in some sense living systems under suitable conditions (see this, this, and this). S ee also this about the recently observed evidence for quantum coherence in mesoscales by Babcock et al that motivated these considerations.

See the article New Results about Microtubules as Quantum Systems or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Intersection forms, exotic smooth structures, and particle vertices in the TGD framework

Gary Ehlenberg sent an email telling about his discussion with GPT  related to exotic smooth structures.     The timing was perfect.  I was just yesterday evening pondering what the exotic smooth structures as ordinary smooth  structures with defects could really mean and how to get a concrete grasp of them.    

Here are some facts about the exotic smooth structures. I do not count myself as a real mathematician but the results give a very useful perspective.

Summary of the basic findings about exotic smooth structures

The study of exotic R⁴'s   has led to numerous significant mathematical developments, particularly in the fields of differential topology, gauge theory, and 4-manifold theory. Here are some key developments:

1. Donaldson's Theorems

   Simon Donaldson's groundbreaking work in the early 1980s revolutionized the study of smooth 4-manifolds. His theorems provided new invariants, known as Donaldson polynomials, which distinguish between different smooth structures on 4-manifolds.

   Donaldson's Diagonalization Theorem: This theorem states that the intersection form of a smooth, simply connected 4-manifold must be diagonalizable over the integers, provided the manifold admits a smooth structure. This result was crucial in showing that some topological 4-manifolds cannot have a smooth structure.

   Donaldson s Polynomial Invariants: These invariants help classify and distinguish different smooth structures on 4-manifolds, particularly those with definite intersection forms.

2. Freedman's Classification of Topological 4-Manifolds

   Michael Freedman's work, which earned him a Fields Medal in 1986, provided a complete classification of simply connected topological 4-manifolds. His results showed that every such manifold is determined by its intersection form up to homeomorphism.

   h-Cobordism and the Disk Embedding Theorem: Freedman's proof of the h-cobordism theorem in dimension 4 and the disk embedding theorem were instrumental in his classification scheme.

3.  Seiberg-Witten Theory

   The development of Seiberg-Witten invariants provided a new set of tools for studying smooth structures on 4-manifolds, complementing and sometimes simplifying the methods introduced by Donaldson.

   Seiberg-Witten Invariants: These invariants are simpler to compute than Donaldson invariants and have been used to prove the existence of exotic smooth structures on 4-manifolds.

4.  Gauge Theory and 4-Manifolds

   Gauge theory, particularly through the study of solutions to the Yang-Mills equations, has provided deep insights into the structure of 4-manifolds.

   Instantons: The study of instantons (solutions to the self-dual Yang-Mills equations) has been crucial in understanding the differential topology of 4-manifolds. Instantons and their moduli spaces have been used to define Donaldson and Seiberg-Witten invariants.

5.  Symplectic and Complex Geometry

   The interaction between symplectic and complex geometry with 4-manifold theory has led to new discoveries and techniques.

   Gompf's Construction of Symplectic 4-Manifolds: Robert Gompf's work on constructing symplectic 4-manifolds provided new examples of exotic smooth structures. His techniques often involve surgeries and handle decompositions that preserve symplectic structures.

   Symplectic Surgeries: Techniques such as symplectic sum and Luttinger surgery have been used to construct new examples of 4-manifolds with exotic smooth structures.

6.  Floer Homology

   Floer homology, originally developed in the context of 3-manifolds, has been extended to 4-manifolds and provides a powerful tool for studying their smooth structures.

   Instanton Floer Homology: This theory associates a homology group to a 3-manifold, which can be used to study the 4-manifolds that bound them. It has applications in understanding the exotic smooth structures on 4-manifolds.

7.  Exotic Structures and Topological Quantum Field Theory (TQFT)

   The study of exotic R⁴'s  has also influenced developments in TQFT, where the smooth structure of 4-manifolds plays a crucial role.  TQFTs are sensitive to the smooth structures of the underlying manifolds, and exotic R⁴'s provide interesting examples for testing and developing these theories.

To sum up, the exploration of exotic R⁴'s has led to significant advances across various areas of mathematics, particularly in the understanding of smooth structures on 4-manifolds. Key developments include Donaldson and Seiberg-Witten invariants, Freedman s topological classification, advancements in gauge theory, symplectic and complex geometry, Floer homology, and topological quantum field theory. These contributions have profoundly deepened our understanding of the unique and complex nature of 4-dimensional manifolds.

How exotic smooth structures appear in TGD

The recent TGD view of particle vertices    relies  on exotic smooth structures emerging in D=4.  For a  background see this, this , and this .

1. In TGD string world sheets are replaced with 4-surfaces in H=M⁴xCP₂ which allow generalized complex structure as also M⁴  and H.

2. The notion of generalized complex structure.

   The generalized complex structure is introduced  for M⁴, for  H=M⁴× CP₂  and for the space-time surface X⁴ ⊂ H.

   1. The generalized complex structure of M⁴  is a fusion of hypercomplex structure and complex structure involving slicing of M⁴ by string world sheets and partonic 2-surfaces transversal to each other.  String world sheets allow  hypercomplex structure and  partonic 2-surface complex structure. Hypercomplex coordinates  of M⁴ consist of a pair of  light-like coordinates as a generalization of a light-coordinate of M² and complex coordinate as a generalization of a complex coordinate for E².  
   2. One obtains a generalized complex structure for H=M⁴×CP₂ with 1 hypercomplex coordinate and 3 complex coordinates.
   3. One  can use a  suitably selected  hypercomplex coordinate and a complex coordinate of H as generalized complex coordinates for X⁴ in regions where the induced metric is Minkowskian. In regions where it is Euclidean one has two complex coordinates for X⁴.    

3. Holography= generalized holomorphy  

   This conjecture gives a general solution of classical field equations.  Space-time surface X⁴ is defined as a zero locus for  two functions of generalized complex coordinates of H,  which are generalized-holomorphic and thus depend on 3 complex coordinates and one light-like coordinate. X⁴ is a minimal surface  apart from singularities at which the minimal surface property fails.  This irrespective of action assuming that it is constructed in terms of the induced geometry.   X⁴   generalizes the complex submanifold of algebraic geometry.  

   At  X⁴ the trace of the second fundamental form, H^k, vanishes. Physically this means that the generalized acceleration for a 3-D particle vanishes i.e one has free massless particle. Equivalently, one has a geometrization of a massless field.  This means particle-field duality.

4. What happens at the interfaces between Euclidean and Minkowskian regions of X⁴ are light-like 3-surfaces X³?  

   The light-like surface  X³ is topologically 3-D but metrically 2-D  and corresponds  to a  light-like orbit of a partonic  2-surface  at which the  induced metric of X⁴ changes its signature from Minkowskian to Euclidean.  At X³  a  generalized complex structure of X⁴ changes from Minkowskian to  its Euclidean variant.

   If the embedding is generalized-holomorphic, the induced  metric  of X⁴ degenerates to an  effective 2-D metric at at X³ so that   the topologically  4-D tangent space is effectively  2-D metrically.

5. Identification of the 2-D singularities (vertices) as regions at which the minimal surface property fails.

   At 2-D  singularities X², which I propose to be counterparts of 4-D smooth structure,  the minimal surface property fails. X² is  a hypercomplex analog  of a pole of complex functions and 2-D.   It is analogous to a source of a massless field.

   At X²  the generalized complex structure fails such that the trace of the second fundamental form generalizing acceleration for a point-like particle develops a delta function like singularity.  This singularity develops for the hypercomplex part  of the generalized complex structure and one has  as an  analog   a pole of analytic function at  which  analyticity fails. At X² the tangent space is 4-D rather than 2-D as elsewhere at the partonic orbit.

   At X²   there is an  infinite generalized acceleration. This  generalizes Brownian motion  of a point-like particle as a piecewise free motion. The partonic orbits could  perform Brownian motion and the 2-D singularities correspond to vertices for particle reactions.

   At least the creation of a   fermion-antifermion pair   occurs at this kind of singularity.    Fermion turns backwards in time. Without  these singularities fermion and antifermion number would be separately conserved and TGD  would be trivial as a physical theory.

6. One can identify the singularity X²  as a defect of the ordinary smooth structure.  

   This is the conjecture that I would like to understand better and here my limitations as a mathematician are the problem.

   I can  only ask questions inspired by the result that the intersection form I (X⁴)  for 2-D homologically non-trivial surfaces of X⁴  detects  the defects of the ordinary smooth structure,  which should correspond to surfaces X², i.e.  vertices for a pair creation.

   1. In  homology, the  defect  should correspond  to an intersection point  of homologically non-trivial 2-surfaces identifiable as wormhole throats, which  correspond to homologically non-trivial 2-surfaces of CP₂. This suggests that  I(X⁴₁)  for X⁴₁ containing the singularity/vertex differs from I(X⁴₂) when X⁴ does  not contain the vertex.  
   2. Singularities  contribute to the intersection form. The creation of  fermion-antifermion pair  has an interpretation in terms of closed monopole flux tubes. A closed monopole flux tube with wormhole contacts at its "ends" splits into two by reconnection.  The  vertex at which the particle is created, should contribute to  the intersection form: the fermion-antifermion vertex as  the intersection point?
   For a summary of earlier postings see Latest progress in TGD.

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Antipodal duality from the TGD point of view: answer to a question by Avi Shrikumar

Avi Shrikumar asked about the antipodal duality (see this), which has been discovered in QCD but whose origin is not well-understood.

Antipodal duality implies connections between strong and electroweak interactions, which look mysterious since in the standard model these interactions are apparently independent. This kind of connections were discovered long before QCD and expressed in terms of the conserved vector current hypothesis (CVC) and partially conserved axial current PCAC hypothesis for the current algebra.

I looked at the antipodal duality as I learned of it (see this) but did not find any obvious explanation in TGD at that time. After that I however managed to develop a rather detailed understanding of how the scattering amplitudes emerge in the TGD framework. The basic ideas about the construction of vertices (see this and this) are very helpful in the sequel.

1. In TGD, classical gravitational fields, color fields, electroweak fields are very closely related, being expressed in terms of CP₂ coordinates and their gradients, which define the basic field like variables when space-time surface 4-D M⁴ projection. TGD predicts that also M⁴ possesses Kähler structure and gives rise to the electroweak U(1) gauge field. It might give an additional contribution to the electroweak U(1) field or define an independent U(1) field.

   There is also a Higgs emission vertex and the CP₂ part for the trace of the second fundamental fundamental form behaves like the Higgs group theoretically. This trace can be regarded as a generalized acceleration and satisfies the analog of Newton' s equation and Einstein's equations. M⁴ part as generalized M⁴ acceeration would naturally define graviton emission vertex and CP₂ part Higgs emission vertex.

   This picture is bound to imply very strong connections between strong and weak interactions and also gravitation.

2. The construction of the vertices led to the outcome that all gauge theory vertices reduce to the electroweak vertices. Only the emission vertex corresponding to Kähler gauge potential and photon are vectorial and can contribute to gluon emission vertices so that strong interactions might involve only the Kähler gauge potentials of CP₂ and M⁴ (something new).
3. The vertices involving gluons can involve only electroweak parity conserving vertices since color is not a spin-like quantum number in TGD but corresponds to partial waves in CP₂. This implies very strong connections between electroweak vertices and vertices involving gluon emission. One might perhaps say that one starts the U(1) electroweak vertex and its M⁴ counterpart and assigns to the final state particles as a center of mass motion in CP₂.

   If this view is correct, then the standard model would reflect the underlying much deeper connection between electroweak, color and gravitational interactions implied by the geometrization of the standard model fields and gravitational fields.

See the article Antipodal duality and TGD or the chapter About TGD counterparts of twistor amplitudes .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Comparison of string model and TGD: answer to the question of Harekhrisna Acharaya

Harekhrisna Acharaya asked how TGD compares with string theory. In the following a short answer.

TGD could be also seen as a generalization of string models. Strings replace point-like particles and in TGD 3-surfaces replace them.

A. Symmetries

1. In string models 2-D conformal symmetry and related symmetries are in key roles. They have a generalization to TGD. There is generalization of 2-D conformal invariance to space-time level in terms of Hamilton-Jacobi structure. This also implies a generalization of Kac-Moody type symmetries. There are also supersymplectic symmetries generalizing assignable to δ M⁴₊×CP₂ and reflecting the generalized symplectic structure for the light-cone M⁴₊ and symplectic structure for CP₂. These symmetries act as Noether symmetries in the "world of classical worlds" (WCW).
2. In string models 2-D conformal invariance solves field equations for strings. This generalizes to the TGD framework.
   1. One replaces string world sheets with 4-D surfaces as orbits of 3-D particles replacing strings as particles. The 2-D conformal invariance is replaced with its 4-D generalization. The 2-D complex structure is replaced with its 4-D analog: I call it Hamilton-Jacobi structure. For Minkowski space M⁴ this means composite of 2-D complex structure and 2-D hypercomplex structure. See this .
   2. This allows a general solution to the field equations defining space-time in H=M⁴×CP₂ realizing holography as generalized holomorphy. Space-time surfaces are analogous to Bohr orbits. Path integral is replaced with sum over Bohr orbits assignable to given 3-surface.

      Space-time surfaces are roots of two generalized holomorphic functions of 1 hypercomplex (light-like) coordinate and 3 complex coordinates of H. Space-time surfaces are minimal surfaces irrespective of action as long as it is expressible in terms of the induced geometry.

   3. There are singularities analogous to poles (and cuts) at which generalized holomorphy and minimal surface property fails and they correspond to vertices for particle reactions. There is also a highly suggestive connection with exotic smooth structures possible only in 4-D. This gives rise to a geometric realization of field particle duality. Minimal surface property corresponds to the massless d'Alembert equation and free theory. The singularities correspond to sources and vertices. The minimal surface as the orbit of a 3-surface corresponds to a particle picture.

B. Uniqueness as a TOE

The hope was that string models would give rise to a unique TOE. In string models branes or spontaneous compactification are needed to obtain 4-D or effectively 4-D space-time. This forced to give up hopes for a unique string theory and the outcome was landscape catastrophe.

1. In TGD, space-time surfaces are 4-D and the embedding space is fixed to H=M⁴×CP₂ by standard model symmetries. No compactification is needed and the space-time is 4-D and dynamical at the fundamental level.

   The space-time of general relativity emerges at QFT limit when the sheets of many-sheeted space-time are replaced with single region of M⁴ made slightly curved and carryin gauge potentials sum of those associated with space-time sheets.

2. How to uniquely fix H: this is the basic question. There are many ways to achieve this.
   1. Freed found that loop spaces have a unique geometry from the existence of Riemann connection. The existence of the Kahler geometry of WCW is an equally powerful constraint and also it requires maximal isometries for WCW so that it is analogous to a union of symmetric spaces. The conjecture is that this works only for H=M⁴×CP₂. Physics is unique from its geometric existence.
   2. The existence of the induced twistor structure allows for the twistor lift replacing space-time surfaces with 6-D surfaces as S² bundles as twistor spaces for H=M⁴×CP₂ only. Only the twistor spaces of M⁴ and CP₂ have Kaehler structure and this makes possible the twistor lift of TGD.
   3. Number theoretic vision, something new as compared to string models, leads to M⁸-H duality as an analog of momentum position duality for point-like particles replaced by 3-surface. Also this duality requires H=M⁴×CP₂. M⁸-H duality is strongly reminiscent of Langlands duality.

      Although M⁸-H duality is purely number theoretic and corresponds to momentum position duality and does not make H dynamical, it brings to mind the spontaneous compactification of M¹⁰ to M⁴×S .

   C. Connection with empiria

   String theory was not very successful concerning predictions and the connection with empirical reality. TGD is much more successful: after all it started directly from standard model symmetries.

   1. TGD predicts so called massless extremals as counterparts of classical massless fields. They are analogous to laser light rays. Superposition for massless modes with the same direction of momentum is possible and propagation is dispersion free.
   2. TGD predicts geometric counterparts of elementary particles as wormhole contacts, that is space-time regions of Euclidean signature connecting 2 Minkowskian space-time sheets and having roughly the size of CP₂ and in good approximation having geometry of CP₂. MEs are ideal for precisely targeted communications.
   3. TGD predicts string-like objects (cosmic strings) as 4-D surfaces. They play a key role in TGD in all scales and represent deviation from general relativity in the sense that they do not have 4-D M⁴ projection and are not Einsteinian space-times. The primordial cosmology is cosmic string dominated and the thickening of cosmic strings gives rise to quasars and galaxies. Monopole flux tubes are fundamental also in particle, nuclear and even atomic and molecular physics, biology and astrophysics. See for instance this .
   4. An important deviation of TGD from string models is the notion of field body. The Maxwellian/gauge theoretic view of fields is replaced with the notion of a field body having flux sheets and flux tubes as body parts. Magnetic monopoles flux tubes require no currents to maintain the associated magnetic fields. This explains the existence of magnetic fields in cosmic scales and of huge cosmic structures. Also the stability of the Earth's magnetic field finds an explanation.

      The dark energy of cosmic strings explains the flat velocity spectrum of stars around galacxies and therefore galactic dark matter.

   5. Number theoretic vision predicts a hierarchy of space-time surfaces defined as roots of pairs of polynomials with increasing degree. This gives rise to an evolutionary hierarchy of extensions of rationals and also behind biological evolution.

      The dimensions of extensions correspond to a hierarchy of effective Planck constants h_eff=nh₀ serving as measure of algebraic complexity and giving rise to a hierarchy of increasing quantum coherence scales. Ordinary particles with h_eff>h behave like dark matter. The identification is not as galactic dark matter but as dark phases residing at field bodies and controlling ordinary matter (h_eff serves as a measure for intelligence). They explain the missing baryonic matter.

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

   For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

An audiofile by Tuomas Sorakivi about TGD view of Biefeld Brown effect

Tuomas Sorakivi prepared an audio file with text about the model of the Biefeld Brown effect in the TGD framework. In the recent experiments of Charles Buhler related to Biefeld Brown effect (emdrive represents earlier similar experiment) are carried out in a vacuum chamber and using a casing of the electrodes of the asymmetric capacitor-like system to prevent the leakage currents. Maximum acceleration of 1 g was detected and the effect increases with the strength of the electric field. Electron currents rather than ionic currents seem to be responsible for the effect.

This strongly suggests new physics. Either the law of momentum fails or there is some third, unidentified, party with which the capacitor-like system exchanges momentum and energy.

In the TGD Universe this third party would be the field body, presumably electric field body (EB). Instead of the ordinary Planck constant, the electric Planck constant at the EB of a single electrode would characterize the electrons. Its value is rather large and makes possible quantum coherence in the scale of the smaller electrode. Generalized Pollack effect would transfer electrons to dark electrons at EBs and its reversal would return them back so a net momentum would be left to the EB.

See the audiofile Topological Geometrodynamics view of Biefeld Brown effect

or the article About Biefeld Brown effect or the chapter About long range electromagnetic quantum coherence in TGD Universe.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

The essay of Marko Manninen about the geometric aspects of TGD

Marko Manninen wrote a very nice essay  about  the geometric aspects of TGD. This has involved a lot of inspiring discussions during a couple of years, which have also helped me to articulate TGD more precisely. Below is Marko's summary of  his essay. 

It is now official. My previous research as a citizen scientist on gamma rays, peer-reviewed and published in 2021, has been followed by a new study, this time in the field of theoretical fundamental physics. It is often said that experiments and empirical data are groping in the dark without theories. In this context, without a unified theory of spacetime, quantum phenomena, and elementary particles, our understanding and applications remain incomplete.

In my new nearly 150-page English essay-research, published by Holistic Science Publications, I aim to present the early history and geometric foundations of the Topological Geometrodynamics (TGD) theory, which Dr. Matti Pitkänen began developing in the 1970s. The text includes accessible introductory material to help a broader audience grasp the basics and motivations of the TGD theory. For specialists, there is in-depth content to engage with.

Current mainstream fundamental physics theories assume elementary particles are geometrically point-like entities. This simplification works when studying phenomena in isolation. However, when attempting to create a unified representation of spacetime, quantum mechanics, and elementary particles, we have been stuck for over 50 years. The problematic concept of time and its nature has also been difficult to resolve.

According to TGD, these issues are overcome by treating elementary particles not as points or even strings, as in string theories, but as 3-surfaces in an eight-dimensional hyperspace. Through a geometric induction process, the required symmetries related to the conservation laws of physics are transferred from the static hyperspace to the dynamic 3-surfaces of the real world. Plato's allegory of the cave is a fitting analogy here: the experienced world with its elementary particles and dynamic laws is a shadowy reflection of higher-dimensional symmetrical mathematical structures. Regarding time, we must also include experienced time and integrate human conscious experience into the unified theory framework, as done in TGD.

These proposals have been surprisingly radical within the community. Despite recognizing the open problems, solutions rarely make it through peer review. This challenge has now been overcome, and we eagerly await to see if new interested researchers will engage with and perhaps apply this work over the years.

My major effort is now behind me, and we can enjoy the fruits of my labor. My most significant long-term endeavor began about four years ago when I first heard about TGD. My interest was piqued by TGD's holistic approach, addressing profound topics without neglecting humanistic and mind-philosophical dimensions. My independent research culminated in a six-month writing marathon, completed at the turn of 2023-24, under Dr. Pitkänen's patient tutorship.

Thanks also go to Antti Savinainen, a physics teacher at Kuopio Lyceum, who thoroughly reviewed and commented on my text. Ville-Veli Einari Saari, Rode Majakka, and Tuomas Sorakivi from our regular Zoom group meetings have been excellent sounding boards for discussions. Dr. Ari J. Tervashonka, who has a Ph.D. in the history of science and ether theories, has been a crucial academic link, guide, and support during the publication process. Heartfelt thanks and humble apologies also go to my loved ones who have often heard these perhaps incomprehensible ideas as I processed the theory aloud.

We would be very grateful if you share the link to my publication and mention it in suitable contexts. Feel free to comment and ask questions; we promise to respond diligently. You can find the essay here or here.

Could the model of Biefeld Brown effect apply to rotating magnetic systems?

Could model of Biefeld Brown be also applied to a rather massive rotating magnetic system studied by Russian researchers Godin and Roschin (see this), which I have tried to understand during years (see this).

1. The system consists of a stator and rollers rotating around it. Also the effect of a radial electric field was studied. The high voltage between stator and electrodes outside the rollers varied in a range 0-20 kV. Therefore a capacitor-like system is in question. Positive potential was associated with the stator so that the force experienced by electrons was towards the electrodes. This generates a strong radial electric field and there is an ionization of air around the rotating magnet, which could be caused by high energy electrons from the surface of the rotor as in coronal discharge.
2. What happens is that the system begins to accelerate spontaneously as the rotation frequency approaches 10 Hz, the alpha frequency of EEG. Rather dramatic weight reduction of 35 per cent and a generation of cylindrical magnetic walls with B=.05 Tesla parallel to the rotation direction are reported. The sign of the effect depends on the direction of rotation.

The situation resembles in many respects to that in the Biefeld Brown effect.

1. Could the Pollack effect feed electrons to the magnetic and/or electric FB of the system. The electrons would also leave some of their angular momentum to the FB and drop back. Otherwise the rotors develop a positive charge Q= ω BS proportional to the rotation frequency ω, magnetic field B and the area S of the vertical boundary of the cylinder, as in the Faraday effect.

   The pumping of electrons to the FB would generate both the momentum and angular momentum as a recoil effect. Now the vertical components of momentum and angular momentum in z-direction would be involved. In the first approximation, the magnetic field can be modelled as a dipole field in Maxwellian theory.

2. Rollers are rotating magnets. What is interesting is that in the Faraday effect a rotating magnet develops a radial voltage proportional to the rotating frequency and magnetic field. One expects that the same occurs for the rollers. This cannot be understood in Maxwell's theory as induction since the motion is not linear and the calculation of the voltage using the same formula requires a generation of a charge density. In TGD, the assumption that the vector potential of the magnetic field rotates with the magnet, explains the effect. Could this charge density be due to a transfer of electrons to the FB of the system? Positive charge density would be generated and create a force opposite to the direction of the Earth's gravitational acceleration so that the Faraday effect for the rollers cannot explain the findings.
3. One expects that the vector potentials for the magnetic fields of rollers rotate as in the Faraday effect. Also the magnetic fields associated with the rollers or rather, their flux tubes should rotate. This could lead to a twisting of the flux tubes. The twisting would suggest that the flux tubes of FBs of the rollers are helical monopole flux tubes (by rotation) emerging from the top and retung back at the bottom of the roller system. There is an obvious analogy with the solar magnetic field.

   Could this generate momentum and angular momentum recoils? The two ends of the rollers should generate different recoils. The only asymmetry between the top and bottom is that the Earth surface bounds the system at the bottom. Could this give rise to a higher degree of quantum coherence at the upper ends of the rollers, which could give rise to a non-vanishing net acceleration and angular acceleration.

4. The observed magnetic walls could correspond to the return flux associated with the magnetic field of the rollers. That they are walls suggests that the flux tubes from the rollers fuse to a single flux wall and this gives rise to a quantum coherence. That the return flux consists of several magnetic walls rather than a single one suggests that the magnetic wall emerging from the roller system decomposes to these walls and the scale of quantum coherence is reduced. If the fluxes of walls return separately to the lower ends of rollers the degree of quantum coherence would be lower and this could give rise to a net effect.
5. Where could the energy of rotation and lift come from? Does it come from some external source, say the MB of the Earth? This could relate to as the 10 Hz cyclotron resonance assignable to the Fe ions in the "endogenous" magnetic field B_end= 2B_E/5 assigned to the monopole flux tubes as the model for the findings of Blackman suggests?

   Does the energy come from the internal magnetic energy of the stator magnet or of rollers? Or does the energy come from the electrostatic energy associated with the horizontal electric field between electrodes and rollers as in the Biefeld Brown effect. This voltage should gradually reduce if this is the case.

See the article About Biefeld Brown effect or the chapter About long range electromagnetic quantum coherence in TGD Universe.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Biefeld Brown effect in TGD Universe

Biefeld Brown effect is one of the effects studied by "free energy" researchers. What happens is that an asymmetry capacitor for which the electrodes are of different size starts to move in the direction of the smaller electrode.  The so called emdrive could be also based on this effect. Recently I learned of the experiments carried out by Buhler's team. An acceleration of 1 g is achieved for a capacitor-like system in vacuum and the effect increases rapidly with the strength of the electric field between the electrodes. This raises the question whether new physics is involved: either as a failure of the momentum conservation or as a presence of an unidentified system with which a momentum transfer takes place. In this article I consider the TGD basic model in which the third system is identified as the electric field body associated with the system.

In the TGD basic model, the third system is identified as the electric field body (FB) associated with the system. The key idea is that electronic momentum is pumped from the electrodes to their FBs: an electron is transferred to the FB, leaves some of its momentum to FB and drops back and in this way gives rise to a recoil. For the smaller electrode the quantum coherence is higher and the pumping is more effective. This gives rise to the Biefeld Brown effect, perhaps even in the situation when the dielectric is present. There is also a net transfer of electrons momentum to the positive electrode, which reduces the voltage while keeping the system neutral and provides in this way electrostatic energy to the kicked electrons. This explains why the effect is stronger when the smaller electrode is positively charged.

See the article About Biefeld Brown effect or the chapter About long range electromagnetic quantum coherence in TGD Universe.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Could entanglement entropy have a role similar to that of thermodynamical entropy?

Gary Ehlenberg sent a link to a popular article (see this) telling about the work of Bartosz Regula and Ludovico Lami published as an article with title "Reversibility of quantum resources through probabilistic protocols" in Nature (see this).

Quantum entanglement gives rise to a density matrix analogous to that of a thermodynamic system. One can assign to it entropy by a standard formula. In thermodynamics entropy characterizes the system besides other state variables. Could entanglement entropy have a role similar to that of thermodynamic entropy and allow us to classify entangled systems?

1. In thermodynamics there are transformations of thermodynamic state, which in the adiabatic case preserve entropy. There is also a theorem giving an upper bound for the efficiency of a thermal engine transforming heat to work, which is ordered energy. Could one consider a similar theorem? Could thermodynamics generalize to a science of entropy manipulation?
2. Could adiabatic transformations preserving entanglement entropy be possible between two systems with the same entanglement entropy? Could the second law stating the increase of entropy generalize and state that transformations decreasing the entanglement entropy are not possible?

Suppose that a system A defined as an equivalence class of systems with the same density matrix can be transformed B. Can one transform B to A in this kind of situation? This would be a counterpart of thermodynamic irreversibility stating that adiabatic time evolutions are reversible.

1. It has been indeed proposed that it is possible to connect any two entangled systems with the same entanglement entropy by a transformation preserving the trace of the density matrix and keeping the eigenvalues positive. This transformation generalizes unitary transformations, which preserve the eigenvalues of the density matrix having probability interpretation. This kind of transformation would consist of basic physical transformations: I must confess that I do not quite understand what this means. If the conjecture is true one could classify entangled systems in terms of the entanglement entropy.
2. There are however systems for which the conjecture fails. A weaker condition would be that this kind of transformation exists in a probabilistic sense: the analog of adiabatic transformation would fail for some system pairs. One can apply the transformations of a specified set O of transformations to an ensemble formed by copies of A. Intuitively it would seem that the number of transformations in the set of O transforming A to B adiabatically divided by the total number of transformations gives a measure for the success. I am not quite sure whether this definition is used.
3. It is also stated the relative entropy for the probability distributions defined by the density matrices for systems A and B would in turn serve as an entanglement measure. Each fixed system B with fixed density matrix (or A) would define such a measure.

This could have application to TGD inspired theory of consciousness.

1. In TGD inspired theory of consciousness, one can assign to any system a number theoretic negentropy as a sum of p-adic entanglement: here the sum is over the ramified primes of an algebraic extension of rationals considered. The p-adic negentropies obey a formula similar to that for the ordinary entanglement negentropy. p-Adic negentropies can be however positive unlike the ordinary entanglement negentropy as a negative of entanglement entropy. The sum of p-adic negentropies measures the information of entanglement unlike ordinary entropy which measures the loss of information about either entangled state caused by the entanglement.
2. The number theoretic vision of TGD predicts that the entanglement negentropy is bound to increase in statistical sense in the number theoretic evolution and that this increase forces the increase of the ordinary entanglement entropy. Could the basic theorems of thermodynamics generalize also to this case and provide a mathematical way to understand the evolution of intelligent systems?

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Equivalence Principle in TGD

Understanding Equivalence Principle has been one of the basic challenges of TGD.

1. If one accepts Einstein's equations as a realization of Equivalence Principle, Equivalence Principle could be seen as a remnant of Poincare invariance of TGD at the quantum field theory limit of TGD. Einstein's equations at QFT limit would express what is left from exact Poincare invariance at the microscopic level. Recall that the microscopic level space-times are 4-D surfaces in H=M⁴×CP₂ realizing holography in terms of generalized holomorphy (see this) implying minimal surface property failing at singularities, which in turn give rise to particle vertices (see this and this).
2. At the quantum level the situation is not so easy. Here the principle should be global and state the identity of gravitational and inertial masses.
   1. Inertial mass is associated with the conserved four momentum labeling the modes of second quantized free spinor fields of H, whose oscillator operators can be used to build all elementary particles, both photons and fermions.
   2. What about gravitational mass? In zero energy ontology one considers causal diamonds CD=cd×CP₂ subset H, where cd is the causal diamond of M⁴. Now it is natural to perform the quantization of spinors inside a CD. Lorentz group with respect to either tip of CD and instead of unitary representations of Poincare group one has unitary irreps of this Lorentz group. Momentum eigenstates are replaced with unitary irreps of SO(1,3) at the Lorentz invariant hyperbolic 3-space H³ and Minkowski time is replaced with light-cone proper time, cosmic time. Mass spectrum is however the same. Hence Equivalence Principle.

The unitary representations of the Lorentz group are something completely new and have direct physical applications.

1. The unitary irreps of Lorentz group and H³ allow symmetry breaking to subgroups just as the representations of translation groups do in condensed matter physics. One obtains an infinite hierarchy of tessellations of H³ as counterparts of condensed matter lattices. There are however only a finite number of regular tessellations.
2. The recent findings of gravitational hum could be understnd in terms of diffraction in a lattice formed by stars (see this).
3. There is a completely unique tessellation allowing tetrahedrons,octahedrons, and icosahedrons as basic components and one can understand genetic code in terms of this tessellation. This suggests that genetic code is universal and biosystems provide only one particular realization of it (see this) .

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Did Moon turn itself inside out?

The group led by Weigang Liang has presented strong evidence that the Moon has turned inside out (see this). The heavy elements, which should be in the core are at the surface. For a popular summary see this. Can TGD explain this mysterious looking finding?

I have proposed that the Moon was formed in an explosion in which Earth lost a spherical layer, which then condensed to form the Moon. These explosions would have occurred on various scales (see this and this). For instance, in the case of the Sun these kinds of explosions could have formed the spherical layers condensing to planets. During the condensation of the spherical layer to the Moon, the gravitational acceleration experienced by the outer parts of the shell was stronger than that experienced by the inner parts. This implied turning inside out. The outer parts containing originally lighter stuff went to the core and the heavier stuff on the inner boundary of the shell remained on the surface.

A more precise calculation shows that the inversion can occur even with the density gradient.

See the article About the recent TGD based view concerning cosmology and astrophysics or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Why the communication of TGD to theoretical physicists is so incredibly difficult?

During these 42 years after my thesis I have often wondered why the communication of TGD to theoretical physicists is so extremely difficult. Communication with people in other fields of science does not seem to be a problem.

A partial answer to the question was provided by a Finnish colleague who pompously stated that not a single colleague would touch anything that I have written, even with a long pole. This might explain the mystery in the case of the community of Finnish colleagues. I of course sent my thesis to Witten and other big names after its publication around 1982 and received no response. Probably they had more important things to do than read my thesis. But is it possible that in the era of the internet my colleagues have never encountered TGD during these 40 years? Even very many lay people know of TGD.

It is difficult to believe that my colleagues would be so stupid as to miss TGD for so long a time. As if my colleagues were afraid to learn what TGD is. If so it would be about egos and losing face. Could they fear that they could not debunk TGD and even that TGD would demonstrate that they have been wrong all these years. It would not be surprising that superstring theorists whose mission ended with a complete catastrophe could suffer from this kind of fear. As a matter of fact, TGD solves the basic problem of quantum theory identified already a hundred years ago. Could the entire clergy of modern theoretical physics suffer the fear of realizing "I have been wrong all these years!"?

What could be the origin of this fear? The answer is short: "Ontology!".

1. The development of quantum theory forced us to ask how to test the predictions of the theory. It turned out that the outcomes of the quantum measurements were not predictable and only the probabilities for the outcomes in the measurement of the sect of selected observables were possible. This was in sharp conflict with the determinism of Schrödinger equation and also with classical determinism. Einstein who had constructed general relativity could not accept this since it would have made his theory pointless. This led to the Einstein-Bohr debate. The classical predictions of general relativity have been repeatedly verified as also the predictions of quantum theory. Both were winners and losers in the battle.
2. Numerous interpretations trying to circumvent the paradox of quantum measurement emerged and Copenhagen interpretation became the text book interpretation. It gave up the notion of ontology altogether. No reality actually exists and quantum mechanics is only a collection of computational recipes to predict the probabilities for the outcomes of quantum measurements. In particular, the notions of quantum states and wave function must be given up.
3. This led to a kind of postmodernism. Inflation theory and superstring models represent the extreme in this sense. In the basic version of the superstring model, the 4-D space-time is replaced by 2-D string world sheets in 10-D target space and 4-D space-time is believed to emerge in a mysterious process known as spontaneous compactification. Heterotic strings are one variant of the theory and for these left- resp. right moving fermions move in 10-D resp. 24-D target space. This is of course complete nonsense unless one takes theory as a mere computational recipe. Landscape catastrophe emerged as an outcome of spontaneous compactification and meant a complete loss of predictivity but even this is not a problem if one gives up ontology algother. The outcome is postmodernism: there is no grand narrative and science reduces to science fiction literature.

In TGD the situation is different. Ontology has played a key role in the construction of the TGD.

Consider first the classical TGD.

1. TGD emerged as a solution to an ontological problem. The notions of energy, momentum and angular momentum are not well-defined in general relativity. Already Emmy Noether realized this but her discovery was put under the rug. My discovery was that the hybrid of general and special relativities obtained by fusing the postulates of general relativity, namely general coordinate invariance and Equivalence principle with the relativity principle of special relativity, one ends up to a theory in which conservation laws are not lost.

   The prediction is that space-time at the fundamental level is not an abstract 4-geometry but corresponds to a 4-D surface in some space M⁴xS. By choosing S to be S=CP₂ one obtains standard model symmetries so that TGD is unique. Einsteinian space-time emerges at the quantum field theory limit at long length scales. The new ontology, I call it zero energy ontology (ZEO) identifying space-times as 4-surfaces,has dramatic implications in all scales, in particular cosmology and astrophysics.

2. There is also a connection with string models. Space-time surfaces can be seen as orbits of 3-surfaces representing particles as a generalization of point-like particles (and also of string). In fact, the conformal invariance of string models extends to a 4-D symmetry implying that space-time surfaces are minimal surfaces with lower-dimensional singularities giving rise to vertices. TGD strongly suggests also other infinite-D symmetries as isometries of WCW. If 3-surfaces are identified as particles, the minimal surface property generalizes wave particle duality massless particles. Minimal surface generalizes light-like geodesic and minimal surface equations generalize massless field equations.
3. Twistorilization involves one more ontological miracle. Twistorialization has shown its power in the construction of supersymmetric quantum field theories and the natural question is whether the twistor lift of TGD is possible using induction as the basic recipe. One would start from the product T(M⁴)xT(CP₂) of twistor spaces of M⁴ and CP₂.

   A good candidate for the twistor space of the space-time surface is as a 6-surface which has the structure of S^2 bundle, where S^2 is sphere. Could one obtain this 6-surfaces as 6-D analog of Bohr orbit for some action. This turns out to be possible and dimensional reduction leads to a 4-D action, which is Kähler action as an analog of Maxwell action plus volume action having an interpretation in terms of cosmological constant which depends on length scale and approach to zero in long length scales. This is possible if the twistor spaces in question allow Kähler structure. This is the case but only for M⁴ and CP₂ so that TGD and physics are unique.

Let us now consider ontology at the level of quantum TGD.

1. The first guess was that scattering amplitudes are defined in terms of a path integral over all 4-surfaces. However, quantum field theories have an ontological problem. The path integral does not exist in a mathematical sense. In TGD this problem is magnified since any general coordinate invariant action is extremely non-linear and there is no hope of the elimination of divergences by renormalization. Therefore the realization of path integral as integral over all 4-surfaces connecting initial and final 3-surfaces makes no sense mathematically.

   The solution is simple: general coordinate invariance is realized by holography assigning to a 3-surface a unique or almost unique 4-surface analogous to Bohr orbit. Path integral disappears. Instead of 3-surfaces, the 4-D Bohr orbits are the basic dynamical objects and quantum TGD reduces to wave mechanics in the space of these Bohr orbits: world of classical worlds (WCW), as I call it.

   WCW must exist mathematically and allow Kähler geometry, otherwise the geometrization of quantum theory is not possible. Again an ontological problem! Dan Freed studied loops spaces and found that their Kähler geometry exists and is unique. There are excellent reasons to expect that the same is true in TGD.

2. Momentum-position duality plays a key role in wave mechanics. It should generalize to TGD. The natural momentum space-counterpart of H is 8-D Minkowski space M⁸ as its cotangent space. M⁸ and H should provide complementary descriptions of physics. I call this M⁸-H duality.

   Here enters a new ontological element to the picture: number theory, which has not has no fundamental role in standard physics. It took a long time to realize that M⁸ can be interpreted as octonions since the real part of octonions squared gives the Minkowskian norm. Dynamics in M⁸ can be formulated as the condition that the normal space of the 4-surface is quaternionic, that is, associative. M⁸-H duality maps these 4-surfaces to H provided the normal spaces contain an integrable distribution of commutative subspaces. The normal space of 4-surface is parametrized by a point of CP₂, which defines the M⁸-H duality. M⁸-H duality has an interpretation as a physical counterpart of Langlands duality.

   Number theoretic vision predicts that the hierarchy of classical number fields, reals, complex numbers, quaternions, and octonions becomes part of the ontology of TGD.

3. Number theoretic view in turn leads naturally to a hierarchy of p-adic number fields and their algebraic extensions as extension of the ontology of the standard physics. A natural assumption is that the space-time surfaces are characterized as roots of two two polynomials which are functions of the 4 generalized complex coordinates of H. If the coefficients are integers, or even integers smaller than the degree of the polynomials, the space-time surfaces exists also in the p-adic sense: ontology is universal in number theoretical sense.

   One obtains hierarchies of extensions of rationals defined by the roots of the polynomials defining evolutionary hierarchies. The dimension for the extension of rationals defines an effective Planck constant and the larger its value, the longer the scale of quantum coherence. The phases of ordinary matter with non-standard value of effective Planck constant behave like dark matter. It turns out that they do not correspond to the galactic dark matter, which corresponds to dark energy in the TGD framework but to missing baryonic matter whose portion has increased during the cosmological evolution. Galactic dark matter is assignable to cosmic strings and monopole flux tubes so that again new ontology solving physical problems is predicted.

   The ramified primes assignable to the polynomials as divisors of its discriminant have physical interpretation as preferred p-adic primes playing a crucial role in TGD, in particular in p-adic mass calculations.

   Number theoretic physics can be regarded as physics of cognition and the common points of the real and p-adic space-time surfaces consisting of algebraic numbers in the intersection of the extensions of polynomials involved define a universal discretization providing a cognitive representation. Therefore cognitive correlates become part of the ontology.

4. The counterparts of Schrödinger amplitudes are identified superpositions of 4-D orbits, that is WCW spinor fields. This requires spinor structure and gamma matrices identifiable as super counterparts of the infinitesimal generators of the WCW isometries identifiable as Noether charges. The gamma matrices are identifiable in terms of oscillator operators of free second quantize spinor fields of H and this makes it possible to calculate correlation functions. Fermion fields of H define the fundamental fields and bosons can be regarded as bound states of fundamental fermions.

   This however creates an ontological problem. Fermion and antifermion numbers are separately conserved. The idea is that fermion pairs are created from classical induced gauge- and gravitational fields which do exist. It turns out that fermion pair creation is possible but only in dimension D=4. This is due to the fact that in the 4-D case there exists an infinite number of exotic smooth structures, which differ from the standard smooth structures by lower-dimensional defects identifiable as these singularities, that is vertices. In general relativity the existence of the exotic smooth structures is lethal. Ontology shows again its marvellous power (see this): realistic quantum theory allowing pair creation is possible only in dimension 4!

It was quantum measurement theory which led to the Copenhagen interpretation and the recent stagnation of theoretical physics. Can TGD solve the problem?

1. WCW spinor fields in the WCW, the space of 4-D Bohr orbits, define the counterparts of wave functions. Quantum jumps occur between these so that the non-determinism of quantum jump is not in conflict with the classical determinism of the Bohr orbit. Therefore the basic problem of quantum measurement theory disappears.

   The basic implication is that there are two times: the geometric time identified as time coordinate of M⁴ or time coordinate of the space-time surface and subjective time presumably identifiable as the sequence of state function reductions. These times are not identical although they must strongly correlate.

2. To understand this in detail, one must generalize the quantum measurement theory. The outcome is actually a quantum theory of consciousness bringing the observer part of the quantum ontology as a conscious entity, self.

   In standard quantum theory one has two kinds of state function reductions (SFRs): the ordinary state function reduction and the sequences of repeated measurement of the same observables producing the same measurement outcome (Zeno effect). In quantum optics the latter measurements are replaced by weak measurements which affect the system slightly. In TGD one has "big" SFRs as counterparts of ordinary SFRs and "small" SFRs associated with the TGD counterpart of the Zeno effect. The dramatic prediction is that the arrow of geometric time changes in BSFRs.

   In small SFRs the arrow of time is preserved but the state of the system changes. The sequence of small SFRs define a conscious entity, self and BSFR means the death of self.

3. A more precise understanding of the relationship between the geometric time and subjective times requires the introduction of further ontology. Perceptive field characterizes a conscious entity. The perceptive field of the conscious entity is identified as causal diamond CD= cdxCP₂, where cd is the causal diamond of M⁴ identified as the intersection of future and past directed light-cones. Causal diamonds form a scale hierarchy. One can apply Poincare transformations, special conformal transformations of M⁴ and scalings on CDs. The finite-D space of CDs obtained in this way (see this) forms the spine of WCW in the sense that WCW decomposes to sub-WCWs consisting of space-time surfaces inside a given CD.

   In zero energy ontology (ZEO) fermionic parts of states are superpositions of products of fermionic Fock states at the opposite boundaries of CD. The size of CD increases in statistical sense in the sequence of SSFRs, which leave the members of the state pairs at the passive boundary of CD invariant (Zeno effect) but change the state at the CD since each SSFR is preceded by an analog of a unitary time evolution.

   The geometric time can be identified as the distance between the tips of the CD and increases in statistical sense since the unitary time evolution in questions corresponds to dispersion in the space of CDs and SSFR means a localization in this space. There is a natural correlation between the subjective time measured using SSFR as a unit and geometric time identified in this way.

4. Still one key idea about ontology must be mentioned. In standard ontology of classical physics one assumes that there is reality behind its mathematical description. The Quantum Platonia of WCW spinor fields as mathematical objects is the reality: there is no need to postulate anything behind them. The quantum jumps of conscious entities hopping around Quantum Platonia give rise to conscious information about this reality and number theoretical evolution forcing the increase of the algebraic complexity creates increasingly complex memory representations about the Quantum Platonia. The Quantum Platonia of TGD differs from that of Tegmark in that it is not a Zombie but teems with life.

To summarize, the basic difference between the standard view of theoretical physics and TGD is that, since TGD solves the quantum measurement paradox, it can accept ontology. Ontology pops up again and again in the development of TGD and has led to a completely unexpected cascade of new deep ideas with strong predictive power since the mere mathematical existence fixes the physics.

What comes to mind is Copernicus who dared to propose that the Universe is full of solar systems like ours and life is everywhere. It had been realized that Earth is not a center of cosmos but the ideas of Copernicus were simply too much and he was burned on the stake. My crime has been the non-Copenhagenian claim that something exists and even worse: there are two kinds of existences: physical and subjective. I was not burnt on stake but learned what it is to be an academic Zombie.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.