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."

These findings will be labelled as mere astrology by the mainstream. In the long run it is however very hard to deny simple facts. 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. In the standard physics framework, this kind of correlations in astrophysical scales are of course impossible.

It is deeply ironic that in the frontier of recent day theoretical physics, fashionable theories which do not have a slightest connection with reality, are sold as breakthroughs, and at the same time data flood from real world phenomena in blatant conflict with the existing views is neglected. This is what happens when career building becomes the main goal of the scientist.

In the TGD framework, 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.

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 are a revolutionary discovery. 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 for instance this, this and this.

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

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

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.