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. 



Ballistic resonance as breaking of second law: TGD view point

The popular article " Scientists have discovered a new physical paradox" (see this) tells about the work of Vitaly A. Kuzkin et al published in Phys Rev E (see this) as article with title " Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature" . The article describes very interesting experimental findings, which could provide a direct application of zero energy ontology (ZEO) based theory of self-organization.

The findings and their explanation provided by experimenters

Researchers from the Peter the Great St. Petersburg Polytechnic University (SPbPU) have discovered a new physical effect: the amplitude of mechanical vibrations can grow without external influence in which system converts its thermal energy to mechanical energy. The phenomenon is known as ballistic resonance. The description of the phenomenon involves also an abnormally high heat conductivity - one speaks of ballistic heat conductivity.

The electromagnetic analogy is very high electric conductivity: the work of Bandyopadhyay related to effects of oscillating voltage on currents flowing along microtubules demonstrates ballistic conductivity possibly reflecting underlying super-conductivity (see this).

This behavior seems to be in conflict with second law of thermodynamics telling that the vibrations should be attenuated. The researchers propose also a theoretical explanation of this paradox (see this) based on a model assuming ballistic heat conduction. One can of course wonder whether the notion of ballistic heat conduction is consistent with second law in its standard form.

Fermi-Past-Ulam-Tsingou problem (see this) relates to the finding about a theoretical model of a vibrating string with a non-linear dynamics. The expectation was that the situation develops ergodic so that energy is evenly divided between the modes of the string. It however turned out that the behavior was essentially periodic. The model explaining the behavior relies on solitons assignable to Korteveg-de-Vries equation. This phenomenon is different from the ballistic resonance observed in the experiments. In Korteveg de-Vries equation there is no dissipative term and the unexpected phenomenon is that wave pattern preserves it shape. Dissipation without energy feed would attenuate the wave.

ZEO based model for the findings

TGD suggests that a genuine explanation requires a profound change in the thinking about time- in particular the relationship between geometric time and experienced time must be updated. I call the new conceptual framework zero energy ontology ZEO) (see this). The identification of these two times in standard ontology is in conflict with simple empirical facts, and leads to a paradox related to state function reduction (SFR) taking place in quantum measurement. The non-determinism of SFR is in conflict with the determinism of Schrödinger equation.

1. According to ZEO in ordinary state function reduction (SFR) the arrow of time subsystem changes: this solves the basic paradox of quantum measurement theory. The experiments of Minev et al (see this) give impressive experimental support for the notion in atomic scales, and sow that SFR looks completely classical deterministic smooth time evolution for the observer with opposite arrow of time. This is just what TGD predicts. Macroscopic quantum jump can occur in all scales but ZEO takes care that the world looks classical! The endless debate about the scale in which quantum world becomes classical would be solely due to complete misunderstanding of the notion of time.
2. Non-standard arrow of time forces a generalization of thermodynamics. For time reversed system generalized second law applies in reverse direction of time. Dissipation with reversed arrow of time extracts energy from environment, in particular thermal energy from internal thermal environment. The energy feed necessary for self-organization reduces to dissipation in reversed arrow of time.

   This explains why self-organization is possible (see this). Standard form of the second flow would imply that also energy flows between systems go to zero: this would mean thermodynamical equilibrium everywhere - heat death. This has led to desperate theoretical proposals such as life as gigantic thermodynamical fluctuation. The recent empirical understanding suggests that this giant fluctuation would have occurred in the scale of the entire Universe and continue forever!

3. Macroscopic quantum coherence is however a necessary prerequisite for macroscopic effects. TGD predicts hierarchy of phases of ordinary matter residing at magnetic body (MB) of the system with value of effective Planck constant h_eff= nh₀ (h=6h₀) of h_effbehaving like dark matter and controlling ordinary matter (see this).. The larger the value of h_eff, the longer the scale of quantum coherence scale at MB. MB acts as master for ordinary matter in the role of slave and induces coherent behaviour. This gives rise to self-organization.

This picture could explain the observations of self induced resonance using thermal energy. A subsystem or its MB in time reversed mode would extract the thermal energy. There are many other applications. The phenomenon of stochastic resonance in which system extracts energy from external noise could have explanation along these lines. Stochastic resonance plays an important role in sensory perception by making possible amplification of weak signal in large background. There is evidence for it even in astrophysical scales. In biology metabolic energy could be extracted from metabolites and maybe also from thermal energy by time reversed dissipation by some subsystems related to metabolism.

TGD picture does not exclude the possibility of delicate models mimicking this behavior in the framework of thermodynamics. The basic challenge in this kind of effective model is to describe the presence time reversed dissipation inducing self-organization and the presence of dark matter at magnetic body phenomenologically. Energy feed as parameter gives rise to states far from thermodynamical equilibria.

For instance, the thermodynamics of ion distributions inside and outside cell is far far from thermodynamical equilibrium and and non-equilibrium thermodynamics has been developed for the modelling of this kind of systems utilizing the notions of ionic pumps and channels. The phenomenological description introduces chemical potentials as parameters to describe the non-equilibrium situation in the framework ordinary thermodynamics. Chemical potentials would model the neglected presence of h_eff>h phases of dark matter at magnetic body of the system.

See the article Ballistic resonance and zero energy ontology or the chapter Zero Energy Ontology and Matrices.

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

Articles and other material related to TGD.

Is there a chronon of time?

This posting was inspired by a popular article telling about the proposal that Planck length is not the fundamental length as often believed but the fundamental chronon is much longer.

The idea about fundamental unit of time - usually assumed to be given by Planck time about 10^-44 seconds - is rather naive if taken to mean discretization of time. The article proposes a variant of this idea and identifies the fundamental chronon as a fundamental periodicity of dynamics and deduces for it a value about 10^-33 seconds from observed bounds to the variation of dynamical periods. This value is by 11 orders of magnitude longer than Planck time. Personally I am a little bit skeptic about reliability of these bounds since very short times are involved.

Planck time is deduced from a mere dimensional analysis argument by feeding in speed of light c, Planck constant h, and Newton's constant G so that it is rather ad hoc noton. Therefore it has been surprising to me at least how seriously people have taken it. Moreover, Planck length appears in theories - in superstring theory in particular - typically as an ad hoc formal parameter with no direct geometric interpretation. In general relativity this leads to non-renormalizability and it is not possible to quantize gravitation in this framework. In superstring theories it led to landscape catastrophe: even a smallest change in the physics at Planck length scales changes completely the physics at long length scales: butterfly effect in the theory space.

For these reasons the question whether there exists some fundamental length-/time unit or several of them is a key problem of recent day physics. Could there be some fundamental length scale or possibly several of them with a clear geometric interpretation? In TGD Universe this is indeed the case.

1. Planck length is derived quantity and CP₂ length scale defines the fundamental length, which from p-adic mass calculation for electron mass roughly 10⁴ times longer than Planck length. Space-time is continuuous but CP₂ length serves as a fundamental unit of length, kind of length stick.
2. p-Adic length scale hypothesis (PLH) predicts actually infinite hierarchy of length/time units as p-adic length scale hypothesis stating that these units are proportional to sqrt(p), p preferred p-adic prime. p-Adic length scale hypothesis in this general form emerges both from M⁸-H duality and p-adic mass calculations.
3. A stronger form of PLH states that certain primes near powers of 2 are physically favored so that in the most general case one obtains a hierarchy of length and time units coming as half octaves. This form of hypothesis not well-understood although it conforms with period doubling in chaotic systems. Also powers of other small primes are possible and there is some evidence for the powers of 3. This would relate the preferred length scales of physics in long scales to CP₂ scale.
4. TGD predicts second length scale hierarchy corresponding to the hierarchy of effective values h_eff=nh₀ of Planck constant (h=6h₀) labelling phases of ordinary matter behaving like dark matter. n corresponds number theoretically to the dimension for extension of rationals. This makes possible a hierarchy of quantum coherence length coming as n/6-multiples of the ordinary Compton length. Quantum coherence in long length scales is the most important implication and the coherence of living matter would be due to quantum coherence at magnetic body - distinguishing between TGD and Maxwellian and QFT view about classical fields. There is considerable evidence for the existence of h_eff hierarchy from various anomalies, in particular from those in living matter.
5. Also now the challenge is testing of this hypothesis in macroscopic length scales: we cannot directly access short scales. The idea is simple: measure ratios of p-adic mass scales. They do not depend on CP₂ scale nor on the value of n. The ratios of dark quantum scales - say dark Compton lengths - are typically given by the ratio n₁/n₂ of integers involved.

   This allows precise tests by measuring mass and length scale ratios rather than masses and length scales. For instance, the possibility of scaled variants of hadron physics and electroweak physics allow to test the hypothesis. There are indeed indications for scaled up variants of mesons with mass scale differing by a factor 512 from that for ordinary hadrons. The Compton lengths would be same as for ordinary hadrons for n/6=512: the dark Compton scale for p-adically scaled up meson be same as ordinary Compton length making possible resonant coupling. If the valence electron of atom is dark, its Bohr radius is scaled up by (n/6)²: these states might be misinterpreted as Rydberg states.

Concerning disretization of space-time TGD allows different view.

1. Physics as number theory vision predicts that hierarchy of extensions of rationals defines evolutionary hierarchy. A generalization of real numbers to adeles labelled by extensions of rationals is assumed. For given extension of rationals adeles form a book like structure having as pages real numbers and extensions of p-adic number fields induced by extension of rationals. The pages are glued together along the back of the book consisting of points in given extension of rationals common to reals and extensions of all p-adic number fields. This hierarchy corresponds to evolutionary hierarchy and the dimension n of extension has identification as effective Planck constant h_eff.
2. Space-time itself becomes a book-like structure. Real space-time surfaces are replaced with adelic surfaces, which containing real sheet and p-adic sheets glued together along the back of a book consisting of points with imbedding space coordinates in given extension of rationals. The points of space-time surface with coordinates in given extension of rationals form a discrete cognitive representaton, which is unique and improves with the dimension of extension so that at the limit of algebraic numbers it is dense set of space-time surface.
3. I call this discretization identifiable as intersection of sensory world (reals) and cognitive worlds as cognitive representation. The discretizaton reflects the limitations of cognition which must always discretize. In M⁸ picture space-time surface are "roots" of octonionic polynomials and the polynomial defines the extension of rationals via its roots. At M⁸ level there are also essentially unique imbedding space coordinates making discretization unique.

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

Articles and other material related to TGD.

What next in TGD?

Last night I was thinking about possible future project in TGD. The construction of scattering amplitudes has been the dream impossible that has driven me for decades. Maybe the understanding of fermionic M⁸-H duality provides the needed additional conceptual tools.

1. M⁸ picture looks simple. Space-time surfaces in M⁸ can be constructed from real polynomials with real (rational) coefficients, actually knowledge of their roots is enough. Discrete data - roots of the polynomial!- determines space-time surface as associative or co-associative region! Besides this one must pose additional condition selecting 2-D string world sheets and 3-D light-like surfaces as orbits of partonic 2-surfaces. These would define strong form of holography (SH) allowing to map space-time surfaces in M⁸ to M⁴×CP₂.
2. Could SH generalize to the level of scattering amplitudes expressible in terms of n-point functions of CFT?! Could the n points correspond to the roots of the polynomial defining space-time region!

   Algebraic continuation to quaternion valued scattering amplitudes analogous to that giving space-time sheets from the data coded SH should be the key idea. Their moduli squared are real - this led to the emergence of Minkowski metric for complexified octonions/quaternions) would give the real scattering rates: this is enough! This would mean a number theoretic generalization of quantum theory.

3. One can start from complex numbers and string world sheets/partonic 2-surfaces. Conformal field theories (CFTs) in 2-D play fundamental role in the construction of scattering string theories and in modelling 2-D statistical systems. In TGD 2-D surfaces (2-D at least metrically) code for information about space-time surface by strong holography (SH) .

   Are CFTs at partonic 2-surfaces and string world sheets the basic building bricks? Could 2-D conformal invariance dictate the data needed to construct the scattering amplitudes for given space-time region defined by causal diamond (CD) taking the role of sphere S² in CFTs. Could the generalization for metrically 2-D light-like 3-surfaces be needed at the level of "world of classical worlds" (WCW) when states are superpositions of space-time surfaces, preferred extremals?

The challenge is to develop a concrete number theoretic hierarchy for scattering amplitudes: R→C→Q→O - actually their complexifications.

1. In the case of fermions one can start from 1-D data at light-like boundaries LB of string world sheets at light-like orbits of partonic 2-surfaces. Fermionic propagators assignable to LB would be coded by 2-D Minkowskian QFT in manner analogous to that in twistor Grassmann approach. n-point vertices would be expressible in terms of Euclidian n-point functions for partonic 2-surfaces: the latter element would be new as compared to QFTs since point-like vertex is replaced with partonic 2-surface.
2. The fusion (product?) of these Minkowskian and Euclidian CFT entities corresponding to different realization of complex numbers as sub-field of quaternions would give rise to 4-D quaternionic valued scattering amplitudes for given space-time sheet. Most importantly: there moduli squared are real! A generalization of quantum theory (CFT) from complex numbers to quaternions (quaternionic "CFT").
3. What about several space-time sheets? Could one allow fusion of different quaternionic scattering amplitudes corresponding to different quaternionic sub-spaces of complexified octonions to get octonion-valued non-associative scattering amplitudes. Again scattering rates would be real. A further generalization of quantum theory?

There is also the challenge to relate M⁸- and H-pictures at the level of WCW. The formulation of physics in terms of WCW geometry leads to the hypothesis that WCW Kähler geometry is determined by Kähler function identified as the 4-D action resulting by dimensional reduction of 6-D surfaces in the product of twistor spaces of M⁴ and CP₂ to twistor bundles having S² as fiber and space-time surface X⁴⊂ H as base. The 6-D Kähler action reduces to the sum of 4-D Kähler action and volume term having interpretation in terms of cosmological constant.

The question is whether the Kähler function - an essentially geometric notion - can have a counterpart at the level of M⁸.

1. SH suggests that the Kähler function identified in the proposed manner can be expressed by using 2-D data or at least metrically 2-D data (light-like partonic orbits and light-like boundaries of CD). Note that each WCW would correspond to a particular CD.
2. Since 2-D conformal symmetry is involved, one expects also modular invariance meaning that WCW Kähler function is modular invariant, so that they have the same value for X⁴⊂ H for which partonic 2-surfaces have induced metric in the same conformal equivalence class.
3. Also the analogs of Kac-Moody type symmetries would be realized as symmetries of Kähler function. The algebra of super-symplectic symmetries of the light-cone boundary can be regarded as an analog of Kac-Moody algebra. Light-cone boundary has topology S²× R₊, where R₊ corresponds to radial light-like ray parameterized by radial light-like coordinate r. Super symplectic transformations of S²× CP₂ depend on the light-like radial coordinate r, which is analogous to the complex coordinate z for he Kac-Moody algebras.

   The infinitesimal super-symplectic transformations form algebra SSA with generators proportional to powers rⁿ . The Kac-Moody invariance for physical states generalizes to a hierarchy of similar invariances. There is infinite fractal hierarchy of sub-algebras SSA_n⊂ SSA with conformal weights coming as n-multiples of those for SSA. For physical states SSA_n and [SSA_n,SSA] would act as gauge symmetries. They would leave invariant also Kähler function in the sector WCW_n defined by n. This would define a hierarchy of sub- WCWs of the WCW assignable to given CD.

   The sector WCW_n could correspond to extensions of rationals with dimension n, and one would have inclusion hierarchies consisting of sequences of n_i with n_i dividing n_i+1. These inclusion hierarchies would naturally correspond to those for hyper-finite factors of type II₁.

   See the article Fermionic variant of M⁸-H duality or the chapter ZEO and matrices.

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

   Articles and other material related to TGD.

M8-H duality for fermions

M⁸-H duality in bosonic sector is rather well understood but the situation is different in the fermionic sector. The basic guideline is that also fermionic dynamics should be algebraic and number theoretical.

1. Spinors should be octonionic. I have already earlier considered their possible physical interpretation.
2. Dirac equation as linear partial differential equation should be replaced with a linear algebraic equation for octonionic spinors which are complexified octonions. The momentum space variant of the ordinary Dirac equation is an algebrac equation and the proposal is obvious: PΨ=0, where P is the octonionic continuation of the polynomial defining the space-time surface and multiplication is in octonionic sense. The masslessness condition restricts the solutions to light-like 3-surfaces m_klP^kP^l=0 in Minkowskian sector analogous to mass shells in momentum space - just as in the case of ordinary massless Dirac equation. P(o) rather than octonionic coordinate o would define momentum. These mass shells should be mapped to light-like partonic orbits in H.
3. This picture leads to the earlier phenomenological picture about induced spinors in H. Twistor Grassmann approach suggests the localization of the induced spinor fields at light-like partonic orbits in H. If the induced spinor field allows a continuation from 3-D partonic orbits to the interior of X⁴, it would serve as a counterpart of virtual particle in accordance with quantum field theoretical picture.

   Addition: A really pleasant surprise that came this morning-9.7.2020 - sounds melodramatic but I do not want to forget it - it could have come more than decade ago but did not. The octonionic inner product for complexified octonionic 8-momenta with conjugation with respect to commuting imaginary unit i gives 8-D Minkowski norm squared. Same about quaternonic norm for complexified quaternionic momenta. Minkowski space with signature of (1,-,-1,-1) for metric follows from number theory alone! This conforms with the very idea of M⁸-H duality that geometry and number theory are dual in physics. Already this single finding makes M⁸-duality a "must".

   See the article Fermionic variant of M⁸-H duality or the chapter Does M⁸ duality reduce classical TGD to octonionic algebraic geometry?: Part III.

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

   Articles and other material related to TGD.

John Baez about Noether's theorem in algebraic approach

John Baez gives very nice summary (see this of the triple of states, observables, and generators of symmetries from purely algebraic point of view. Jordan Banach algebra with commutative product A*B= (AB+BA)/2 would play the role of observables. The operators correspond to symmetry generators commuting with observables and the unitary evolutions generated by them in this algebra are trivial. One could say that this defines analogy of Noether's theorem usually deduced for the symmetries of action principle.

To me the weakness of the algebraic approach is that it says very little about the dynamics- it woul be just unitary evolution generation by some generator of symmetry. Second problem is mentioned at the end of the posting is how classical relates to quantal. And there is nothing about quantum measurement problem so that basically an attempt to reproduce wave mechanics using operators is in question.

My own view - zero energy ontology (ZEO) - goes much beyond quantum mechanics of simple systems.

1. The basic problem of quantum measurement theory is the starting point. The notion of quantum state is modified. In wave mechanics and quantum field theories it is based on initial value problem in configuration space (space of positions for particle). Initial state is wave function - a superposition of possible initial values in configuration space. Time evolution is formulated in terms of unitary evolution defined by exponential of Hamiltonian and reduces to Schroedinger equation.

   What happens in quantum measurements is not consistent with this time evolution. This is the problem.

   1. In TGD one replaces initial value problem with a boundary value problem with boundary corresponding to values at times t1 and t2 (this is a simplification).
   2. One defines states as superpositions of deterministic classical time evolutions - preferred extremals - analogous to Bohr orbits having the property that boundary value problem is equivalent to initial value problem. Once on knows configuration at t1, one knows it at t2.
   3. Quantum states are superpositions of these preferred extremals from t1 to t2 and quantum jump replaces this kind of superposition with a new one. I call this approach zero energy ontology (ZEO).
   4. The basic problem of quantum measurement theory disappears since there are two causalities: that of quantum jump and classical causality, and there is no need to break the deterministic time evolution analogous to that given by Schroedinger equation in quantum jump. There are two times: experienced time as sequence of quantum jumps and geometric time. One ends up with a theory of consciousness without moment of consciousness identifiable as state function reduction. Also a ZEO theory of self-organization emerges. This is of course only the basic idea. For instance, one must understand how correlation between experienced time and geometric time emerges.
   5. One the many implications of ZEO is new view about quantum tunnelling: it must have classical time evolution as quantum correlate. This leads to a new view about tunnelling in nuclear reactions relying essentially on the change of the arrow of time in ordinary state function reduction. Just today I received link telling about strange phenomenon occurring in what is believe to be ordinary electron tunnelling. The electron getting through the barrier radiates energy which increases with the height of the barrier. I discusse the ZEO based explanation in a related posting New support for TGD view about quantum tunnelling .

   How to realize this picture and how unique is it? Here one must leave the realm of wave mechanics.

   1. Loosely speaking, in TGD framework point-like particle is replaced with 3-D surface in M⁴× CP₂ and its orbit as preferred extremal of action principle, whatever it might be, defines space-time region. A generalization of string model is in question. Also a generalization of general relativity solving its problem due to the loss of Poincare symmetries is in question.
   2. This leads to a generalization of Einstein's geometrization program: replace configuration space with the "world of classical worlds" (WCW) and give it Kaehler geometry to realize geometrization of quantum theory. Points of WCW are 3-surfaces or equivalently 4-surfaces: this reduces holography and reduces it to general coordinate invariance.

      WCW spinor fields would represent physical states as "wave functions". Configuration space gamma matrices would be superpositions of fermionic oscillator operators so tha also fermions are geometrized. The mere existence of WCW Kaehler geometry requires maximal isometries and this fixes TGD highly uniquely. Freed realize the uniqueness for loop spaces.

   How to realize the crucial preferred extremal property making initial value problem equivalent with boundary value problem?
   1. Here the maximal isometry group of WCW enter the game. The symmetry algebra is replaced with an analog of infinite-D symplectic algebra acting as isometries of WCW induced from symplectic transformations at delta M⁴₊xCP₂ labelled by integer valued conformal weights assignable to the radial light-like coordinate of light-cone bounary δM⁴₊ defining second boundary of causal diamond cd identified as the intersection of future and past directed light-cones.
   2. The crucial point is that this algebra - call it A - has fractal hierarchy of sub-algebras A_n with conformal weights coming as multiples of n=1,2,... which very probably corresponds to a hierarchy of hyper-finite factors of type II₁ forming inclusion hierarchies labelled by sequences ...n₁ divides n₂ divides....
   3. Infinite-D sub-algebra A_n appears would have vanishing classical Noether charges in the class of preferred extremals associated with A_n. Also [A_n,A] would have the same property.This is like posing the condition that analogous sub-algebra of say Kac-Moody algebra annilates physical states. The space-time surfaces in question would be minimal surfaces satisfying the additional condition that they extremize also what I call Kaehler action, and being analogous to Maxwell action.

   See the article Some comments related to Zero Energy Ontology (ZEO).

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

   Articles and other material related to TGD.

New support for TGD view about quantum tunnelling

There is a popular article in Phys.org describing a highly interesting finding made by condensed matter physicist Doug Natelson and his colleagues at Rice and the University of Colorado Boulder. Light emission associated with the tunnelling of electrons through a nano-scaled potential barrier between Gold electrodes has been observed. The intensity of emission is larger by factor 10,000 than predicted and this suggests new physics.

By definition tunnelling means that electrons get through the potential barrier without getting energy - classical picture would require this. Now it however seems that electron receives energy and the higher the barrier the larger energy is needed. Should one challenge the notion of tunnelling? Could it have a classical counterpart?

In TGD framework all quantum phenomena should have classical counterparts, also tunnelling. Therefore the tunnelling electron would actually get energy to get over the potential barrier classically.

1. In zero energy ontology (ZEO) solving the basic problem of quantum measurement theory quantum states are superpositions of deterministic classical time evolutions, preferred extremals. Classical physics is exact part of quantum theory. The key prediction is that in ordinary, "big", state function reductions (BSFRs) the arrow of time is changed. In small SFRs (SSFRs) - analogs of weak measurements - this does not happen. ZEO leads to a theory of self-organization in which energy feed to self-organizing system corresponds to dissipation for time reversed state associated with dissipating system.
2. Tunnelling could mean in TGD framework that electrons make a BSFR reversing the arrow of time. In time reversed state they dissipate in reverse time direction: in standard time direction of observer they receive energy in standard time direction allowing them to get over the potential barrier. In the second BSFR establishing the original arrow of time they would liberate the energy and the higher the barrier, the larger the liberated energy and the brighter the light emission. The tunnelling in this sense would be already self-organization phenomenon involving BSFR. This dynamical tunnelling does not exclude the analog of wave-mechanical tunnelling as a non-dynamical process. The dynamical tunnelling could actually lead to asymptotic states in which particle is at both sides of the forbidden region or even in forbidden region but with extraction of classical energy from environment so high that this is not forbidden anymore.

For ZEO see the article Some comments related to Zero Energy Ontology (ZEO). For TGD view about tunnelling in nuclear reactions see Solar Metallicity Problem from TGD Perspective

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

Articles and other material related to TGD.

New ideas about the transition to ferromagnetic phase

I received a link to a highly interesting article about ferromagnetism. According to the article, Yi Li, a physicist working at John Hopkins University and his two graduate students, Eric Bobrow and Keaton Stubis, seem to have made a considerable progress in understanding how the system of electron spins in lattice ends up to a ferromagnetic sate. This ferromagnetism is known as itinerant ferromagnetism and involves vacancies, sites without electron, which can be moved freely without affecting the energy of the state.

1. The ideas related to the work of Li et al

The problem considered by Li et al how the ferromagnetic state could emerge from an arbitrary state with some numbers of spin up and spin down states at lattice sites connected by edges.

1. Permutation of electrons with same spin leave the ferromagnetic state invariant and does not cost energy while permutations in arbitrary configuration can do so.
2. Li et al considered a simple 4× 4 lattice with single vacancy and noticed a connection with so called 15-puzzle involving 15 tiles and single vacancy with neighboring tiles of vacancy able to move to its position. The observation is following. If one has spin lattice containing single vacancy, one can number the sites by a number running from 1 to N (now 15) in arbitrary manner. If so called connectedness condition holds true one can realize any permutation of these numbers. This means that 15-puzzle has always a solution. In particular, one can arrange the situation that the numbers form an ordered sequence from 1 to N so that numbers n and n+1 are nearest neighbors.

   The result found by Li et al first for 2-D 4× 4 lattice with single vacancy generalizes to lattices, which are non-separable in the sense the removal of a lattice site does not separate any pair of spins - they are still connected by an edge-loop.

3. The curve solving the 15-puzzle goes through all points of the 4× 4 lattice and is generally known as Hamiltonian curve. It becomes Hamiltonian cycle if the numbers 1 and N are nearest neighbors.
4. The basic problem of this approach is that the theorem is true only for single vacancy and does not allow generalization to a larger number of vacancies. It is however known that ferromagnetism is possible up to fraction 1/3 for vacancies. The challenge is to generalize the result of Li et al.

2. Some reasons to get interested

In TGD framework there are good reasons of getting interested on these results.

1. The result of Li et al states that ferromagnetic phase transition might be understood in terms of shifting of lattice vacancy if the lattice with single defect allows deformaton of any configuration of spin labelled by numbers n running from 1 to N to a closed curve connecting nearest neighbors along which n increases. Could there be a connection with Hamiltonian curves making sense for lattice like structures (actually all graphs)? Could Hamiltonian curve have some deeper physical meaning or is it only an auxialiary notion useful for representing the possibility to realize all points of the lattice with vacancy by shifting it suitably?

   Hamiltonian curve connects neighboring points of a lattice and goes through all points without self-intersections. Icosahedral geometry appears in biology and one can ask whether this kind of cycles could be actually realized physically - say as flux tubes at icosahedron and tetrahedron, which play key role in TGD inspired biology. Flux tube are actually fundamental objects in TGD Universe in all scales. For instance, final states of stars could correspond to flux tube spaghettis consisting of single volume filling flux tube (see this).

2. If the Hamiltonian cycle is something physical it could correspond to flux tube. The notion of magnetic flux tube central in TGD might allow application to ferromagnetism. TGD predicts two kinds of flux tubes: Maxwellian ones and monopole flux tubes with magnetic fields requiring no currents to generate them: they are not not allowed by Maxwell's theory.

   The preservation of the Earth's magnetic field predicted to decay rather rapidly as currents generating it dissipate supports the view that it contains monopole flux part which from biological input would correspond to endogenous magnetic field B_end, which is a fraction 2/5 about the nominal value of B_E=.5 Gauss. The presence of magnetic fields in cosmological scales is also a mystery finding a solution in terms of monopole flux tubes.

3. Monopole flux tubes must be closed. Closed non-intersecting flux tubes connecting nearest neighbors in lattice would correspond to Hamiltonian cycles. In TGD inspired biology Hamiltonian cycles associated with icosahedron and tetrahedron provide a realization of the vertebrate genetic code (see this) but it is still somewhat of mystery why the points of icosahedron and tetrahedron, which are lattices (tesselations) at sphere, would be connected by a curve. Quantum classical correspondence suggests that magnetization corresponds to flux tubes connecting magnetic dipoles as formal analogs of monopole-antimonopole pairs. Could magnetic flux tubes provide a concrete realizaon for these Hamiltonan cycles?
4. Closed monopole flux tubes seem to be unrealistic for the description of ferromagnetism, which suggests the presence of N parallel flux tubes carrying magnetization M and defining a braid connecting opposite ends of ferromagnet. The monopole fluxes could arrive as single flux along parallel space-time sheet carrying field H defined by single thick flux tube. Test particle would experience B= M+H.

The following considerations are not much more than first impressions and probably require updating.

3. TGD based view

Flux tubes are the new element of condensed matter physics predicted by TGD. Could they provide insights into ferromagnetism?

3.1 Starting from text book picture about ferromagnetism

To develop TG view about ferromagnetism it is best to start from the text book picture.

1. In the standard model of ferromagnetism one assumes the presence of field B identified as sum B= M+H of magnetization and field H equal to B outside the magnet. M is due to magnetic dipoles besides magnetic field B and the interaction of spins with H is important. B is usually regarded as the fundamental field M and and H appear as auxiliary notions and their relation to B requires a model for the system: typically H, B, and B are assumed to be linearly related.
2. The field M could be naturally assigned with a flux tube connecting the spins - perhaps at nearest neighbor lattice points. What about H? In standard model H and B are parallel for the ferromagnetic configuration. If B is assigned with the flux tube connecting the magnetic moments and B is parallel to H, this would suggest a flux tube consisting of long straight portions parallel to each other.

   In the many-sheeted space-time of TGD M and B can reside at different space-time sheets, which are parallel so that they are on top of each other in M⁴×CP₂. The decomposition to sum would have representation as a set theoretic union.

   The test particle would experience the sum of the magnetic fields associated with the two sheets. Could M and H as the return flux associated with M and superpositing with the external contribution to H correspond to these two space-time sheets so that particle would experience their sum B=M+H? If so, ferromagnetism could be seen as a direct signature of many-sheeted space-time.

3.2 Could also monopole flux tubes be important?

There is still one important aspect related to the TGD view about magnetic field which might play important role. TGD predicts two kind of flux tubes. The first kind of flux tubes could be called Maxwellian and the corresponding magnetic fields require current to generate them. There are also flux tubes having closed cross section and carrying monopole fluxes. No currents are required to generate corresponding magnetic fields. Could also these flux tubes having no current as sources be present? This would mean new physics.

1. The first thing to notice is that the interpretation of magnetization M is as a magnetic field generated by magnetic moments. The usual interpretation is that spins are analogous to magnetic moments created by currents consisting of rotating charge. Now there is no such rotating charge. Second interpretation is as magnetic moments identifiable as infinitesimal monopole pairs.
2. Could one think that the flux tubes containing sequence magnetic moments correspond to monopole flux and that closing this loop could give rise to monopole magnetic field? Ordinary Maxwellian part could be also present and have current as source. How M and H would relate to these. Could M correspond to the monopole part and H the Maxwellian part?

   Are spins necessary for the existence of a monopole flux tube? Could quantum classical correspondence require this? Could dark charged matter assigned with the monopole flux tubes correspond to the magnetic moments of say dark valence electrons with non-standard value of h so that M would be represented by monopole flux tubes classically? If the return flux represented by H is absent, flux tube must give rise to a Hamiltonian cycle. If H is present, it would be enough to have flux tubes representing N braid strands fusing to single monopole flux carrying the return flux.

   Formation of a flux loop defining Hamiltonian cycle would be a new kind of phenomenon analogous to spontaneous magnetization requiring no external field H. Spontaneous magnetization would be however something different. A trivial braid consisting of N parallel strands representing M and parallel to it locally with return flux arriving along single large flux tube carrying H would be formed in ferromagnetic transition and also in spontaneous magnetization.

3.3 Bringing in thermodynamics

One can try to make this more concrete by bringing in thermodynamics.

1. Assume that there exists single flux tube - connecting all the lattice points (magnetic moments) or possibly N flux tubes parallel to local magnetization M and giving rise to a braid like structure representing the topology of flux lines of M connecting opposite boundaries of magnet.
2. In the general case, the points of the lattice could be connected by a flux tubes connecting points, which need not be nearest neighbors. The first guess is that the magnetic interaction energy of spins at the ends of the flux tube portion connecting them decreases with the distance between spins. There sould be also magnetic energy associated with the field H at the space-time sheet carrying the return flux. Thermodynamics would bring in entropy and free energy F=E-TS would be mimimized. Entropy maximization would favor long random flux tubes and energy minimization short flux tubes.

   One expects that flux tube has free energy F increasing with flux tube length. If one does not allow self-intersections - as suggested by repulsive Coulomb interaction and Fermi statistics - the flux tube could be either Hamiltonian cycle or consist of analogs of braid strands: in the case of ferromagnetism the strands would be parallel to each other. The interacton energy would be same for all Hamiltonian cycles if determined by nearest neighbour interactions.

3. In the general case with lattice replaced by graph one expects that a large number of Hamiltonian cycles not related by rotation to each other exists so that one would have large number of states with same minimum energy. Could this somehow correspond to spin glass state allowing large number of degenerate states? The flux tube need not be closed. In ferromagnetic configuration this would be the case.
4. How would the assignment of spin direction to the lattice points affect the situation? Could the numbers N₊ and N_- of spin up and spin down electrons determine the flux tube configuration by (Gibbs) energy minimization?

3.4 Could 2-braid describe the transition to ferromagnetism?

In the work of Li et al discussed in the article, the permutations of lattice points are induced by moving the vacancy around. This picture inspired the considerations above but is too limited. In fact the work of Li et al only directed attention to Hamiltonian cycles and braids formed by the non-closed analogs.

1. TGD picture brings in mind braid-knot connection. One can replace braid assiciated with M with a knot by connecting the magnetic moments at the opposite ends of the braid by trands of a trivial braid at parallel space-time sheet. This trivial braid would carry the return flux having interpretation in terms of field H.

   The flux tubes of trivial braid could also fuse to single thicker flux tube carrying the total return flux associated with M. This would conform with the idea that H provides a description of the system in longer length scale being analogous to a smoothed out total magnetic field acting as self-consistent background.

   This stimulates a critical question. Could one assume that only H assignable to big flux tube has constant direction and magnitude and that M is represented as flux tubes connecting dipoles can in principle correspond to any permutation of atoms. For this option the spontaneous magnetization would correspond to a superposition of different configurations with same weights and would be invariant under permutations as in the argument of Li et al involving no flux tubes. This option does not look attractive.

2. What braid picture allows to say about the transition to ferromagnetism? Could the transition be realized by deforming the flux tubes associated with M and forming a non-trivial braid be induced by permutation of the lattice points taking the non-trivial braid to trivial one? This would be like opening the braid. The lattice points in the initial and final state would correspond to the ends of a dynamical evolution. The permutation would be realized as a time-like braiding with braid strands in time direction.

   Mathematically braid group corresponds to the covering group of permutation group and quantum group representations correspond to the representations of braid groups. The description of the transition could provide a new application of quantum groups.

The description as time-like braiding is not however complete since there isan additional structure involved: the flux tubes connecting the magnetic dipoles in lattice and defining a braid or even more complex configuration having flux tube connections between non-neighboring poins.

1. If there is no return flux assignable to H, M corresponds to a closed flux tube carrying monopole flux the dynamical time-like dynamcial braiding would lead to a Hamiltonian cycle in this case and the number of final state configurations would be finite, there is degeneracy. Could spin glass phase correspond to this situation?
2. In ferromagnetism final state would contain N parallel strands carrying the monopole flux assignable to M and the return flux H would arrive along parallel thick flux tube. In general configuration these strands can be braided. The transition to ferromagnetism would represent time-like braiding of an ordinary 3-D braiding of flux tube strands connecting the opposite boundaries of ferromagnetic. In the initial state braid would be non-trivial and the flux tubes of braid would not have minimal length and minimum energy. In the the final ferromagnetic state braid would be trivial with parallel flux tubes.

   Mathematically this process would correspond to what is called 2-braiding: I have proposed that 2-braidings are important in TGD inspired biology as a topological description of dynamical processes. An interesting interpretation is as a topological analog for problem solving. I have also proposed that in biosystems topological quantum computation programs are represented as this kind 2-braidings for flux tubes (see this and this.)

   Ferromagnetism would correspond to an opening of a non-trivial braid. If the return flux arrives along flux tubes this is possible smoothly only if the knot defined in this manner is trivial. To achieve opening, the 2-braiding must involve reconnections, which correspond to cutting the knot strand and reconnecting the pieces in new manner: this is how Alexander opened his knot. Fermi statistics and repulsive Coulomb interaction do not fabour this mechanism. If the return flux arrives along single flux tube, the opening could correspond to a smooth deformation without reconnections transferring the braidedness to the parallel space-time sheet, where it is "neutralized" by fusing the flux tubes to single flux tube.

See the article TGD based view about ferromagnetism or the chapter Quantum criticality and dark matter: part I.

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

Articles and other material related to TGD.