Surfaceology and TGD

The inspiration coming from the work of Nima Arkani-Hamed and colleagues concerning the twistor Grassmannian approach provided a strong boost for the development of TGD. I started from the problems of the twistor approach and ended up with a geometrization of the twistor space in terms of sub-manifold geometry with twistor space represented as a 6-surface. Also the twistor space of CP₂ played a key role.

This led to rather dramatic results. Most importantly, the twistor lift of TGD is possible only for H=M⁴× CP₂ since only M⁴ and CP₂ allow twistor space with Kähler structure: TGD is unique. The most recent result is that one can formulate the twistor-lift in terms of 6-surfaces of H (rather than 6-surfaces in the product of the twistor spaces of M⁴ and CP₂). These twistor surfaces represent twistor spaces of M⁴ and CP₂ or rather their generalizations, their intersection would define the space-time surface. Therefore one can formulate the twistor lift without the the 12-D product of twistor spaces of M⁴ and CP₂.

During last years I have not followed the work of Nima and others since our ways went in very different directions: Nima was ready to give up space-time altogether and I wanted to replace it with 4-surfaces. I was also very worried about giving up space-time since twistor is basically a notion related to a flat 4-D Minkowski space.

However, in Quanta Magazine there there was recently a popular article telling about the recent work of Nima Arkani Hamed and his collaborators (see this). The title of the article was "Physicists Reveal a Quantum Geometry That Exists Outside of Space and Time". The article discusses the notions of amplituhedron and associahedron which together with the twistor Grassmann approach led to considerable insights about theories with N=4 supersymmetry. These theories are however rather limited and do not describe physical reality. In the fall of 2022, a Princeton University graduate student named Carolina Figueiredo realized that three types of particles lead to very similar scattering amplitudes. Some kind of universality seems to be involved. This leads to developments which allow to generalize the approach based on N=4 SUSY.

This approach, called surfaceology, still starts from the QFT picture, which has profound problems. On the other hand, it suggests that the calculational algorithms of QFT lead universally to the same result and are analogous to iteration of a dynamics defined in a theory space leading to the same result irrespective of the theory from which one starts from: this is understandable since the renormalization of coupling constants means motion in theory space.

How does the surfaceology relate to TGD?

1. What one wants are the amplitudes, not all possible ways to end up them. The basic obstacle here is the belief in path integral approach. In TGD, general coordinate invariance forces holography forcing to give up path integral as something completely unnecessary.
2. Surfaceology and brings strongly in mind TGD. I have talked for almost 47 years about space-time as surfaces without any attention from colleagues (unless one regards the crackpot label and the loss of all support as such). Now I can congratulate myself: the battle that has lasted 47 years has ended in a victory. TGD is a more or less mature theory.

   It did not take many years to realize that space-times must be 4-surfaces in H=M⁴×CP₂, which is forced by both the standard model symmetries including Poincare invariance and by the mathematical existence of the theory. Point-like particles are replaced with 3-surfaces or rather the 4-D analogs of their Bohr orbits which are almost deterministic. These 4-surfaces contain 3-D light-like partonic orbits containing fermion lines. Space-time surfaces can in turn be seen as analogs of Feynman graphs with lines thickened to orbits of particles as 3-surfaces as analogs of Bohr orbits.

3. In holography=holomorphy vision space-time surfaces are minimal surfaces realized as roots of function pairs (f₁,f₂) of 4 generalized complex coordinates of H (the hypercomplex coordinate has light-like coordinate curves). The roots of f₁ and f₂ are 6-D surfaces analogous to twistor spaces of M⁴ and CP₂ and their intersection gives the space-time surface. The condition f₂=0 defines a map between the twistor spheres of M⁴ and CP₂. Outside the 3-D light-like partonic orbits appearing as singularities and carrying fermionic lines, these surfaces are extremals of any general coordinate invariant action constructible in terms of the induced geometry. In accordance with quantum criticality, the dynamics is therefore universal.

   Holography=holomorphy vision generalizes ordinary holomorphy, which is the prerequisite of twistorialization. Now light-like 4-D momenta are replaced with 8-momenta which means that the generalized twistorialization applies also to particles massive in 4-D sense.

This indeed strongly resembles what the popular article talks about surfaceology: the lines of Feynman diagrams are thickened to surfaces and lines are drawn to the surfaces which are however not space-time surfaces. Note that also Nima Arkani-Hamed admits that it would be important to have the notion of space-time.

The TGD view is crystallized in Geometric Langlands correspondence is realized naturally in TGD and implying correspondence between geometric and number theoretic views of TGD.

1. Space-time surfaces form an algebra decomposing to number fields so that one can multiply, divide, sum and subtract them. The classical solution of the field equations can be written as a root for a pair of analytic functions of 4 generalized complex coordinates of H. By holography= holomorphy vision, space-time surfaces are holomorphic minimal surfaces with singularities to which the holographic data defining scattering amplitudes can be assigned.
2. What is marvelous is that the minimal surfaces emerge irrespective of the classical action as long as it is general coordinate invariant and constructed in terms of induced geometry: action makes itself visible only at the partonic orbits and vacuum functional. This corresponds to the mysterious looking finding of Figueiredo.

   There is however a unique action and it corresponds to Kähler action for 6-D generalization of twistor space as surface in the product of twistor spaces of M⁴ and CP₂. These twistor spaces of M⁴ and CP₂ must allow Kahler structure and this is only possible for them. TGD is completely unique. Also number theoretic vision as dual of geometric vision implies uniqueness. A further source of uniqueness is that non-trivial fermionic scattering amplitudes exist only for 4-D space-time surfaces and 8-D embedding space.

3. Scattering amplitudes reduce at fermionic level to n-point functions of free field theory expressible using fermionic propagators for free leptonic and quark-like spinor fields in H with arguments restrict to the discrete set of self-intersections of the space-time surfaces and in more general case to intersections of several space-time surfaces. This works only for 4-D space-time surfaces and 8-dimensional H. Also pair creation is possible and is made possible by the existence of exotic smooth structures, which are ordinary smooth structures with defects identifiable as the intersection points. Therefore there is a direct correspondence with 4-D homology and intersection form (see this). One can say that TGD in its recent form provides an exact construction recipe for the scattering amplitudes.
4. There is no special need to construct scattering amplitudes in terms of twistors although this is possible since the classical realization of twistorialization is enough and only spin 1/2 fermions are present as fundamental particles. Since all particles are bound states of fundamental fermions propagating along fermion lines associated with the partonic orbits, all amplitudes involve only propagators for free fermions of H. The analog of twistor diagrams correspond to diagrams, whose vertices correspond to the intersections and self-intersections for space-time surfaces.

For the the recent view of TGD see this and this. For the Geometric Langlands duality in the TGD framework see this .

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

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

How a rubbing with a microfiber manages to shatter the "bullet proof" windshield of Musk Cybertruck?

I learned from Heikki Hirvonen an about Musk Cybertruck windshield that was told to be "bullet proof" but turned out to be quite not so (see this). Even worse, it has been found that interaction with microfiber and the material of Musk Windshield creates some specific style of resonance that would then shatter that material. This brings to mind opera sopranos shattering wine glasses. One might think that the system considered must be critical so that very small periodic perturbations can induce very large changes if they are of the right kind and have a correct frequency.

1. Why should one worry about sopranos shattering wine glasses?

One might wonder what the point is in building complex new physics scenarios for how sopranos manage to break wine glasses. This has been understood a long time ago.

But is this really the case? We are used to thinking that physics somehow mysteriously transforms from quantum to classical on some scale. Quantum coherence, which is not possible above atomic scales, would be replaced by classical coherence on long scales. If this is assumed, glass-breaking sopranos cease to look mysterious. This thinking has actually no justifications but only restates what is a fact. When you give this thinking up, the imagined self-evidences collapse. Phenomena that were undeniably a bit strange become impossible.

In TGD, a new view of spacetime comes to rescue. The spacetime surface defines the coherence region in both classical and quantum sense. Field bodies make long-scale quantum coherence and, as its correlate, classical coherence, possible. The entire scale of the space-time surface corresponds to the scale of classical coherence and quantum coherence (i.e. related to the magnetic body). Long-scale quantum coherence accompanies classical coherence.

Classical long-scale coherence has a quantum counterpart and would be related to classical long-range gravitational and electromagnetic fields. Gravitational and em Planck's constant, whose values can be enormous compared to h, quantify the hypothesis. This windshield effect is just one example of many.

2. Background observations and assumptions

It is good to start with some background observations.

1. The super strength of the glass could mean that it does not break under the deformations studied. Throwing piece rock and rubbing with microfiber do not belong to the class of allowed deformations. So what could be the deformations that do not break the glass?

   Could it be that only deformations have been tested where pressure is applied to the windshield, i.e. an impulse current in the direction of the impulse, but not deformations involving shear, i.e. the direction of the impulse current is perpendicular to the transferred impulse. The second difference is that there is a direct contact with the microfiber.

   Rubbing creates a shear. The microfiber is pressed against the surface and pushed horizontally at the same time: both pressure in the normal direction and shear in the direction of the surface are created. For example, in hydrodynamics, the very poorly understood generation of vortices at the interface (turbulence is due to shear). The creation of vortices is forced by the conservation of angular momentum. In TGD based quantum hydrodynamics, this process is essentially a quantum critical process on macroscales (see this).

   Could it be that the strength of the glass, as defined in the way I guessed, was exactly the reason for the breakage. Would the glass be too rigid in this sense and unable to flex and break?

   Or could the glass be fragile in terms of certain types of deformations that have not been taken into account? Pressure wouldn't create them, but shear could do so. The characteristics of the microfiber could also be important.

3. What kind of model could be imagined for the phenomenon?

The TGD based model for the phenomenon relies on gravitational quantum coherence predicted to be possible in astrophysical scales and also possible quantum criticality. The gravitational magnetic bodies of both the Sun and Earth are assumed to play a key role. The reason is that macroscopic quantum coherence requires very large values of the effective Planck constant. It is assumed that the gravitational Compton frequency of the Sun defines a gravitational quantum coherence scale and sets a lower bound for the frequencies assignable to the acoustic oscillations inducing the instability of the windshield .

One can also consider other mechanisms of macroscopic quantum coherence. Cyclotron frequencies for the endogenous magnetic field of Earth are in EEG range and would correspond to energies above thermal energy and play a key role in the TGD inspired quantum biology and might be involved with the microfibers. This would require transformation of dark cyclotron radiation to sound waves and require a ferro electret property typical for organic materials. Quantum criticality making possible a generation of large $h_{eff}$ phases is involved and warping deformations possible for planar or nearly planar systems are considered as a possible realization of the quantum criticality.

1. Could the strength of the glass be defined so that when a weight is placed on the glass plate, it does not develop dent: this would mean that no curvature is generated. For example, a planar sheet of metal is a good example. It does not break easily.

   However, a flat metal or glass plate (flatness is important!) is very sensitive against development of warping, which only bends but does not curve the flat surface so that it remains flat (curvature tensor vanishes). The fluttering of a metal plate is a good example of this. Another kind example is a sheet of paper unstable against fluttering. Such time-dependent warpings would decompose to 1-dimensional plane waves propagating along the surface of the metal of glass. They would be very much like transversal sound waves.

   What is important is that warping is a critical phenomenon due the large number of flat warped surfaces (the warping profile can correspond to any differentiable function). In TGD criticality involves the development of large h_eff phases and long-range quantum correlations, which gives strong clues concerning the understanding of the situation.

2. Already Euler thought about what happens when a weight is placed on a bar bent upwards (Euler buckling) (see this). At a critical weight, a collapse occurs. This is one of the basic applications of catastrophe theory. The critical amplitude of the warping wave would be analogous to the critical weight for which the glass would break.
3. One might think that the action principle contains an energy density term that is proportional to the square of the 2-D curvature (see this) for the induced metric and vanishing for warped configurations. There would be an enormous vacuum degeneracy. Stability against deformations generating curvature requires that the coefficient of this term is very large. A lot of energy would be needed to produce a dent. But bending without curving brings in the Troyan horse.

   Action would of course also contain a term proportional to the surface area, which would correspond to the normal tension that tends to oppose the increase of the surface area. For warping, the energy would be only needed to increase the surface area. Could warping waves, possibly created by the rubbing with microfiber, lead to the breakage? Shear should provide the needed momentum and energy resonance should strengthen the warping wave.

4. What happens when the window shield breaks?

1. A catastrophe theorist might state that the system is characterized by, for example, a cusp catastrophe. When the critical shear is reached, the system undergoes a sudden transition: the system breaks down.
2. If one starts from the quantum level, the reduction of quantum coherence comes first to mind. In collapse the quantum coherence length would decrease dramatically from the size of the whole system to the size of the fragments. If the quantum coherence with the magnetic body of the glass surface takes care of the coherence of the glass, then it would have to decrease. In the h_eff distribution, the average value of h_eff would decrease.

   This is however only the outcome, not the primary cause. Long-scale quantum coherence and quantum criticality together with energy feed occurring at resonance frequency and increasing the value of h_eff would be the reasons leading to the limit at which the system collapses.

3. Why would rubbing with microfiber induce a critical shear leading to the breaking and loss of quantum coherence? Warping waves are a good candidate. The windglass would start to shake in the vertical direction. When the amplitude of the warping wave would exceed the critical limit, the result would be collapse and breaking into pieces. Rubbing with microfiber would feed into the system the necessary energy needed to generate h_eff phases and this would occur at quantum criticality associated with the warping waves.

5. Identifying the resonance frequency

This should include a frequency resonance that would correspond to the wavelength of the wave identifiable as a natural length scale for microfiber and/or glass. One would expect the flutter frequency to be on the Hertz scale and the acoustic resonance frequency of the windshield is a good guess. The sequel will certainly arouse academic head shaking, but it is based on the fact that in the TGD world, the planets and the sun form a quantum-coherent system, the effect of which can be seen on Earth at all levels, especially in biology. Second justification was given already in the beginning: our belief that we understand the classical world is based on an illusion about a mysterious transition from quantum to classical.

1. Microfiber has a wavelength λ ≈ 1 micrometer as a natural scale. The IR energy scale 1 eV of infrared photons would correspond to that and it can be assumed to be the basic scale. Could photons with this energy transform into bundles of dark photons with much longer wavelength; they, in turn, would eventually end up via intermediate steps into bundles of ordinary phonons or even into a Bose-Einstein condensate or a coherent state as a quantum analog of classical state.
2. Let's start with the Earth's gravitation (see this, this and this). The gravitational Compton length Λ_gr related to the Earth's gravitation Planck's constant is .5 cm (half of the Schwartschild radius), independently of particle mass, and the associated frequency is f_gr= 67 GHz. The frequency is quite too big. Furthermore, the Earth's gravitation is now not decisive because the warping is not in the vertical direction but closer to the tangential direction. In any case Earth's gravitation is not enough.
3. One must follow the example of Icarus and hope for better luck. The Sun's gravitational constant gives a frequency of f_gr=50 Hz, which is the average EEG frequency and important resonance frequency of the EEG central in communications between the brain and its magnetic body (see this and this). This is a reasonable frequency. The corresponding gravitational wavelength Λ= c/f_gr is half the radius of the Earth.

   Needless to emphasize that this makes no sense unless one accepts the astrophysical quantum coherence assigned with gravitation and that the oscillation takes place on the magnetic body of the glass plate on the scale of the Earth's radius.

4. A strong objection is that f_gr does not depend at all on the geometry of the glassy system, in particular on the size scale of the windshield. A reasonable expectation is that the model should apply also to shattering of wine glasses.

   A more general assumption is that the allowed frequencies are above the threshold defined by f_gr= 50 Hz defining the gravitational quantum period. At frequencies above f_gr gravitational quantum coherence would make itself visible. However, the frequencies coming as harmonics of f_gr could be especially interesting. This assumption is analogous to that appearing in the proposal for how gravitational quantum coherence could become important in classical computers (see this). In any case, the assumption f≥f_gr is rather strong and gives lower bounds for the quantal resonance frequencies.

Could the resonance (basically acoustic warping wave) correspond to a frequency above f_gr or be identifiable as the frequency of dark photons generated at the magnetic body of the Sun?

1. The phonons of the acoustic wave would couple to the dark photons, produced by shear, at the magnetic body. This is where microfiber would take the role of a Trojan horse. Note that in liquid flow for which shear occurs near boundaries, the conservation of angular momentum forces the production of vortices which in TGD based hydrodynamics would be associated with dark monopole flux tubes. Also now, Z⁰ magnetic vortices could be created.]
2. The frequencies above f_gr would be the same, but the energy of a dark photon would correspond to the energy of many "warping phonons": a Bose-Einstein condensate/coherent state analogy of phonons would be created. Assuming proton-Earth pair, one has ℏ_gr(Earth,proton) proportional to m_pM_E. This gives 1 eV energy scale, which corresponds to 1 micrometer wavelength for ordinary photons.

   The critical reader has probably noticed that the magnetic bodies of both the Sun and the Earth are included, characterized by ℏ_gr(Sun,proton) and ℏ_gr(earth,proton) respectively. The gravitational Compton length Λ_gr(Sun,proton) of Sun is R_E/2, which is the size scale for the Earth's magnetic body. Also ℏ_gr(Earth,proton) is required. Could one think that dark photons for which h_eff= h_gr(Sun,proton) are created first, and that these break up into bunches of dark photons with h_eff= h_gr(Earth,proton). The frequency would remain the same. These in turn break up into bunches of "warping phonons" with the same frequency.

3. If the propagation speed of the warping wave is roughly estimated to be the sound velocity in glass, that is v=4540m/s, then the wavelength would be Λ = v/f= 90.6 m if one assumes that the value of f is smallest possible that is f=f_gr= 50 Hz. The wavelength is quite too long as compared to the dimensions of the windshield. v should be 2 orders of magnitude smaller, coincidentally(?) the same order as the conduction velocity of the nerve impulse. Note also that a micrometer is the scale of a cell nucleus. However, f_gr=50 Hz defines only a lower bound for the quantum resonance frequency. A resonance frequency dictated by the geometry is in question and roughly scales like the inverse size of the system.

   In the case of wine glass, one expects a frequency scale, which is by two orders of magnitude larger, in the kHz scale. The E note at the hearing threshold corresponds to 20.6 Hz and, according to net source, for a wine glass some octave of E is a reasonable estimate for the resonance frequency. The resonance frequency is k:th octave of this frequency and assuming that λ is of order .1 m, one obtains an estimate that 7:th octave is a reasonable guess. This is of order kHz. In the case of a windshield, one would expect λ to be 5 to 10 times longer so that the frequency could be around 3 or 4 octaves.

6. Summary

Microfiber rubbing would induce warping waves, whose amplitude would increase in resonance and lead to shattering.

1. First, dark photons (piezoelectricity) would be generated at the solar magnetic body and then decay to bunches of dark photons at the magnetic body of Earth with energy of order eV, corresponding to the scale of the basic structure of the microfiber. Their frequency would be abe f_gr=50 Hz corresponding to the gravitational Compton wavelength of the Sun, which is of the order of the Earth's radius/2. The dependence of the resonance frequency on the geometry requires that f_gr defines only a lower bound for f and its interpretation of f_gr is as a quantum coherence period.
2. h_eff= h_gr(Earth,proton) photons would in turn decay to a "warping phonon" beam with frequency above f_gr=50 Hz. Phonons would form a coherent state or BE condensate. This could lead to an acoustic laser effect and amplification, and the result would be resonance and catastrophe, analogous to Euler buckling, when the warping amplitude becomes too large. Here, quantum criticality, which is naturally associated with warping waves, would be essential, it would make the Trojan horse effect possible.

See the article How a rubbing with a microfiber manages to shatter the "bullet proof" windshield of Musk Cybertruck? or the chapter TGD and Condensed 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.

Space-time surfaces as numbers: what could this mean from the point of view of metamathematics?

These comments were inspired by the links to Bruno Marchal's posts by Jayaram Bista (see this). The comments compare the world views behind two Platonisms, the Platonism based on integers or rationals and realized by the Turing machine as a Universal Computer and the quantum Platonism of TGD. Marchal also talks about Digital Mechanism and claims that it is not necessary to assume a fixed physical universe "out there". Marschal also speaks of mathematical theology and claims that quantum theory and even consciousness reduce to Digital Mechanism.

In the TGD Universe, the space-time surfaces form an algebra with respect to multiplication and that this algebra decomposes to a union of number fields means a dramatic revision of what computation means. The standard view of computation as a construction of arithmetic functions is replaced with a physical picture in which space-times as 4-surfaces have interpretation as almost deterministic computations. Space-time surfaces allow arithmetic operations and also the counterparts of functional composition and iteration are well-defined.

Replacement of the static universe with a Universe continuously recreating itself

It seems to me that the problems of computationalism emerge from a single ontological assumption: the "system", be it Universe in some sense or God, is fixed. In quantum TGD this is not the case. The Quantum Universe, which could be seen as a counterpart for God, is continually recreating itself and this means the unavoidable increase of algebraic complexity since the dimensions associated with extensions of rationals defining space-time regions unavoidably increase. This in turn implies evolution.

In zero energy ontology (ZEO) "small" state function reductions (SSFRs), whose sequence generalizes Zeno effect, which has no effect on physical state. SSFRs have and their sequence gives rise to conscious entities, selves. This makes possible memory: the outcome of SSFR has classical information about the initial state and also about the transition. Therefore the Universe remembers and learns consciously: one can talk about Akashic records.

This dynamical view of the Universe recreating itself and becoming more intelligent by learning about what it was before the previous SSFR is very different from the view of the Universe as a Turing machine or Universal Computer. These notions are static notions (Universe "out there") and computation is based on integers. In the TGD view one obtains an entire hierarchy of computationalisms based on the hierarchy of extensions of rationals. Even transcendental extension can be considered. TGD Universe as a counterpart of the Turing machine is also conscious and has free will.

A generalization of number

Also the notion of number generalizes from integers N to space-time surfaces. Space-time surfaces can be multiplied and summed and form an algebra. This algebra decomposes to a union of number fields with product,division, sum and subtraction. One can identify space-time surfaces forming analogs for hierarchies of algebraic integers, algebraic rationals, etc... So that the mathematics performed by Quantum Platonia is considerably more complex than counting by 5+5 fingers!

These structures are defined by the corresponding structures for function algebras and fields defined in terms of analytic functions of 8 generalized complex coordinates of H=M⁴×CP₂. One of the coordinates is a hypercomplex coordinate with light-like coordinate curves.

1. In TGD space-time surfaces are numbers. Their dynamics is almost deterministic (at singularities the determinism fails and this forces us to take space-time surfaces instead of 3-surfaces as basic objects). The space-time surface as an almost deterministic time evolution is analogous to a proof of a theorem. The assumptions correspond to the initial state 3-surface and the outcome of the theorem to the final 3-surface. Second interpretation is as analogs of deterministic computer programs. Space-time surface as a proof of a theorem is analogous to its own Gödel number as a generalized number.
2. Cognition always requires a discretization and the space of space-time surfaces ("world of classical worlds", WCW) allows a hierarchy of discretizations. The Taylor coefficients of the two analytic functions defining space-time belong to some extension of rationals forming a hierarchy. Therefore a given space-time surface corresponds to a discrete set of integers/rationals in an extension so that also WCW is discretized. For polynomials and rational functions this set is discrete. This gives a hierarchy. At the level of the space-time surface an analogous discretization in terms of an extension of rationals takes place.
3. Gödel number for a given theorem as almost deterministic time evolution of 3-surface would be parametrized by the Taylor coefficients in a given extension of rationals. Polynomials are simplest analytic functions and irreducible polynomials define polynomial primes having no decomposition to polynomials of a lower degree. They might be seen as counterparts of axioms.
4. One can form analogs of integers as products of polynomials inducing products of space-time surfaces. The space-time surfaces are unions for the space-time surfaces defined by the factors but an important point is that they have a discrete set of intersection points. Fermionic n-point functions defining scattering amplitudes are defined in terms of these intersection points and give a quantum physical realization giving information of the quantum superpositions of space-time surfaces as quantum theorems.

Could space-time surfaces replaced as integers replace ordinary integers in computationalism?

It is interesting to play with the idea that space-time surfaces as numbers, in particular integers, could define counterparts of integers in ordinary computationalism and metamathematics.

What might be the counterpart for the possibility to represent theorems as integers deduced using logic and for the Gödel numbering for theorems by integers?

1. In TGD space-time surfaces are numbers. Their dynamics is almost deterministic (at singularities the determinism fails and this forces us to take space-time surfaces instead of 3-surfaces as basic objects). The space-time surface as an almost deterministic time evolution is analogous to a proof of a theorem. The assumptions correspond to the initial state 3-surface and the outcome of the theorem to the final 3-surface. Second interpretation is as analogs of deterministic computer programs. Space-time surface as a proof of a theorem is analogous to its own Gödel number as a generalized number.
2. Cognition always requires a discretization and the space of space-time surfaces ("world of classical worlds", WCW) allows a hierarchy of discretizations. The Taylor coefficients of the two analytic functions defining space-time belong to some extension of rationals forming a hierarchy. Therefore a given space-time surface corresponds to a discrete set of integers/rationals in an extension so that also WCW is discretized. For polynomials and rational functions this set is discrete. This gives a hierarchy. At the level of the space-time surface an analogous discretization in terms of an extension of rationals takes place.
3. Gödel number for a given theorem as almost deterministic time evolution of 3-surface would be parametrized by the Taylor coefficients in a given extension of rationals. Polynomials are simplest analytic functions and irreducible polynomials define polynomial primes having no decomposition to polynomials of a lower degree. They might be seen as counterparts of axioms.
4. One can form analogs of integers as products of polynomials inducing products of space-time surfaces. The space-time surfaces are unions for the space-time surfaces defined by the factors but an important point is that they have a discrete set of intersection points. Fermionic n-point functions defining scattering amplitudes are defined in terms of these intersection points and give a quantum physical realization giving information of the quantum superpositions of space-time surfaces as quantum theorems.

Adeles and Gödel numbering

Adeles in TGD sense inspire another interesting development generalizing the Gödelian view of metamathematics.

1. p-Adic number fields are labelled by primes and finite fields induced by their extensions. One can organize the p-adic number fields to adele and the same applies to their extensions so that one has an infinite hierarchy of algebraic extensions of the rational adele. TGD brings something new to this picture.
2. Two p-adic number fields for which elements are power series in powers of p₁ resp. p₂ with coefficients smaller than p₁ resp. p₂, have common elements for which expansions are in powers of integers n(k₁,k₂)= p₁^k₁×p₂^k₁, k₁>0, k₂>0. This generalizes to the intersection of p₁,p₂,..., p_n. One can decompose adeles for a union of p-adic number fields which are glued together along these kinds of subsets. This decomposition is general in the description of interactions between p-adic sectors of adeles. Interactions are localized to these intersections.
3. Mathematical cognition would be based on p-adic numbers. Could one think that ordinary integers should be replaced with the adelic integers for which the p_i:th factor would consist of p-adic integers of type p_i.

   These integers are not well-ordered so that the one cannot well-order theorems/programs/etc... as in Gödel numbering.

   The number of p-adic integers is much larger than natural numbers since the pinery expansion can contain an infinite number of terms and one can map p-adic integers to real numbers by what I call canonical identification. Besides this one has fusion of various p-adic number fields.

An interesting question is how this changes the Gödelian views about metamathematics. It is interesting to play with the idea that space-time surfaces as numbers, in particular generalized integers, could define counterparts of integers in ordinary computationalism and metamathematics.

Numbering of theorems by space-time surfaces?

What might be the counterpart for the possibility to represent theorems as integers deduced using logic and for the Gödel numbering for theorems by integers?

1. In TGD space-time surfaces are numbers. Their dynamics is almost deterministic (at singularities the determinism fails and this forces us to take 4-D space-time surfaces instead of 3-surfaces as basic objects). The space-time surface as an almost deterministic time evolution is analogous to a proof of a theorem. The assumptions correspond to the initial state 3-surface and the outcome of the theorem to the final 3-surface. Second interpretation is as an analog of a deterministic computer program. The third interpretation as a biological function. Space-time surface as a proof of a theorem is analogous to its own Gödel number, but now as a generalized number. One can define the notions of prime , integer , rational and transcendental for the space-time surfaces.

   The counterparts of primes, determined by pairs of irreducible polynomials, could be seen as axioms. The product operation for space-time surfaces generates unions of space-time surfaces with a discrete set of intersection points, which appear as arguments of fermionic n-point functions allowing to define fermionic scattering amplitudes. Also other arithmetic operations are possible.

   Also functional composition, essential in computationalism, is possible. One can take any analytic h(z) function of a complex coordinate z and form a functional composite h(f₁(...)) or h(f₂(...)). One can also iterate this process. This would make it possible to realize recursion, essential in computationalism. This iteration leads also to fractals.

2. Cognition always requires a discretization and the space of space-time surfaces ("world of classical worlds", WCW) allows a hierarchy of discretizations. The Taylor coefficients of the two analytic functions f₁,f₂ defining space-time belong to some extension E of rationals forming a hierarchy. Therefore a given space-time surface corresponds to a discrete set of integers/rationals in an extension of rationals so that also WCW is discretized for given E. For polynomials and rational functions this set is discrete. This gives a hierarchy. At the level of the space-time surface an analogous discretization in terms of E takes place.
3. Gödel number for a given theorem as almost deterministic time evolution of 3-surface would be parametrized by the Taylor coefficients in a given extension of rationals. Polynomials are simplest analytic functions and irreducible polynomials define polynomial primes having no decomposition to polynomials of a lower degree. Polynomial primes might be seen as counterparts of axioms. General analytic functions are analogous to transcendentals.
4. One can form analogs of integers as products of polynomials inducing products of space-time surfaces as their roots. The space-time surfaces are unions for the space-time surfaces defined by the factors but an important point is that they have a discrete set of intersection points. Fermionic n-point functions defining scattering amplitudes are defined in terms of these intersection points and give a quantum physical realization giving information of the quantum superpositions of space-time surfaces as quantum theorems.

See the articles TGD as it is towards end of 2024: part I, TGD as it is towards end of 2024: part II, and About Langlands correspondence in the TGD framework.

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.

More precise views about some aspects of the icosa tetrahedral realization of the genetic code

The article Progress in the understanding of the icosa tetrahedral realization of the genetic code provides an answer to the question how many icosahedrons, octahedrons and tetrahedrons meet at the vertex of ITT: the answer comes by studying the vertex figure of ITT: these numbers are 12, 30, and 20. The study of the vertex figure of ITT suggests that the ITT can be constructed as a "blow-up" of the icosahedral tessellation (IT) by replacing icosahedral vertices with tetrahedra and dodecahedral vertices by pentagons and adding between icosahedral tetrahedra and dodecahedra octahedra as analogs of edges. Icosahedral and dodecahedral bioharmonies correspond to 12-note resp. assignable to Western resp. Eastern music. One can ask whether octahedral 4-codons should also be allowed.

The picture provided by RID is consistent with the earlier notion of "super-icosahedron". The model of the genetic code generalizes: besides the icosahedral Hamilton cycles (HCs) and codons for the three icosahedral codes and the tetrahedral HC and corresponding codons, also a unique dodecahedral HC and associated 5-codons plus pentahedral HC and codons are in principle possible. The fundamental region deduced from RID corresponds to a sequence of 10 or 12 DNA codons as proposed already earlier on the basis "super-icosahedron model" (see this).

The model allows us to understand the symmetry breaking of genetic codons. In particular, tetrahedral codons correspond to 3 stop codons and the codon coding for trp. A given codon corresponds either to I/T or D/pentahedron. The fundamental region represents a sequence of 10 or 12 DNAs so that all codons of the Hamiltonian cycle are used and the HC corresponds to a section of DNA. Fundamental region represents both DNA strands.

See the article Progress in the understanding of the icosa tetrahedral realization of the genetic code or the chapter About honeycombs of hyperbolic 3-space and their relation to the genetic code.

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.

Some strange astrophysical and cosmological findings from the TGD point of view

Anomalies in both astrophysics and cosmology have been rapidly accumulating during the last years. The most recent astrophysical anomalies that I have encountered relate closely also to the TGD view of biology and consciousness.

1. There is evidence that Earth had a ring before the Cambrian Explosion (see this). The proposed explanation based on the TGD variant of the Expanding Earth hypothesis (see this). The ring would have existed already before the Cambrian Explosion along the equator but the rapid expansion of the Earth (radius was doubled) implied that Earth catched the ring resulting in a large number of meteor craters along the equator and a temporary cooling of the climate caused by the shadow of the ring.
2. The scent of space (see this) is a strange phenomenon reported by astronauts. It is now known that olfaction involves at the fundamental level infrared light. The scent could relate to so called PAHs (see this and this), which are aromatic molecules with several rings. PAH are known to produce the so called unidentified infrared bands (UIBs) for a radiation arriving from the interstellar space, even from regions containing no stars or involving no star formation. The mechanism could be non-chemical and involve the generalization of Pollack effect transferring ordinary protons to dark protons at the magnetic body and their dropping back back and in this way producing the infrared photons.
3. The surprisingly strong evidence that the position of Mars (see this) correlates strongly with the stock market crashes is in conflict with the basic beliefs of physicalist. During the crashes the distance of Mars tends to be at the other side of the Sun than Earth. The TGD based explanation would rely on the loss of the predicted quantum coherence in astrophysical scales due the splitting of the monopole flux tubes connecting Earth and Mars (this mechanism might be at work although Mars has no large scale magnetic field). This would lead to a partial loss of quantum control at the level of collective consciousness and lead to panic reactions.

There are also numerous cosmological anomalies.

1. The surplus of deuterium nuclei in the cosmic ray spectrum (see this) is difficult to understand in the standard physics framework. There is also evidence for pairs of deuterium-anti-deuterium nuclei and even helium-antihelium nuclei (see this).

   The TGD inspired model of stars discussed in (see this) deviates dramatically from the standard model and proposes that M₈₉ hadron physics with mass scale 512 times that for the ordinary hadron physics might be involved. There is some evidence for M₈₉ mesons from LHC. The decay of the monopole flux tubes carrying dark M₈₉ nuclei as analogs of ordinary nuclei along the solar surface would produce solar energy and solar wind. The flux of M₈₉ nuclei arriving along monopole flux tubes from the Milky Way giant blackhole would provide the energy and serve also as a metabolic energy source of dark nuclei in the solar interior forming a state analogous to a cell.

   The decay of M₈₉ mesons could produce nucleon-antinucleon pairs but the production of deuterium-anti-deuterium pairs and even Helium-anti-Helium pairs is not so plausible. An alternative explanation is the decay of monopole flux tubes containing sequences of ordinary or M₈₉ anti-nuclei.

2. There is evidence that galaxies rotating in different directions relative to the Milky Way have different redshifts and that the difference increases with distance: the explanation in terms of tired light (see this) does not seem plausible. This suggests that they have slightly different Hubble constants. TGD suggests explanation in terms of the variation of slightly different values of effective Planck constant h_eff near to h in both cases: also the fluctuations of CMB temperature and accelerated expansion could be understood in this way (see this).

See the article Some strange astrophysical and cosmological findings from the TGD point of view.

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

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

Did Earth have a ring before the Cambrian Explosion and did the rapidly expanding Earth catch the ring?

I encountered a link to a very interesting popular article "Did Earth have a ring like Saturnus?" (see this) telling about the article "Evidence suggesting that earth had a ring in the Ordovician" of Tomkins et al published in Earth and Planetary Science Letters (see this).

The proposal is that the ring would have formed as a large asteroid was caught by the Earth. The tidal forces of Earth would have destroyed the asteroid so that it became a ring along the equator of the Earth. The ring created a shadow. If it formed along the equator, it could have initiated global cooling about 465 years ago: the so-called Hirnantian Icehouse followed 20 million years later. There are as many as 21 meteor strikes along the equator and this is very implausible if the meteors would have arrived from random directions.

This is a highly interesting finding from the point of view of the Expanding Earth hypothesis inspired by TGD. About 524 million years ago the so-called Cambrian Explosion occurred. Highly evolved multicellular life forms suddenly emerged. A possible explanation of this mystery could be a fast expansion of the Earth: radius would have increased by about factor 2: these fast expansions could be the TGD counterpart of smooth cosmic expansion. This would have led to the bursting of underground oceans containing the multicellular life to the surface of the Earth. It is not difficult to invent objections against the idea but the new physics predicted by TGD allows to circumvent them and the model explains a large number of anomalies to the evolution of Earth. For the TGD view of the Expanding Earth Hypothesis see for instance this and this .

The ring would have formed about 60 million years later and existed for a time measured 10 million years as a natural unit. Could one think that Earth had already before this time a ring and the Expanding Earth caught the ring? This could explain why the ring was along the equator, something not obvious if the ring was formed by the asteroid rotating around the Earth. This would have produced the 21 meteor strikes along the equator, a phenomenon which is extremely implausible if the meteors did not originate from the same source. The expansion of the Earth would have gradually increased the width of the shadow and the collision with the ring would have generated dust in the atmosphere and caused an additional shadowing effect causing the cooling of the climate.

See the article Expanding Earth Hypothesis and Pre-Cambrian Earth 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.

Is it really possible to formulate all geometric statements as statements of algebraic geometry?

The TGD view of the geometric Langlands correspondence states that there is a correspondence between the algebraic, essentially linguistic view of physics and the geometric view of physics relying on vision. This leads to a kind of language game. The highly non-trivial challenge is to find whether the geometric picture can be formulated using the language of algebraic geometry involving generalized complex variables of which one is hypercomplex and real.

First of all, one must find out whether the known algebraically universal extremals appearing for practically any conceivable action, deduced by geometric and symmetry arguments, have a simple algebraic description as the roots (P,Q)=(0,0) where P and Q are analytic functions of generalized complex coordinates of H=M⁴× CP₂. This is not at all obvious. One should carefully check whether CP₂ type extremals, cosmic strings and monopole flux tubes, and massless extremals allow this kind of formulation.

Inequalities are part of geometric description and involve in an essential manner the notion of distance. The representation of topological boundaries gives rise to inequalities. In TGD a long standing question is whether one should allow boundaries and whether the boundary conditions guaranteeing conservation laws indeed allow space-time boundaries. For instance, could one eliminate CP₂ type extremals defining wormhole contacts glued to the Minkowskian background and leaving partonic orbits as boundaries (see this).

1. The problem is that well-ordering required by inequalities characterizes only real numbers: the notion of inequality is not algebraically universal. Inequalities have no natural place in pure algebraic geometry involving complex numbers or p-adic numbers. In TGD, the natural variables are generalized complex coordinates and inequalities cannot be represented for the complex numbers using only complex analytic functions.

   In TGD, the light-like hypercomplex coordinate u is however an exception. u is real and inequalities make sense for it. For instance, the segment u₁≤u≤u₂ can be defined in the semialgebraic context and the simplest situation corresponds to a position dependent time interval x-u₁≤ t ≤ x+u₂ or propagating pulse. The real part Re(w) of the complex coordinate w of the space-time surface defining the analog of the real axis in complex analysis would be a second coordinate of this kind and could be assigned to the partonic 2-surface.

2. Also in the p-adic topology well-ordering is absent and inequalities would be represented in terms of norm but this is not a notion of algebraic geometry. Only the discrete subsets of p-adic numbers defined by powers of p are well-ordered and inequalities can be defined for them. The hierarchy of discretizations as cognitive representations defined by extensions of rationals could however allow to overcome this problem by reducing them to inequalities.

The notion of semi-algebraic geometry makes it possible to represent these observations formally.

1. In semi-algebraic geometry inequalities are allowed in the real case but do not make sense for complex and p-adic numbers. In TGD, semialgebraic geometry would make sense for the regions of space-time surface for which the generalized complex coordinates of H or space-time surface are real.

   All inequalities should be formulated for the real sub-manifolds, which for ordinary complex 4-manifolds are 2-D. This is the case now. String world sheets parameterized by light-like coordinates u and v, would be naturally 2-D surfaces of this kind but the coordinate v does not appear as the argument of the functions (P,Q). Only the inequalities relating to u seem to make sense.

2. Hamilton-Jacobi structure (see this) means a slicing of M⁴ by pairs of strings world sheets and partonic 2-surfaces and would allow to generalize this representation to the interior of the space-time surface. Could the inequalities related to the geometry of preferred extremals implied by holography=holomorphy correspondence reduce to this kind of inequalities? The two real coordinates u and x= Re(w) could have interpretation as local choices of light-like direction and polarization direction and inequalities in this sense would be consistent with the notion of semialgebraic geometry.

   An interesting question is whether symplectic structure, which is basic element of the WCW geometry and can be seen as a companion of the generalized complex structure, could correspond to the decomposition of the complex space-time coordinate as w= P+iQ and hypercomplex coordinate as (u,v) such that (P,Q) and (u,v) define canonically conjugate coordinate pairs is consistent with the Hamilton-Jacobi structure. Note that the two real coordinates u and x= Re(w) could have interpretation as local choices of light-like direction and polarization direction and inequalities in this sense would be consistent with the notion of semialgebraic geometry.

Could one get rid of inequalities altogether by a suitable choices of the real coordinate variants (u,x)? There is indeed a well-known trick allowing to get rid of an inequalities representable in the form t≥ 0 by a change of the coordinate variable as a replacement t → T= t². Only the points with t≥ 0 are allowed by mere reality conditions. This trick might work to inequalities involving u and x.

See the article About Langlands correspondence in the TGD framework 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.

Scent of space

Heikki Hirvonen sent a link to a FB post about the scent of space (see this). He is the content of the FB post.

"Astronauts say that space smells like gunpowder and burnt steak. It being a vacuum and all, space isn't often thought of as having a scent of its own. And while no one has directly smelled outer space, exposure without a helmet would be fatal. Many astronauts have reported that it smells like a mix of gunpowder and burnt steak. The odor is most noticeable after an astronaut returns to their spacecraft through the airlock and removes their helmet, at which point the lingering scent can be detected by both the astronaut who had been outside the ship and their crewmates who remained aboard.

It has been theorized that the source of space's scent is dying stars, which release molecules called polycyclic aromatic hydrocarbons, a chemical compound also found in coal, oil, and food as they near the end of their existence.

There's even a cologne named Eau de Space based on the smell, which was originally synthesized by biochemist Steve Pearce at NASA's behest to better prepare astronauts for every aspect of the job. Based on his interviews with astronauts who had been to space, Pearce described the aroma as hot metal, burnt meat, burnt cakes, spent gunpowder, and welding of metal."

PAHs (polycyclic aromatic compounds) look like a possible explanation. They would produce IR radiation assigned with unidentified infrared bands (UIBs) and since the odour sensation at the fundamental level is based on IR light, UIBs could produce the sensation.

Consider first PAHs. I have considered PAHs several times while developing TGD view of quantum biology.

1. PAHs are obtained by fusing together organic molecules involving aromatic rings and are produced in burning and are often poisonous. The list of the basic properties of PAHs \cite{bbio/PAH,PAH1} (see this) can be found for instance in (see this).

   The properties of PAHs have led to the PAH world hypothesis stating that PAHs are predecessors of the recent basic organic molecules. For instance, the distances of aromatic molecules appearing as basic building bricks are the same as distances of DNA base pairs.

2. So called Unidentified Infrared Bands (UIBs) of radiation around IR energies E ∈ {.11 , .20, .375} eV arriving from the interstellar space are proposed to be produced by PAHs. The UIBs can be mimicked in the laboratory in reactions associated with photosynthesis producing PAHs (see this and this).
3. PAHs are detected in interstellar space. James Webb telescope found that PAHs exist in the very early cosmology 1 billion years before they should be possible in the standard cosmology! Furthermore, PAHs exist in regions, where there are no stars and no star formation (see this).

The interpretation of the findings in the TGD framework is discussed in (see this) and this)!

1. In the TGD framework, a possible explanation would be that the nuclei involved are not produced by hot fusion in stars but by dark fusion occurring at rather low temperatures. PAH world as a predecessor of recent chemical life would have developed in interstellar space.
2. The original TGD inspired proposal was that dark fusion preceded "cold fusion" associated with prestellar objects preceded ordinary nuclear and ignited hot fusion leading to the formation of the stellar core (see this). The numerous anomalies related to the standard model of the Sun assuming that the energy is produced in the core of the Sun suggest that something in the nuclear physics of the Sun is badly misunderstood. The analysis of the anomalies in the TGD framework leads to a rather radical proposal assuming that also the interior of the Sun is at a rather low temperature and dark fusion prevails in this region. The core would be a quantum system analogous to the cell interior or even cell nucleus (see this). Needless to say this would completely change our views about the Sun and of life and consciousness.

   Sun would be in a well-defined sense a living system needing metabolic energy feed. Solar surface would contain a layer producing both solar wind and solar energy and would receive metabolic energy feed from outside, for instance from galactic black holes along monopole flux tubes. This view requires taking seriously the prediction of TGD that ordinary hadron physics is accompanied by several scaled variants of hadron physics. In particular, M₈₉ hadron physics with a mass scale which is 512 times higher than for ordinary hadron physics (see this). The transformation of M₈₉ nuclei to ordinary nuclei would produce solar energy and also provide the Sun itself with metabolic energy.

3. In the TGD framework, this picture suggests that PAHs might have been created as an outcome of dark fusion in interstellar space. PAHs might have made possible a primitive form of metabolism and photosynthesis (see this and this) at relatively low temperatures prevailing in interstellar space. This would have made it possible for plasmoids as primitive life forms to store metabolic energy chemically. The hypothesis about plasmoids as predecessors of the recent chemical life forms in the Earth's ionosphere is discussed in (see this).
4. Dark proton sequences, providing a universal representation of the genetic code, based on a completely unique hyperbolic tessellation known as icosa tetrahedral tessellation (see this), would have realized the genetic code for the plasmoids and the chemical code would have emerged later. Also the recent realization of the genetic code would involve sequences of dark protons, with genetic codons represented as dark protons triplets. The triplets of dark cyclotron photons forming quantal units would induce resonant transitions between the dark codons: 3-resonance would be in question. Genes with N codons would give rise to 3N-resonances and a universal addressing in the communications by dark 3N-photons with the message coded to frequency scale modulation.

This does not yet say anything about how PAHs and UIBs could relate to the scent of space.

1. Luca Turin (see this) discovered that the absorption of infrared light produces odour perception. The earlier view was that a purely chemical mechanism involving the attachment of odorant molecules to the odour receptors is the mechanism of the odour perception. At the basic level the odour sensation would be however produced by infrared light. In particular, space odout might be produced by the infrared light emitted by PAHs. This makes possible remote odour perception.
2. In principle, also the solar radiation at infrared wavelengths could induce the sensation of odour. The odorant molecules could be present in the air inside the helmet. They would be excited by UIB light arriving from interstellar space and emit IR photons as they return to the ground state. This would generate the sensation of the scent of space. In the long run sensory adaptation would lead to the situation in which the scent of space is not perceived anymore. When the astronaut is outside the aircraft sensory adaptation takes care that the sensation is not felt. The sensation is most intense when the helmet is removed after the return to the spacecraft.

Whether the UIBs are produced by ordinary chemical transitions associated with photosynthesis or its predecessor or whether they involve new physics suggested by TGD, is an interesting question to ponder.

1. This relates interestingly also to the Pollack effect, which is most effectively induced by infrared light. Pollack effect is indeed central in the TGD inspired quantum biology and is a non-chemical transition in which photons provide the energy kicking protons to the "magnetic body" of the molecule. It is also essential in photosynthesis and in a temporary non-chemical storage of metabolic energy to the magnetic body of the system.

   In the Pollack effect and its TGD inspired generalizations, the photon would increase the value of effective Planck constant h_eff for the protons. This could make the Compton length of the radiation, emitted as a dark photon as the proton transforms to ordinary proton, very long.

2. Could the large value of h_eff make possible space scent even without the presence of PAHs in the nearby environment? Smell is usually regarded as a sense restricted to rather short scales. Basically it would be infrared vision. Could this make it possible to smell over astrophysical distances?!

   In fact, insects are known to be able to smell over distances measured in tens of kilometers. Could the real reason be that the smell sensation is also now mediated by (dark) infrared photons rather than by diffusing odorant molecules? I learned from my chemist friend that the odour of vanilla cannot be produced artificially. Could one understand this in terms of dark IR photons?

See the article About long range electromagnetic quantum coherence 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.

Why the redshifts of galaxies rotating in opposite directions relative to Milky Way should have different redshifts?

In his Youtube video, Anton Petrov (see this) talks about the notion of tired light proposed by Lior Shamir (see this) as an explanation for some strange findings about galactic redshifts. The observation that the redshifts of distant galaxies are different depending on whether they rotate in the same or opposite direction to the Milky Way is very interesting and unexpected. Asymmetry also increases with distance. Rotation affects the redshift, but the effect should be very small.

Tired light as a mechanism producing cosmological redshift is suggested as a possible explanation of the findings. As described by Anton Petrov, this mechanism leads to many long-known contradictions with cosmological observations, and in my opinion it can be safely forgotten. However, the effect may be real, even though it has been reported by only one researcher hitherto.

Redshift is real and in general relativity it would most naturally be interpreted as a direct evidence that energy is not conserved. In TGD, where spacetimes are surfaces, the explanation for the cosmological redshift is much simpler and consistent with conservation of energy. The 4-D tangent spaces of the 4-D surfaces related to the 3-surfaces corresponding to the detector and the source differ from each other by the Lorentz transformation and this produces an analogy of the Doppler effect. The energy of the photons is preserved, but one could say that they are perceived as if from systems in different states of motion. The projections of the three-surface tangent spaces M⁴ to the sender and the receiver differ by the Lorentz transformation and this results in a redshift.

A possible TGD based explanation for the observed effect relies on many-sheeted spacetime. The galaxies rotating in opposite directions could correspond to space-time sheets for which Hubble constants are slightly different at the moment of the emission of the radiation. In the GRT framework this would mean that the density of matter is slightly different for these space-time regions.

I have proposed that the fluctuations of h_eff at quantum criticality induce fluctuations of density and temperature. If the regions of many-sheeted space-time tend to contain galaxies with the same direction of rotation, one can imagine that the h_eff depends on the direction of rotation. The CMB temperature behaves as T(a)=T₀(a₀/a) and a naive dimensional guess for the dependence of h_eff is T₀(h_eff)= (h_eff/h)T₀. This would scale the energy density of radiation by a factor (h_eff/h)⁴ and the following little calculations show that the value of H increases.

Using Einstein's equations, Hubble constant can be expressed as

H²== [(da/dt)/a]²=(8πG/3)ρ -k/a²+Λ/3 ,

The expression for Hubble constant reads as

H(a)=H₀X^1/2 ,
X=Ωka^-2+Ω_m a^-3 + Ω_ra^-4 +Ω_DEa^-3(1+w) .

Here parameter w depends on the model of dark energy and w=1 is a possible value. From this formula one sees that if the temperature of CMB background is proportional to h_eff, regions of larger h_eff have a large Hubble constant.

The critical density and density parameter are defined

ρ_c=3H²/8πG, Ω =ρ/ρ _c .

The parameters Ω_k (k∈{0,-1,1}, Ω_m, Ω_r, and Ω_DE refer to various contributions to the density corresponding to the curvature of 3-space (k=0 corresponds to flat space), matter, radiation and dark energy. If dark energy corresponds to the cosmological constant, one obtains

ρ_c= 3H₀²/8πG ,
Ω_m== ρ_m0/ρ_c = (8π G/3H₀²)ρ_m0 , Ω_k== -k/a₀²H₀², Ω_Λ== Λ/3H₀² .

The question is whether the measured two different values of H could reflect slightly different temperatures for the Hubble constant in some space-time regions induced by different values of h_eff and whether these regions could correspond to regions containing preferentially galaxies, which rotate in the same or opposite direction as the Milky Way. Some kind of parity violation in cosmic scales is suggestive.

This mechanism could also provide insights to two other cosmological problems.

1. The proposal might explain the observed two values of the Hubble constant. The two Hubble constants could correspond to stars of galaxies rotating in different directions as compared to the Milky Way.

   Note that TGD suggests the formula for G in terms of the fundamental length scale as G= kR²/h_eff. This would induce factor 1/h_eff to Ω_m and Ω_r but the conclusions would not be changed in the radiation dominated phase.

2. Could the accelerated expansion of the Universe could relate to the increase of h_eff suggested by the number theoretic evolution possibly explaining the apparent disappearance of the baryonic matter. One expects that the average value of h_eff increases and that this corresponds to the gradual transformation of the baryonic matter to dark matter in the TGD sense.

   From the formula for the Hubble constant one can calculate the dH/dt as

   dH/dt= -H²(1+q) , q== -[d²a/dt²]a/ (da/dt)² .

   From this one can estimate the change of the parameter q caused by the time evolution of h_eff. The additional term Δ q in q due to T₀ ∝ h_eff dependence would be

   Δ q=H₀²/H²T₀× 4Ω_r a^-4 (dh_eff/dt)/h_eff .

   If h_eff increases, the sign of Δ q= -a(d²a/dt²)/(da/dt)² is positive so that the acceleration is positive.

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

Could the position of Mars have an effect on stock market?

In the group Unifying Physics, Anthony Moore (see this) sent an extremely interesting link to his article published in Academia.edu (see this).

I glue below his own summary of his claimed findings.

"Before reading the content, it is important to take into account a recent study published in Nature Communications in March of 2024, roughly 5 years after this idea was first introduced to the public. In that study published in March of 2024, researchers discovered that Mars is exerting a gravitational pull on earth's tilt, exposing earth to warmer temperatures and more sunlight, all within a 2.4 million year cycle. I assert that this allows us to surmise that, even within smaller timeframes, Mars is still exerting a gravitational pull on earth's axial tilt, enough to raise temperatures and affect human behavior, even investor sentiment. Citing the fact of numerous studies that link irritability and negative mood states to warmer temperatures, I can establish an axiom. This perspective should help the reader move beyond the preconceived notion of absurdity and realize that this has scientific merit. This paper lays out the 25 major stock market crashes and downturns in US history.

The data shows a 100 percent correlation between such events and Mars position in relation to earth. Every stock market crash and major stock downturn in US history has happened when Mars was orbiting behind the sun from earth s point of view. When Mars is going further out from earth, it is also when Mars's gravity is pulling Earth s axial tilt towards the sun, possibly bringing warmer temperatures, which should affect investor sentiment most negatively, presuming that warmer temperatures relative to the mean affect cognitive function and trigger some variant of irritability or pessimism. There are studies that corroborate this dynamic between warmer temperatures and negative mood states. As Mars gets closer to earth, Mars s gravity is pulling earth s axial tilt away from the sun, bringing presumably cooler temperatures, and less negative mood outcomes, which may explain why major stock market crashes never happen during that phase of Mars s orbit."

Let us first look at the data.

1. The article discusses 25 stock crashes including also short 1 day long events in the financial history of the US. The article gives the year, month and day of month for the events and also links to the tables containing the basic data about crashes. Besides this the data about the relative position of Mars and Earth are given for each case.

   In one case (March 12, 13, and 16, 2020) the are 3 mini crashes within few days and in another case 2 crashes within 2 months (October 9 and December 1, 2008) so that from the perspective of the hypothesis one must to them as a single event so that there are 22 rather than 25 crashes.

2. If the crashes occur randomly, half of them would occur when the planar angle φ for positions of Mars and Earth is larger than π2: this means that the distance between Mars and Earth is above a critical value whose geometric interpretation is rather obvious. This criterion is applied in the examples discussed in the article and can be formulated as a condition for the distance of the Mars and Earth (1/4:th of the orbit length). The claim is that all the studied 25 crashes in the economic history of the US satisfy the claim. Professional statisticians should check the claimed correlations between the position of Mars relative to Earth and stock market crashes to find whether they are genuine.
3. The data used seems to cover the history rather well. Indeed, in the web one can find is a list of 12 contemporary stock market crashes in the US beginning from year 1929 (see this). The events have occurred 1929, 1937, 1962, 1987, 1989 (mini crash), 1990, 2999, 20008, 2010, 2011, 2015, 2020 (corona). The number of events studied in the article is 22 and roughly twice the number of events listed in the table.

   This page contains also a list of global events that also affected the US. This list contains 6 cases (1772, 1796-1797, 1873 and 2001, 2002, 2018) of which 3 have occurred after millenia.

4. In Wikipedia there is a list of 55 stock market crashes, which are fast events and bear markets, which are slow and long lasting (see this) starting from year 1637: this list contains also the events that have occurred outside US.

The reason for why I take these claims seriously is that there is a lot of earlier data about unexpected correlations between planetary physics and human collective behavior. For instance, Russian physicist Shnoll carried his entire life's work by charting this kind of correlations at molecular and even nuclear physics level (see this, this, this, this, this, and this). I have discussed the Shnoll effect in the TGD framework in (see this): quantum gravitation in planetary scale is predicted to be crucial for understanding life and consciousness and could explain the claimed correlations.

Anyone can check whether the claims are indeed true and also check whether the claim holds true for a more extensive global data including 55 events. It is enough to consider the time evolution of the azimuthal angle difference defining the angular distance φ of Mars and Earth using a simple model assuming circular orbits in the same plane. φ corresponds to the actual distance of Mars and Earth. From this model one can check whether the claim holds true for the events listed in the above mentioned tables.

1. One must first make clear the difference between sidereal and synodic periods. Sidereal period is defined in a system for which the rest system is defined by distant stars. In what follows, the Sun is approximated as a system at rest so that an approximation to the sidereal period would be in question. I will use the term orbital period and keep in mind that an approximation is in question.

   Synodic period is the time required for a body within the solar system, such as a planet, the Moon, or an artificial Earth satellite, to return to the same or approximately the same position relative to the Sun as seen by an observer on the Earth. Therefore the rest system is defined with respect to Earth-Mars system. For Earth-Mars system the synodic period is 780 days and 50 days longer than 2 years. For instance, the closest approach configuration repeats with the synodic period (Earth, Mars and Sun are at the same line). The synodic period is what matters in the recent case.

2. The orbital period of Mars is T_M=1.882 years (687 days) and roughly twice the period T_E=1 year of Earth (365 days): one has T_E/T_M= .5313. The eccentricities of the orbits of Mars resp. Earth are .093 resp. 0.017. If the crashes occur completely randomly and if the condition for the critical distance between Mars and Earth is φ≥ π2 is larger than π/2 then the number roughly 1/2 of all crashes should have φ≥ π/2. Also the orbital planes fail to be quite identical.

   Mars comes closest to Earth every other year, around the time of its opposition, when Earth is sweeping between the Sun and Mars. The eccentricity of the orbit of Mars implies exceptionally close oppositions of Mars happen every 15 to 17 years, when we pass between Mars and the Sun around the time of its perihelion (closest point to the Sun in orbit). Also the situation in which the distance of Mars and Sun are largest from the Sun are especially interesting from the perspective of the hypothesis. Intriguingly, the time t=17τ corresponds to 17+14.994 ≈ 32 years and defines in a good approximation a period for the system. This periodicity might be visible in the time series for the crashes.

3. One can test the hypothesis by using an approximation in which Mars and Earth have circular orbits in the same plane. The azimuthal angles φ_M and φ_E for the positions of Mars in the plane with respect to Sun are in this approximation given by

   φ_M= ω_Mt = 2π t/T_M ,
   φ_E= ω_Et = 2πt/T_E .

   Note that these angles are defined modulo 2π.

4. The condition that the difference φ of these angles is larger than π/2 reads as

   φ= |φ_M-φ_E|= 2π |t/T_E-t/T_M| ≥ π/2 .

   This translates to the condition

   t/T_E |1-T_E/T_M| ≥ 1/4 .

5. It is useful to study the approximate situation in which one has T_E=T_M/2. In this case the situation is strictly periodic and synodic and sidereal periods are identical. In this case, the dynamics is periodic with period T_M= 2T_E and one has

   t/T_E≥ 1/2.

   If Mars and the Earth are closest to each other in the initial situation (φ=0), the critical period for which the condition φ≥ π2 corresponds to the range [1/2,3/2] year. These critical periods repeat with 2-year periodicity. T_M is slightly smaller than 2T_E so that the growth of the angular distance between Mars and Earth is slower and the synodic period is larger than 2T_E.

6. In the more general case the situation is not quite periodic and the points at which Mars and Earth are nearest to each other repeat with period τ= T_E/(1-T_E/T_M) ≤ 2T_E. Earth catches Mars before it has rotated a full period. For a given value t=nτ the critical period is

   1/4(1-x) ≤ t_R/t_E ≤ 3/4(1-x) ,
   x= T_E/T_M ≈ .5313 .

   Here t_R= t-nτ refers to the reduced time coordinate having values in the interval [0,τ]= [0, T_E/(1-T_E/T_M)]. One can use τ as a unit of time and check whether the crashes tend to occur in the intervals [τ/4,3τ/4] for t= nτ, n=1,2....

7. Assuming that orbital rather than synodic period matters and in the approximation considered, the period τ has length about τ=780 days and indeed corresponds to the synodic period.

   The duration of the critical period is t_cr=τ/2 ≈ 1 year 25 days. The critical period starts at τ_cr/4= 6 months 12.5 days. The distance of Mars and Earth is largest for t_R=τ_cr/2≈ 1 year 25 days. The closest approach of Mars and Earth will be January 12 2025. In the simple approximation used, one finds that the critical period starts 24 June, the distance between Mars and Earth will be maximal February 7 2026, the critical period ends at July 20 2026, and the next closest approach would be on March 2 2027.

   The synodic view of the apparent closest approaches of Mars and Sun \href{https://www.nao.ac.jp/en/astro/sky/2016/05-topics03.html}{this}) between April 14 2014 and February 20 2027. From the table one finds that the time intervals between the closest approaches correspond to the synodic period and are indeed longer than 2 years.

How could one understand the observations in the TGD framework?

1. In the, the notion of field body (FB), which can be magnetic (MB) or electric (EB), changes the situation completely. Number theoretic view of TGD predicts that FB carries phases of the ordinary matter with very large values of effective Planck constant implying quantum coherence in astrophysical scales. Gravitational and electric fields in long scales are accompanied by a long length scale quantum coherence. There is evidence that the FBs of the Sun, planets and even the FB of the galaxy have effects on the behavior of biological systems and humans as conscious entities (see for instance this, this and this).
2. A long list of numerical miracles involving the masses of astrophysical objects appear in fundamental biology, supporting this view. For instance, EEG would be responsible for the communications to and control by the magnetic body of Earth. It is indeed difficult to understand why the organisms as master energy savers would spend a very large amount of metabolic energy to send information to outer space without any receiver. Furthermore, resonant EEG frequencies correspond to cyclotron frequencies for the associated "endogenous" magnetic field.

If really true, the findings of Moore would provide a further support for findings of Shnoll and other researchers. They would fit very nicely with the TGD view of quantum biology, which predicts that the magnetic bodies of the Sun and planets, in particular Mars, can affect biology and consciousness.

1. Although Mars has no large-scale magnetic field, the monopole tubes of the gravitational magnetic body of Mars could connect Earth and Mars.
2. The gravitational magnetic bodies of the Sun and planets carrying h_eff phases of ordinary particles behaving like dark matter, would control biomatter and receive information from it. The large distance of Mars when behind the Sun relative to Earth might reduce this control action.
3. The reconnection of U-shaped flux tubes is the fundamental interaction mechanism in all scales and plays a key role for instance in bio-catalysis. Also now this mechanism would be naturally involved and it would become less probable when the distance of Mars from the Earth increases (it is roughly 5AU at the backside of the Sun and 1 AU at the front side). Therefore the quantum coherence scale for the Mars-Earth system would be reduced and could affect even collective behavior of humans and of biology in general.
4. This explanation conforms with the intuition of Moore that the gravitational field of Mars is involved if gravitation is mediated by the radial U-shaped monopole flux tubes, for which the average density decreases as 1/r², i.e. like gravitational flux. Now however the effect would be based on astrophysical quantum coherence of the gravitational field making possible effects on biology and consciousness.

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.

Extension of Langlands geometric duality to trinity involving also physics-geometry correspondence

The master formula for TGD allowing construction of quantum states using the interpretation of space-time surfaces as numbers realizes the analog of geometric Langlands duality and generalizes it to a trinity. Geometric Langlands correspondence assigns to a pair of elements of a function field, which is a number theoretic object, a geometric object as algebraic surface having interpretation also as a Riemann surface with K\"ahler structure, twistor structure and spinor structure. This extends the number-theory-algebraic geometry duality to trinity and physics becomes the third part of a trinity.

1. The most high level form of number theory corresponds to function fields, which are infinite-D structured. In TGD, the pairs (f₁,f₂) of two functions of generalized complex coordinates of H=M⁴×CP₂ define a linear space and the functions f_i are elements of a function field. This is the number theoretic side of the Langlands geometric duality.
2. A function pair, whose root (f₁,f₂)=(0,0) defines a space-time surface in H and induces the number field structure of the function field to the space of space-time surfaces, "world of classical worlds" (WCW). Basic arithmetic operations of the number field apply to the component functions f_i and induce corresponding operations for space-time surfaces in WCW. The notion of induction, which is the basic principle of TGD, is central also here. It is missing from standard physics and also string models.
3. The root as a space-time surface obeys holography =holomorphy principle and is a minimal surface (as classical representation of generalized massless particle and massless field equations) and represents the geometry side of the geometric Langlands duality. This connection represents geometric Langlands duality in TGD. Riemannian geometries restricted to algebraic geometries is what makes the geometric Langlands duality possible.

   It is still unclear whether the choice of the classical action defining space-time surfaces and producing, apart from singularities, a minimal surface as an outcome, is only analogous to a choice of the coordinates and whether the recent choice (volume action + Kaehler action) is only the most convenient choice. If so, the laws of physics boil down to a completely action independent form, that is to the construction of quantum states induced by the products for space-time surfaces regarded as generalized numbers.

4. Space-time surfaces as minimal surfaces with generalized complex structure and are extremals for any variational principle constructible in terms of the induced geometry since extremal property reduces to the generalized complex structure. The action makes itself visible only at the singularities.
5. Langlands geometric duality becomes actually a trinity: number theory<-->geometry<--->physics. The number theory<-->geometry part of this trinity duality corresponds to Langlands geometric duality. The geometry<--->physics part is the TGD counterpart of Einstein's equations identifying geometry and physics.

See the article About Langlands correspondence in the TGD framework 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.

Space-time surfaces as numbers and construction of quantum states in terms of products of space-time surfaces

The exact solution of field equations of TGD in terms of holography=holomorphy vision and the recent progress in the understanding of the TGD view of Langlands correspondence allows to propose an explicit recipe, a kind of a master formula, for the construction of states describing the interaction in terms of generalized holomorphic algebraic geometry.

Space-time surfaces have the structure of number field

As I wrote the most recent article about the recent TGD view of Langlands correspondence (see this), I become convinced that the space-time surfaces indeed have a structure of a number field, induced by the structure of the function field formed by the analytic functions with respect to the four generalized complex coordinates of H= M⁴× CP₂ (one of the coordinates is hypercomplex light-like coordinate). Function fields are indeed central in the geometric Langlands correspondence.

1. This function field also has a hierarchical structure. There are hierarchies of polymials of various degrees and also rational functions with coefficient fields in different extensions of rationals. Analytical functions for which the Taylor coefficients are in extensions of rationals in the expansion is the next step. At the ultimate limit one has algebraic numbers as coefficients. Also transcendental extensions can be thought of and in this way one eventually ends up with complex numbers.
2. For H=M⁴× CP₂, this would correspond to the lowest level of the hierarchy of infinite primes but the Cartesian powers of H=M⁴× CP₂ correspond to the higher levels in the hierarchy of infinite primes. Again, this hierarchy is be analogous to the hierarchy used in the description of condensed matter, 3N-dimensional spaces, N number of particles.

In zero energy ontology (ZEO) (see this), quantum states corresponds to spinor fields of WCW, which consists of space-time surfaces satisfying holography and therefore being analogous to Bohr orbits, and also having interpretation as elements of number field so that one can multiply them (see this and this). WCW spinor fields assign to a given space-time surface a pair of fermionic Fock states at its 3-D ends located at the opposite light-like boundaries of the causal diamond (CD). Could one multiply two WCW spinor fields so that the space-time surfaces appearing as their arguments are multiplied

X⁴₁ ∪ X⁴₂ → X⁴₁*X⁴₂ ,

and the tensor product of the fermionic states at the boundaries of CD is formed. This would give

Ψ(X⁴₁)⊗ Ψ(X⁴₂) (X⁴₁∪ X⁴₂) → Ψ(X⁴₁)⊗ Ψ(X⁴₂)(X⁴₁*X⁴₂) .

Here X⁴₁*X⁴₂ would be the product of surfaces induced by the function algebra and the product of fermion states would be tensor product. Could Gods compute using spacetime surfaces as numbers and could our arithmetics be a shadow on the wall of the cave.

So: could a believer of TGD dream conclude that these meta-levels and perhaps even mathematical thinking could be described within the framework of the mathematics offered by the infinite dimensional number field formed by the space-time surfaces. This quite a lot more complicated than binary math with a cutoff of the order of 10³⁸!

Product of space-time surfaces as geometric counterpart of the tensor product

What could the product of space-time surfaces mean concretely? The physical intuition suggest that t corresponds to ae creation of an interacting pair of 3-D particles identified as they 4-D Bohr orbits. The product would be the equivalent of a tensor product, but now with interaction. If so, this product could provide a geometric and algebraic description of the interactions.

What would you get?

1. Let's examine the function pairs (f₁,f₂) and (g₁,g₂) defined in H=M⁴× CP₂ and the corresponding space-time surfaces for which (f₁,f2)=(0,0) and (g₁,g₂)=(0,0) apply. It should be noted that, for example, that the condition f₁=0 defines the analog of a 6-D twistor space, and the space-time surface X⁴ is the intersection of the analogs of the twistor bundles of M⁴ and CP₂, i.e., its base space.
2. The product of the function pairs is (f₁g₁,f₂g₂). Its components vanish in four cases.
   1. The cases (f₁,f₂)= (0,0) and (g₁,g₂)=(0,0) correspond to the union of the incoming surfaces. The corresponding particles are free.
   2. The cases (f₁,g₂)= (0,0) and (f₂,g₁)=(0,0) could define space-time regions having an interpretation in terms of the interaction of the particles. Under what conditions could this interpretation makes sense geometrically?

      Physical intuition suggests that for interacting particles, which do not form a bound state, the product reduces near the passive boundary (initial state) of the CD to the union of the surfaces associated with the free particles. The surfaces (f₁,g₂)= (0,0) and (f₂,g₁)=(0,0) would not temporally extend to the passive boundary of the CD. which correspond to the initial state of the particle reaction.

      This imposes some conditions on the functions involved. f₁=0 and g₂=0 (f₂=0 and g₁=0) are not satisfied near to the passive boundary of the CD simultaneously , so that the intersection of the corresponding 6-D surfaces (analogous to twistori space) is empty near the boundary of the CD.

      If this condition is not true, the interpretation would be as a bound state. TGD view of nuclei, atoms, and molecules assume that particles forming the bound state are indeed connected by monopole flux tubes (see this).

What about the product of spinors fields?

The WCW spinor field assigns multifermion states to the 3-D ends of a given spacetime surface at the boundaries of the CD. If one can define what happens to the multifermion states associated with the zero energy states in the interaction, then one has a universal construction for the states of WCW as spinor fields of WCW providing a precise description of interactions analogous to an exact solution of an interacting quantum field theory. At the geometric level, the product of the surfaces corresponds to the interaction. At the fermion level, essentially the ordinary tensor product of the multifermion states should correspond to this interaction.

Under what conditions does this vision work for fermionic states as WCW spinors, identified in ZEO as pairs of the many-fermion states at the 3-surfaces at the boundaries of the CD? It is obvious that the definition of the fermion state should be universal in the sense that at the fundamental level the fermion state is defined without saying anything about space-time surfaces involved.

Induction is a basic principle of TGD and the induction of spinor fields indeed conforms with this idea. The basic building bricks of WCW spinor fields are second quantized spinor fields of H restricted to the 3-surfaces defining the ends of the space-time surfaces at the boundaries of CD. Therefore the multifermion states are restrictions of the multifermion states of H to the spacetime surfaces. The Fourier components (in the general sense) for the second quantized spinor field Ψ of H (not WCW!) and its conjugate Ψ{†} would only be confined to the ends of X⁴ at the light-like boundaries of CD.

The oscillator algebra of H spinor fields makes it possib le to calculate all fermionic propagators and fermionic parts of N-point functions reduce to free fermionic field theory in H but arguments restricted to the space-time surfaces. The dynamics of the formally classical spinor fields of WCW would very concretely be a "shadow" of the dynamics of the second quantized spinor fields of H. One would have a free fermionic field theory in H induced to space-time surfaces!

In this way, one could construct multiparticle states containing an arbitrary number of particles. The construction of quantum spaces would reduce to a multiplication in the number field formed by space-time surfaces, accompanied by fermionic tensor product!

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.