What topics are allowed for a decent finnish stargazer?

I wrote yesterday a comment to the blog article by Syksy Räsänen, which appeared at the page of Ursa, an organization of finnish stargazers. The blog article of Räsänen is in finnish (see this). The comment was on the question whether standard model could be much more than people have though hitherto. Here is my comment which was originally in finnish:

"I made this question for 43 years ago. I asked whether standard model symmetries are much more profound that we have imagined. I also answered the question. The geometry of complex projective space CP₂ codes for the gauge symmetries, quantum numbers, and classical gauge fields if space-times are 4-D surfaces in 8-D embedding space M⁴× CP₂ at the fundamental level .

TGD generalized string model 7 years before its breakthgouh 1984. 2-D world sheet of string became 4-D space-time surface. Iy was possible to avoid the horrors of spontaneous compactivitation and brane world: as we know they eventually led to the collapse of superstring empire.

TGD also provided a solution to the energy problem of general relativity, which in turn closely relates to the failure of the quantization of general relativity. The lost Poincare incariance of general relativity is lifted to Poincare invariance at the level of H=M⁴× CP₂. This gives back the basic conservation laws. Colleagues still refuse to take TGD seriously. Could the time be ripe for using common sense. The funding agencies get nervous when no heurekas have been heard from the workshops of theoreticians for half century."

I could have added also the following lines of text demonstrating that CP₂ is unique also for mathematical reasons.

1. CP₂ follows by M⁸-M⁴× CP₂ duality from the number theoretic vision dual to geometric vision of physcs (geometrization of entire quantum theory). In number theoretical vision, complexified M⁸ corresponds to complexified octonions. Associativity is the number theoretical counterpart of variational principle. Color SU(3) corresponds to SU(3) subgroup of octonionic automorphisms.
2. M⁴ and CP₂ are the only 4-D manifolds that allow twistor space with Kaehler structure so that TGD is unique. Twistorialization of TGD means geometrization of also twistor fields as 6-D surfaces in the product of twistor spaces T(M⁴) and T(CP₂) and relies on 6-D Kähler action having as preferred extremals 6-D surfaces having interpretation as the twistor space of space-time surface (S² bundle structure induced from T(M⁴)×T(CP₂)).

It was not surprising that Syksy Räsänen did not publish comment but expressed strong words of caution suggesting that I my critical comment was hate speech. Anyone can decide whether this is the case. It was also encouraged the decent participants should make only questions about starry sky, space, and star hobby. I admit that my comment did not satisfy the latter criterion.

Reader can wonder what might be the real reason for the censorship? Last 40 years after the publication of my thesis 1982 (maded possible by very positive referee statement by Wheeler), I have been on blacklist in Finland (no financial support, no research jobs). At this moment I can however relax: I was right.

The censorship has now badly failed and references to TGD appear on daily basis from several reaseach fields. The reason is that in the TGD framework quantum measurement theory extends to a theory of consciousness and cognition having quantum biology as an application. Most of finnish colleagues are however strangely silent and censorship continues.

TGD can be found at my homepage and most of the material have been published in the journals founded by Huping Hu.

Latest progress in TGD.

Articles related to TGD.

Goedel's theorem and TGD

The following is a response to Lawrence Crowell in the discussion group "The Road to unifying Relativistic and Quantum Theories". The topic of discussion related to Gödel's theorem and its possible connection with consciousness proposed by Penrose.

My own view is that quantum jump as state function reduction (SFR) cannot reduce to a deterministic calculation and can be seen as a moment of re-creation or discovery of a new truth not following from an existing axiomatic system summarizing the truths already discovered. My emphasis in the sequel is on how the number theoretic vision of the TGD proposed to provide a mathematical description of (also mathematical) cognition could allow us to interpret the unprovable Gödel sentence and its negation.

I decided to look more precisely at the Gödel number for polynomials with integer coefficients (no common factor coefficients) to which all rational polynomials can be scaled without changing the roots. Most of the classical physical content, if not all of it, can be coded by the coefficients [a₀,...,a_N] of the polynomial.

The Gödel numbering assigning to P an Gödel number G would be

G=p₁^a₀p₂^a₁...p_N+1^a_N,

where p_i is i:th prime and is an injection.

The discriminant D is the determinant of an (2N-1)×(2N-1) matrix defined by P and its derivative dP/dx ([a₁,2a₂,...,Na_N]) and is an integer decomposing to a product of ramified primes of P.

The first guess for Gödel's undecidable statement would that there exist polynomial P for which one has G=D. The number D coding a sentence, whatever it is, would be its own Gödel number. Why this guess? At least this statement is short;-). Can this statement be undecidable?

1. The equation involves both D as a polynomial of a_i and G involving transcendental functions p_i^a_i (essentially exponential functions) so that one goes outside the realm of rationals and algebraic numbers.
2. D=G is analog of Diophantine equation for a₁,....,a_N and both powers and exponential p_i^a_i appear. If the coefficients a_i are allowed to be a complex numbers, one can ask whether the complex solutions of G=D could form an N-1-D manifold. One can however assume this since p_i^a_i leads outside the realm of algebraic numbers and one does not have a polynomial equation.
3. The existence of an integer solution to D=G would mean that the primes p_i for which a_i are non-vanishing, correspond to ramified primes of P with multiplicity a_i so that the polynomials would be very special if solutions exist.
4. It might be possible to solve the equation for any finite field G_p, that is in modulo p approximation. Here one can use Fermat's little theorem p_i^p= p_i mod p. If integer solutions exist, they exist for every G_p.

What about the physical interpretation?

1. The polynomials P define space-time surfaces and one possible interpretation is that the ramified primes of P define external particles for a space-time region representing particle scattering. The polynomials P which reduce to single ramified prime would represent forward scattering of a single "elementary" particle.
2. In zero energy ontology, ordinary quantum states are replaced by superpositions of almost deterministic time evolutions so that also "elementary" particle would correspond to a scattering event. What exists would be events and TGD would predict not only scattering events but densities of particles as single particle scattering events inside a given causal diamond causal diamond representing quantization volume.
3. What kind of scattering events would these analogues of Godel sentences correspond? Representations of new mathematical axioms as scattering events, not provable from existing axioms?

Exactly what we cannot prove to be true or not true for these special polynomials? What does the sentence labelled by G= D state?

1. Integer D would express the sentence. D codes for the ramified primes. Their number is finite and we know them once we know P. Does the unprovable Gödel sentence say that there exists a polynomial P of some degree N, whose ramified primes are the primes p_k associated with a_i? Or dös it say that there exists polynomial satisfying G=D in the set of polynomials of fixed degree N.
2. Is it that we cannot prove the existence of integer solution a_i to P=G using a finite computation. Is this due to the appearance of the functions p_i^a_i or allowance of arbitrarily large coefficients a_i? The p-adic solutions associated with finite field solutions have an infinite number of coefficients and can be p-adic transcendentals rather than rationals having periodic pinary expansions.
3. Polynomials of degree N satisfying D=G are very special. The ramified primes are contained in a set of N+1 first primes p_i so that D is rather small unless the coefficients a_i are large. D is a determinant of 2N-1×2N-1 matrix so that its maximum value increases rapidly with N even when one poses the constraint a_i< N. Rough estimates and explicit numerical calculations demonstrate that determinants involving very large primes are possible, in particular those involving single ramified prime identified as analogues of elementary particles, D can reduce to single large prime: D=P.

   What about the polynomials P in the vicinity of points of the space of polynomials of degree N satisfying D=0: they correspond to N+1 ramified primes, which are minimal (note that the number of roots is N). D is a product of the root differences and 2 or more roots coincide for D=0. D is a smooth function of real arguments restricted to the integer coefficients. The value of D in the neighborhood of D=0 can be however rather large. Note that the proposed Gödel numbering fails for D=0, and therefore makes sense only for polynomials without multiple roots.

4. For D(P)=0 one has a problem with the equation G=D. G(P) is well-defined also now. The condition D(P)=0=G(P) does not however make sense. The first guess is that for 2 identical roots, P is replaced with dP/dx in the definition of D: D(P)-->D(dP/dx). D is nonvanishing and the ramified primes p_i do exist for dP/dx. Therefore the condition D(dP/dx)=G(P) makes sense. For n identical roots one must use have D(d^n-1P/dx^n-1)=G(P).
5. Interestingly, in TGD the hypothesis that the coefficients of polynomials of degree N are smaller than N, is physically very natural (see this) and would make the number of polynomials to be considered finite so that in this case one can check the existence of a G=D sentence in a finite time. It seems rather plausible that for given N, no G=D sentence, which satisfies the conditions a_i< N, does exist.

   One can of course criticize the hypothesis a_i< N implying a strong correlation between the degree N of P and the maximal size of ramified primes of P identified as p-adic primes characterizing elementary particles. One can argue that in absence of this correlation predictivity is lost. This hypothesis also makes also finite fields basic building bricks of number theoretic vision of TGD (see this).

6. Could this give rise to a realization of undecidability at the level of conscious experience and cognition relying on number theoretic notions. How?

   Quantum states are superpositions of space-time surfaces determined by polynomials P and if the holography of consciousness is true, conscious experience reflects the number theoretic properties of these polynomials if associated to a localization to a given polynomial P in a "small" state function reduction (SSFR). This would be position measurement in the "world of classical worlds" (WCW)? The proof of the statement D=G would mean that a cognizing system becomes conscious of the D=G space-time surface by a localization to it.

   Suppose that for a given finite N and condition a_i< N, G=D sentences do not exist. Hence one can say that G=D sentences go outside the axiomatic system realized in terms of the polynomials considered. Even the space of all allowed polynomials identified as a union of spaces with varying value for degree N would not allow this. G=D sentences would be undecidable by the condition a_i< N.

   One can of course criticize the hypothesis a_i< N implying a strong correlation between the degree N of P and the maximal size of ramified primes of P identified as p-adic primes characterizing elementary particles. One can argue that in absence of this correlation predictivity is lost. This hypothesis also makes also finite fields basic building bricks of number theoretic vision of TGD (see this).

See the articles Gödel's Undecidability Theorem and TGD and Finite Fields and TGD.

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

Articles related to TGD.

How do AdS/CFT- and TGD based holographic dualities relate?

The article "Traversable wormhole dynamics on a quantum processor" by Jafferis et al published in Nature received a lot of media attention. My original reaction was due to frustration caused by the media hype. What was done was a quantum computer simulation of the so-called SYK (Sachdev-Ye-Kitaev) model proposing AdS/CFT duality for a particular condensed matter system.

The attempts to understand what is involved soon led to a realization that since TGD predicts the analog of AdS/CFT holographic duality, the quantum computational aspects of the experiment should be understandable also using the holographic duality of TGD. This raises the question whether can one translate the notions of AdS holography to TGD holography. In particular, what could be the TGD counterparts for the notions of wormhole and negative energy shock waves needed to stabilize the wormhole. This article deals with these kinds of questions and leads to a rather detailed view of TGD based holography.

See the article How do AdS/CFT- and TGD based holographic dualities relate? or the chapter with the same title.

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

Articles related to TGD.

Comparison of AdS/CFT duality with the TGD based holographic duality

This posting is a continuation to an earlier posting Are wormholes really created in quantum computer? (see this).

In the experiment considered, the so-called SYK model (Sachev-Ye-Kitaev) was simulated using a quantum computer. The model is constructed to realize AdS₂/CFT correspondence and the quantum computer simulates the 1-D quantum system dual to wormhole in 2-D AdS₂.

The problem is that the AdS₂ is completely fictitious so that the physics at this side cannot be tested. However, TGD also predicts holographic duality between 3-D surfaces as boundaries of space-time surfaces and identifiable as outer boundaries of physical objects and the interior of the space-time takes the role of AdS. In particular, the system considered in the experiment should allow TGD based dual description.

The thesis Holographic quantum matter : toy models and physical platforms (see this) of Etienne Lantagne-Hurtubise gives a nice description of the SYK model and the following comments are based on the introduction of the thesis.

TGD should give a classical description of quantum dynamics coded by 3-D data holographically in terms of classical physics in the interior of space-time surface. Therefore the challenge is to also describe the reported findings.

How do AdS/CFT holography and TGD holography relate to each other?

There are obvious questions to be answered. How closely AdS/CFT and TGD holography could relate and how do they differ? Could there exist some kind of AdS/CFT-TGD dictionary?

1. AdS/CFT correspondence predicts 4→ 5 holography for AdS₅ interpreted as an emergent 5-D space-time. M⁴ would carry gauge fields and theory would be conformally invariant. The gravitational holography is 3→ 4. Skeptics could argue that there is total mess: for instance, what happens to the general relativistic description of gauge fields using 4-D space time? Should one have 3→ 4 gravitational holography followed by 4→ 5 for gauge fields?
2. TGD predicts 3→4 holography. Instead of AdS one has space-time but is realized as a 4-D surface. Light-like 3-surfaces with extended conformal symmetries due to their metric 2-dimensionality defined boundaries of 4-D space-time surfaces and contain holographic data defining the space-time surface and also the data defining the fermionic part of quantum state.

   4-D general coordinate invariance implies almost exact holography and classical deterministic dynamics becomes an exact part of quantum TGD: one has what one might call Bohr orbitology. This picture has a number theoretic counterpart at the level of M⁸: associativity assigns 4-D surface of M⁸_c to the roots of rational polynomials represented as 3-D mass shells in M⁴_c ⊂ M⁸_c.

Do the time loops of AdS has time-like loops have a TGD counterpart?

AdS time loops have indeed TGD countepart. The reason is that 4-D space-times are completely exceptional.

1. 4-D, and only 4-D, space-times allow exotic smooth structures (see this)! A continuum of exotic smooth structures are possible. Exotic smooth structure can be always regarded as ordinary smooth structure apart from a discrete set of points.

   Exotics break cosmic censorship so that global hyperbolicity fails and the initial-value problem becomes ill-defined because of time-like loops. Time-like loops are a heavy counter argument against AdS/CFT duality. They are however encountered also for the TGD variant of the holographic duality.

   Could it be that time-like loops are not a nuisance but something fundamental forcing the space-time dimension to be D=4.

2. In the TGD framework holography predicts the smooth structure of the space-time surface so that the non-uniqueness is not a problem.

   The discrete set of points spoiling the standard smooth structure is an analogue for a set of point-like defects. Outsides this set the standard smooth structure fails. The proposal (see this) is that this set of points is assignable to particle reaction vertices in TGD and have a topological interpretation. Two partonic 2-surfaces with opposite homology charges (monopole fluxes) touch at defect point and fuse together to a single particle 2-surface.

3. This makes possible time loops which are essential for understanding pair creation in TGD. It is essential that the interiors of the orbits of wormhole contacts have an Euclidian signature: this is obviously a completely new element when compared to AdS/CFT. The boundaries between these Euclidian regions and Minkowskian regions of the space-time surface are light-like and correspond to the orbits of wormhole throats at opposite Minkowskian sheets (see this). The creation of a fermion pair would correspond to a change of the time direction of the fermion at the defect point of the exotic smooth structure.
4. Could exotic smooth structures make possible quantum computations as evolution forth-and-back in space-time in the TGD framework? Could the time loops serve as microscopic classical correlates for this and could the defects give a topological realization for what happens. Note that wormhole throats can in principle have large sizes and scale like h_eff and can be very large for gravitational Planck constant h_gr.
5. Could wormholes correspond in the TGD framework to magnetic flux tubes? Or could they correspond to light-like orbits of wormhole throats/partonic 2-surfaces appearing analogous to lines of topological counterparts of Feynman diagrams? Orbits of wormhole contacts identified as orbits of pairs of wormholes give rise to light-like orbits of wormhole throats, which are always paired. Fermionic quantum numbers are associated with the light-like lines of the wormhole throat. They represent building bricks of elementary particle orbits. Could these structures be seen as analogues of wormholes?

What is the TGD counterpart of time reversal of the SYK model?

Time reversal is central in the SYK model.

1. Time reversed of time evolution as unitary time evolution with Hamiltonian having opposite sign is central in the model. This notion is somewhat questionable since usually one requires that the energy eigenvalues are positive. In TGD, this time evolution could correspond to a sequence of SSFRs in the reversed time direction following BSFR.
2. Shock wave in the wormhole appears as a negative energy signal. This could correspond to time reversed classical signals having effectively negative energy and propagating along the flux tube or the counterpart of the wormhole in TGD. Time reversal would be induced by BSFR.
3. One could also interpret reversed time evolution as a generation of Hawking radiation. Negative energy particles falling to the blackhole would correspond to the time reversed signal propagating from right to left after BSFR has occurred in the experiment considered.

TGD counterparts of scrambling time evolution and descrambling as its time reversal

Scrambling means generation of quantum chaos. Descrambling does the opposite and is in conflict with the second law unless the arrow of time changes. For a unitary time evolution descrambling can be considered if the negative of Hamiltonian makes sense.

Scrambling corresponds to a random unitary time evolution inducing mixing as dispersion of entanglement in the entire system. Actually a sequence of scramblings characterized by scrambling times described by random Hamiltonians is assumed to take place. Scrambling time is assumed to depend on blackhole entropy S= A/4Gℏ= 4π GM²/ℏ roughly as

T_s= r_s ×O(S^1/2 log(S)) ,

where r_s = 2GM is Schwarzschild time. Blackholes are assumed to be very fast scramblers.

1. A sequence of "small" state function reductions (SSFRs) as the TGD counterparts of "weak" measurements of quantum optics, generalizes Zeno effect to a subjective time evolution of self. The sequence of SSFRs as analog of a sequence of unitary time evolutions

   Scrambling could correspond to a sequence of SSFRs as an analogue for a sequence of random unitary evolutions in TGD. Since one has a sequence of SSFRs, scrambling might correspond to the emergence of thermodynamics chaos.

   An alternative interpretation of chaos is an increase of complexity. Mandelbrot fractal is complex but not chaotic in the thermodynamic sense. Could scrambling correspond to an effective increase of the extension of rationals during the sequence of SSFRs? More and more roots of polynomials defining light-cone propertime a=constant hyperboloids become visible at the increasing space-time surface inside the CD. This option does not look plausible.

2. De-scrambling time evolution is in conflict with intuition. In TGD, de-scrambling could correspond to scrambling with an opposite arrow of time emerging in "big" SFR (BSFR) and therefore dissipation with a reverse arrow of time looking like self-organization for an observer with an opposite arrow of time. This process is fundamental in biology and would correspond to processes like healing. BSFR corresponds to a "death" or falling asleep in TGD inspired theory of consciousness and self lives forth and back in geometric time.
3. It is interesting to look for the TGD counterpart of scrambling time T_s. For hbar →, where ℏ_gr = GM²/β₀ is the gravitational Planck constant and β₀≤1 is a velocity parameter, one obtains

   T_s= r_s ×4π β₀log(β₀) .

   Scrambling time would be negative for β₀< 1: could the interpretation be that scrambling takes place with opposite arrow of time? Blackhole entropy is equal to β₀ and smaller than 1 and practically zero. One must of course take this expression for the scrambling time with a big grain of salt and as found in the previous posting, TGD allows us to consider a more general picture in which the correspondence with blackholes is not so concrete.

See the earlier posting Are wormholes really created in quantum computer? and the article How do AdS/CFT- and TGD based holographic dualities relate?.

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

Articles related to TGD.

Ignition of nuclear fuel achieved in hot fusion: what does this mean?

The claimed breakthrough in hot fusion (see this) is the latest hype in our hype-filled world. Ignition, which initiates energy production, must be achieved but this is only a small step in the ladder leading to a real fusion.

One of the problems of which I learned only some time ago is the following: the energy feed does not appear to raise the temperature as expected but goes somewhere. The underlying physics is poorly understood. The TGD inspired solution could be in terms of Hagedorn temperature predicted for flux tube like objects, analogs of strings, predicted by TGD. New degrees open and the temperature does not increase and reactions do not start to produce energy. This problem should be solved (see this) .

The hypish news tells that ignition has been achieved. This is certainly a big achievement. There is an energy feed by hundreds of lasers on an energy pellet and this system indeed ignites and starts to produce more energy than the input energy from lasers. However, the entire system however needs energy input, which is exponentially higher than the energy required by lasers so that there is a long way to go for nuclear fusion.

This one little step in progress, which one can hope to lead to real hot fusion. But is it so?

In the TGD Universe, "cold fusion" using dark nuclei (in the TGD sense) would be an alternative solution to the problem. The huge energy feed needed in heating the system to the required temperature would be overcome. As a matter of fact, cold fusion could actually heat the system to the temperature required by hot fusion. Also stellar nuclei as fusion reactors could have emerged in this way. But this is not the time for new theoretical physics so it will take decades before cold fusion can be taken seriously.

See the article Could TGD provide new solutions to the energy problem?.

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

Articles related to TGD.

The findings of the James Webb telescope from the TGD point of view

The findings of the James Webb telescope have revolutionized the views about galaxy formation in the early Universe. I have commented these findings briefly in "Some anomalies of astrophysics and cosmology" (see this) but have not found time for further comments.

Scientific American has an article with title "JWST's First Glimpses of Early Galaxies Could Break Cosmology" (see this), which provide a nice summary of the first findings of the telescope. This gave an opportunity to sharpen the somewhat fuzzy view of how the findings of James Webb telescope relate to TGD.

What was found first, was a galaxy dubbed as "GLASS-z13". It was found by Rohan Naidu and led to an article published within a few days. The discovery of the GLASS-z13 was followed by a discovery by numerous even more distant galaxies. The very existence and the properties of these galaxies came as a total surprise.

1. From the redshift of about z=13, the GLASS-z13 was dated back 300 million years after the big bang that is thought to have occured 13.8 billion years ago. According to the standard view of galaxy formation (so called Lambda CDM model involving dark matter as exotic particles), galaxies with such a large distance are not expected to even exist. According to the standard model, the formation of galaxies should have begun at the cosmic age of about 400 million years. The galaxy found by Naidu would have emerged more than 70 millions years too early.
2. The images of the galaxies from so early era were expected to be extremely dim. The galaxies discovered were however anomalously bright.
3. The large size of the galaxies came as a total surprise. The age of the galaxies increases with its age and the conclusion was that the galaxies had to be much more mature than the standard model for the formation of galaxies allows. This leads to a paradox since the first galaxies should be very young.

During the years, I have developed a TGD based model of galaxy formation. The model is supported by the ability to explain the increasing number of anomalies of the standard model (see for instance this and this).

Monopole flux tubes are the basic element of the TGD view of galaxy formation. They are present in all length scales in the TGD Universe and distinguish TGD from both Maxwell's electrodynamics and general relativity.

1. Flux tubes can carry monopole flux, in which case they are highly stable. The cross section is not a disk but a closed 2-surface so that no current is needed to create the magnetic flux. The flux tubes with vanishing flux are not stable against splitting.
2. Flux tubes relate to the model for the emergence of galaxies (see this and this) and explain galactic jets propagating along flux tubes (see this). Dark energy and possible matter assignable to the cosmic strings predicts correctly the flat velocity spectrum of stars around galaxies.
3. In the MOND model it is assumed that the gravitational force transforms for certain critical acceleration from 1/r² to 1/r force. In TGD this would mean that the 1/ρ force caused by the cosmic string would begin to dominate over the 1/ρ² force (ρ denotes transversal distance from string). The predictions of MOND TGD are different since in TGD the motion takes place in the plane orthogonal to the cosmic string.
4. The flux tubes can appear as torus-like circular loops. Also flux tube pairs carrying opposite fluxes, resembling a DNA double strand, are possible and might be favoured by stability. Flux tubes are possible in all scales and connect astrophysical structures to a fractal quantum network. The flux tubes could connect to each other nodes, which are deformations of membrane-like entities having 3-D M^4 projections and 2-D E³ projections (time= constant) (also an example of "non-Einsteinian" space-time surface).
5. Pairs of monopole flux tubes with opposite direction of fluxes can connect two objects: this could serve as a prerequisite of entanglement. The splitting of a flux tube pair to a pair of U-shaped flux tubes by a reconnection in a state function reduction destroying the entanglement. Reconnection would play an essential role in bio-catalysis.
6. Flux tube pairs can form helical structures and stability probably requires helical structure. Cosmic analog of DNA could be in question: fractality and gravitational quantum coherence in arbitrarily long scales are a basic prediction of TGD so that monopole flux tubes should appear in all scales. Also flux tubes inside flux tubes inside and hierarchical coilings as for DNA are possible.

Could one understand the paradoxical findings in the TGD view of galaxy formation?

1. According to the standard model, these galaxies were formed quite too early. The standard mechanism of formation is a gravitational condensation of stars and interstellar to form galaxies. Dark matter halo plays a key role in the process. The model is however plagued by several contradictions. As a matter of fact, empirical facts suggest that there is no dark halo. The MOND model explains many of the anomalies but is in conflict with the Equivalence Principle and in conflict with standard Newtonian gravitation. The TGD based model replaces dark matter halo with long cosmic strings carrying dark energy and possibly also dark matter. One does not lose either Equivalence Principle or Newtonian gravitation.

   The TGD based view of galaxy formation is diametrically opposite to the standard view, being analogous to the generation of ordinary matter via the decay of the inflation field in the inflationary cosmology. Ordinary matter would have been created by the decay of the energy of cosmic strings to ordinary matter as they formed tangles. This led to a thickening of cosmic strings to monopole flux tubes and to a reduction of string tension so that energy was liberated as ordinary matter. In particular, galactic dark matter and the flat velocity spectrum of distant stars find an elegant explanation.

   In this view galaxies started to emerge already during the TGD analogue of the inflationary period.

2. The high apparent luminosity of these galaxies is the second mystery. Are the galaxies indeed so luminous as they seem to be? Or could it be that the standard view of how light emitted by galaxies is distributed is somehow wrong?

   In the TGD framework, the space-time of general relativity is replaced with a fractal network of nodes defined by various structures including galaxies, stars, planets,... Monopole magnetic flux tubes connect the nodes and the light propages as beams of dark photons (in the TGD sense) along these flux tubes. A light beam travelling along a flux tube is not attenuated at all if the cross section of the flux tube stays constant. Therefore the intensity of the light beam is not reduced with distance. In GRT it would be reduced since there would be no splitting to beams. This would explain why the apparent luminosities of the galaxies are anomalously high.

3. The unexpectedly large size of the galaxies implies a long age if one believes in the standard view of galactic evolution. This paradox finds a solution in zero energy ontology (ZEO), which defines the ontology of quantum TGD. ZEO solves the basic paradox of quantum measurement theory and is forced by the holography implied in the TGD framework by 4-D general coordinate invariance.

   In ZEO, the arrow of time changes in ordinary quantum jumps ("big" state function reductions, BSFRs). The repeated change of the arrow of time in the sequence of BSFRs implies that the system can be said to live forth and back in geometric time. Aging does not correspond to "center of mass motion" in time direction but this forth and back motion. In the TGD inspired biology, BSFR is analogous to death or falling asleep.

   In "small" SFRs (SSFRs) the arrow of time is not changed and they are counterparts of weak measurements introduced by quantum opticians. They generalize the quantum measurements associated with the Zeno effect, in which a system is frozen and its state does not change. Now the sequence of SSFRs would define a conscious entity, self.

   In TGD, gravitational quantum coherence is possible in all scales and galaxies would be astrophysical quantum systems performing BSFRs. Even astrophysical objects such as galaxies would live forth and back in time. This would give rise to galaxies and stars older than the Universe if one tries to explain their age using the standard view of the relationship between experienced time and geometric time.

See the article Some anomalies of astrophysics and cosmology or the chapter TGD View of the Engine Powering Jets from Active Galactic Nuclei .

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

Articles related to TGD.

Are wormholes really created in quantum computer?

The most recent really heavy hype is by "Quantum gravity in Lab" movement and involves publicity stunt related to the article "Traversable wormhole dynamics on a quantum processor" (see this) by Jafferis et al published in Nature.

Already the title of the article has a very high hype content. The tweet of the journal Quanta (a popular journal usually satisfying very high standards) published the following tweet:

Physicists have built a wormhole and successfully sent information from one end to the other. The stormy reception of the article and of the tweet forced Quanta to change the tweet to Experimental physicists built the mathematical analog of a wormhole inside a quantum computer by simulating a system of entangled particles.

The Quanta article contains for instance the statement The team developed quantum software that could reproduce wormhole inspired teleportation on both quantum computers. This was later corrected to Experimental physicists built the mathematical analog of a wormhole inside a quantum computer by simulating a system of entangled particles. This statement is still far from an honest statement telling only what was actually done: Physicists have simulated a model of wormhole in a system argued but not proven to obey AdS/CFT duality by using a quantum computer. This simulation had been done already earlier using an ordinary computer.

Peter Woit commented in Not Even Wrong this hyper-superhype rather critically in two postings (see this and this).

Also Scott Aaronson, who is one of quantum computation gurus, expressed his very critical views shared by most quantum computation professionals (see this and this). Here is the core part of Scott Aarronson's commentary.

Tonight, David Nirenberg, Director of the IAS and a medieval historian, gave an after-dinner speech to our workshop, centered around how auspicious it was that the workshop was being held a mere week after the momentous announcement that a wormhole had been created on a microchip (!!) in a feat that experts were calling the first-ever laboratory investigation of quantum gravity, and a new frontier for experimental physics itself. Nirenberg speculated that, a century from today, people might look back on the wormhole achievement as today we look back on Eddington s 1919 eclipse observations providing the evidence for general relativity.

I confess: this was the first time I felt visceral anger, rather than mere bemusement, over this wormhole affair. Before, I had implicitly assumed: no one was actually hoodwinked by this. No one really, literally believed that this little 9-qubit simulation opened up a wormhole, or helped prove the holographic nature of the real universe, or anything like that. I was wrong.

There is also a very nice popular article "The truth about wormholes and quantum computers" representing a harsh criticism (see this).

The combination of all kinds of fashionable pop science related to quantum computation, quantum gravitation, wormholes, EPR= EP, AdS/CFT, etc.... yields this kind of pseudo-science. It has been extremely frustrating to witness the stagnation of theoretical physics to pop science during these more than four decades.

Critics however agree that when one drops all this irrelevant hype away, the work of quantum computer pioneers satisfies the highest standards. It is regrettable that excellent experimentation and engineering is not enough for funding but must be iced with a sugar layer of bad theoretical physics.

Although this kind of hyperhypes do not deserve the attention they receive, I decided to look what I could learn and how could I relate this picture to TGD where holography forced by 4-D general coordinate invariance is also central and defines quantum classical correspondence. Classical physics is an exact part of quantum theory and is realized as Bohr orbitology equivalent to holography. Could I perhaps analyze the experiment in the TGD framework or suggest something analogous and maybe learn something new from TGD?

A. Traversable wormhole dynamics on a quantum processor

In the following I summarize my understanding of the various notions involved with the experiment. These include AdS/CFT duality, traversable wormholes, and negative energy shock waves mentioned in the abstract.

A.1 Abstract of "Traversable wormhole dynamics on a quantum processor"

Here is the abstract of the article "Traversable wormhole dynamics on a quantum processor" Daniel Jafferis et al. (see this).

The holographic principle, theorized to be a property of quantum gravity, postulates that the description of a volume of space can be encoded on a lower-dimensional boundary.

The anti-de Sitter (AdS)/conformal field theory correspondence or duality is the principal example of holography. The Sachdev Ye Kitaev (SYK) model of N >> 1 Majorana fermions has features suggesting the existence of a gravitational dual in AdS2, and is a new realization of holography.

We invoke the holographic correspondence of the SYK many-body system and gravity to probe the conjectured ER=EPR relation between entanglement and spacetime geometry through the traversable wormhole mechanism as implemented in the SYK model.

A qubit can be used to probe the SYK traversable wormhole dynamics through the corresponding teleportation protocol. This can be realized as a quantum circuit, equivalent to the gravitational picture in the semiclassical limit of an infinite number of qubits.

Here we use learning techniques to construct a sparsified SYK model that we experimentally realize with 164 two-qubit gates on a nine-qubit circuit and observe the corresponding traversable wormhole dynamics. Despite its approximate nature, the sparsified SYK model preserves key properties of the traversable wormhole physics: perfect size winding, coupling on either side of the wormhole that is consistent with a negative energy shockwave, a Shapiro time delay causal time-order of signals emerging from the wormhole, and scrambling and thermalization dynamics.

Our experiment was run on the Google Sycamore processor. By interrogating a two-dimensional gravity dual system, our work represents a step towards a program for studying quantum gravity in the laboratory. Future developments will require improved hardware scalability and performance as well as theoretical developments including higher-dimensional quantum gravity duals and other SYK-like models.

There is a long list of questions to be answered.

1. What does the term AdS₂ holography mean? How does it relate to ordinary, "real" quantum gravitation?
2. What does "traversable wormhole" mean?
3. What does the negative energy shockwave, argued to open the wormhole for quantum teleportation, mean in general relativity? What interaction between quantum computers, modelled as blackholes, does the negative energy shock wave correspond at the level of the quantum computer system?

A.2 What does AdS/CFT duality mean?

My non-specialist's view of AdS/CFT is the following.

1. AdS/CFT is not part of string theory.
2. It is not a proposal to describe gravitation but gauge interactions in terms of effective gravitation assigned to effective AdS_nx S^10-n. AdS₅ would have 4-D Minkowski space as boundary and standard model would be dua to a theory of effective gravitation in AdS₅.
3. The fact that has been forgotten is that the list of physics successes of AdS/CFT duality is very short, actually non-existing. Even if AdS/CFT is regarded as mathematically well-defined, this does not save it from the ultimate fate of wrong theories.
4. The basic statement is a field theory with conformal symmetries at the boundary of AdS (say 4-D Minkowski space) is dual to a theory of gravitation in the interior of Ads.
5. In the beginning AdS/CFT with n=5 so that the boundary is 4-D Minkowski space, was tried to apply QCD, to nuclear physics and many other cases. The idea was to deduce predictions from the physics of the AdS side. The attempts failed.

Why did AdS/CFT fail?

1. The probable reason is that the basic mathematical framework, although probably correct using physics standards, does not correspond to the physical situation.

   For instance, AdS is a pathological space-time geometry having time-like loops violating cosmic censorship and spoiling the initial value problem. In fact, the SYK model simulated in the experiments, postulates an interaction between blackholes which prevents the occurrence of these time-like loops and this interaction would make possible quantum teleportation!

2. The real reasons for the failure are at a much deeper level. Quantum field theory (QFT) itself is to be blamed. QFT relies on the notion of a point-like particle and fails (of course divergence problems and the non-existence of path integral have tried to tell this to us for more than half a century).
3. The idea that 4-D conformally symmetric field theory is something fundamental rather than mere approximate QFT limit is probably wrong. Also the Einsteinian view of gravitation has a fundamental problem: one loses basic conservation laws of special relativity. Both sides of the duality are sick.

A.3 What are traversable wormholes in GRT

One can learn of traversable wormholes from the thesis of Alex Simpson (see this). The thesis describes a family of solutions of Einstein' equations characterized by one parameter a. The solutions have time translations and rotations as symmetries and in contrast to naive expectations the radial coordinate varies from -∞ to +∞ rather than from 0 to ∞.

The general solution of the family is given by

ds²= (1-2GM/X)dt²- dr²/(1-2GM/X)+ (r² +a²)(dθ²) +sin²(θ)dθ²)

X=(1-2GM/(r² +a²)^1/2)

The coordinates r and t vary in the range (-∞,+∞). r_S= 2GM is Schwartschild radius.

Some comments are in order.

1. a=0 gives Schwarzschild solution and in this case r=0 corresponds to a single point as the singularity of spherical coordinates. For ²>0 r=0 corresponds to a sphere with radius a. In the TGD framework this would be obtained if there is a magnetic monopole flux through the monopole throat.
2. The study of radial geodesics provides information about the object. One has for the radial velocity dr/st= +/- (1-X). For a>r_s this is always nonvanishing. Therefore the radiation from r< 0 blackhole can propagate to r> 0 blackhole.

   For Schwartschild solution with a=0, dr/dt it vanishes at Schwartschild radius so that light cannot classically escape from blackhole interior.

   For a< r_s, dr/dt vanishes for two radii r_+/-= +/- (r_S² -a²)^1/2 so that these acts as horizons and the two blacholes are isolated.

It is known that the traversable wormholes are not stable without a condition requiring a negative vacuum energy density. I must admit that in the case of AdS/CFT duality I do not really know what wormholes mean. One should deform the AdS metric just like one deforms the Minkowski metric to obtain the analogues of blackhole and wormhole.

A.4. What negative energy shock waves are and why they are needed?

The basic problem against the idea of quantum gravitational view of teleportation is that wormholes are unstable against splitting. Neither space ships nor information can travel from blackhole to another one. Wormholes are not traversable: geodesics cannot connect the blackholes but stop at the horizon.

The introduction of the article "Traversable wormholes via a double trace deformation" by Gao et al (see this) describes how traversable wormholes might be generated.

The conditions making wormholes traversable would look like follows.

1. Negative energy shock waves are needed to open the wormhole throat so that classical signals can propagate between the blackholes and make quantum teleportation possible. One says that the wormhole becomes traversable. What do these negative energy shock waves mean in GRT context?
2. Quantum gravitational deformation of metric such would make the average energy density negative. The first problem is that QFTs do not allow this. Second problem is that after the sad fate of superstring theory, no generally accepted theory of quantum gravitation exists. The third problem, not usually noticed, is that the notions of energy and other Poincare charges are lost in GRT.
3. One can however forget these little nuisances and assume that negative energy densities are possible. Negative energy condition means technically that there exists an infinitely long null geodesic going through the blackhole-like entity such that the integral of the trace of energy momentum tensor is negative along the geodesic. This means that negative energy signals can propagate through the entire geodesic and make possible classical communications necessary for teleportation.

   The averaged null energy condition (ANEC), presumably stating the vanishing or even non-negativity of the trace, is said to fail if this is the case. It is also mentioned that physically the failure of the condition implies that light-rays focused at the other end of the wormhole, defocus at the other end. I didn't quite understand how this follows from the condition.

4. The existence of this kind of defocusing of geodesics is excluded by several conditions. As already noticed, the averaged null energy condition (ANEC) denies their existence. Neither does the generalized second law, stating that the area of blackhole horizon increases, allow them. It would seem that the hopes for traversable wormholes are rather meager.
5. Here comes however AdS₂ to rescue. Before continuing, note that AdS₂ duality is not for gravitation but for the gravitational dual of a conformally invariant QFT, such as gauge theories. Therefore we can allow things which we would condemn as nonsense in real quantum gravitation.

   The problem is that AdS has 1 time-like dimension but has closed time-like directions. AdS is imbedded in 3-D space with line element ds²= dt₁² + dt₁² -dz² containing 2 time-like directions and a 2-sphere with time-like metric containing circular time loops.

   These time-like loops imply at the level of AdS a violation of cosmic censorship and global hyperbolicity. As a consequence, the standard initial value problem with 3-D initial data is ill-defined since the signals from the surface of initial data return back. This also implies that the blackhole horizon extending through the wormhole intersects itself. The radial direction for AdS between the blackholes is time-like and effectively a compact circle. One has a causal anomaly. This is bad.

6. One must save the causality. The postulated interaction between the boundaries of the wormhole comes to rescue at this time and solve the problem that we have created. This interaction would save causality and would also imply failure of ANEC, and therefore make the wormhole traversable. Negative energy is interpreted in terms of effective Casimir effect.

   Of course, it would be much easier to not postulate at all the AdS duality and be satisfied with the fact that we have been able to simulate certain quantum model using quantum computers.

It must be added that the cosmic censorship hypothesis is in conflict with the existence of exotic smooth structures even in flat R⁴ since they imply the existence of closed time-like geodesics. Dimension D=4 is indeed completely exceptional in this sense and this should be important as an important message by theoreticians. I have discussed the possible existence and interpretation of smooth exotics in the TGD framework where holography also fixes the smooth structure (see this).

B. The TGD view

In the following I describe a possible TGD based interpretation of the experiment.

B.1 The TGD counterpart of AdS/CFT duality

TGD is a proposal, which solves the energy problem of general relativity and generalizes string models by replacing strings with 3-D surfaces. In the TGD framework, 4-D general coordinate invariance leads to a holography and analog of AdS/duality having interpretation as quantum classical correspondence.

1. In the TGD framework, light-like 3-surfaces appear as fundamental objects and are metrically effectively 2-D. These 3-D objects are related by holography (forced by general coordinate invariance in TGD) to 4-D objects defining space-time as a 4-D surface in M⁴×CP₂.

   These 3-D surfaces possess a conformal symmetry which is much larger than the usual 2-D conformal conformal symmetry, which is already infinite-D.

2. In AdS/CFT, the dimensions related by holography are wrong: in AdS₅ one has 4→5 and AdS₅ is non-physical unless one wants to believe in the emergence of space-time, a second fashionable but remarkably unsuccessful idea.
3. In TGD holography corresponds 3→4, which is the case also in the duality proposed between blackhole horizon and interior. In this duality everything has precise physical meaning and both sides of the duality can be tested unlike in AdS/CFT duality for which AdS is a completely fictitious notion.

The basic aspects of the TGD counterpart of horizon-interior duality are as follows.

1. The TGD variant of duality is forced by the 4-D general coordinate invariance and is not a separate principle.
2. In quantum theory this duality corresponds to Bohr orbitology: space-time surface is analogous to Bohr orbit of 3-surface generalizing the notion of particle.
3. Classical theory as Bohr orbitology becomes an exact part of quantum theory.
4. This duality leads to what I call zero energy ontology in which these 4-D Bohr orbits replace time=constant snapshots as basic objects. This leads to a solution of the basic paradox of quantum measurement theory and has profound implications in quantum biology and theory of consciousness.

B.2 Magnetic flux tubes as TGD counterparts of traversable wormholes

Could these traversable wormholes have TGD analogues and could they provide a dual or a classical correlate for the quantum description as the quantum classical correspondence suggests. One must remember that the TGD view of quantum differs considerably from the standard view.

1. New view of space-time predicts topological field quantization and the notion of field body. The notion of magnetic body (MB) consisting of flux tubes and flux sheets is in a central role in TGD.
2. The number theoretic vision of TGD predicts hierarchy of dark matter as phases of ordinary matter labelled by values of effective Planck constant: dark matter would reside at the field body, in particular at MB. These phases of ordinary matter might be highly relevant for quantum computation, which is in the standard framework formulated using standard quantum mechanics.
3. TGD predicts zero energy ontology (ZEO) one predict is that the TGD counterpart of ordinary state function reduction (SFR) reverses the arrow of time. Could this give rise to quantal versions of time loops.

In TGD monopole flux tubes are natural candidates for traversable wormholes.

1. Wormholes of GRT are not stable. Monopole flux tubes are stabilized by the very fact that monopole flux is conserved so that the flux tube cannot be split.
   1. If one has a pair of flux tubes with opposite fluxes connecting two systems, reconnection can split the flux tubes to U-shaped flux loops. This is a basic mechanism in the TGD inspired quantum biology.
   2. Could the reconnection of flux tube loops to a flux tube pair generate an analogue ofa traversable wormhole with classical signals propagating along the flux tube pair and making possible quantum teleportation?
   3. Could thus involve "big" SFR changing the arrow of time and giving rise to effective negative energy signals propagating backwards in time?

   Could this correspondence be made more quantitative and detailed.

   1. Could one assign to wormhole throats and entire wormhole scales which are analogous to r_S and the parameter a and could the condition a>r_S have an analog in TGD framework.

   2. The first trial starts from wormhole contacts as they are identified in TGD.
      1. The expectation is that at least for elementary particles the wormhole throat radius is of order CP₂ radius and therefore extremely small. Mass would be of order m=10^-4 m_Pl and could correspond to a mesoscopic mass: a water blob of size 10^-4 meters has mass of order Planck mass. Could this give some idea how to proceed?
      2. The Earth's gravitational field has a key role in TGD inspired quantum biology since gravitational quantum coherence is possible in even astrophysical scales. Gravitational Compton length Λ_gr= h_gr/m GM0 defines a fundamental quantum gravitational size scale. One has scale of about .45 cm for Earth mass M_E and v₀=c.
   3. The condition a> r_s for the traversability need not generalize as such in the TGD framework. Therefore it is better to start from the physical picture rather than trying to mimic wormhole physics.

      This kind condition should tell when the reconnection for U-shaped flux tubes is possible. This suggests that the parameter a characterizes the length of the U-shaped flux tubes connecting the two quantum systems. This length scale should be more than half the distance between the two systems as analogues of blackholes so that the half-distance d/2 would be the counterpart of r_S and the typical length of the U-shaped flux tube would be the counterpart of a. The length of the flux tube depends on the value of h_eff and on the p-adic length scale assignable to the flux tube.

   4. For an ordinary wormhole, the formula a> r_s guaranteeing traversability can be satisfied if the mass parameter m decreases below a but remains unaffected. This would correspond to a generation of negative vacuum energy reducing m below a/2G. Does this have any generalization to the TGD framework?
   5. Now the distance d between the systems could decrease or the length of U-shaped flux tubes could increase so that reconnection becomes possible.

      The postulated interaction between the two systems should give rise to the analog of negative energy shock wave. In the TGD framework, the reconnection of flux tubes could be the interaction generating a communication line. The GRT analogue for the negative energy shock wave could be Hawking radiation. The system at right would receive positive energy and the system at left would receive negative energy, or equivalently, send positive energy.

      The positive energy received by the system at right would serve as a metabolic energy feed and would change the distribution of values of h_eff in it. The values of h_eff would tend to increase. Therefore also entanglement negentropy content of the system would increase meaning that entanglement is created. At the left side the opposite would occur. One can say that information is transformed between the two systems.

   6. Could "big" SFR (BSFR, that is the ordinary SFR) in the scale of the entire system be involved. This would imply a change of the arrow of time and one could say that the system at right sends negative energy signals to the system at left and gets positive energy as a recoil acting as metabolic energy. This mechanism is a basic metabolic mechanism in the TGD inspired quantum biology. The energy for dark analogs of Hawking photons would be large although frequencies could be small. For h_gr = GM_Em/v₀, the energies would be in visible range for frequencies of order 10 Hz.

      The simplest interpretation is that the first BSFR creates reconnection and maximal entanglement between the systems and and the second BSFR corresponds to de-reconnection and the quantum measurement destroying the entanglement. The classical communication during the entanglement period induces the transfer of internal entanglement. One could see this phenomenon also as quantum tunneling for information.

   Is this very rough picture consistent with the more detailed description of the simulation (see this)?
   1. Prepare an entangled state between two copies of H: one is the left side of the wormhole, and the other is the right side of the wormhole. This entangled state is dual to a wormhole at time t=0. The devised through learning small SYK-like system has 7 Majorana fermions on the left and 7 Majorana fermions on the right; encoding all 14 fermions in superconducting qubits requires 7 qubits. Comment: the initial state is quantum entangled. If the existence of flux tube pair(s) serves as a correlate for entanglement, BSFR changing the arrow of time must take place.
   2. Evolve the wormhole backwards in time according to H. This moves the horizons of the left and right mouths of the wormhole. Comment: The time evolution backwards in time could correspond to a sequence of SSFR ("small" SFRs as counterparts for "weak" measurements preserving the arrow of time). What could the motion of the wormhole mouths correspond to? Suppose that the half-distance d between the two systems corresponds to the parameter r_S and the length L of flux U-shaped flux tubes corresponds to the parameter a. If this is the case, the motion would correspond to the decrease of the length of the U-shaped flux tubes so that the condition L> d/2 as an analogy for the condition a>r_S ceases to be true and the U-shaped flux tubes become too short to reconnect anymore.
   3. Prepare two maximally entangled qubits: call the first one the reference qubit and the other the probe qubit. We later attempt to send the probe qubit through the wormhole, and we will be able to check if it made it through by comparing against the reference qubit. These two additional qubits bring the total circuit size to 9 qubits. Comment: this step could correspond to a BSFR changing the arrow of time to normal.
   4. Swap the probe qubit with one of the qubits in the left quantum system of the wormhole. This inserts the entanglement probe qubit into the wormhole. Evolve the wormhole forwards in time according to H. As this happens, the information of the probe qubit gets chaotically scrambled throughout the entire quantum system. Comment: This step could correspond to a time evolution by SSFRs with the standard arrow of time.
   5. Apply an entangling interaction between the two sides of the wormhole. In the gravitational dual, this corresponds to sending an energy shockwave through spacetime. We can apply an interaction that gives this shockwave negative energy to prop open the wormhole and make it traversable, or we can choose a positive energy shockwave to close the wormhole and prevent information from getting across. Comment: This step could correspond to a reconnection of U-shaped flux tubes to a flux tube pair connecting the two systems and entangling them, perhaps maximally.
   6. Evolve the wormhole forwards in time according to H. As this happens, information of the probe qubit undergoes further chaotic dynamics. The dynamics refocus the information onto the right side of the wormhole. Comment: this step could be a BSFR followed by a sequence of SSFRs with standard arrow of time. The entanglement would become visible for the observer with a standard arrow of time.
   7. Measure the amount of entanglement between the rightmost qubit of the right system and the reference qubit. More entanglement means more information was transferred from the left system to the right system. In our experiment, we observed more entanglement when a negative energy shockwave was used compared to a positive energy shockwave, which is consistent with the interpretation that some quantum information was transferred via the traversable wormhole mechanism.

   There are still some questions to be considered.

   1. In the TGD framework, the expectation is that the radial coordinate r for the family of blackhole-like objects corresponds to the radial coordinate of E³ ⊂M⁴ and is therefore non-negative.

      One can however ask whether one could have parallel space-time sheets connected by a wormhole contact with an Euclidean signature of the induced metric and whether the negative values of r could be natural for the radial coordinate at the other space-time sheet. Both sheets would be covered by a single coordinate.

   2. One must also ask whether the connected space-time sheets could correspond to opposite arrows of time, which changes in "big" state function reductions (BSFRs). If so, wormhole contacts would mediate interaction between quantum states with opposite arrows of time. The wormhole throat with Euclidean signature does not have a definite arrow of time and could mediate this interaction. In this picture elementary particles, at least bosons, would be pairs of particles with opposite arrows of time. Can this make sense?

   See the article How do AdS/CFT- and TGD based holographic dualities relate?.

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

   Articles related to TGD.

The ultrametric topology of discretized "world of classical worlds"

For more than a month, I have been preparing an article related to the role of finite fields in TGD. I however feel that "Finite Fields and TGD" is now in rather "final" shape and I expect that no new ideas will emerge anymore.

One of the unexpected outcomes is that I now understand the discretization of the "world of classical worlds" (WCW) using polynomials P(x) with integer coefficients as representations of 4-surfaces.

1. The polynomials P(x) satisfy strong additional conditions implying that finite fields can be regarded as basic building bricks in the mathematical structure of TGD besides all other basic number fields.
2. Polynomial P(x) defines a set of mass shells via its roots containing a set of 3-surfaces and defining holographic data fixing space-time surface almost uniquely by M⁸-H correspondence. The holography is forced by general coordinate invariance and there is no need to postulate it as a separate principle.
3. The discretization of WCW by polynomials, assumed to correspond to extrema or even maxima for the exponent of K\"ahler function of WCW, replaces WCW with a discrete set. Polynomials P(x) can have fixed degree k, or degree smaller than some maximum degree k_max, or satisfy some more general condition.
4. These restrictions reflect the basic feature of the spin glass energy landscape, namely that annealing as repeated heating and cooling allows to build localized thermodynamic equilibria localized inside some valley since thermal excitations are not able to kick the system out of the valley (failure of ergodicity).

   Elementary particles with D=P would result during cosmic evolution as repeated annealing when the degeneracy d(D) of polynomials with fixed value of D=P is fixed.

WCW has the fractal structure of a spin glass energy landscape containing valleys inside valleys inside... valleys. This discretized WCW is expected to have ultrametric topology and to decompose to sectors with p-adic topologies. The challenge is to understand what this means.

1. Spin glass energy landscape is realized number-theoretically in terms of the polynomials P(x) with integer coefficients and contains always a finite number of space-time surfaces, a hierarchy of analogues of Riemann zeta emerge defined as

   ζ= ∑ d(D) D^-k .

   These zeta functions have interpretation as partition functions and provide probabilistic description of the number-theoretically discretized WCW. The analogy with Riemann zeta suggests that k=1 corresponds to point at which convergence fails.

   1. Discriminant D provides a concrete realization for how the ultrametric distance function emerges.
   2. d(D) is the number of space-time surfaces with the same D, degeneracy.
   3. k corresponds naturally to the degree or more generally, maximal degree, of polynomials contributing to sub-WCW. k can be also interpreted as an analogue of inverse temperature. k=1 would correspond to linear polynomials defining trivial algebraic extensions.
2. p-Adic length scale hypothesis P ≈ p^k, p=2 or small prime, for preferred ramified primes D=P turns out to be equivalent with the proposal for logarithmic coupling constant evolution for Kähler coupling strength fixed to high degree by number theoretical constraints. Therefore two separate hypotheses fuse to a single one.

Consider now the structure of WCW as an analogue of the spin glass energy landscape.

1. Number theoretically, WCW decomposes to subsets for which a given ramified prime P appears as a prime factor of discriminant D characterizing the polynomial and coding information of ordinary primes that split or are ramified in the extension defined by P(x).
2. D=P space-time regions correspond to particles and those with several ramified primes to interaction regions with external particles corresponding to various primes P_i dividing D: these interaction regions are shared by several regions characterized by P as a factor of D.
3. Elementary particles correspond to D=P regions for which one has an especially large number of 4-surfaces with D=P: that is the degeneracy factor d(D) appearing in the analog of Riemann zeta is large so that annealing leads with a high probability to this state. One can say that space-time surfaces define number theoretical analogs of Feynman graphs consisting of particle lines and vertices.

In the standard ontology, one can predict scattering rates but particle densities cannot be predicted without further assumptions. In ZEO both can be predicted since there is a complete democracy between particles and particle reactions. Physical event as a superposition of deterministic time evolutions becomes the basic notion and both particles and particle reactions correspond to physical events.

The statistical model represents the probabilities of physical events within the quantization volume defined by CD. Particle characterized by D=P and corresponds to a scattering event with a single incoming and outgoing particle, and the statistical model predicts the densities of various particles as probabilities of D=P events. Genuine particle reaction corresponds to D= ∏ P_i and the model gives the probabilities of observing these events within CD.

See the article Finite Fields and TGD or the chapter with the same title.

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

Articles related to TGD.