Thursday, May 23, 2024

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, Hk, 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 guv=0 at X3 outside X2. The condition   Juv= 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 Juv=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 DuΨ=0 and DvΨ=0 in X3 except at the 2-D singularities X2, where  the induced  metric would have diagonal components guu and gvv. This would give rise to the gauge boson vertices involving emission of fermion-antifermion pairs.

  5.  By the generalized holomorphy, the second fundamental form Hk  vanishes   outside X2. At X2,  Hk   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 Hk 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.

Wednesday, May 22, 2024

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 2BE/5=.2 Gauss would realize 12-note spectrum for the bioharmony. The spectrum of Josephson energies assignable to cell membrane is independent of heff (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.

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).

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.

Monday, May 20, 2024

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, Hk, 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.

Sunday, May 19, 2024

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 heff=nh0 serving as measure of algebraic complexity and giving rise to a hierarchy of increasing quantum coherence scales. Ordinary particles with heff>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 (heff 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.

Saturday, May 18, 2024

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.

Thursday, May 09, 2024

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 Bend= 2BE/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.

Tuesday, May 07, 2024

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.