4-D generalization of holomorphies and Kac-Moody symmetries as isometries of the "world of classical worlds"?

Quite recently, I learned that the generalized holomorphies of space-time surfaces define non-trivial conserved charges. Also a generalization of Super-Kac-Moody charges associated with certain embedding space isometries emerges. This suggests a very close connection with string models and provides a possibility to provide answers to the longstanding questions relating to the identification of the isometry group of the "world of classical worlds" (WCW). Generalized holomorphy not only solves explicitly the equations of motion but, as found quite recently, also gives corresponding conserved Noether currents and charges.

1. Generalized holomorphy algebra generalizes the Super-Virasoro algebra and the Super-Kac-Moody algebra related to the conformal invariance of the string model. The corresponding Noether charges  are conserved. Modified Dirac action allows to construct the supercharges having interpretation as WCW gamma matrices. This suggests an answer to a longstanding question related to the isometries of the "world of the classical worlds" (WCW).
2. Either the generalized holomorphies or the symplectic symmetries of H=M⁴× CP₂ or both together define WCW isometries and corresponding super algebra. It would seem that symplectic symmetries induced from H are not necessarily needed and might actually correspond to symplectic symmetries of WCW. This would give a close similarity with the string model, except that one has half-algebra for which conformal weights are proportional to non-negative integers and gauge conditions only apply to an isomorphic subalgebra. These are labeled by positive integers and one obtains a hierarchy.
3. By their light-likeness, the light cone boundary and orbits of partonic 2-surfaces allow an infinite-dimensional isometry group. This is possible only in dimension four. Its transformations are generalized conformal transformations of 2-sphere (partonic 2-surface) depending on light-like radial coordinate such that the radial scaling compensates for the usual conformal scaling of the metric. The WCW isometries would thus correspond to the isometries of the parton orbit and of the boundary of the light cone! These two representations could provide alternative representations for the charges if the strong form of holography holds true and would realize a strong form of holography. Perhaps these realizations deserve to be called inertial and gravitational charges.

   Can these transformations leave the action invariant? For the light-cone boundary, this looks obvious if the light-cone is sliced by a surface parallel to the light-cone boundary. Note however that the tip of this surface might produce problems. A slicing defined by the Hamilton-Jacobi structure would be naturally associated with partonic orbits.

4. What about Poincare symmetries? They would act on the center of mass coordinates of causal diamonds (CDs) as found already earlier (see this). CDs form the "spine" of WCW, which can be regarded as fiber space with fiber for a given CD containing as a fiber the space-time surfaces inside it.

The super-symmetric counterparts of holomorphic charges for the modified Dirac action and bilinear in fermionic oscillator operators associated with the second quantization of free spinor fields in H, define gamma matrices of WCW. Their anticommutators define the Kähler metric of WCW. There is no need to calculate either the action defining the classical Kähler action defining the Kähler function or its derivatives with respect to WCW complex coordinates and their conjugates. What is important is that this makes it possible to speak about WCW metric also for number theoretical discretization of WCW with space-time surfaces replaced with their number theoretic discretizations.

See the article About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe or the chapter About Platonization of Nuclear String Model and of Model of Atoms or .

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 proposal of Phil Gibbs that energy is conserved in General Relativity

I have been analyzing the basic visions of TGD trying to identify weak points. Symmetries are central in TGD. The basic motivation for TGD was the loss of Poincare invariance in GRT and in the following I will analyze the claim of Phil Gibbs that one obtains energy conservation in GRT.

The following considerations are inspired by discussions with Marko Manninen and related to whether in general relativity it could be possible to define conserved quantities associated with at least some general coordinate transformations as proposed by Phil Gibbs (see this). This is certainly in conflict with the general idea that the choice of coordinates cannot have any physical effect and personally I am skeptic. I however decided to analyze the proposal in detail and found that it relates to a possible generalization of the notion of Newtonian gravitational flux, which gives the gravitational mass of the system.

1. Gibbs' proposal for Noether charges associated with general coordinate transformations says nothing detailed about charges and the straightforward application of the basic formula gives vanishing charges since these currents turn out to be proportional to T-G which vanishes by Einstein's equations. However, the action includes a term containing second derivatives of the metric. Could this give an anomalous contribution to the Noether charge?
2. In electrodynamics and gauge theories, charges are obtained in connection with gauge transformations that become constant at a distance. The gauge charge density is a total divergence and gauge charge can be expressed as an electric flux across a very large sphere. On the other hand, in Newton's theory, the gravitational flux far enough from the system gives its mass. Could mass correspond to a time translation as a symmetry? Could the transformation of the charge into total divergence generalize to other general coordinate transformations?
3. Einstein action (curvature scalar) contains terms proportional to the second order partial derivatives of the metric: these terms come from the part of the curvature scalar linear in Riemann connection, which serves as analogs of non-abelian gauge potentials. However, this does not give third derivatives to the equations of motion. The reason is that the second derivatives occur linearly. If the square of the curvature tensor would define the action as an analog of Yang-Mills action, the situation would be different. This term is analogous to a dissipative and might relate to the general features of GRT dynamics (blackholes as asymptotic states).

   Is the divergence term taken into account automatically in the straightforward Noetherian guess for the conserved currents? Or could the charges associated with the general coordinate transformations emerge as analogs of electric charge as a flux integral over a very large sphere. This would certainly contradict the fact that general coordinate transformations do nothing to the system, so that they cannot relate physically non-equivalent configurations.

4. The deduction of field equations involves transformation of the terms containing derivatives for the variation of the metric so that only terms involving only the variation of the metric remains besides total divergences, which must vanish for symmetries leaving the action invariant. This gives an explicit formula for the conserved Noether currents. In the case of the curvature scalar, the first term in the conserved current comes from the variation of the first derivatives of the metric. The second term comes from the variation of the second derivatives of the metric tensor and an explicit expression can be deduced for it. This gives a total divergence. Is this term automatically included in the term proportional to T-G? This seems very likely.

Just for curiosity, let us consider the possibility that the total divergence term is not included in T-G and must be included as an additional term.

1. As a total divergence this term can be transformed into a surface integral and is proportional to the vector field generating the transformation. This term could give a non-vanishing contribution as an integral over the boundary at infinity which can be regarded as an infinitely large sphere. If the space-time is asymptotically Minkowskian, the counterparts of 4-momentum, angular momentum and also charges associated with Poincare transformation are obtained. Also the charges associated with arbitrary general coordinate transformations are obtained but these are not in general conserved.
2. The explicit form for the conserved current associated with infinitesimal general coordinate transformation generated by the vector field j^μ is

   J^μ(j)= L^αβμ Dg_αβ + L^αβμν ∂_νDg_αβ

   =L^αβμDg_αβ +∂_νL^αβμ Dg_αβ - ∂_ν[L^αβμDg_αβ],

   where one has

   L^αβμ= ∂ L/∂(∂_μ g_αβ),
   L^αβμν= ∂ L/∂(∂_μνg_αβ),
   Dg_αβ=j^ρ∂_ρ g_αβ .

   The third term at the second line is a total divergence and this contribution, call it Q₃(j) to the expression for the charge as a 3-D integral of the μ=t component of the current can be transformed to a surface integral.

   Q₃(j)= -∫_S²[∂ L/∂(∂_tr ∂_ρ g_αβ)] ∂_ρ g_αβj^ρ]dS .

3. In the stationary case, the g_tt component of the metric includes the gravitational potential and its radial derivative gives a 1/r² term whose flux over the spherical surface is non-vanishing and gives the same result as gravitational flux in Newton's theory. Therefore there is a 1/r² term in the curvature tensor, which is analogous to the electric field. This interpretation requires that the space is asymptotically Minkowski space, so it is possible to talk about Poincare symmetry as an asymptotic symmetry. Constant time shift corresponds to mass.
4. The flux contribution to the charge must be linear in Christoffel symbols and involve the indices t and r. A good guess is that the charge is proportional to

   Q₃(j)= ∫_S² C(t,rρ) j^ρdS .

   where C(t,rρ) denotes Christoffel symbol. For Schwarzschild metric this gives Newtonian gravitational flux for time translation j^ρ=δ^ρ,t.

5. One must pose additional conditions guaranteeing that these charges do not flow radially out of the infinite sphere. This becomes a condition that the second derivatives of the metric with respect to the radial coordinate r approach zero faster than 1/r². This holds true very generally. Note however that the flux associated with arbitrary j need not be conserved. Consider as an example generalized coordinate transformations which approach trivial transformations in the future and non-trivial transformations in the past.
6. Year or two ago there was a lot of talk about an infinite number of charges that can be connected in this way as asymptotic charges to conformal transformations of an infinitely large sphere. These charges could be a special case of pseudo charges described above.

To conclude, there are two options. Personally I am convinced that the T-G option is the right one. For the T-G=0 option, the total charges related to general coordinate transformations are therefore zero. One could however say that the total Noether charges are always zero but that they can be divided into interior and flux parts according to holography and cancel out each other. Flux part would correspond to what is called gravitational charge. In this sense the charges related to general coordinate transformations or at least Poincare transformations can be assigned with the system via holography as flux integrals. This could perhaps be interpreted within the framework of holography. These fluxes would characterize the asymptotic behavior giving in turn information about the dynamics in the interior.

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

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

The age of the Universe is twice the usual estimate for the age suggests James Webb: what does this mean?

James Webb telescope has reported stars and galaxies older than the Universe. This finding is new but cannot be put under the rug anymore. It has been proposed (see this that the age of the Universe is about 26.7 billion years and rather precisely twice the standard age about 13.2 billion years.

In TGD the time arrow changes in ordinary "big" state function reductions (BSFRs) which can take place in arbitrarily long time scales. This means that the system lives forth and back in time. One must distinguish between ordinary age and developmental age.

Remarkably, the total evolutionary time spent per given ordinary time interval is roughly twice(!) this time interval! This view explains stars and galaxies older than the Universe and might also explain why the researchers have concluded that the age of the universe is twice the standard age.

The most dramatic implications relate to living systems. BSFR means death and reincarnation with a reversed arrow of time. Similar doublings might occur in biology: for instance, the developmental age of the genome could be twice the age deduced from, say fossiles.

See the article TGD view of the paradoxical findings of the James Webb telescope.

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.

Questions related to the generalized holomorphies and fundamental vertices according to TGD

We had very inspiring discussions with Marko Manninen at a birthday meal with wine. During the trip back home some questions and ideas emerged. Could the 4-D generalization of holomorphy realizing holography allow an infinite hierarchy of conserved charges generalizing the Super Virasoro algebra. Could the only particle vertex in TGD correspond to a creation of fermion-antifermion pair: in this 2-vertex fermion state and fermionic line, partonic orbit, or Bohr orbit turns back in time.? Can one identify the graviton emission vertex?

4-D generalization of the holomorphy allows conserved charges associated with the generalized holomorphies

Does the 4-D analogy of holomorphy as a realization of holography give rise to conserved quantities? Now the symmetries would not be isometries, nor some other symmetries of the action, but dynamic symmetries satisfied only by the Bohr orbits. A little calculation that one can do in your head shows that one obtains conserved currents: the reason is the same as in the case of field equations. The divergence of the Noether current is a contraction of tensors with no common index pairs for the generalization of complex coordinates.

Unlike those associated with the general coordinate invariance, these conserved quantities do not vanish. They correspond to the 4-D generalization of conformal transformations and give rise to a generalization of the Virasoro algebra and also of Super Virasoro algebra realized in terms of the modified Dirac action for the induced spinor fields obtained from the free second quantized spinor fields of H.

In the string model, these conformal charges are assumed to annihilate the physical states. In TGD, I have proposed that only a subalgebra that is isomorphic to the whole algebra, having conformal weights which are integer multiples of the entire algebra, does this. In TGD framework, the conformal weights are necessarily non-negative and ZEO allows this. One obtains a whole hierarchy of subalgebras and a sub-hierarchy of algebras for which conformal symmetry as gauge symmetry is "broken" to dynamical Lie symmetries for physical states having conformal weight below some maximum value. These hierarchies could correspond to the hierarchies of algebraic extensions for rationals defined by composite polynomials.

Are fermionic 2-vertices all that is needed in TGD?

In quantum field theories, already the interaction vertex for 3 particles leads to divergences. In a typical 3-vertex, fermion emits a boson or boson decays to a fermion-antifermion pair. In TGD, the situation changes.

1. Fermions are the only fundamental particles in TGD. Since fundamental bosons are missing, there is no vertex representing emission of a fundamental boson emission from fermion or a vertex producing fermion antifermion pair from a fundamental boson. In TGD, bosons as elementary particles (distinguished from fundamental bosons) are fermion-antifermion pairs, and the emission of elementary bosons is possible. However, the problem is that the total fermion and antifermion numbers are separately conserved. Unless it is possible to create fermion pairs from classical fields!
2. In the standard theory fermion-antifermion pairs can be indeed created in classical gauge fields. This creation is an experimental fact but it is thought that this description is only a convenient approximation. In TGD however, the classical fields associated with the Bohr orbits of 3-surfaces are an exact part of quantum theory. Could this description be accurate in TGD? In the classical induced fields associated with particles, pairs could arise. Approximation would become exact in TGD.

A 2-vertex for creation of fermion-antifermion pair (or corresponding boson) is needed. In this vertex, the fermion turns must turn backwards in time.

1. I managed to identify the fermionic 2-vertex was specified towards the end of this year as I realized the connection to the problem of general relativity, which arises from the existence of GRT space-times for which the 4-D diffeo structure is non-standard. There are a lot of these. For an exotic diffeo structure, the standard diffeo structure can be said to have point-like defects analogous to lattice defects.
2. Remarkably, this problem is encountered only in the space-time dimension 4 (see this)! Physical intuition suggests that it must be possible to turn this problem from a disaster to victory. In TGD, this is what actually happens: these point-like diffeo-defects can be identified as interaction vertices, the fermion turns back in the direction of time. Pair creation would be possible only in space-time dimension 4!

   A generalization of the classical fermion pair creation vertex has the same general form as in QFT. As a special case the pair can correspond to a boson as a fermion-antifermion bound state. This vertex also has geometric variants in different dimensions. A fermion line, string world sheet, the orbit of a partonic 2-surface and also the Bohr orbit of 3 surface can turn backwards in time and the fermion states associated with the induced spinor fields do the same.

This inspires two questions.

1. Is the creation of a pair actually the only vertex or is it possible to have a geometric 3-vertex and is it really needed? At the fermion level only the 2-vertex described above is not possible, but for the topological reactions of surfaces one could think of 3-vertices and in the earlier picture I thought these are needed. They do not seem to be necessary however.

   If so, the theory would be extremely simple compared to quantum field theories. There dangerous genuine 3-vertices would be absent and diffeo defects defining 2 vertices, which give all that is needed! At the geometric level, monopole fluxes would replicate and break and join. Intriguingly, this is what would happen at the magnetic bodies of DNA and induce similar reactions at the level of DNA molecules! Maybe biology has been doing its best to tell us what the fundamental particle dynamics is!

2. Since only the induced electroweak gauge potentials couple to fermions, the question arises whether color and strong interactions are obtained. How is it possible to have strong interactions without parity violation when basic vertices involve weak parity violation? I have already discussed this question (see this).

Vertex for graviton emission

There is still one crucially important question left. Is it possible and what would happen in it? Can one obtain a vertex, where the analog for a contraction T^αβδ g_αβ of energy-momentum tensor with the deviation of the metric from the Minkowski metric appears?

1. In TGD all elementary particles, also gravitons, are identified as closed 2-sheeted monopole flux tubes with two wormhole contacts at its "ends" and opposite wormhole throats carrying fermions and antifermions (see this and this). For gravitation one has 1 fermion or antifermion for each wormhole throat.

   The graviton emission vertex should correspond to a splitting of flux tubes. Mopole flux tubes with fermion-antifermion pairs assignable to both wormhole contacts should appear. The fermion and antifermion should reside at the opposite throats of each wormhole contact. This should happen in the splitting of a monopole flux tube and second monopole flux tube would correspond to graviton. That two bosonic vertices are involved with the emission, brings to mind the proposal that gravitation is in some sense a square of gauge theory.

2. The vertex is the same as for gauge boson emission and for a creation of a fermion-antifermion pair. The definition of the modified gamma matrices as Γ^α= T^α_kΓ^k appearing in the modified Dirac action (see this), involving the modified Dirac operator Γ^μD_μ makes it possible to identify the gravitational part of the vertex. Here the quantities T^α_k=∂ L/∂(∂_αh^k) are canonical momentum currents associated with the action defining the space-time surface and also the analog of the energy-momentum tensor.

   Modified gamma matrices are required by hermiticity forcing the vanishing of the divergence of the vector Γ^α giving classical field equations for space-time surfaces. This implies a supersymmetry between the dynamics of fermions and 3-surfaces. The gravitational interaction would correspond to the deviation of the induced metric from the induced metric defined by induced CP₂ metric. CP₂ radius must correspond to Planck length l_P. This requires that the CP₂ as R≈ 10⁴l_P must correspond to h= nh₀, n≈ 10⁷ as found already earlier.

3. The cosmological term in GRT has coefficient 1/8π GΛ== 1/R⁴ so that the modified gamma matrices would contain a term proportional to 1/R⁴ plus a term coming from the Kähler action. In the TGD framework (see this and this) cosmological constant Λ depends on the p-adic length scale, which is assumed to correspond to a ramified prime for an extension of rationals associated with the polynomial P determining to high degree the space-time surface and approaches to zero in cosmic scales. The cosmological value corresponds to R≈ 10^-4 meters, i.e. cell length scale and a scale near neutrino Compton length.

   In the general coordinate invariant formalism, one does not assign dimension to the coordinates or to covariant derivative D_α. Metric has dimension 2. The scale dimension of T^α_kg^1/2 is the same dimension of Lg^1/2 and thus vanishes. Γ^α has scale dimension -1. The modified Dirac action must be dimensionless so that the induced spinors must have scale dimension 1/2.

4. The cosmological constant as the coefficient of the action depends on the p-adic length scale unlike. This term contributes to the string tension of string-like objects an additional term, which among other things can explain hadronic string tension. This term is visible also in the interaction vertices. The Kähler part of the bosonic action terms comes from the deviation of the induced metric from the flat metric and should give the usual gravitational interactions with matter.
5. Holomorphy hypothesis allows any general coordinate invariant action constructible in terms of the induced geometry. Although preferred extremals are always minimal surfaces, the properties of the action are visible via classical conservation laws, via the field equation at singular 3-surfaces involving the entire action, and via the vertices.

See the article About the Relationships Between Weak and Strong Interactions and Quantum Gravity in the TGD Universe or the chapter About Platonization of Nuclear String Model and of Model of Atoms

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.

Gravitational hum and the precise form of M8-H duality

The precise form of M⁸-H duality (see this, this, and this) has remained open since one consider several variants for the duality in M⁴⊂ M⁸=M⁴× E⁴ degrees of freedom mapped to M⁴⊂ M⁴× CP₂ degrees of freedom.

The model for gravitational hum involves diffraction in the tessellation of H³ formed by stars or rather, by their magnetic bodies (see this). Reciprocal lattice is closely related to diffraction and equals to the lattice only in the case of cubic lattice. This is expected to be true also for tessellations and suggests that M8-H duality mapping momentum 4-surfaces of M⁸ to space-time surfaces of H maps the tessellation of H³⊂ M⁴⊂ M⁸ to a reciprocal tessellation of H³⊂ M⁴⊂ H.

The first problem is that the momenta at the M⁸ side are complex unlike the space-time points at the H side. The basic condition comes from the Uncertainty Principle in semiclassical form but does not complettessellationely fix the duality.

1. If the momenta are real, the simplest option is that the mass shell is mapped to a time shell a=h_eff/m, where a is light-cone proper time. For physical states the momenta are real by Galois confinement and have integer components when the momentum scale is defined by causal diamond (CD). For virtual fermions, the momenta are assumed to be algebraic integers and can be complex. The question is whether one should apply M⁸-H a duality only too the real momenta of physical states or also the virtual momenta.
2. For virtual momenta the M⁸-H duality must be consistent with that for real momenta and the simplest option is that one projects the real part of the virtual momentum and applies M⁸-H duality to it. The square m_R² for the real part Re(p) of momentum however varies for the points on the complex mass shell since only the real part Re(m²) of mass squared is constant at the complex mass shell. If 3-momenta are real, one has (Re(p))²= Re(p₀)² -p₃²=m_R² and is not constant and in general larger than Re(p²)=Re(p₀)²-Im(p₀)² -p₃²=Re(m²). Should one use m_R² or Re(m²) in M⁸-H duality?

   Re(m²) is constant at M⁸ side in accordance with the definition of mass shell. The value of a²= h_eff²(m_R²)/Re(m²)² at H side varies and has a width defined by the variation of Im(p₀) at the points of the mass shell. m_R² is not constant at the M⁸ side. This might relate to the fact that particle masses have a width and would relate Im(p₀ to a physical observable. The time shell is given by a²= h_eff²/m_R² and is genuine H³.

The model for the gravitational hum (see this) provides another problem, which can serve as an additional guideline.

1. The model was based on gravitational diffraction in the tessellation defined by a discrete subgroup of SL(2,C). This tessellation is a hyperbolic analog of a lattice in E³ with a discrete translation group replaced with a discrete subgroup Γ of the Lorentz group or its covering SL(2,C). The matrix elements of the matrices in Γ should belong to the extension of rationals defined by the polynomial P defining the space-time surface by M⁸-H duality.
2. For ordinary lattices, the reciprocal lattice assigns to a spatial lattice a momentum space lattice, which automatically satisfies the constraint from the Uncertainty Principle. Could the notion of the reciprocal lattice generalize to H³? What is needed are 3 basis vectors (at least) characterizing the position of a fundamental region and having components that must belong to the algebraic extension of rationals considered. The application of Γ would then produce the entire lattice. In this case a linear superposition of lattice vectors is not possible.
3. The (at least) 3 basic 3-vectors p_3,i need not be orthogonal or have the same length. They should have components, which are algebraic integers in the extension of rationals defined by P. M⁴⊃ H³ is a subspace of complexified quaternions with the space-like part of momentum vector, which is imaginary with respect to commuting imaginary unit i to transform the algebraic scalar product (no conjugation with respect to i). The ordinary cross product appearing in the definition of the reciprocal lattice appears in the quaternionic product. This suggests that the (at least) 3 reciprocal vectors p_3,i as M⁴ projections of four-momentum vectors are proportional to the cross products of the basis vectors apart from a normalization factor determined by the condition that the light-cone proper time is proportional to the inverse of mass. One would have x₃ⁱ∝ ε^ijkp_3,j× p_3,k.

   Physical intuition suggests that the components of spatial momentum are real for the basis vectors p_3,i so that only the energy has imaginary part. For their discrete Lorentz books by Γ this cannot be the case in M⁸_c.

4. Mass shell condition p_i²= M² must be replaced with x²_i= h_eff/M². The precise identification of M² will be considered below. The image of the real part of the energy p_0,i is the time coordinate t⁰_i= h_effRe(p_0,i)/M². For the naive option considered earlier, the time shell condition is satisfied if the dual position vectors x₃ⁱ are of form x₃ⁱ =h_eff [(p²_i)^1/2/M²]eⁱ, where eⁱ is a unit vector in the direction of p_i. This option is correct if the momenta p_i are orthogonal since in this case the reciprocal unit vectors and vectors co-insider. In the general case, one must replace the unit vector eⁱ with the unit vector associated with the vector Xⁱ= ε^ijkp_3,j p_3,j given by eⁱ=Xⁱ/((Xⁱ)²)^1/2 so that the formula for x³_i remains otherwise the same.
5. The conditions have a similar form independently of whether one takes the mass squared parameter M² to be M²=Re(m²) or M²=m_R². The time components of the momentum vectors p_i associated with p_3,i, which are assumed to be real, are determined by the mass shell condition Re(p_0,i)²-Im(p_0,i)²- Re(p_i²) =Re(m²). Spatial coordinates in M⁴ must obey similar formula, which implies the length of the image vector is r= h_eff×p_3,i/M² so that time shell condition t²-r²= h_eff²/M² conforms with the Uncertainty Principle.
6. Which option is correct: M²=Re(m²) or M²=m_R²? For the first option the discretized real mass shell m_R² is deformed and might be essential for having a non-trivial number theoretical holography implying by M⁸-H duality a non-trivial holography at H side. One can however defend the second option by non-trivial holography at H side.

Note that the proposed definition of M⁸-H duality is indirect in that it is applied only to the (at least) 3 basic vectors and the action of Γ gives the tessellation in H³⊂ H. One can also apply the entire Lorentz group to these vectors to obtain the time shell.

The proposed construction assumed that the basis of 3 vectors is essential for the definition of tessellation and that it is possible to assign a set of reciprocal vectors to it in the proposed way involving cross product, which is essentially 3-D notion and relates to quaternions. Is this really the case for all tessellations of H³?

1. In Euclidian 3-space E³ only cubic lattice defines a regular tessellation and for the reciprocal lattices is well-defined. In this case, the linear combinations of 3 basic vectors define the lattice. Note that Platonic solids have duals but this duality has nothing to do with the reciprocal lattice.
2. In hyperbolic 3-space H³ one can have cubic tessellation, 2 icosahedral tessellations, and dodecahedral tessellation as regular tessellations. There is also icosa tetrahedral tessellation involving both tetrahedra, octahedra and icosahedra (see this). Linear combinations of the basic vectors do not exist now. If 3 basic vectors of the tessellation are known, it would be their orbit under the discrete group Γ which defines the tessellation. If Γ is transitive, a single point in principle defines the tessellation as the orbit of Γ.
3. One can assign to tessellations of H³ what might be called a fundamental tetrahedron as a 4-simplex, which is not a regular tetrahedron in the general case. Could the loci of its vertices with respect to a selected vertex define the fundamental tetrahedron? For a cube in E³ this tetrahedron would correspond to a tetrahedron defined by the 3 nearest vertices of a selected vertex of the cube. At least in E³ one can select the vertex by requiring that the distances of the neighbouring vertices from the selected vertex are minimal. In this case, the remaining 3 vertices form an orbit under Z³ .
4. Could one reduce the situation from H³ to E³ by considering the 3 basic vectors for the projection of the fundamental tetrahedron from H³ to t=constant hyperplane E³?The 3 basic vectors would be from the selected vertex defining the origin to neighboring vertices: note that here the submanifold property H³⊂ M⁴ is essential.

   Could one assign to this triplet a reciprocal in the same way as in the Euclidian case using the cross product induced by the quaternion structure? The notion of dual basis is also behind the bra-ket formalism of quantum mechanics and is based on the notion of vector space and its dual? The following considerations rely on this optimistic assumption.

See the article The TGD view of the recently discovered gravitational hum as gravitational diffraction, the articles Magnetic Bubbles in TGD Universe: Part I and Magnetic Bubbles in TGD Universe: Part II or the chapter Quantum 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.

Pollack effect and evaporation without heating

It has been found (see this) that irradiation by visible light can evaporate water without any heat, this is called photomolecular effect.

What comes first in mind for a habitant of the TGD Universe is that the photomolecular effect reduces to Pollack effect in which light in visible and infrared wavelength range induces a formation of negatively charged regions, exclusion zones (EZs) containing fourth phase of water, as Pollack calls them. These regions have very strange properties suggesting time reversal: for instance, they clean themselves from impurities which suggests diffusion with a reversed arrow of time at the magnetic body of EZ. EZs are layered structures with effective stoichiometry H_1.5O. Pollack talks of EZs fourth phase of water and the ordered water at the surfaces of say biomolecules like DNA and folded proteins could consist of this kind of phase.

TGD proposes a model of Pollack effect based on the TGD view of dark matter. Part of protons would go to the magnetic body of water and form dark proton sequences. Pollack effect has become one of the key mechanisms of the TGD inspired quantum biology and would appear in metabolism (ATP), biocatalysis, and nerve pulse generation: see https://tgdtheory.fi/public_html/articles/np2023.pdf .

Quite recently it has been learned that the water-air boundary has a thin cover, which consists of a phase analogous to ice (see this). A reasonable hypothesis is that this phase consists of the fourth phase of water and is responsible for the surface tension of the bulk water. I have developed a TGD based model for the anomalies related to freezing in nanoscales explaining also this phenomenon (see this).

How could the Pollack effect induce evaporation without heat? Evaporation occurs at criticality. In TGD it could be accompanied by quantum criticality meaning the presence of dark matter at the magnetic body of the water, in particular dark protons generated in the Pollack effect. At quantum criticality the bulk water is unstable against the formation of water droplets surrounded by layers of fourth phase. The surface area of droplets plus bulk water is larger than that of bulk. More of the fourth phase of water must be created and this requires energy. The irradiation would provide this energy by the Pollack effect.

See the article TGD inspired model for freezing in nano scales 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 Relationship Between Strong and Weak Interactions in the TGD Universe

In the TGD view, classical electroweak interactions are basically local and only classical electroweak gauge potentials appear in the TGD analogs of fundamental interaction vertices describing splitting and reconnection of monopole flux tubes, which describe also strong interactions at the particle level. The basic problem is to understand how strong interactions can be parity conserving while the classical electroweak dynamics violates parity conservation.

The proposed model, argued to overcome this problem, involves several topological elements.

1. The topological explanation of the family replication phenomenon in terms of the genus of partonic 2-surface carrying fermion lines as boundaries of string world sheets.
2. The view of holography as a 4-D analog of holomorphy reducing to 2-D holomorphy for partonic 2-surfaces. This predicts two kinds of partonic 2-surfaces as complex 2-surfaces in CP₂ with a spherical topology. Tor the homologically non-trivial geodesic sphere induced weak fields vanish (no parity violation classically) and for the second complex sphere they do not. A natural working hypothesis is that these two spheres explain the difference between strong and weak interactions.
3. The homology (Kähler magnetic) charge h of the partonic 2-surface correlates with the genus of the partonic 2-surface. For complex partonic 2-surfaces in CP₂, the genus is given g=(h-1)(h-2)/2-s, where s is the number of singularities. Only the genera g=(h-1)(h-2)/2 are free of singularities. For g=0, this includes h=1 and h=2. Already for g=2 there would be singularity. It is however possible to overcome this problem since partonic 2-surfaces can be deformed to M⁴ degrees of freedom and one can add handles in this way. A rather detailed picture of partonic 2-surfaces and monopole flux tubes emerges.

See the article About the Relationship Between Strong and Weak Interactions in the 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.

About the difference between electroweak and strong interactions assuming generalized holomorphy

It is interesting to find what the holography as a generalization to 4-D holomorphy hypothesis predicts when combined with the proposed explanation of the family replication phenomenon and the proposal for how parity violation is avoided in strong interactions despite the fact that only electroweak induced fields appear in the fundamental vertices for the creation of various particles identified as closed 2-sheeted monopole flux tubes. This includes mesons and gluons. The following considerations force a rather detailed picture about partonic 2-surfaces.

1. Monopole flux tubes can be regarded as M⁴ deformations of cosmic strings representable as Cartesian products of string world sheet X²⊂ M⁴ and 2-surface Y²⊂ CP₂. Partonic 2-surfaces would appear as "ends" of 2-sheeted monopole flux tubes. If the holomorphic realization of holography makes sense, the space-time surfaces are complex algebraic surfaces. In the simplest situation the 2-D cross section of a cosmic string is a complex surface of CP₂. A more general option is as a complex algebraic curve in E²× CP₂.
2. Riemann-Roch theorem (see this) allows to define geometric genus (see this) of a complex algebraic curve in CP₂ as

   g= (d-1)(d-2)/2-s ,

   where s is the number of singularities, which are cones and as a special case cusps (infinitely sharp cones). According to the Wikipedia article, this formula generalizes to algebraic surfaces in higher than 2-D complex manifolds, or at least projective space.

   From this one can conclude for the (d-1)(d-2)/2 ≥ g for partonic 2-surfaces as complex surfaces in CP₂, there are always singularities. For s=0, g=0 allows d=0 and d=1. For s=0, g=1 allows d=3 related to elliptic functions. Already for g=2 one has s≥ 1. The genera g=(d-1)(d-2)/2 are special in that they also allow s=0.

3. It is known (see this) that for s=0 the topological genus, algebraic genus and arithmetic genus are identical. This might be relevant for the definition of genus for the p-adic counterparts of partonic 2-surfaces, where the topological genus does not make sense. This could make g ∈ {0,1, (d-1)(d-2)2} cognitively special. It would seem that p-adic variants of g=2 partonic 2-surfaces do not make sense unless one can eliminate the singularities by a deformation of Y² to a complex 2-surface in E²× Y². One should also be able to represent g>0 surfaces as surfaces in E²× CP₂, where CP₁ corresponds to either d=1 of d=2.

Generalized holomorphy, difference between strong and weak interactions, and family replication phenomenon

It is instructive to consider the CP₂ option and its generalization in more detail from the perspective of weak and strong interactions and family replication phenomenon.

1. g=0 option is the most natural one for cosmic strings and allows polynomials of degree d=1 and d=2. d=1 would correspond to the homologically non-trivial geodesic sphere of CP₂ and d=2 a more complex surface. For the homologically non-trivial sphere only the Kähler form would contribute to the vertex related to the splitting of the cosmic string. This could explain why the generation of hadronic and gluonic monopole strings does not lead to a parity violation.

   For d=2 and g=0 induced electroweak fields are non-vanishing and parity violations are predicted. Could photons and gluons correspond to cosmic strings with cross section as d=1 surface of CP₂? Could parity violating weak bosons relate to cosmic strings with a d=2 spherical cross section so that the difference between strong and weak interactions would reduce to algebraic geometry?

2. The genus g=1 could be also realized for cosmic strings with d=3 to which elliptic functions. In this case, the induced weak fields would be present for the CP₂ option. This does not conform with the idea that parity breaking effects do not depend on the genus (generation of fermion).

   Could the deformations of partonic 2-surfaces in M⁴ degrees of freedom come in rescue? For partons as complex 2-surfaces in E²× S² ⊂ E²× CP₂, S² homologically non-trivial geodesic sphere, no charged weak fields would be present. If this picture is correct the deformations in E² degrees of freedom would distinguish between fermion families but the difference should be subtle. I do not know whether the formula for algebraic surfaces in projective spaces still holds true.

3. Genus g=2 partonic 2-surface in CP₂ would have at least one singular point. Is this physically acceptable? Is it possible to avoid the singularity for the Y² ⊂ E²× S² ⊂ E²× CP₂ option? Blowing up of the singularities by removing a small disk of S² around the singularity and gluing back a disk of E²× S² is what comes to mind. Blowup, in particular a blowup at a given point of complex manifold, such as a cone singularity of complex surface, is described in the Wikipedia article (see this).

   Topologically this means construction of a connected sum with the projective space CP₁ by removing a small disk around the singularity. The realization of this operation would now occur in E²× Y². If the genus g= d(d-1)/2-s is preserved in the blowup so that one would obtain non-singular representatives also in g=2 case. Obviously the formula for the genus would not hold anymore.

4. Since all quark genera g ≤ 2 appear in strong interactions, which do not violate parity, one should have a way of constructing g>0 surfaces from the homologically non-trivial sphere CP₁⊂ CP₂ with n=1 complex surface in E²⊂ CP₁. Addition of handles should be the way. These surfaces would be associated with quarks, gluons and mesons, which all would correspond to 2-sheeted monopole fluxe tubes.

   This operation should be possible also for the d=2 complex sphere carrying induced weak gauge fields. The predicted higher families of weak bosons as analogs of mesons could be obtained from d=2 monopole flux tubes. The existence of strong and weak interactions would reflect the existence of d=1 and d=2 complex spheres of CP₂. In particular, one obtains non-singular g=2 fermions. Also leptons could correspond to d=2 spheres.

About the relationship between Kähler magnetic charge and genus

What can one say about the homological (Kähler magnetic) charge of a partonic 2-surface with a given genus. At least homological charges +/- 1 and +/- 2 should be realized for the partonic 2-surfaces. For about 4 decades ago, my friend Lasse Holmström, who is a mathematician, gave me as a gift a Bulletin of American Mathematical Society containing articles about 4-D topology and also about topology of CP₂. At page 124 there were interesting results related to the realization of homologically non-trivial 2-surfaces in CP₂, in particular there were conditions on the minimal genus of these surfaces.

The basic result was that a surface with homology charge h can be realized as a surface with genus g=(h-1)(h-2)/2 and there are no known realizations with a smaller genus. For d=h, this sequence would correspond to the sequence g= (d-1)(d-2)/2 for complex surfaces without singularities. This correlation between genus and homology charge troubled me since in the TGD framework h∈{+/- 1,+/- 2} should be possible for all genera. The addition of handles to d=1,2 complex spheres of CP₁⊂ CP₁⊂ E² would solve the problem. An interesting question is whether the sequence 0,1,6,10,15,... of homologically special genera could have a physical interpretation and perhaps predict a hierarchy of analogs of strong and weak interactions.

About the number of complex deformations of a given partonic 2-surface

It is interesting to ask about how many deformations a given partonic 2-surface represents as a complex surface in E²× CP₁, where CP₁ corresponds to the surface of CP₂ with d∈{1,2}. For the deformations of CP₁ with d=1,2, one can express E² complex coordinate as a meromorphic function of CP₁ complex coordinate. More generally, one can consider the partonic 2-surface in E²× S² as a surface with given genus g and consider the complex deformations of this surface. The dimension of the space of these deformations is of obvious physical interest if generalized holomorphy is accepted.

In the case of a pole, the E² point would go to infinity so that poles are not allowed. If the notion of Hamilton-Jacobi structure (see this) makes sense, one can slice M⁴ also using closed partonic 2-surfaces with complex coordinates so that meromorphic functions with poles are allowed. In TGD, rational functions with rational coefficients of corresponding polynomials are favoured.

These functions can be characterized by so-called principal divisors expressible as formal superpositions D=∑ ν_kP_k. Here P_k are the singular points (zeros for ν_k>0 and poles for ν_k<0). One can assign also to complex one-forms divisors: this kind of divisor is known as canonical divisor and is unique apart from addition of principal divisor, which corresponds to a multiplication of the 1-form with a meromorphic function. The degree of the divisor can be defined as deg(D)= ∑ ν_k.

Riemann-Roch theorem applies also to algebraic surfaces such as complex surfaces in E²× CP₁, and allows to get grasp about the numbers of the surfaces obtained as deformations of CP₁ with a given divisor D for a surface with a given genus g. These numbers correspond to the dimensions of the linear spaces of rational functions, whose poles are not worse than the coefficients of D, where P_k are the singular points (zeros for n_k>0 and poles for n_k<0). The Riemann-Roch formula reads as

l(D)-l(K-D)=deg(D)-g+1 .

Here l(D) is the dimension of the space of meromorphic functions h for which all the coefficients of (h)+D are non-negative (no poles). The term -l(K-D) is a correction term present only for low degrees deg(D) defining the analog of polynomial degree characterizing the winding number of h. Because l(K-D) is a dimension of vector space, it cannot be negative and vanishes for large enough degrees. For large values of deg(D) the formula reads therefore as l(D)=deg(D)-g+1.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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 could strong interactions emerge at the level of scattering amplitudes?

The above considerations are dangerous in that the intuitive QFT based thinking based is applied in TGD context where all interactions reduced to the dynamics of 3-surfaces and fields are geometrized by reducing them to the induced geometry at the level of space-time surface. Quantum field theory limit is obtained as an approximation and the applications of its notions at the fundamental level might be dangerous. In any case, it seems that only electroweak gauge potentials appear in the fermionic vertices and this might be a problem.

1. By holography perturbation series is not needed in TGD. Scattering amplitudes are sums of amplitudes associated with Bohr orbits, which are not completely deterministic: there is no path integral. Whether path integral could be an approximate approximation for this sum under some conditions is an interesting question.
2. It is best to start from a concrete problem. Is pair creation possible in TGD? The problem is that fermion and antifermion numbers are separately conserved for the most obvious proposals for scattering amplitudes. This essentially due to the fact that gauge bosons correspond to fermion-antifermion pairs. Intuitively, fermion pair creation means that fermion turns backwards in time. If one considers fermions in classical background fields this turning back corresponds to a 2-particle vertex. Could pair creation in classical fields be a fundamental process rather than a mere approximation in the TGD framework. This would conform with the vision that classical physics is an exact part of quantum physics.

   The turning back in time means a sharp corner of the fermion line, which is light-like elsewhere. M⁴ time coordinate has a discontinuous derivative with respect to the internal time coordinate of the line. I have propoeed (see this and this) that this kind of singularities are associated with vertices involving pair creation and that they correspond to local defects making the differentiable structure of X⁴ exotic. The basic problem of GRT would become a victory in the TGD framework and also mean that pair creation is possible only in 4-D space-time.

One can imagine two kinds of turning backs in time.

1. The turning back in time could occur for a 3-D surface such as monopole flux tube and induce the same process the string world sheets associated with the flux tubes and for the ends of the string world sheets as fermion lines ending at the 3-D light-like orbits of partonic 2-surfaces.
2. In the fusion of two 2-sheeted monopole flux tubes along their "ends" identifiable as partonic 2-surfaces wormhole contacts, the ends would fuse instantaneously (this process is analogous to "join along boundaries". The time reversal of this process would correspond to the splitting of the monopole flux tube inducing a turning back in time for a partonic 2-surface and for fermionic lines as boundaries of string world sheets at the partonic 2-surface.

   This would be analogous to a creation of a fermion pair in a classical induced gauge field, which is electroweak. The same would occur for the partonic 2-surfaces as opposite wormhole throats and for the string world sheets having light-like boundaries at the orbits of partonic 2-suraces.

3. The light-like orbit of a partonic 2-surface contains fermionic lines as light-like boundaries of string world sheets. A good guess is that the singularity is a cusp catastrophe so that the surface turns back in time in exactly the opposite direction. One would have an infinitely sharp knife edge.

What one can say about the scattering amplitudes on the basis of this picture? Can one obtain the analog for the 2-vertex describing a creation of a fermion pair in a classical external field?

1. The action for a geometric object of a given dimension defines modified gamma matrices in terms of canonical momentum currents as Γ^α= T^αkΓ_k, T^α_k= ∂ L/∂(∂_α h^k). By hermiticity, the covariant divergence D_αΓ^α of the vector defined by modified gamma matrices must vanish. This is true if the field equations are satisfied. This implies supersymmetry between fermionic and bosonic degrees of freedom.

   For space-time surfaces, the action is Kähler action plus volume term. For the 3-D light-partonic orbits one has Chern-Simons-Kähler action. For string world sheets one has area action plus the analog of Kähler magnetic flux. For the light-boundaries of string world sheets defining fermion lines one has the integral ∫ A_μdx^μ. The induced spinors are restrictions of the second quantized spinors fields of H=M⁴× CP₂ and the argument is that the modified Dirac equation holds true everywhere, except possibly at the turning points.

2. Consider now the interaction part of the action defining the fermionic vertices. The basic problem is that the entire modified Dirac action density is present and vanishes if the modified Dirac equation holds true everywhere. In perturbative QFT, one separates the interaction term from the action and obtains essentially ΨbarΓ^α D_αΨ. This is not possible now.

   The key observation is that the modified Dirac equation could fail at the turning points! QFT vertices would have purely geometric interpretation. The gamma matrices appearing in the modified Dirac action would be continuous but at the singularity the derivative ∂_μΨ= ∂_μm^k ∂_kΨ of the induced free second quantized spinor field of H would become discontinuous. For a Fourier mode with momentum p^k, one obtains

   ∂_μΨ_p= p_k∂_μ m^kΨ_p == p_μΨ_p.

   This derivative changes sign in the blade singularity. At the singularity one can define this derivative as an average and this leaves from the action Ψbar Γ^α D_αΨ only the term ΨbarΓ^α A_αΨ. This is just the interaction part of the action!

3. This argument can be applied to singularities of various dimensions. For D=3, the action contains the modified gamma matrices for the Kähler action plus volume term. For D=2, Chern-Simons-Kähler action defines the modified gamma matrices. For string world sheets the action could be induced from area action plus Kähler magnetic flux. For fermion lines from the 1-D action for fermion in induced gauge potential so that standard QFT result would be obtained in this case.

How does this picture relate to perturbative QFT?

1. The first thing to notice is that in the TGD framework gauge couplings do not appear at all in the interaction vertices. The induced gauge potentials do not correspond to A but to gA. The couplings emerge only at the level of scattering amplitudes when one goes to the QFT limit. Only the Kähler coupling strength and cosmological constant appear in the action.
2. The basic implication is that only the electroweak gauge potentials appear in the vertices. This conforms with the dangerous looking intuition that also strong interactions can be described in terms of electroweak vertices but this is of course a potential killer prediction. One should be able to show that the presence of WCW degrees of freedom taken into account minimally in terms of fermionic color partial waves in CP₂ predicts strong interactions and predicts the value of α_s. Note that the restriction of spinor harmonics of CP₂ to a homologically non-trivial geodesic sphere gives U(2) partial waves with the same quantum numbers as SU(3) color partial waves have.
3. TGD approach differs dramatically from the perturbative QFT. Since 1/α_s appears in the vertex, the increase of h_eff in the vertex increases it: just the opposite occurs in the perturbative QFT! This seems to be in conflict with QFT intuition suggesting a perturbation series in α_s ∝ 1/ℏ_eff. The explanation is that 1/α_K appears as a coupling parameter instead of α_s.

   This reminds of the electric-magnetic duality between perturbative and non-perturbative phases of gauge theories, where magnetic coupling strength is proportional to the inverse of the electric coupling strength. The description in terms of monopole flux tubes is therefore analogous to the description in terms of magnetic monopoles in the QFT framework. In TGD, it is the only natural description at the fundamental level. The decrease of α_K by increase of h_eff would indeed correspond to the QFT type description reduction of α_s.

   Could the description based on Maxwellian non-monopole flux tubes correspond to the usual perturbative phase without magnetic monopoles? In the Maxwellian phase there is huge vacuum degeneracy due to the presence of vacuum extremals with a vanishing Kähler form at the limit of vanishing volume action. Could this degeneracy allow path integral as a practical approximation at QFT limit.

4. h_eff/h₀ = n is proposed to correspond to the dimension of algebraic extension of rationals associated with the space-time surface and serve as a measure for algebraic complexity. The increase of algebraic complexity of the space-time region defining the strong interaction volume would also make interactions strong. In TGD, the fundamental coupling strength would be α_K and the increase of α_K for ordinary value of h would force the increase of h. This should happen below the electroweak scale or at least the confinement scales and make perturbation theory describing strong interactions possible. This description would involve monopole flux tubes and their reconnections.

   See the article About Platonization of Nuclear String Model and of Model of Atoms 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 identification of the Schrödinger galaxy

The latest mystery related created by the observations of James Webb (see this and this).

It has been found that the determination of the redshift 1+ z = a_now/a_emit gives two possible space-time positions for the Schrödinger galaxy CEERS-1749 a_now resp. a_emit corresponds to the scale factor for the recent cosmology resp. cosmology when the radiation was emitted. Note that for not too large distances the recession velocity β satisfies the Hubble law β= HD. The nickname "Schrödinger galaxy" comes from the impression that the same galaxy could have existed in two different times in the same direction.

Accordingly, CHEERs allows two alternative identifications: either as an exceptionally luminous galaxy with z≈ 17 or as a galaxy with exceptionally low luminosity with z≈ 5. Both these identifications challenge the standard view about galaxy formation based on Λ CDM cosmology.

1. The first interpretation is that CHEERS is very luminous, much more luminous than the standard cosmology would suggest, and has the redshift z≈ 17, which corresponds to light with the age of 13.6 billion years. The Universe was at the moment of emission t_emit=220 million years old.

   In the TGD framework, the puzzlingly high luminosity might be understood in terms of a cosmic web of monopole flux tubes guiding the radiation along the flux tubes. This would also make it possible to understand other similar galaxies with a high value of z but would not explain their very long evolutionary ages and sizes. Here the zero energy ontology (ZEO) of TGD could come in rescue (see this, this and this).

2. Another analysis suggest that the environment of the CHEERS contains galaxies with redshift z≈ 5. The mundane explanation would be that CHEERS is an exceptionally dusty/quenched galaxy with the redshift z≈ 5 for which light would be 12.5 billion years old.

   Could TGD explain the exceptionally low luminosity of z≈ 5 galaxy? Zero energy ontology (ZEO) and the TGD view of dark matter and energy predict that also galaxies should make "big" state function reductions (BSFRs) in astrophysical scales. In BSFRs the arrow of time changes so that the galaxy would become invisible since the classical signals from it would propagate to the geometric past. This might explain the passive periods of galaxies quite generally and the existence of galaxies older than the Universe. Could the z≈ 5 galaxy be in this passive phase with a reversed arrow of time so that the radiation from it would be exceptionally weak.

TGD seems to be consistent with both explanations. To make the situation even more confusing, one can ask whether two distinct galaxies at the same light of sight could be involved. This kind of assumption seems to be unnecessary but one can try to defend this question in the TGD framework.

1. In the TGD framework space-times are 4-surfaces in M⁴× CP₂. A good approximation is as an Einsteinian 4-surface, which by definition has a 4-D M⁴ projection. The scale factor a corresponds to the light-cone proper time assignable to the causal diamond CD with which the space-time surface is associated. a is a very convenient coordinate since it has a simple geometrical interpretation at the level of embedding space M⁴× CP₂. The cosmic time t assignable to the space-time surface is expressible as t(a).
2. Astrophysical objects, in particular galaxies, can form comoving tessellations (lattice-like structures) of the hyperbolic space H³, which corresponds to a=constant, and thus t(a) constant surfaces. The tessellation of H³ is expanding with cosmic time a and the values of the hyperbolic angle η and spatial direction angles for the points of the tessellation do not depend on the value of a. The direction angles and hyperbolic angle for the points of the tessellation are quantized in analogy with the angles characterizing the points of a Platonic solid and this gives rise to a quantized redshift.

   A tessellation for stars making possible gravitational diffraction and therefore channelling and amplification of gravitational radiation in discrete directions, could explain the recently observed gravitational hum (see this).

   These tessellations could also explain the mysterious God's fingers, discovered by Halton Arp, as sequences of identical look stars or galaxies of hyperbolic tessellations along the line of sight (see this and this. Maybe something similar is involved now.

This raises two questions.

1. Could two similar galaxies at the same line of sight be behind Schrödinger galaxy and correspond to the points of scaled versions of the tessellation of H³ having therefore different values of a and hyperbolic angle η? The spatial directions characterized by direction angle would be the same. Could one think that the tessellation consists of similar galaxies in the same way as lattices in condensed matter physics? The proposed explanation for the recently observed gravitational hum indeed assumes tessellation form by stars and most stars are very similar to our Sun (see this).

   The obvious question is whether also the neighbours of the z≈ 5 galaxy belong to the scaled up tessellation. The scaling factor between these two tessellations would be a₅/a₁₇= 17/5. Could it be that the resolution does allow to distinguish the neighbors of the z≈ 17 galaxy from each other so that they would be seen as a single galaxy with an exceptionally high luminosity? Or could it be that the z≈ 5 galaxy is in a passive phase with a reversed arrow of time and does not create any detectable signal so that the signal is due to z≈ 17 galaxy.

2. Could one even think that the values of hyperbolic angles are the same for the two galaxies in which case the z≈ 5 galaxy could correspond to z≈ 17 galaxy but in the passive phase with an opposite arrow of time? The ages of most galaxies are between 10 and 13.6 billion years so that this option deserves to be excluded. Could the hyperbolic tessellation explain why two similar galaxies could exist at the same line of sight in a 4-dimensional sense?

   This option is attractive but is actually easy to exclude. The light arriving from the galaxies propagates along light-like geodesics. Suppose that a light-like geodesic connects the observer to the z≈ 17 galaxy. The position of the z≈ 5 galaxy would be obtained by scaling the H³ of the older galaxy by the ratio a(young)/a(old). Geometrically it is rather obvious that the geodesic connecting it to the observer cannot be lightlike but becomes space-like. If one approximates space-time with M⁴ this is completely obvious.

   For more detailed analysis, see the article TGD view of the paradoxical findings of the James Webb telescope or the chapter TGD View of the Engine Powering Jets from Active Galactic Nuclei.

   For a summary of earlier postings see Latest progress in TGD. For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Pollack effect as a universal energy transfer mechanism?

The proposal of the recent article Some New Aspects of the TGD Inspired Model of the Nerve Pulse is that nerve pulse generation relies on the flip-flop mechanism using the energy liberated in the reversal of Pollack effect at one side of cell membrane to induce Pollack effect at the opposite side. The liberate energy would be channelled along the pair of monopole flux tubes emerging by re-connection from two U-shaped monopole flux tubes. The flip-flop mechanism is highly analogous to a seesaw in which the gravitational binding energy at the first end of the seesaw is reduced and transforms to kinetic energy reducing gravitational binding energy at the second end of the seesaw.

All biochemical processes involve a transfer of metabolic energy. Could the flip-flop mechanism serve as a universal mechanism of energy transfer accompanying biochemical processes?

The first example is TGD based view of biocatalysis according to which a phase transition reduces the value of h_eff and thus the length for the monopole flux tube pair connecting the reactants liberates energy, which kicks the reactants over the potential energy wall and in this way increases dramatically the rate of the reaction. Also now, the liberated energy could propagate as dark photons along the flux tube pair raise the system above the reaction wall or at least reduce its height.

Also the ADP→ ADP process could involve the Pollack effect and its reversal. In this process 3 protons are believed to flow through the cell membrane and liberate energy given to the ADP so that the process ADP+P_i → ATP takes place. This system has been compared to an energy plant. This raises heretic questions. Does the flow of ordinary protons through the mitochondrial membrane really occur? Could the charge separation be also now between the cell interior and its magnetic body?

1. The protons believed to flow through the mitochondrial membrane would be in the initial situation gravitationally dark and generated by Pollack effect for which the energy would be provided as energy liberated by biomolecules in a process which could be a time reversal for its storage in photosynthesis.
2. The reverse Pollack effect inside the mitochondrial membrane could transform the dark protons to ordinary protons and liberate energy, which is carried through the membrane as dark photons to the opposite side. This would allow the high energy phosphate bond of ATP to form in the reaction ADP+P_i → ATP. According to the TGD proposal (see this and this), the liberated energy could be used to kick the proton to the gravitational monopole flux tube, which would have length of order Earth size scale so that gravitational potential energy would of the same order of magnitude as the metabolic energy quantum with a nominal value .5 eV. This dark proton would be the energy carrier in the mysterious high phosphate energy bond, which does not quite fit the framework of biochemistry.
3. ATP would donate the phosphate ion P^- for the target molecule, which would utilize this temporarily stored metabolic energy as the dark proton transforms to an ordinary one. Depending on the lifetime of the dark proton, this could occur as the target molecule receives P or later. In any case, this should involve the transformation P^-→ P_i. This could correspond to the transformation of the gravitationally dark proton to ordinary proton so that the charge separation giving rise to P^- would be between P_i and its magnetic body.

In the chemical storage of the metabolic energy in photosynthesis, ATP provides the energy for the biomolecule storing the energy. This process should be accompanied by the transformation of P^- to P_i. It is instructive to consider two options that come immediately into mind.

Option I: The realistic looking option is that the energy is stored as the energy of an ordinary chemical bond.

1. Hydrogen bond, which can form between a proton and other electronegative atoms such as O or N, is a natural candidate. Hydrogen bond indeed has an energy, which is of the order of metabolic energy quantum .5 eV. The simplest option is that the metabolic energy provided by the gravitational flux tube of ATP is liberated and used to generate a hydrogen bond of the protein. The dark gravitational flux tube loop would be nothing but a very long hydrogen bond.
2. For negatively charged molecules, the proton of a hydrogen bond could be gravitationally dark. For dark positively charged ions, some valence electrons could be gravitationally dark. In the electronic case the reduction of the gravitational binding energy would be roughly by a factor m_e/m_p∼ 2^-11 smaller and this leads to a proposal of electronic metabolic energy quantum (see this and this and this) for which there is some empirical support from the work of Adamatsky (see this.

Option II: The less realistic looking option is that the molecule stores the metabolic energy permanently as a gravitationally dark proton. The motivation for its detailed consideration is that it provides insights to the Pollack effect.

1. The dark proton associated with P^- should become a dark proton associated with the molecule. In this case the length of hydrogen bond would become very long, increasing the ability to store metabolic energy.

   The hydrogen bonded structure would be effectively negatively charged but this is just what happens in the EZ in Pollack effect! This supports the view that the Pollack effect for water basically involves the lengthening of the hydrogen bonds to U-shaped gravitational monopole flux tubes.

2. The Pollack effect requires a metabolic energy feed since the value of h_gr tends to decrease spontaneously. This suggests that the dark gravitational hydrogen bonds are not long-lived enough for the purpose of long term metabolic energy storage. Rather, they would naturally serve as a temporary metabolic energy storage needed in the transfer of metabolic energy. The temporary storage of the metabolic energy to ATP would be a quantum variant of the seesaw.
3. The first naive guess for the scale of the life-time of the gravitationally dark proton would be given as a gravitational Compton time determined by the gravitational Compton length Λ_gr= GM/β₀ =r_S(M)/2β₀. For the Earth with r_S∼ 1 cm, one has T_gr=1.5 × 10^-11 s corresponding to the energy .6 meV for the ordinary Planck constant and perhaps related to the miniature membrane potentials. For the Moon with mass M_M=.01M_E, this time is about T_gr∼ 1.5× 10^-13 ns. For the ordinary Planck constant, this time corresponds to energy of .07 eV and is not far from the energy assignable to the membrane potential. For the Sun, one would gravitational Compton length is one half of the Earth's radius, which gives T_gr= .02 s, which corresponds to 50 Hz EEG frequency.

   Note that the rotation frequency for the ATP synthase analogous to a power plant is around 300 Hz which is the cyclotron frequency of the proton in the endogenous magnetic field .2 Gauss interpreted in TGD as the strength of the monopole fluz part of the Earth's magnetic field.

See the article Some New Aspects of the TGD Inspired Model of the Nerve Pulse or the chapter with the same title.

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

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

Could a new kind of period of exploration save our civilization from collapse?

Everyone knows Columbus and other heros from the period of exploration journeys when we learned the geography of our home planet. After Newton emerged planetary physics and we started to learn of our planetary system. Eventually astrophysics and cosmology emerged. Now the James Webb telescope is shattering our views about the foundations of cosmology and astrophysics and a profound revolution of the world of physics seems unavoidable. This revolution also relates to our view of time. Also in biology, brain science, and science of consciousness, we are at the verge of revolution. After half a century our recent physics based world view might be regarded as childish as the world of Flat-Earthers.

In a sharp contrast to this progress in science and technology, our civilization has fallen in a state of stagnation. Its materialism based world view has deprived us of ethics and morals and the highest goal of modern man is to consume even more. Success in our society means money, fame and power. The uncontrolled application of various technological breakthroughs have created lethal looking environmental problems and social structures are breaking down. It is quite possible that our civilization is doomed to collapse. This is nothing new and actually fits with the fact that all complex systems are born, flourish, and eventually die. It is good to remember that after the collapse of the Roman empire there was a period of stagnation of about 1500 years before the development of mathematics created in antiquity could continue. In fact, they almost discovered calculus and computers before the collapse.

In this kind of situation one can wonder whether a new exploration period could be possible and give a deeper meaning for the existence of our western civilization and make us something more than consumers? Could the striking findings of the James Webb telescope and equally striking discoveries from other branches of natural science, which do not gain attention of popular media, inspire a new period of exploration of our world allowing us to update our lethally wrong world view?

What could be the big question?

What could be the question catching the attention of adventurous minds in the future? What dark matter and energy really are? This is certainly one of the deepest mysteries of recent day science. Could the attempt to understand dark matter give meaning to the existence of the society?

Dark matter is indeed an excellent candidate for the problem of the next century. The mainstream science knows only of the existence of dark matter and energy. The particle physics inspired models have repeatedly failed the tests and also the halo model for galactic dark matter relying on particle physics models is in deep difficulties. Same can be said about the o MOND model, which denies the existence of dark matter and postulates that Newtonian gravitation fails for weak fields. This is a rather paradoxical looking assumption, which very few can consider seriously.

TGD answer to the big question

The TGD explanation of dark matter relies on a new view of both space-time and quantum theory. TGD predicts the existence of a dark matter hierarchy as phases of the ordinary matter labelled by the values h_eff of the (effective) Planck constant, which is a multiple of its minimal value. Dark matter would be simply ordinary matter in a phase with a nonstandard value of Planck constant. If the value of the h_eff is large enough, this phase of matter is quantum coherent in even macroscopic scales. This would explain the mysterious ability of living matter to behave coherently in macroscales, impossible to understand in the biology as nothing but chemistry approach. The quantum coherence of the dark matter would induce the ordinary coherence of biomatter.

This view also revolutionizes the views about elementary particle-, hadron-, nuclear-, atomic- and molecular physics. The same basic topological mechanisms appear in all these physics and a lot of new physics is predicted (see this ). The dark matter would reside at space-time sheets (a new notion forced by the TGD view of space-time) characterized by the value of h_eff. The value of h_eff would characterize algebraic complexity of the space-time sheet, which in turn is a natural measure for the capacity to represent conscious information. The h_eff hierarchy would define an evolutionary hierarchy.

The most natural candidates for the space-time sheet carrying dark matter would be what I call magnetic bodies. The TGD view of space-time predicts that the electromagnetic fields of a system define a kind of field body of the system as a well-defined geometric anatomy and having body parts, motor actions, etc... In particular, the magnetic body consisting of monopole flux tubes would serve as the "boss", controller of the system because its IQ characterized by the high value of h_eff would be high.

The predicted values of Planck constant are largest for the monopole flux tubes mediating quantum gravitation. This conforms with the facts that gravitation has infinite range, is un-screened and the fact that quantum coherence scale increases with h_eff. The highest values of Planck constant would be associated with gravitational monopole flux tubes of Earth, Moon, other planets, Sun, and even galaxies. The unavoidable prediction is that the magnetic bodies of these astrophysical objects could play a key role in the quantum biology of the Earth. Horoscopes make no sense but astrologers might have not been completely wrong. Hard science must rely on numbers and the number of numerical miracles supporting this view has been accumulating (see for instance this, this, this).

The field bodies as the target of the new period of exploration?

These considerations suggest that the new period of exploration could have the electromagnetic environment of the Earth as its target. What do the magnetic and electric bodies of Earth, planets, Sun, galaxy,... look like? How dow they interact? This would be also exploration of the inner world, not only ours: the prediction is that life and consciousness are universal. This is so because the h_eff hierarchy plays a central role in understanding of conscious experience and intelligence.

Could the predicted new atomic physics kill the Platonic vision?

The Platonic vision connecting hadron physics, nuclear physics and atomic physics predicts a lot of new atomic physics and this could turn out to be fatal. I hasten to confess that the following speculations reflect my rudimentary knowledge of details of atomic physics. The new conceptual element are flux tubes, which can be regarded as springs with mass and elastic constant (string tension).

The first question concerns electric fields in the flux tube picture.

1. If there are only flux tubes present, the electric fluxes must run along them (a more conservative option is that fluxes flow to a large space-time sheet). Perhaps the most natural interpretation is that the localization of electric fluxes to flux tubes induces a constraint force due to the space-time geometry, something completely new. If so, one can argue that the dynamics for the flux tubes carrying also electric flux automatically describes the repulsive Coulomb force subject to geometrodynamic constraints.

   An important implication is that the Hamiltonian cycles of j-blocks must reconnect to the Hamiltonian cycles of other j-blocks and to the nucleus. The Hamiltonian cycles of the entire atom must fuse to a single large cycle, which can be closed for a neutral atom, and would correspond to closed monopole flux tube starting from the atomic nucleus. Each charge along the cycle contributes to the electric flux flowing in the monopole flux tube.

   It has been proposed (see this) that molecular bonds could be interpreted as electric flux tubes. This proposal is discussed from TGD point of view in \cite{allb/qcritdark3}. If the atoms of the molecule are ionized the Hamiltonian cycles of atoms must be reconnect by U-shaped tentacles and ionic bonds would correspond to flux tubes and presumably all chemical bonds.

Consider next the mass of the flux tube.

1. Flux tubes connecting neighboring charges could be p-adically scaled electropions with mass smaller than the mass 1 MeV of electropions and would contribute to the mass of the atom. In the case of nuclei scaled hadronic pions between nucleons having mass of order MeV are replaced by p-adically scaled elctropions. Note that electropions have mass of 1 MeV. In the case of atoms, their scaled variants should have a considerably smaller mass, which would naively correspond to the atomic p-adic length scale and mass scale of 1-10 keV. Note that 10 keV would be the scale of proposed nuclear excitation energies supported by nuclear physics X-ray anomalies. One can argue that the mass corresponds to the atomic p-adic length scale L(137) as a natural length scale for the flux tube gives and would be of order m\sim keV.
2. One the other hand, one could argue that the mass should be very small because, to my best knowledge, standard atomic physics works very well. However, the additive contribution of these masses does not affect the electronic bound state energies but only the total mass of the system. I do not know whether anyone has studied the possible dependence of the total mass of atom on the number of electrons? Does it contain an additive contribution increasing by one unit at each step along the row of the periodic table as an additional flux tube appears to the Hamilton cycle. These contributions could be also interpreted as contributions of the repulsive interactions of electrons to the energy.

As in the case of nuclei, the atomic flux tubes would act as springs, i.e. harmonic oscillators. This predicts a spectrum of excited states with scale determined by the elastic constant k or equivalent ground state oscillation frequency ω₀.

1. If ω₀ is large enough, the excitation energies would be greater than the ionization energy and there would be no detectable effects. The naive argument that ω₀ corresponds to the atomic length scale L(137) as a natural length scale for the flux tube gives ω₀\sim 1 keV. This energy scale would be for light atoms with Z≤ 9 (Oxygen) larger than the ionization energy E= Z²× 13.7 eV so that photons causing excitation would cause ionization.
2. An equally naive scaling from nuclear scale to atomic scale would suggest that the value of ω₀ is scaled from ℏω₀= 1 MeV by the ratio L(113/L(137)=2^-12 of nuclear and atomic length scales to about ω₀=.25 keV. This is not far from the above estimate.
3. How to deal with atoms with a small number of electrons, in particular Helium with 2 electrons? j=2 j-blocks are special in the sense that they do not allow sub-Hamiltonian cycle. Could the flux tube connecting the electrons be absent in this case so that only the repulsive electronic contribution would be present? Note also that the repulsive interaction energy between electrons would be smaller than the attractive interaction energy of electrons for atoms with Z=2. If this picture is correct, new atomic physics would emerge when j-block contains more than 2 electrons.

   One can also consider the possibility that the coupling to photons is weak enough, perhaps by the condition that the photon must transform first to dark photon. The behavior of multi-electron atoms in a radiation field whose photons have a low energy must have been studied.

One could also imagine that the flux tubes form a h_eff≥ h quantum coherent state, in which there are n=h_eff/h flux tubes forming the sub-tessellation of Platonic tessellation for a given j-block with vertices connected by flux tubs. Here n would be the number electrons in the j-block. The excitation energy E= ℏ_effω₀ is scaled by ℏ_eff/ℏ=n.

1. If all flux tubes associated with atom were excited at once as a phase transition, the required excitation energy would be rather large for large enough n and the excitations by photons might be possible without ionizing the atom.
2. The atoms at the left end of the row are the problem for this option and more generally, the atoms at the left end of each j-blocks. One expects that the flux tube length depends on the value of the principal quantum number N labelling the rows since the size of Platonic solid must increase with n like n². Can one assume that the mass of the spring does not depend on the row? If the elastic constant k does not depend on the row, one could consider a simultaneous collective excitation of all flux tubes so that the binding energy could increase enough.

See the article About Platonization of Nuclear String Model and of Model of Atoms 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.