Why the dark energy density is inversely proportional to the surface area of the volume studied?

Sabine Hossenfelder commented in her posting "Surprise Comeback: Dark Energy Could Be Holographic After All" (see this) the idea that the mysterious dark energy might not be real but an outcome of holography and assignable to the 3-D surface which in holography contains the information determining the dynamic in the interior of the space-time. The comments were inspired the the article "Evolution of perturbations in the model of Tallis holographic dark energy" (see this) by Artyom et al.

The starting point observation is that the dark energy density is in a good approximation found to be proportional to 1/S, where S is the surface area of a large sphere surrounding the region studied. By the way, Sabine makes a little mistake here: she talks about dark energy rather than dark energy density. The reader can check this from the article of Artyom et al. The model of Tallis has been given up long ago but the authors represent an argument that since dark energy is not ordinary cosmic fluid, ordinary perturbation theoretic analysis does not apply.

TGD suggests however a much simpler explanation of the finding. In TGD, dark energy is identifiable as a galactic dark matter and consists of magnetic and volume energy assignable to very long monopole flux tubes with a huge string tension. No galactic dark matter halo nor exotic dark matter particles are needed. The galactic velocity spectrum is correctly predicted from the string tension which is also predicted.

To see whether TGD can explain the finding that dark energy density is proportional to 1/S, one must estimate the average density of dark energy in a large cylindrical volume around a long cosmic string. The dark energy is proportional to the length L of the string. The volume is roughly V=SL, where S, is the surface area of the cross section of the cylinder. Therefore one has that dark energy density satisfied E/V= E/SL= 1/S. Just as has been found.

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 possible to have objective laws of physics?

Daniel Oriti (see this) concludes that there are no objective laws of physics. 40 years ago, the views were very different but the dramatic failure of the superstring approach together with the multiverse catastrophe changed the optimistic opinions. It is of course psychologically much easier to conclude that there are no objective laws than to admit that my generation failed to find them.

The answer to the question depends on what one means with objective reality. If space-time is taken as objective reality, there is no such thing as objective reality, something existing independent of observer. In the TGD framework, one can speak only of space-time surfaces in H=M⁴×CP₂ as analogs of Bohr orbits for particles as 3-surfaces, and obeying almost deterministic holography forcing the Bohr orbits to be basic dynamical objects. Zero energy ontology (ZEO) is the new ontology solving the basic paradox of quantum measurement theory. Quantum states are quantum superpositions of these "Bohr orbits".

There are global objective laws: they reduce the mathematical existence of TGD. H isn fixed by existence of the twistor lift and number theory-geometry duality (M⁸-H duality) and holography= holomorphy principle giving holomorphic 4-surfaces, as minimal surfaces, and as extremals of any general coordinate invariant action constructible in terms of the induced geometry. Action makes itself visible only at singularities. Induction fixes the dynamics for fermions: second quantized free spinor fields in H. Fermion pair creation is possible thanks to the 4-D spacetime allowing exotic smooth structures as defects of standard one. The point-like defects define vertices and are also identifiable as (self-) intersections of space-time surfaces. Dimensions D=4 and D=8 for space-time and H are crucial for non-trivial physics.

Space-time surfaces are expressible as roots for pairs(f₁,f₂) of analytic functions of 4 generalized complex coordinates of H (one is hypercomplex coordinate with light-like coordinate curves). They form an algebra induced by the arithmetic operators for f_i. This algebra decomposes to a union of number fields with f₂ fixed. Space-times are thus generalized numbers: this realizes geometric Langlands correspondence (see this and this) .

The space of space-time surfaces defining a number system, the world of classical worlds (WCW) and it exists objectively. Subjective existence means sequence of quantum jumps between states defined by WCW spinor fields, this means hopping around in WCW. ZEO allows a realization of conscious memory so that the system learns all the subjective time about physics (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.

Negative group delay and Zero Energy Ontology

Paul Kirsch sent an interesting link to a finding to an article "Experimental evidence that a photon can spend a negative amount of time in an atom cloud" (see this).

The finding is very interesting from from the point of view of zero energy ontology (ZEO) defining the ontology of classical and quantum TGD (see this, this, this, this). Could the negative group delay be understood in terms of a time period with a reversed arrow of time spent by the photon around an atom?

1. Absorption and re-emission by atom would correspond to two types of "big" state function reductions (BSFRs) taking place. In the first BSFR photon would "die" by absorption by an atom. Photon would however reincarnate with an opposite arrow of time. The same would happen in the second BSFRs and photon would reincrane with the original arrow of time.

   According to the recent view of ZEO, after the second BSFR the photon would emerge geometrically later than it was absorbed in the first BSFR. The photon wave packet would come out as less entropic, that is younger. This effect would be like waking up as a less entropic, in this sense a younger person after a well slept night.

2. Does the group delay measure this effect? If the aging of the wave packet means widening then this might be the case. Free photon wave packet keeps its shape since it does not disperse. The widening must be of thermodynamic origin and would be due to SSFRs replacing the wave packet gradually with a wider one.
3. In TGD, the shape preserving wave packet has as a classical geometric correlates a "massless extremal" (ME) representing a pulse propagating in a precise direction. The shape of pulse does not change but "small" state function reductions (SSFRs) would replace ME with a new one representing in general a wider pulse. This would be dissipation: ME would age. The pair of BSFRs induced by atomic absorption would lead to a reincarnation as a younger ME. This would be the counterpart for the group delay.

The finding creates a tongue in cheek consideration related to my personal life. I suffer from bad sleep and wake-up continually. BSFR means falling asleep or in an extreme case death at some level of the personal self hierarchy. Temporary reversals of the arrow of time in pairs of BSFRs would provide a universal trial and error mechanism in conscious information processing and quantum biology. For instance, homeostasis as a way to stay near quantum criticality would be based on continual change of the arrow of time. If the temporary deaths indeed provide a way to fight against the second law, they might slow down aging. The personal curse would be actually a blessing?

See the article TGD and Condensed Matter or a 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.

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.