Monday, June 10, 2024

Holography=holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M8-H duality

Holography=holomorphy vision generalizes the realization of quantum criticality in terms of conformal invariance. Holography=holomorphy vision provides a general explicit solution to the field equations determining space-time surfaces as minimal surfaces X4⊂ H=M4× CP2. The space-time surfaces are roots of two generalized analytic functions defined in H.

By holography, the space-time surfaces correspond to not completely deterministic orbits of particles as 3-surfaces and are thus analogous to Bohr orbits. This implies zero energy ontology (ZEO) and to the view of quantum TGD as wave mechanics in the space of these Bohr orbits located inside a causal diamond (CD), which form a causal hierarchy. Also the consruction of vertices for particle reactions has evolved dramatically during the last year and one can assign the vertices to partonic 2-surfaces.

M8-H duality is a second key principle of TGD. M8-H duality can be seen a number theoretic analog for momentum-position duality and brings in mind Langlands duality. M8 can be identified as octonions when the number-theoretic Minkowski norm is defined as Re(o2). The quaternionic normal space N(y) of y∈ Y4⊂M8 having a 2-D commutative complex sub-space is mapped to a point of CP2. Y4 has Euclidian signature with respect to Re(o2). The points y\∈ Y4 are lifted by a multiplication with a co-quaternionic unit to points of the quaternionic normal space N(y) and mapped to M4⊂ H inversion.

This article discusses the relationship of the holography=holomorphy vision with the number theoretic vision predicting a hierarchy heff=nh0 of effective Planck constants such that n corresponds to the dimension of extension rationals. How could this hierarchy follow from the recent view of M8-H duality. The proposed realization relies on the idea that quantum criticality implies that the two polynomials P1,P2 defining space-time surfaces as their roots have rational coefficients at the partonic 2-surfaces X2 appearing as generalized vertices. Partonic 2-surfaces would represent a stronger form of quantum criticality than generalized holomorphy so that the islands of algebraic extensions from the ocean of complex numbers would be selected. Adelic physics would emerge at quantum criticality and scattering amplitudes would become number-theoretically universal. In particular, the hierarchy of Planck constants and the identificaiton of p-adic primes as ramified primes would emerge as a prediction.

Also a generalization of the theory of analytic functions to the 4-D situation is suggestive. The poles of cuts of analytic functions would correspond to the 2-D partonic surfaces as vertices at which holomorphy fails and 2-D string worlds sheets could correspond to the cuts. This provides a general view of the breaking of the generalized conformal symmetries and their super counterparts as a necessary condition for the non-triviality of the scattering amplitudes.

See the article Holography=holomorphy vision in relation to quantum criticality, hierarchy of Planck constants, and M8-H duality 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.

Tuesday, June 04, 2024

About the origin of multicellularity in the TGD Universe

A living organism consists of cells that are almost identical and contain DNA that is the same for all of them but expresses itself in different ways. This genetic holography is a fundamental property of living organisms. Where does it originate?

Dark DNA associated with magnetic flux tubes is one or the basic predictions of the TGD inspired biology. One can say that the magnetic body controls the ordinary biomatter and dictates its development. Could one have a structure that would consist of a huge number of almost identical copies of dark DNA forming a quantum coherent unit inducing the coherence of ordinary biomatter? Could this structure induce the self-organization of the ordinary DNA and the cell containing it.

Could one understand this by using the TGD based spacetime concept. There are two cases to be considered. The general option is that fi are analytic functions of 3 complex coordinates and 1 hypercomplex (light-like) coordinate of H and (f1,f2)=(0,0) defines the space-time surface.

A simpler option is that fi are polynomials Pi with rational or even algebraic coefficients. Evolution as an increase of number theoretic complexity (see this) suggest that polynomials with rational coefficients emerged first in the evolution.

  1. For the general option (f1,f2), the extension of rationals could emerge as follows. Assume 2-D singularity X2i at a particular light-like partonic orbit (mi such orbits for fi) defining a X2i as a root of fi. If f2 (f1 ) is restricted to X21 resp. X22 is a polynomial P2i with algebraic coefficients, it has m2 resp. m1 discrete roots, which are in an algebraic extension of rationals with dimension m2 resp. m1. Note that m2 can depend on X2i. Only a single extension appears for a given root and can depend on it. The identification of heff=nih0 looks natural and would mean that heff is a local property characterizing a particular interaction vertex. Note that it is possible that the coefficients of the resulting polynomial are algebraic numbers.

    For the polynomial option (f1,f2)=(P1,P2), the argument is essentially the same except that now the number of roots of P1 resp. P2 does not depend on X22 resp. X21. The dimension n1 resp. n2 of the extension however depends on X22 resp. X21 since the coefficients of P1 resp. P2 depend on it.

  2. The proposal of the number theoretic vision of TGD is that the effective Planck constant is given by heff=nh0, h0<h is the minimal value of heff and n corresponds to the dimension of the algebraic extension of rationals. As noticed, n would depend on the roots considered and in principle m=m1m2 values are possible. This identification looks natural since the field of rationals is replaced with its extension and n defines an algebraic dimension of the extension. n=m1m2 can be also considered. For the general option, the degree of the polynomial P1 can depend on a particular root X22 of f2 .
  3. The dimension nE of the extension depends on the polynomial and typically seems to increase with an exponential rate with the degree of the polynomials. If the Galois group is the permutation group Sm it has m! elements. If it is a cyclic group Zm, it has m elements.
For the original view of M8-H duality, single polynomial P of complex variable with rational coefficients determined the boundary data of associative holography (see this, (see this, and this). The iteration of P was proposed as an evolutionary process leading to chaos (see this) and led to an exponential increase of the degree of the iterated polynomial as powers mk of the degree m of P and to a similar increases of the dimension of its algebraic extension.

This might generalize to the recent situation (see this) if the iteration of polynomials P1 resp. P2 at the partonic 2-surface X22 resp. X21 defining holographic data makes sense and therefore induces a similar evolutionary process by holography. This could give rise to a transition to chaos at X2i making itself manifest as the exponential increase in the number of roots and degree of extension of rationals and heff. One can consider the situation also from a more restricted point of view provided by the structure of H.

  1. The space-time surface in H=M4× CP2 can be many-sheeted in the sense that CP2 coordinates are m1-valued functions of M4 coordinates. Already this means deviation from the standard quantum field theories. This generates a m1-sheeted quantum coherent structure not encountered in QFTs. Anyons could be the basic example in condensed matter physics (see this). m1 is not very large in this case since CP2 has extremely small size (about 104 Planck lengths) and one would expect that the number of sheets cannot be too large.
  2. M4 and CP2 can change the roles: M4 coordinates define the fields and CP2 takes the role of the space-time. M4 coordinates could be m2 valued functions of CP2 coordinates: this would give a quantum coherent system acting as a unit consisting of a very large number m2 of almost identical copies at different positions in M4. The reason is that there is a lot of room in M4. These regions could correspond to monopole flux tubes forming a bundle and also to almost identical basic units.

    If mi corresponds to the degree of a polynomial, quite high degrees are required. The iteration of polynomials would mean an exponential increase in powers dk of the degree d of the iterated polynomial P and a transition to chaos. For a polynomial of degree d=2 one would obtain a hierarchy m=2k.

  3. Lattice like systems would be a basic candidate for this kind of system with repeating units. The lattice could be also realized at the level of the field body (magnetic body) as a hyperbolic tessellation. The fundamental realization of the genetic code would rely on a completely unique hyperbolic tessellation known as icosa tetrahedral tessellation involving tetrahedron, octahedron and icosahedron as the basic units (see this and this). This tessellation could define a universal genetic code extending far beyond the chemical life and having several realizations also in ordinary biology.
  4. The number of neurons in the brain is estimated to be about 86 billions: 1012≈ 240. If cell replications correspond to an iteration of a polynomial of degree 2, morphogenesis involves 40 replications. Human fetal cells replicate 50-70 times. Could the m almost copies of the basic system define a region of M4 corresponding to genes and cells? Could our body and brain be this kind of quantum coherent system with a very large number of almost copies of the same basic system. The basic units would be analogs of monads of Leibniz and form a polymonad. They could quantum entangle and interact.
  5. If n=heff/h0 corresponds to the dimension nE of the extension, it could be of the order 1014 or even larger for the gravitational magnetic body (MB). The MB could be associated with the Earth or even of the Sun: the characteristic Compton length would be about .5 cm for the Earth and half of the Earth radius for the Sun).
Could this give a recipe for building geometric and topological models for living organisms? Take sufficiently high degree polynomials f1 and f2 and find the corresponding 4-surface from the condition that they vanish. Holography=holomorphy vision would also give a model for the classical time evolution of this system as classical, and not completely deterministic realization of behaviors and functions. Also a quantum variant of computationalistic view emerges.

See the article TGD view about water memory and the notion of morphogenetic field 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.

See the article TGD view about water memory and the notion of morphogenetic field 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.

About the generation of matter-antimatter asymmetry in the TGD Universe?

I have developed a rather detailed view of interaction vertices (see this). Everything boils down to the question of what the creation of a fermion-antifermion pair is in TGD. Since bosonic fields are not primary fields (bosons are bound states of fermions and antifermions), the usual view about generation of fermion antifermion pairs does not work as such and the naive conclusion seems to be that fermions and antifermions are separately conserved.

Holography=holomorphy identification leading to an explicit general solution of field equations defining space-time surfaces as minimal surfaces with 2-D singularities at which the minimal surface property fails, is the starting point. A generalized holomorphism, which maps H to itself, is characterized by a generalized analyticity, in particular by a hyper-complex analyticity. The analytic function from H to H in the generalized sense depends on the light-like coordinate or its dual ( say -t+z and t+z in the simplest case) and the 3 remaining complex coordinates of H=M4/ti,esCP2.

Let's take two such functions, f1 and f2, and set them to zero. We get a 4-D space-time surface that is a holomorphic minimal surface with 2-D singularities at which the minimal surface property and holomorphy fails. Singularities are analogs of poles. Also the analogs of cuts can be considered and would look like string world sheets: they would be analogous to a positive real axis along which complex function z^(i/n) has discontinuity unless one replaces the complex plane with its n-fold covering. The singularities correspond to vertices. and the fundamental vertex corresponds to a creation of fermion-antifermion pair.

There are at least two types of holomorphy in the hypercomplete sense, corresponding to analyticity with respect to -t+z or t+z as a light-like coordinate defining the analogs of complex coordinates z and its conjugate. Also CP2 complex coordinates could be conjugated.

These two kinds of analyticities would naturally correspond to fermions and to antifermions identified as time-reflected (CP reflected) fermions. This time reflection transforms fermion to antifermion. This is not the reversal of the arrow of time occurring in a "big" state function reduction (BSFR) as TGD counterpart of what occurs in quantum measurement, which corresponds to interchange of the roles of the fermionic creation and annihilation operators.

When a fermion pair, which can also form a boson as a bound state, is created, the partonic 2-surface to which the fermion line is assigned, turns back in time. At the vertex, where this occurs, neither of these two analyticities applies: holomorphy and the minimal surface property are violated because at the vertex the type of analyticity changes.

Now comes the crucial observation: the number theoretic vision of TGD predicts that quantum coherence is possible in macroscopic and even astrophysical and cosmological scales and corresponds to the existence of arbitrarily large connected space-time regions acting as quantum coherence regions: field bodies as counterparts of Maxwellian fields can indeed be arbitrarily large.

For a given region of this kind one must choose the same kind of generalized analyticity, say -t+z or t+z even at very long scales. Only fermions or antifermions but not both are possible for this kind of space-time sheets! Does this solve the mystery of matter-antimatter asymmetry and does its presence demonstrate that quantum coherence is possible even in cosmological scales?

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.

Monday, June 03, 2024

Comparing TGD- and QFT based descriptions of particle interactions

Marko Manninen made interesting questions related to the relationship between TGD and quantum field theories (QFTs). In the following, I will try to summarize an overview of this relationship in the recent view about quantum TGD. I have developed the latest view of quantum TGD in various articles (see this, this, this, this, this, and this).

Differences between QFT and quantum TGD

Several key ideas related to quantum TGD distinguish between TGD and QFTs.

  1. The basic problem of QFT is that it involves only an algebraic description of particles. An explicit geometric and topological description is missing but is implicitly present since the algebraic structure of QFTs expresses the point-like character of the particles via commutation and anticommutation relations for the quantum fields assigned to the particles.

    In the string models, the point-like particle is replaced by a string, and in the string field theory, the quantum field Ψ(x) is replaced by the stringy quantum field Ψ(string), where the "string" corresponds a point in the infinite-D space of string configurations.

    In TGD, the quantum field Ψ(x) is replaced by a formally classical spinor field Ψ (Bohr orbit). The 4-D Bohr orbits are preferred extremals of classical action satisfying holography forced by general coordinate invariance without path integral and represent points of the "world of classical worlds" (WCW). The components of Ψ correspond to multi-fermion states, which are pairs of ordinary 3-D many-fermion states at the boundaries of causal diamond (CD).

    The gamma matrices of the WCW spinor structure are linear combinations of fermionic oscillator operators for the second quantized free spinor field of H. They anticommute to the WCW metric, which is uniquely determined by the maximal isometries for WCW guaranteeing the existence of the spinor connection. Physics is unique from its existence, as implied also by the twistor lift and number theoretic vision and of course, by the standard model symmetries and fields.

  2. In TGD, the notion of a classical particle as a 3-surface moving along 4-D "Bohr orbit" as the counterpart of world-line and string world sheet is an exact aspect of quantum theory at the fundamental level. The notions of classical 3-space and particle are unified. This is not the case in QFT and the notion of a Bohr orbit does not exist in QFTs. TGD view of course conforms with the empirical reality: particle physics is much more than measuring of the correlation functions for quantum fields.

    Quantum TGD is a generalization of wave mechanics defined in the space of Bohr orbits. The Bohr orbit corresponds to holography realized as a generalized holomorphy generalizing 2-D complex structure to its 4-D counterpart, which I call Hamilton-Jacobi structures (see this). Classical physics becomes an exact part of quantum physics in the sense that Bohr orbits are solutions of classical field equations as analogs of complex 4-surfaces in complex M4×CP2 defined as roots of two generalized complex functions. The space of these 4-D Bohr orbits gives the WCW (see this), which corresponds to the configuration space of an electron in ordinary wave mechanics.

  3. The spinor fields of H are needed to define the spinor structure in WCW. The spinor fields of H are the free spinor fields in H coupling to its spinor connection of H. The Dirac equation can be solved exactly and second quantization is trivial.

    This determines the fermionic propagators in H and induces them at the space-time surfaces. The propagation of fermions is thus trivialized. All that remains is to identify the vertices. But there is also a problem: how to avoid the separate conservation of fermion and antifermion numbers. This will be discussed below.

  4. At the fermion level, all elementary particles, including bosons, can be said to be made up of fermions and antifermions, which at the basic level correspond to light-like world lines on 3-D parton trajectories, which are the light-like 3-D interfaces of Minkowski spacetime sheets and the wormhole contacts connecting them.

    The light-like world lines of fermions are boundaries of 2-D string world sheets and they connect the 3-D light-like partonic orbits bounding different 4-D wormhole contacts to each other. The 2-D surfaces are analogues of the strings of the string models.

  5. In TGD, classical boson fields are induced fields and no attempt is made to quantize them. Bosons as elementary particles are bound states of fermions and antifermions. This is extraordinarily elegant since the expressions of the induced gauge fields in terms of embedding space coordinates and their gradients are extremely non-linear as also the action principle. This makes standard quantization of classical boson fields using path integral or operator formalism a hopeless task.

    There is however a problem: how to describe the creation of a pair of fermions and, in a special case, the corresponding bosons, when there are no primary boson fields? Can one avoid the separate conservation of the fermion and the antifermion numbers?

Description of interactions in TGD

Many-particle interactions have two aspects: the classical geometric description, which QFTs do not allow, and the description in terms of fermions (bosons do not appear as primary quantum fields in TGD).

  1. At the classical level, particle reactions correspond to topological reactions, where the 3-surface breaks, for example, into two. This is exactly what we see in particle experiments quite concretely. For instance, a closed monopole flux tube representing an elementary particle decomposes to two in a 3-particle vertex.

    There is field-particle duality realized geometrically. The minimal surface as a holomorphic solution of the field equations defines the generalization of the light-like world line of a massless particle as a Bohr orbit as a 4-surface. The equations of the minimal surface in turn state the vanishing of the generalized acceleration of a 3-D particle identified as 3-surface.

    At the field level, minimal surfaces satisfy the analogs of the field equations of a massless free field. They are valid everywhere except at 2-D singularities associated with 3-D light-like parton trajectories. At singularities the minimal surface equation fails since the generalized acceleration becomes infinite rather than vanishing. The analog of the Brownian particle experiences acceleration: there is an "edge" on the track.

    At singularities, the field equations of the whole action are valid, but are not separately true for various parts of the action. Generalized holomorphy breaks down. These 2-D singularities are completely analogous to the poles of an analytic function in 2-D case and there is analogy with the 2-D electrostatics, where the poles of analytic function correspond to point charges and cuts to line charges.

    This gives the TGD counterparts of Einstein's equations, analogs of geodesic equations, and also the analogy Newton's F=ma. Everything interesting is localized at 2-D singularities defining the vertices. The generalized 8-D acceleration Hk defined by the trace of the second fundamental form, is localized on these 2-D parton surfaces, vertices. One has a generalization of Brownian motion for a particle-like object defined by a partonic 2-surface or equivalently for a particle as 3-surface. Intriguingly, Brownian motion has been known for a century and Einstein wrote his first paper after his thesis about Brownian motion!

    Singularities correspond to sources of fermion fields and are associated with various conserved fermion currents: just like in QFTs. For a given spacetime surface, the source- vertex - is a discrete set of 2-D partonic surface just as charges correspond to poles of analytic function in 2-D electrostatics.

    At the classical level, the 2-D singularities of the minimal surfaces therefore correspond to vertices and are localized to the light-like paths of parton surfaces where the generalized holomorphy breaks down and the generalized acceleration Hk is there non-vanishing and infinite.

Description of the interaction vertices

  1. How to get the TGD counterparts of the QFT vertices?

    Vertices typically contain a fermion and an antifermion and the gauge potential, which is second quantized. Now, classical gauge potentials are not second quantized. How to obtain the basic gauge theory vertices?

    This is where the standard approximation of QFTs helps intuition: replace the quantized boson field with a classical one. This gives the vertex corresponding to the creation of a pair of fermions. Thanks to that, only the fermion and the sum of the antifermion numbers are conserved and the theory does not reduce to a free field theory. One should be able to do the same now. However, the precise formulation of this vision is far from trivial.

  2. The modified Dirac action should give elementary particle vertices for a given Bohr trajectory.

    There are two options:

    1. Modified gammas are defined as contractions of ordinary gamma matrices of H with the canonical momentum currents associated with the classical action defining the space-time surface. Supersymmetry is now exact: besides color and Poincare super generators there is an infinite number of conserved super symplectic generators and infinitesimal generalized superholomorphisms.

      This option does not work: the modified Dirac equation implies that the Dirac action and also vertices vanish identically. Although one has partonic 2-surfaces as singularities of minimal surfaces defining vertices, the theory is trivial because the usual perturbation theory does not work.

    2. Modified gamma matrices are replaced by the induced gamma matrices defined by the volume term (cosmological term of the classical action). Supersymmetry is broken but only at the 2-D vertices. The anticommutator of the induced gammas gives the induced metric. This is not true for the modified gammas defined by the entire action: in this case the anticommutators are rather complex, being bilinear in the canonical momentum currents. Is it possible to have a non-trivial theory despite the breaking of supersymmetry at vertices or or does the supersymmetry breaking make possible a non-trivial theory? This seems to be the case.
      1. In 2-D vertices, the generalized acceleration field Hk is proportional to the 2-D delta function and gives rise to the graviton and Higgs vertices. One obtains also the vertices related to gauge bosons from the coupling of the induced spinor field to induced spinor connection. Only the couplings to electroweak gauge potentials and U(1) K&aum;hler gauge potential of M4 are obtained. The failure of the generalized holomorphy is absolutely essential.
      2. Color degrees of freedom are completely analogous to translational degrees of freedom since color quantum numbers are not spin-like in TGD. Strong interactions are vectorial and correspond to Kähler gauge potentials.
      3. Generalized Brownian motion gives the vertices. One obtains the equivalents of Einstein's and Newton's equations at the vertices. The M4 part Mk of the generalized acceleration is related to the gravitons and the CP2 part Sk to the Higgs field. Spin J=2 for graviton is due to the rotational motion of the closed monopole flux tube associated with the gravitation giving an additional unit of spin besides the spin of Hk, which is S=1.
    3. Consider now the description of fermion pair creation.
      1. Intuitively, the creation of a fermion pair (and thus also a boson) corresponds to the fermion turning backwards in time. At the level of the geometry of the space-time surface, this corresponds to the partonic 2-surface turning backwards in time, and the same happens to the corresponding fermion line. Turning back in time means that effectively the fermion current is not conserved: if one does not take into account that the parton surface turns in the other direction of time, the fermion disappears effectively and the current must has a singular divergence. This is what the divergence of the generalized acceleration means.
      2. This implies that the separate conservation is lost for fermion and antifermion numbers. This means breaking of supersymmetry, of masslessness, of generalized holomorphy and also the generation of the analog of Higgs vacuum excitation as CP2 part Sk of the generalized acceleration Hk. The Higgs vacuum expectation is only at the vertices. But this is exactly what is actually wanted! No separate symmetry breaking mechanism is needed!
      3. The failure of the generalized holomorphy at the 2-D vertex means that the holomorphic partonic orbit turns at the singularity to an antiholomorphic one. For the annihilation vertex it could occur only for the hypercomplex part of the generalized complex structure.
      4. Remarkably, the states associated with connected 4-surfaces consist of either fermions or antifermions but not both. This explains matter antimatter asymmetry if quantum coherence is possible in arbitrarily long scales. In TGD, space-time surfaces decompose to regions containing either matter or antimatter and, by the presence of quantum coherence even in cosmological scales, these regions can be very large. The quantum coherence in large scales is implied by the number theoretic vision predicting a hierarchy of Planck constants labelling phases of ordinary matter behaving like dark matter (see for instance this).
      5. What is the precise mathematical formulation of this vision? This is where a completely unique feature of 4-dimensional manifolds comes in: they allow exotic smooth structures. Exotic smooth structure is the standard smooth structure with lower-dimensional defects. In TGD, the defects correspond in TGD to 2-D parton vertices as "edges" of Brownian motion. In the exotic smooth structure, the edge disappears and everything is soft. Pair creation and non-trivial theory is possible only in dimension D=4 (see this and 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.

Friday, May 31, 2024

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.

Thursday, May 30, 2024

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 heff>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 heff. heff 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 heff>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.

Wednesday, May 29, 2024

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.

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.

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 Bend=2BE/5, where BE=.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 ℏgr/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 ℏeff/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× 1011-3x, x=-1,0,2,... Could there exist a hierarchy of biological scales differing by powers of 103? Could these scaling factors correspond to various values of heff/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 ℏgr= GMEm/β0, β0=v0/c≈ 1, where ME 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 RE≈ .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 ℏgr 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 ℏem= Qe20, 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.

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.