Summary of TGD (a lot of figures!)

I wrote an article summarizing the recent situation in TGD. Without Reza Rastmanesh's encouragement and help this article would not have been written. I particular, the help of Reza in preparing figures making it easier to grasp the general structure of TGD was irreplaceable. As usual, the boring duty of writing about things already done transformed to a creative process and several new mathematical and physical results about TGD emerged.

Reza even persuaded me to send the article to Nature. That the editors do not even bother to send it to the proposed referees, was not a surprise, but I managed to overcome the deep professorphobia induced during the painful 42 years by academic arrogance and stupidity. Thanks to the therapeutical efforts of Reza, I even sent the article to big names like Witten, Maldacena, Susskind, Penrose, Arkani-Hamed, etc... I received a message about receival from Susskind and Penrose only. Probably the secretaries insulate most names into their academic bubble and probably they also want to live in the bubbles they have created.

I added the article "Summary of Topological Geometrodynamics" to both Research Gate and my homepage.

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

Articles and other material related to TGD.

What MIP*= RE could mean in TGD Universe?

The MIP*=RE means that quantum entanglement has a crucial role in quantum computation. One the other hand, HFFs would not allow to use quantum entanglement to quantum computational purposes. This looks at first a bad news from the TGD point of view, where HFFs are highly suggestive. One must be however very careful with the basic definitions.

1. The notion of finite measurement resolution in TGD

Measurement resolution is one of the basic notions of TGD.

1. There are intuitive physicist's arguments demonstrating that in TGD the operator algebras involved with TGD are HFFs provides a description of finite measurement resolution. The inclusion of HFFs defines the notion of resolution: included factor represents the degrees of freedom not seen in the resolution used (see this) and this).

   Hyperfinite factors involve new structures like quantum groups and quantum algebras reflecting the presence of additional symmetries: actually the "world of classical worlds" (WCW) as the space of space-time surfaces as maximal group of isometries and this group has a fractal hierarchy of isomorphic groups imply inclusion hierarchies of HFFs. By the analogs of gauge conditions this infinite-D group reduces to a hierarchy of effectively finite-D groups. For quantum groups the infinite number of irreps of the corresponding compact group effectively reduces to a finite number of them, which conforms with the notion of hyper-finiteness.

   It looks that the reduction of the most general quantum theory to TGD-like theory relying on HFFs is not possible. This would not be surprising taking into account gigantic symmetries responsible for the cancellation of infinities in TGD framework and the very existence of WCW geometry.

2. Second TGD based approach to finite resolution is purely number theoretic (see this) and involves adelic physics as a fusion of the real physics with various p-adic physics as correlates of cognition. Cognitive representations are purely number theoretic and unique discretizations of space-time surfaces defined by a given extension of rationals forming an evolutionary hierarchy: the coordinates for the points of space-time as a 4-surface of the imbedding space H=M⁴× CP₂ or of its dual M⁸ are in the extension of rationals defining the adele. In the case of M⁸ the preferred coordinates are unique apart from time translation. These two views would define descriptions of the finite resolution at the level of space-time and Hilbert space. In particular, the hierarchies of extensions of rationals should define hierarchies of inclusions of HFFs.

For hyperfinite factors the analog of MIP*=RE cannot hold true. Doesn't the TGD Universe allow a solution of all the problems solvable by Turing Computer? There is however a loophole in this argument

1. The point is that for the hierarchy of extensions of rationals also Hilbert spaces have as a coefficient field the extension of rationals!. Unitary transformations are restricted to matrices with elements in the extension. In general it is not possible to realize the unitary transformation mapping the entangled situation to an un-entangled one! The weakening of the theorem would hold true for the hierarchy of adeles and entanglement would give something genuinely new for quantum computation!
2. A second deep implication is that the density matrix characterizing the entanglement between two systems cannot in general be diagonalized such that all diagonal elements identifiable as probabilities would be in the extension considered. One would have stable or partially stable entanglement (could the projection make sense for the states or subspaces with entanglement probability in the extension). For these bound states the binding mechanism is purely number theoretical. For a given extension of p-adic numbers one can assign to algebraic entanglement also information measure as a generalization of Shannon entropy as a p-adic entanglement entropy (real valued). This entropy can be negative and the possible interpretation is that the entanglement carries conscious information.

2. What about the situation for the continuum version of TGD?

At least the cognitively finitely representable physics would have the HFF property with coefficient field of Hilbert spaces replaced by an extension of rationals. Number theoretical universality would suggest that HFF property characterizes also the physics of continuum TGD. This leads to a series of questions.

1. Does the new theorem imply that in the continuum version of TGD all quantum computations allowed by the Turing paradigm for real coefficients field for quantum states are not possible: MIP*⊂ RE? The hierarchy of extensions of rationals allows utilization of entanglement, and one can even wonder whether one could have MIP*= RE at the limit of algebraic numbers.
2. Could the number theoretic vision force change also the view about quantum computation? What does RE actually mean in this framework? Can one really assume complex entanglement coefficients in computation. Does the computational paradigm makes sense at all in the continuum picture?

   Are both real and p-adic continuum theories unreachable by computation giving rise to cognitive representations in the algebraic intersection of the sensory and cognitive worlds? I have indeed identified real continuum physics as a correlate for sensory experience and various p-adic physics as correlates of cognition in TGD: They would represent the computionally unreachable parts of existence.

   Continuum physics involves transcendentals and in mathematics this brings in analytic formulas and partial differential equations. At least at the level of mathematical consciousness the emergence of the notion of continuum means a gigantic step. Also this suggests that transcendentality is something very real and that computation cannot catch all of it.

3. Adelic theorem allows to express the norm of a rational number as a product of inverses of its p-adic norms. Very probably this representation holds true also for the analogs of rationals formed from algebraic integeres. Reals can be approximated by rationals. Could extensions of all p-adic numbers fields restricted to the extension of rationals say about real physics only what can be expressed using language?

Also fermions are highly interesting in the recent context. In TGD spinor structure can be seen as a square root of Kähler geometry, in particular for the "world of classical worlds" (WCW). Fermions are identified as correlates of Boolean cognition. The continuum case for fermions does not follow as a naive limit of algebraic picture.

1. The quantization of the induced spinors in TGD looks different in discrete and continuum cases. Discrete case is very simple since equal-time anticommutators give discrete Kronecker deltas. In the continuum case one has delta functions possibly causing infinite vacuum energy like divergences in conserved Noether charges (Dirac sea).
2. I have proposed (see this) how these problems could be avoided by avoiding anticommutators giving delta-function. The proposed solution is based on zero energy ontology and TGD based view about space-time. One also obtains a long-sought-for concrete realization for the idea that second quantized induce spinor fields are obtained as restrictions of second quantized free spinor fields in H=M⁴× CP₂ to space-time surface. The fermionic variant of M⁸-H-duality (see this) provides further insights and gives a very concrete picture about the dynamics of fermions in TGD.

These considerations relate in an interesting manner to consciousness. Quantum entanglement makes in the TGD framework possible telepathic sharing of mental images represented by sub-selves of self. For the series of discretizations of physics by HFFs and cognitive representations associated with extensions of rationals, the result indeed means something new.

3. What about transcendental extensions?

During the writing of this article an interesting question popped up.

1. Also transcendental extensions of rationals are possible, and one can consider the generalization of the computationalism by also allowing functions in transcendental extensions. Could the hierarchy of algebraic extensions could continue with transcendental extensions? Could one even play with the idea that the discovery of transcendentals meant a quantum leap leading to an extension involving for instance e and π as basic transcendentals? Could one generalize the notion of polynomial root to a root of a function allowing Taylor expansion f(x)= ∑ q_n xⁿ with rational coefficients so that the roots of f(x)=0 could be used define transcendental extensions of rationals?
2. Powers of e or its root define and infinite-D extensions having the special property that they are finite-D for p-adic number fields because e^p is ordinary p-adic number. In the p-adic context e can be defined as a root of the equation x^p-∑ pⁿ/n!=0 making sense also for rationals. The numbers log(p_i) such that p_i appears a factor for integers smaller than p define infinite-D extension of both rationals and p-adic numbers. They are obtained as roots of e^x-p_i=0.
3. The numbers (2n+1)π (2nπ) can be defined as roots of sin(x)=0 (cos(x)=0. The extension by π is infinite-dimensional and the conditions defining it would serve as consistency conditions when the extension contains roots of unity and effectively replaces them.
4. What about other transcendentals appearing in mathematical physics? The values ζ(n) of Riemann Zeta appearing in scattering amplitudes are for even values of n given by ζ(2n)= (-1)ⁿ⁺¹ B_2n (2π)²ⁿ/2(2n+1)!. This follows from the functional identity for Riemann zeta and from the expression ζ(-n)= (-1)ⁿ B_n+1/(n+1) ( (B(-1/2)=-1/2) (see this). The Bernoulli numbers B_n are rational and vanish for odd values of n. An open question is whether also the odd values are proportional to πⁿ with a rational coefficient or whether they represent "new" transcendentals.

4. What does one mean with quantum computation in TGD Universe?

The TGD approach raises some questions about computation.

1. The ordinary computational paradigm is formulated for Turing machines manipulating natural numbers by recursive algorithms. Programs would essentially represent a recursive function n→ f(n). What happens to this paradigm when extensions of rationals define cognitive representations as unique space-time discretizations with algebraic numbers as the limit giving rise to a dense in the set of reals.

   The usual picture would be that since reals can be approximated by rationals, the situation is not changed. TGD however suggests that one should replace at least the quantum version of the Turing paradigm by considering functions mapping algebraic integers (algebraic rational) to algebraic integers.

   Quite concretely, one can manipulate algebraic numbers without approximation as a rational and only at the end perform this approximation and computations would construct recursive functions in this manner. This would raise entanglement to an active role even if one has HFFs and even if classical computations could still look very much like ordinary computation using integers.

2. ZEO brings in also time reversal occurring in "big" (ordinary) quantum jumps and this modifies the views about quantum computation. In ZEO based conscious quantum computation halting means "death" and "reincarnation" of conscious entity, self? How the processes involving series of haltings in this sense differs from ordinary quantum computation: could one shorten the computation time by going forth and back in time.
3. There are many interesting questions to be considered. M⁸-H duality gives justifications for the vision about algebraic physics. TGD leads also to the notion of infinite prime and I have considered the possibility that infinite primes could give a precise meaning for the dimension of infinite-D Hilbert space. Could the number-theoretic view about infinite be considerably richer than the idea about infinity as limit would suggest (see this). The construction of infinite primes is analogous to a repeated second quantization of arithmetic supersymmetric quantum field theory allowing also bound states at each level and a concrete correspondence with the hierarchy of space-time sheets is suggestive. For the infinite primes at the lowest level of the hierarchy single particle states correspond to rationals and bound states to polynomials and therefore to the sets of their roots. This strongly suggests a connection with M⁸ picture.

See the article MIP*= RE: What could this mean physically? or the chapter Evolution of Ideas about Hyper-finite Factors in TGD.

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

Articles and other material related to TGD.

MIP*=RE: What it could possibly mean?

I received a very interesting link to a popular article (see this) explaining a recently discovered deep result in mathematics having implications also in physics. The article (see this) by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen has a rather concise title " MIP*=RE". In the following I try to express the impressions of a (non-mainstream) physicist about the result. In the first posting I discuss the finding and the basic implications from the physics point of view. In the second posting the highly interesting implications of the finding in the TGD framework are discussed.

The result expressed can be expressed by using the concepts of computer science about which I know very little at the hard technical level. The results are however told to state something highly non-trivial about physics.

1. RE (recursively enumerable languages) denotes all problems solvable by computer. P denotes the problems solvable in a polynomial time. NP does not refer to a non-polynomial time but to "non-deterministic polynomial acceptable problems" - I hope this helps the reader- I am a little bit confused! It is not known whether P = NP is true.
2. IP problems (P is now for "prover" that can be solved by a collaboration of an interrogator and prover who tries to convince the interrogator that her proof is convincing with high enough probability. MIP involves multiple l provers treated as criminals trying to prove that they are innocent and being not allowed to communicate. MIP* is the class of solvable problems in which the provers are allowed to entangle.

The finding, which is characterized as shocking, is that all problems solvable by a Turing computer belong to this class: RE= MIP*. All problems solvable by computer would reduce to problems in the class MIP*! Quantum computation would indeed add something genuinely new to the classical computation.

Two physically interesting applications

There are two physically interesting applications of the theorem which are interesting also from the TGD point of view and force to make explicit the assumptions involved.

1. About the quantum physical interpretation of MIP*

To proceed one must clarify the quantum physical interpretation of MIP*.

1. Quantum measurement requires entanglement of the observer O with the measured system M. What is basically measured is the density matrix of M (or equivalently that of O). State function reduction gives as an outcome a state, which corresponds to an eigenvalue of the density matrix. Note that this state can be an entangled state if the density matrix has degenerate eigenvalues.
2. Quantum measurement can be regarded as a question to the measured system: "What are the values of given commuting observables?". The final state of quantum measurement provides an eigenstate of the observables as the answer to this question. M would be in the role of the prover and O_i would serve as interrogators.

   In the first case multiple interrogators measurements would entangle M with unentangled states of the tensor product H₁⊗ H₂ for O followed by a state function reduction splitting the state of M to un-entangled state in the tensor product M₁⊗ M₂.

   In the second case the entire M would be interrogated using entanglement of M with entangled states of H₁⊗ H₂ using measurements of several commuting observables. The theorem would state that interrogation in this manner is more efficient in infinite-D case unless HFFs are involved.

3. This interpretation differs from the interpretation in terms of computational problem solving in which one would have several provers and one interrogator. Could these interpretations be dual as the complete symmetry of the quantum measurement with respect to O and M suggests? In the case of multiple provers (analogous to accused criminals) it is advantageous to isolate them. In the case of multiple interrogators the best result is obtained if the interrogator does not effectively split itself into several ones.

2. Connes embedding problem and the notion of finite measurement/cognitive resolution

Alain Connes formulated what has become known as Connes embedding problem. The question is whether infinite matrices forming factor of type II₁ can be always approximated by finite-D matrices that is imbedded in a hyperfinite factor of type II₁ (HFF). Factors of type II and their HFFs are special classes of von Neumann algebras possibly relevant for quantum theory.

This result means that if one has measured of a complete set of for a product of commuting observables acting in the full space, one can find in the finite-dimensional case a unitary transformation transforming the observables to tensor products of observables associated with the factors of a tensor product. In the infinite-D case this is not true.

What seems to put alarms ringing is that in TGD there are excellent arguments suggesting that the state space has HFFs as building bricks. Does the result mean that entanglement cannot help in quantum computation in TGD Universe? I do not want to live in this kind of Universe!

2. Tsirelson problem

Tsirelson problem (see this) is another problem mentioned in the popular article as a physically interesting application. The problem relates to the mathematical description of quantum measurement.

Three systems are considered. There are two systems O₁ and O₂ representing observers and the third representing the measured system M. The measurement reducing the entanglement between M and O₁ or O₂ can regarded as producing correspondence between state of M and O₁ or O₂, and one can think that O₁ or O₂ measures only obserservables in its own state space as a kind of image of M. There are two manners to see the situation. The provers correspond now to the observers and the two situations correspond to provers without and with entanglement.

Consider first a situation in which one has single Hilbert space H and single observer O. This situation is analogous to IP.

1. The state of the system is described statistically by a density matrix - not necessarily pure state -, whose diagonal elements have interpretation as reduction probabilities of states in this bases. The measurement situation fixes the state basis used. Assume an ensemble of identical copies of the system in this state. Assume that one has a complete set of commuting observables.
2. By measuring all observables for the members of the ensemble one obtains the probabilities as diagonal elements of the density matrix. If the observable is the density matrix having no- degenerate eigenvalues, the situation is simplified dramatically. It is enough to use the density matrix as an observable. TGD based quantum measurement theory assumes that the density matrix describing the entanglement between two subsystems is in a universal observable measure in state function reductions reducing their entanglement.
3. Can one deduce also the state of M as a superposition of states in the basic chosen by the observer? This basis need not be the same as the basis defined by - say density matrix if the system has interacted with some system and this ineraction has led to an eigenstate of the density matrix. Assume that one can prepare the latter basis by a physical process such as this kind of interaction.

   The coefficients of the state form a set of N² complex numbers defining a unitary N× N matrix. Unitarity conditions give N conditions telling that the complex rows and unit vectors: these numbers are given by the measurement of all observables. There are also N(N-1) conditions telling that the rows are orthogonal. Together these N+N(N-1)=N² numbers fix the elements of the unitary matrix and therefore the complex coefficients of the state basis of the system can be deduced from a complete set of measurements for all elements of the basis.

Consider now the analog of the MIP* involving more than one observer. For simplicity consider two observers.

1. Assume that the state space H of M decomposes to a tensor product H=H₁⊗ H₂ of state spaces H₁ and H₂ such that O₁ measures observables X₁ in H₁ and O₂ measuresobservables X₂ in H₂. The observables X₁ and X₂ commute since they act in different tensor factors. The absence of interaction between the factors corresponds to the inability of the provers to communicate. As in the previous case, one can deduce the probabilities for the various outcomes of the joint measurements interpreted as measurements of a complete set of observables X₁ ⊗ X₂.
2. One can also think that the two systems form a single system O so that O₁ and O₂ can entangle. This corresponds to a situation in which entanglement between the provers is allowed. Now X₁ and X₂ are not in general independent but also now they must commute. One can deduce the probabilities for various outcomes as eigenstates of observables X₁ X₂ and deduce the diagonal elements of the density matrix as probabilities.

Are these manners to see the situation equivalent? Tsirelson demonstrated that this is the case for finite-dimensional Hilbert spaces, which can indeed be decomposed to a tensor product of factors associated with O₁ and O₂. This means that one finds a unitary transformation transforming the entangled situation to an unentangled one and to tensor product observables.

For the infinite-dimensional case the situation remained open. According to the article, the new result implies that this is not the case. For hyperfinite factors the situation can be approximated with a finite-D Hilbert space so that the situations are equivalent in arbitrary precise approximation.

See the article MIP*= RE: What could this mean physically? or the chapter Evolution of Ideas about Hyper-finite Factors in TGD.

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

Articles and other material related to TGD.

Why not publish a book about TGD?

I was asked why I do not publish a book about TGD. Some people also ask why I have not considered the idea of applying TGD to some real problem in physics. I glue below the reply which should explain why not.

Some people have also informed me that Einstein said that any big idea must be so simple that even a child can understand. Why I do not publish a picture book for children about TGD explaining the big idea using a couple of pictures? My answer could be the following: Einstein made only a single big blunder in his life. It was not the proposal of the cosmological constant but the above statement: fools around the world really take it literally. I appreciate people writing for children but I am a different kind of writer.

So: why don't I publish a book for adult readers or even colleagues about TGD? I actually have 24 online books almost ready for printing. Basic theory and lots of applications covering all branches of physics and also biology and neuroscience, which the people making these questions have not noticed since just seeing a link to my homepage - no time for more than this - does not give any idea about what TGD really is. These books can be published posthumously as collected works when the time is ripe for this. The reasons are many-fold.

There are overlapping topics and colleagues would not lose the opportunity to blame me for self-plagiarism as happened with the previous book about TGD. There was some ridiculous counting of words mechanism used to reveal my criminal character. For two years I spent a lot of useful working time with totally irrelevant activities having very little to do with the contents of the book. The compensation is so small that bank costs would make me the net payer. No one reads books nowadays and no-one even considers buying a book by non-name.

I do not have too many years left and I want to use them to develop TGD. This is for purely selfish reasons: it marvellous to live in full swing still at this age and do history of science.

I have also given up the hopes of explaining TGD understandably: 42 years distance to colleagues is so long that I feel myself being on a mountain top covered by clouds. They refuse even to believe that there is some-one there. 24 books as a climbing guide telling also about all wrong tracks is too much for anyone, and it is not inspiring to passively follow the instructions. It is much more motivational for them to rediscover TGD by themselves.

I have hoped that I could help them in this process and perhaps shorten 42 years to a decade. I have explained again and again what the deep problems are and what would be the TGD solution to them hoping that it would be more motivational to use their own brain to solve the key problems. They are not interested even in this option. They prefer to follow the wrong paths shown by names and repeat the mistakes already made. Or alternatively, to build totally non-sensible one-line theory based on mere pictures. So: let them discover all by themselves. Trial and error is the most effective manner to learn.

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

Articles and other material related to TGD.

Fast Radiowave bursts in TGD framework

I encountered a highly interesting popular article (thanks for my friend Asta for the link) with title "Mysterious 'fast radio burst' detected closer to Earth than ever before" (this).

Fast radio wave bursts (FRBs) arrive from a distance of hundreds of millions of light years - the scale of a large void. If the energy of FRBs is radiated isotropically in all directions - an assumption to be challenged below - the total energy is of the same order of magnitude that the energy of the Sun produced during a century. There are FRBs repeating with a period of 16 days located to a distance of 500 million light years from Earth.

The latest bursts arrive from a distance of only about 30 thousand light years from our own galaxy Milky Way described in the popular article can be assigned with magnetar (see this) which is a remnant of neutron start and has extremely strong magnetic field of about 10¹¹ Tesla.

Below is the abstract of the article (this) reporting the discovery.

We report on International Gamma-Ray Astrophysics Laboratory (INTEGRAL) observations of the soft γ ray repeater SGR 1935+2154 performed between 2020 April 28 and May 3. Several short bursts with fluence of ∼ 10^-7-10^-6 erg cm^-2 were detected by the Imager on-board INTEGRAL (IBIS) instrument in the 20-200 keV range. The burst with the hardest spectrum, discovered and localized in real time by the INTEGRAL Burst Alert System, was spatially and temporally coincident with a short and very bright radio burst detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME) and Survey for Transient Astronomical Radio Emission 2 (STARE2) radio telescopes at 400-800 MHz and 1.4 GHz, respectively.

Its lightcurve shows three narrow peaks separated by ∼ 29 ms time intervals, superimposed on a broad pulse lasting ∼ 0.6 s. The brightest peak had a delay of 6.5 +/- 1.0 ms with respect to the 1.4 GHz radio pulse (that coincides with the second and brightest component seen at lower frequencies). The burst spectrum, an exponentially cutoff power law with photon index Γ =0.7_-0.2^+0.4 and peak energy E_p=65+/- 5 keV, is harder than those of the bursts usually observed from this and other magnetars.

By the analysis of an expanding dust-scattering ring seen in X-rays with the Neil Gehrels Swift Observatory X-ray Telescope (XRT) instrument, we derived a distance of 4.4_-1.3^+2.8 kpc for SGR 1935+2154, independent of its possible association with the supernova remnant G57.2+0.8. At this distance, the burst 20-200 keV fluence of (6.1+/- 0.3)× 10^-7 erg cm^-2 corresponds to an isotropic emitted energy of ∼ 1.4× 10³⁹ erg. This is the first burst with a radio counterpart observed from a soft γ ray repeater and it strongly supports models based on magnetars that have been proposed for extragalactic fast radio bursts.

What could be the interpretation of the finding in the TGD framework? The weirdest feature of the FRB is its gigantic total energy assuming that the radiation is isotropic during the burst. This assumption can be challenged in the TGD framework, where the stellar systems are connected to a monopole flux tube network and radiation flows along flux tubes, which can also branch. This brings strongly in mind the analog of a nervous system in cosmic scales and this analogy is used in what follows.

1. The duration of pulses is few milliseconds: the duration of nerve pulses is the same. Is this a wink-wink to the Poirots of astrophysics?
2. Bursts can arrive regularly for instance with a period of T=16.35 days (see this). This brings in the mind of astro-Poirot biorhythm, in particular EEG rhythms. This would not be the only such rhythms: also the period of T_alpha=160 minutes, for which have proposed an interpretation as a cosmic analog of alpha rhythm is known (see this). The ratio T/T_α=147.15 would give for the analogous brain rhythm the value of 14.7 seconds.
3. Let us assume that stellar systems indeed form an analog of neural network connected by flux and assume that the topology of this network is analogous to that defined by axons. In TGD framework neural communications between neurons occur actually by using dark photons with effective Planck constant h_eff=nh₀ along the flux tubes with the velocity of light so that feedback from brain and even from the magnetic body of brain back to sensory organs as a virtual sensory input becomes possible. The function of nerve pulses is to connect the outgoing branch of the flux tube associated with the axon and those associated with dendrites of the post-synaptic neuron toa longer flux tubes by using neurotransmitters as relays.
4. The stellar object as an analog of a neuron would send its dark photon signals along the flux tube assignable to a single axon. Axon would later branch to dendrites arriving to other stellar systems and eventually perhaps to planets as analogs of synaptic contacts. An interesting question is whether also the analogs of nerve pulses and neurotransmitters acting as relays in the synaptic contacts defined by planets could make sense. What could nerve pulses propagating along the flux tube correspond to?

   Remark: In the TGD based model of brain there would be also flux tube network analogous to the meridian system of Eastern medicine and responsible for the holistic and spatial aspects of consciousness since more than one flux tube can emanate from a given node making possibly non-linear networks (see this). Nervous system with tree- like structure would be responsible for the linear and temporal aspects of conscious experience. Tree-like structure would be crucial for the understanding of Alzheimer disease (see this). Meridian system would be a predecessor of the neural system.

5. The distances of FRBs are of the order of large voids having galaxies at their boundaries and forming lattice-like networks possibly assignable to the tesselations of 3-D hyperbolic space defining cosmic time= constant surfaces. This kind of tesselations could accompany also brain (see this). In the fractal Universe of TGD one can wonder whether these voids are analogs of cells or even neurons and form cosmic biological organisms with flux tubes forming a network allowing communications.

The basic implication is that the energy of the emitted radiation could be dramatically smaller than that predicted by an isotropic radiation burst. It is interesting to look whether the proposed picture survives quantitative modelling.

1. The reduction factor r for the total emitted energy would be essentially r= S/A, where S is the area of the "axonal" flux tube and A=4π R² is the surface area of the magnetar. One must estimate the value of r.
2. Flux quantization for a single sheet of the many-sheeted magnetic flux tube involved would give eBS= hbar₀ h=6h₀ (see this and this). The general order of magnitude estimate is eB ∼ hbar₀/S. If each sheet carries out the same energy, the number of sheets is n=h_eff/h₀ and the effective area of a flux tube is S= hbar₀/eB. Does the magnetic field assigned with magnetar correspond to a single sheet or to all sheets? If the field is measured from cyclotron energies assuming h_eff=h it would correspond to all sheets and the measured magnetic field would be the effective magnetic field B_eff= nB/6 for h= 6h₀.
3. The branching of the flux tube could correspond to the splitting of the many-sheeted flux tube to tubes with smaller number of sheets and involve reduction of h_eff. This would give the estimate r= hbar₀/eBA. Magnetic field of 1 Tesla corresponds to a unit flux quantum with radius - magnetic length . about 2.6× 10^-8 meters. Assuming magnetar radius R=20 km one has r∼ 10^-25/6.
4. The estimate for the total emitted energy assuming isotropic radiation is the energy radiated by the Sun during a century. Sun transforms roughly E₁₀₀=1.3× 10¹⁹ kg of mass to radiation during a century. This gives for the energy emitted in FRB the estimate E= r E₁₀₀∼ 10^-6/6 kg which is roughly 7.5 Planck masses m_Pl≈ 2.2× 10^-8 kg. The order of magnitude is Planck mass. The estimate is of course extremely rough.

   In any case, the idea that pulses could have mass of order few Planck masses is attractive. Note that a large neuron with radius about 10^-4 meters has a mass of order Planck mass (see this).

5. From the total detected energy dE/dS= 6.1× 10^-7 erg m^-2= 3.8× 10⁹ eVm^-2 and total radiated energy E= 7.5 m_Pl one can estimate the total area S covered by the branched energy flux if it covers the entire area with a shape of disk of radius R. This gives some idea about how wide the branching is. The total energy is E =(dE/dS)× π R² giving R= [E/π (dE/dS)]^1/2∼ .9× 10⁹ m. The equitoral radius of the Sun is R_Sun= .7× 10⁹ m. R_Sun∼ .78 R. This conforms with the idea that the radiation arrives along the axon-like flux tube connecting Sun and the magnetar branching so that it covers the entire Sun.

The ratio h_eff/h should be of the same order of magnitude as the ratio X=E/E_rad, where E_rad is the energy of the radio wave photon with frequency 1.4 GHz for h_eff=h: X∼ h_eff/h. The ratio Y= X/(h_eff/h) should satisfy Y∼ 1.

1. To proceed further, one can use the TGD variant of Nottale's hypothesis. The hypothesis states that one can assign to gravitational flux tubes gravitational Planck constant hbar_gr. The original hypothesis was ℏ_eff=ℏ_gr and the more recent form inspired by the adelic vision states that h_gr corresponds to a large integer factor of h_eff. One has ℏ_gr= GMm/v₀= r_Sm/2v₀. Here M is the mass of the large object - now that of magnetar. m is the mass of the smaller quantum coherent object in contact with the gravitational flux tube mediating gravitational interaction as dark graviton exchanges.

   v₀ is a velocity parameter. For Sun would have β_0,S=v₀/c≈ 2^-11 from the model for the inner planets as Bohr orbits (see this).

2. The Planckian educated guess is m∼ m_Pl so that one would have hbar_gr/hbar= r_S(M)/(2L_Plβ₀), where L_Pl is Planck length and r_S(M) is the Schwartshild radius of magnetar. This would give Y= X/(ℏ_gr/ℏ)= .4 if one has r_S=3 km as for the Sun. r_S is probably large but smaller than magnetar radius about 20 km. The masses of the magnetars are in the range 1-2 solar masses. For M= 2M_S one obtains Y=.8.

   The rough estimate is not far from Y=1 and suggests that the interacting quantum units at the receiving end have mass of order Planck mass. Interestingly, the mass of a large neuron with radius 10^-4 m is about Planck mass (see this), which supports the view that quantum gravitation in the TGD sense is fundamental for life - even in the cosmic scales.

The physical interpretation of the velocity parameter v₀ is one of the key challenges.

1. The order of magnitude of v₀ is the same as for the rotational velocities in the solar system. I have considered a geometry based interpretation (see this and this).
2. The analogy with the neural system encourages the question whether v₀ could have a concrete interpretation as the analog of the nerve pulse conduction velocity assignable to the dark magnetic flux tubes connecting distant systems.

   In TGD framework nerve pulses (see this) are proposed to be induced by the perturbations of Sine-Gordon soliton sequences for the generalized Josephson junctions assignable to the cell membrane and identifiable as transversal flux tubes assignable to various membrane proteins such as ion channels and pumps. The dark variants of the biologically important ions would give rise to the supra currents.

   Could the gravitational flux tubes analogous to axons have this kind of structure and give rise to generalized Josephson junctions with ions serving also in this case as current carriers?

To sum up, the proposed interpretation as cosmic neural networks conforms with the basic assumptions of TGD. Most importantly, quantitative predictions are correct. The picture is of course not deduce from axioms: this is pattern recognition with basic principles predicting a lot of new physics.

I encountered a highly interesting popular article (thanks for my friend Asta for the link) with title "Mysterious 'fast radio burst' detected closer to Earth than ever before" (this).

Fast radio wave bursts (FRBs) arrive from a distance of hundreds of millions of light years - the scale of a large void. If the energy of FRBs is radiated isotropically in all directions - an assumption to be challenged below - the total energy is of the same order of magnitude that the energy of the Sun produced during a century. There are FRBs repeating with a period of 16 days located to a distance of 500 million light years from Earth.

The latest bursts arrive from a distance of only about 30 thousand light years from our own galaxy Milky Way described in the popular article can be assigned with magnetar (see this) which is a remnant of neutron start and has extremely strong magnetic field of about 10¹¹ Tesla.

Below is the abstract of the article (this) reporting the discovery.

We report on International Gamma-Ray Astrophysics Laboratory (INTEGRAL) observations of the soft γ ray repeater SGR 1935+2154 performed between 2020 April 28 and May 3. Several short bursts with fluence of ∼ 10^-7-10^-6 erg cm^-2 were detected by the Imager on-board INTEGRAL (IBIS) instrument in the 20-200 keV range. The burst with the hardest spectrum, discovered and localized in real time by the INTEGRAL Burst Alert System, was spatially and temporally coincident with a short and very bright radio burst detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME) and Survey for Transient Astronomical Radio Emission 2 (STARE2) radio telescopes at 400-800 MHz and 1.4 GHz, respectively.

Its lightcurve shows three narrow peaks separated by ∼ 29 ms time intervals, superimposed on a broad pulse lasting ∼ 0.6 s. The brightest peak had a delay of 6.5 +/- 1.0 ms with respect to the 1.4 GHz radio pulse (that coincides with the second and brightest component seen at lower frequencies). The burst spectrum, an exponentially cutoff power law with photon index Γ =0.7_-0.2^+0.4 and peak energy E_p=65+/- 5 keV, is harder than those of the bursts usually observed from this and other magnetars.

By the analysis of an expanding dust-scattering ring seen in X-rays with the Neil Gehrels Swift Observatory X-ray Telescope (XRT) instrument, we derived a distance of 4.4_-1.3^+2.8 kpc for SGR 1935+2154, independent of its possible association with the supernova remnant G57.2+0.8. At this distance, the burst 20-200 keV fluence of (6.1+/- 0.3)× 10^-7 erg cm^-2 corresponds to an isotropic emitted energy of ∼ 1.4× 10³⁹ erg. This is the first burst with a radio counterpart observed from a soft γ ray repeater and it strongly supports models based on magnetars that have been proposed for extragalactic fast radio bursts.

What could be the interpretation of the finding in the TGD framework? The weirdest feature of the FRB is its gigantic total energy assuming that the radiation is isotropic during the burst. This assumption can be challenged in the TGD framework, where the stellar systems are connected to a monopole flux tube network and radiation flows along flux tubes, which can also branch. This brings strongly in mind the analog of a nervous system in cosmic scales and this analogy is used in what follows.

1. The duration of pulses is few milliseconds: the duration of nerve pulses is the same. Is this a wink-wink to the Poirots of astrophysics?
2. Bursts can arrive regularly for instance with a period of T=16.35 days (see this). This brings in the mind of astro-Poirot biorhythm, in particular EEG rhythms. This would not be the only such rhythms: also the period of T_alpha=160 minutes, for which have proposed an interpretation as a cosmic analog of alpha rhythm is known (see this). The ratio T/T_α=147.15 would give for the analogous brain rhythm the value of 14.7 seconds.
3. Let us assume that stellar systems indeed form an analog of neural network connected by flux and assume that the topology of this network is analogous to that defined by axons. In TGD framework neural communications between neurons occur actually by using dark photons with effective Planck constant h_eff=nh₀ along the flux tubes with the velocity of light so that feedback from brain and even from the magnetic body of brain back to sensory organs as a virtual sensory input becomes possible. The function of nerve pulses is to connect the outgoing branch of the flux tube associated with the axon and those associated with dendrites of the post-synaptic neuron toa longer flux tubes by using neurotransmitters as relays.
4. The stellar object as an analog of a neuron would send its dark photon signals along the flux tube assignable to a single axon. Axon would later branch to dendrites arriving to other stellar systems and eventually perhaps to planets as analogs of synaptic contacts. An interesting question is whether also the analogs of nerve pulses and neurotransmitters acting as relays in the synaptic contacts defined by planets could make sense. What could nerve pulses propagating along the flux tube correspond to?

   Remark: In the TGD based model of brain there would be also flux tube network analogous to the meridian system of Eastern medicine and responsible for the holistic and spatial aspects of consciousness since more than one flux tube can emanate from a given node making possibly non-linear networks (see this). Nervous system with tree- like structure would be responsible for the linear and temporal aspects of conscious experience. Tree-like structure would be crucial for the understanding of Alzheimer disease (see this). Meridian system would be a predecessor of the neural system.

5. The distances of FRBs are of the order of large voids having galaxies at their boundaries and forming lattice-like networks possibly assignable to the tesselations of 3-D hyperbolic space defining cosmic time= constant surfaces. This kind of tesselations could accompany also brain (see this). In the fractal Universe of TGD one can wonder whether these voids are analogs of cells or even neurons and form cosmic biological organisms with flux tubes forming a network allowing communications.

The basic implication is that the energy of the emitted radiation could be dramatically smaller than that predicted by an isotropic radiation burst. It is interesting to look whether the proposed picture survives quantitative modelling.

1. The reduction factor r for the total emitted energy would be essentially r= S/A, where S is the area of the "axonal" flux tube and A=4π R² is the surface area of the magnetar. One must estimate the value of r.
2. Flux quantization for a single sheet of the many-sheeted magnetic flux tube involved would give eBS= hbar₀ h=6h₀ (see this and this). The general order of magnitude estimate is eB ∼ hbar₀/S. If each sheet carries out the same energy, the number of sheets is n=h_eff/h₀ and the effective area of a flux tube is S= hbar₀/eB. Does the magnetic field assigned with magnetar correspond to a single sheet or to all sheets? If the field is measured from cyclotron energies assuming h_eff=h it would correspond to all sheets and the measured magnetic field would be the effective magnetic field B_eff= nB/6 for h= 6h₀.
3. The branching of the flux tube could correspond to the splitting of the many-sheeted flux tube to tubes with smaller number of sheets and involve reduction of h_eff. This would give the estimate r= hbar₀/eBA. Magnetic field of 1 Tesla corresponds to a unit flux quantum with radius - magnetic length . about 2.6× 10^-8 meters. Assuming magnetar radius R=20 km one has r∼ 10^-25/6.
4. The estimate for the total emitted energy assuming isotropic radiation is the energy radiated by the Sun during a century. Sun transforms roughly E₁₀₀=1.3× 10¹⁹ kg of mass to radiation during a century. This gives for the energy emitted in FRB the estimate E= r E₁₀₀∼ 10^-6/6 kg which is roughly 7.5 Planck masses m_Pl≈ 2.2× 10^-8 kg. The order of magnitude is Planck mass. The estimate is of course extremely rough.

   In any case, the idea that pulses could have mass of order few Planck masses is attractive. Note that a large neuron with radius about 10^-4 meters has a mass of order Planck mass (see this).

5. From the total detected energy dE/dS= 6.1× 10^-7 erg m^-2= 3.8× 10⁹ eVm^-2 and total radiated energy E= 7.5 m_Pl one can estimate the total area S covered by the branched energy flux if it covers the entire area with a shape of disk of radius R. This gives some idea about how wide the branching is. The total energy is E =(dE/dS)× π R² giving R= [E/π (dE/dS)]^1/2∼ .9× 10⁹ m. The equitoral radius of the Sun is R_Sun= .7× 10⁹ m. R_Sun∼ .78 R. This conforms with the idea that the radiation arrives along the axon-like flux tube connecting Sun and the magnetar branching so that it covers the entire Sun.

The ratio h_eff/h should be of the same order of magnitude as the ratio X=E/E_rad, where E_rad is the energy of the radio wave photon with frequency 1.4 GHz for h_eff=h: X∼ h_eff/h. The ratio Y= X/(h_eff/h) should satisfy Y∼ 1.

1. To proceed further, one can use the TGD variant of Nottale's hypothesis. The hypothesis states that one can assign to gravitational flux tubes gravitational Planck constant hbar_gr. The original hypothesis was ℏ_eff=ℏ_gr and the more recent form inspired by the adelic vision states that h_gr corresponds to a large integer factor of h_eff. One has ℏ_gr= GMm/v₀= r_Sm/2v₀. Here M is the mass of the large object - now that of magnetar. m is the mass of the smaller quantum coherent object in contact with the gravitational flux tube mediating gravitational interaction as dark graviton exchanges.

   v₀ is a velocity parameter. For Sun would have β_0,S=v₀/c≈ 2^-11 from the model for the inner planets as Bohr orbits (see this).

2. The Planckian educated guess is m∼ m_Pl so that one would have hbar_gr/hbar= r_S(M)/(2L_Plβ₀), where L_Pl is Planck length and r_S(M) is the Schwartshild radius of magnetar. This would give Y= X/(ℏ_gr/ℏ)= .4 if one has r_S=3 km as for the Sun. r_S is probably large but smaller than magnetar radius about 20 km. The masses of the magnetars are in the range 1-2 solar masses. For M= 2M_S one obtains Y=.8.

   The rough estimate is not far from Y=1 and suggests that the interacting quantum units at the receiving end have mass of order Planck mass. Interestingly, the mass of a large neuron with radius 10^-4 m is about Planck mass (see this), which supports the view that quantum gravitation in the TGD sense is fundamental for life - even in the cosmic scales.

The physical interpretation of the velocity parameter v₀ is one of the key challenges.

1. The order of magnitude of v₀ is the same as for the rotational velocities in the solar system. I have considered a geometry based interpretation (see this and this).
2. The analogy with the neural system encourages the question whether v₀ could have a concrete interpretation as the analog of the nerve pulse conduction velocity assignable to the dark magnetic flux tubes connecting distant systems.

   In TGD framework nerve pulses (see this) are proposed to be induced by the perturbations of Sine-Gordon soliton sequences for the generalized Josephson junctions assignable to the cell membrane and identifiable as transversal flux tubes assignable to various membrane proteins such as ion channels and pumps. The dark variants of the biologically important ions would give rise to the supra currents.

   Could the gravitational flux tubes analogous to axons have this kind of structure and give rise to generalized Josephson junctions with ions serving also in this case as current carriers?

To sum up, the proposed interpretation as cosmic neural networks conforms with the basic assumptions of TGD. Most importantly, quantitative predictions are correct. The picture is of course not deduce from axioms: this is pattern recognition with basic principles predicting a lot of new physics.

See the article Fast radio wave bursts: is life a cosmic fractal? or the chapter About the Nottale's formula for h_gr and the relation between Planck length and CP₂ length R .

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

Articles and other material related to TGD.

Nature Physics and physics

Did quite hard work to prepare summary of TGD for a possible publication in Nature Physics. They had not even sent it to referees. Incredible stupidity. I however sent the artice to Witten, Maldacena, Susskind, and Arkani-Hamed. One might hope that the message managed to bypass their secretaries.

Dear Dr Pitkänen,

Thank you for submitting your manuscript "Summary of Topological Geometrodynamics" which we are regretfully unable to offer to publish.

It is Nature Physics' policy to return a substantial proportion of manuscripts without sending them to referees. Decisions of this kind are made by the editors of Nature Physics according to the demanding editorial criteria of the journal.

In the present case, while your findings may well prove stimulating to others' thinking about such questions, we regret that we are unable to conclude that the work provides the sort of firm advance in general understanding that would warrant publication in Nature Physics.

We are sorry that we cannot respond more positively on this occasion.

Regards,

Nature Physics

I will add the article to my homepage when communications to it work again.

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

Articles and other material related to TGD.