## Monday, January 31, 2022

### Some useful critical questions concering the twistorial construction of scattering amplitudes

The details of the proposed construction of the scattering amplitudes (see this) starting from twistors are still unclear and the best way to proceed is to invent objections and critical questions.

How do the quark momenta in M8 and H relate to each other?

The relationship between quark momenta in M8 and H is not clear. M8-H duality suggests the same momentum and mass spectrum for quarks in M8 and H. However, the mass spectrum of color partial waves for quark spinors for the Dirac operator D(H) is very simple and characterized by 2 integers labeling triality t=1 representations of SU(3) (see this). D(H) does not allow a mass spectrum as algebraic roots of polynomials.

What can one conclude if M8-H duality holds true in a strong sense so that these spectra are identical?

The only possible conclusion seems to be that the propagator in both M8 and H is just the M4 Dirac propagator D(M4) and that the roots of the polynomial P give the spectrum of off-mass-shell masses. Also tachyonic mass squared values are allowed as roots of P. The real on-shell masses would be associated with Galois singlets.

Consider first the nice features of the D(M4) option.

1. The integration over momentum space reduces to a finite summation over virtual mass shells defined by the roots of P and one avoids divergences. This tightens the connection with QFTs.
2. Massless propagation conforms with the twistorialization since mass one obtains holomorphy in twistor variables.
3. Massless quarks are consistent with the QCD picture about quarks.
4. The consideration of problems related to right-handed neutrino (see this) led to the proposal that the quark spinor modes in H are annihilated only by the H d'Alembertian but not by the H Dirac operator. The assumption that on mass shell H-spinors are annihilated by the M4 Dirac operator leads to the same outcome.

This allows different M4 chiralities to propagate separately and solves problems related to the notion of right-handed neutrino νR (assumed to be 3-antiquark state and modellable using leptonic spinors in H. This also conforms with the right and left-handed character of the standard model couplings.

5. A further argument favoring the D(M4) option comes from the following observation. Suppose that one takes seriously the idea that the situation can be described also by using massless M8 momenta. This implies that for some choices of M4⊂ M8 the momentum is parallel to M4 and therefore massless in 4-D sense. Only the quarks associated with the same M4 can interact. Hence M4 can be always chosen so that the on mass-shell 4-momenta are light-like.
Consider next the objections against the D(M4) option.

1. For some years ago I found that the space-time propagators for points of H connected by a light-like geodesic behave like massless propagators irrespective of mass. CP2 type extermals have a light-like geodesic as an M4 projection. This would suggest that quarks associated with CP2 type extremals effectively propagate as massless particles even if one assumes that they correspond to modes of the full H Dirac operator. This allows us to consider D(H) as an alternative. For this option most quarks would be extremely massive and practically absent from the interior of the space-time surface.
2. Since the color group acts as symmetries, one can assume that spinor modes correspond to color partial waves as eigen states of CP2 spinor d'Alembertian D2(CP2). One would get rid of the constraint on masses but the correlation between color and electroweak quantum numbers would be still "wrong".
3. If M4 Kähler form is trivial, there would be no need for the tachyonic ground state in p-adic mass calculations to reduce the mass quark squared to zero. On the other hand, this might lead also to the problems since the earlier calculations were sensitive to the negative ground state conformal weight and would not work as such. This conformal weight could be generated by conformal generators with weights h coming as roots of P with a negative real part.
4. If leptons are allowed as fundamental fermions, D(H) allows νR as a spinor mode which is covariantly constant in CP2. If leptons are not allowed, one can argue that νR as a 3-quark state can be modeled as a mode of H spinor with Kähler coupling giving correct leptonic charges.

The M4 Kähler structure favored by the twistor lift of TGD (see this) implies that νR with negative mass squared appears as a mode of D(H). This mode allows the construction of tachyonic ground states.

For D(M4) with a M4 Kähler coupling, one obtains for all spinor modes states with both positive and negative mass squared from the JklΣkl term. Physical on-mass- shell states with negative mass squared cannot be allowed. These would however allow to construct tachyonic ground states needed in the p-adic mass calculations. Note that a pair of these two modes gives massless ground state.

Can one allow "wrong" correlation between color and electroweak quantum numbers for fundamental quarks?

For CP2 harmonics, the correlation between color and electroweak quantum numbers is wrong (see this). Therefore the physical quarks cannot correspond to the solutions of D2(H)Ψ=0. The same applies also to the solutions of D(M4)Ψ=0 if one assumes that they belong to irreducible representations of the color group as eigenstates of D(CP2).

How to construct quark states, which are physical in the sense that they are massless and color-electroweak correlation is correct?

1. For D(H) option, the reduction of quark masses to zero requires a tachyonic ground state in p-adic mass calculations (see this). Also for D(M4) with M4 Kähler structure ground states can be tachyonic.

Colored operators with non-vanishing conformal weight are required to make all quark states massless color triplets. This is possible only if the ground state is tachyonic, which gives strong supper for M4 Kähler structure.

2. This is achieved by the identification of physical quarks as states of super-symplectic representations. Also the generalized Kac-Moody algebra assignable to the light-like partonic orbits or both of these representations can be considered. These representations could correspond to inertial and gravitational representations realized at "objective" embedding space level and "subjective" space-time level.

Supersymplectic generators are characterized by a conformal weight h completely analogous to mass squared. The conformal weights naturally correspond to algebraic integers associated with P. The mass squared values for the Galois singlets are ordinary integers.

3. It is plausible that also massless color triplet states of quarks can be constructed as color singlets. From these one can construct hadrons and leptons as color singlets for a larger extension of rationals. This conforms with the earlier picture about conformal confinement. These physical quarks constructed as states of super-symplectic representation, as opposed to modes of the H spinor field, would correspond to the quarks of QCD.

One can argue that Galois confinement allows to construct physical quarks as color triplets for some polynomial Q and also color singlets bound states of these with extended Galois group for a higher polynomial P\circ Q and with larger Galois group as representation of group Gal(P)/Gal(Q) allowing representations of a discrete subgroup of color group.

Are M8 spinors as octonionic spinors equivalent with H-spinors?

At the level of M8 octonionic spinors are natural. M8-H duality requires that they are equivalent with H-spinors. The most natural identification of octionic spinors is as bi-spinors, which have octonionic components. Associativity is satisfied if the components are complexified quaternionic so that they have the same number of components as quark spinors in H. The H spinors can be induced to X4⊂ M8 by using M8-H duality. Therefore the M8 and H pictures fuse together.

The quaternionicity condition for the octonionic spinors is essential. Octonionic spinor can be expressed as a complexified octonion, which can be identified as momentum p. It is not an on-mass shell spinor. The momenta allowed in scattering amplitudes belong to mass shells defined by the polynomial P. That octonionic spinor has only quaternionic components conforms with the quaternionicity of X4⊂ M8 eliminating the remaining momentum components and also with the use of D(M4).

Can one allow complex quark masses?

One objection relates to unitarity. Complex energies and mass squared values are not allowed in the standard picture based on unitary time evolution.

1. Here several new concepts lend a hand. Galois confinement could solve the problems if one considers only Galois singlets as physical particles. ZEO replaces quantum states with entangled pairs of positive and negative energy states at the boundaries of CD and entanglement coefficients define transition amplitudes. The notion of the unitary time evolution is replaced with the Kähler metric in quark degrees of freedom and its components correspond to transition amplitudes. The analog of the time evolution operator assignable to SSFRs corresponds naturally to a scaling rather than time translation and mass squared operator corresponds to an infinitesimal scaling.
2. The complex eigenvalues of mass squared as roots of P be allowed when unitarity at quark level is not required to achieve probability conservation. For complex mass squared values, the entanglement coefficients for quarks would be proportional to mass squared exponents exp(im2λ), λ the scaling parameter analogous to the duration of time evolution. For Galois singlets these exponentials would sum up to imaginary ones so that probability conservation would hold true.
See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Saturday, January 29, 2022

### The mass spectrum for an iterate of polynomial and chaos theory

Suppose that the number theoretic interaction in the scattering corresponds to a functional composition of the polynomials characterizing the external particles. If the number of the external particles is large, the composite can involve a rather high iterate of a single polynomial. This motivates the study of the scattering of identical particles described by the same polynomial P at the limit of a large particle number. These particles could correspond to elementary particles, in particular IR photons and gravitons. This situation leads to an iteration of a complex polynomial.

If the polynomials satisfy P(0)=0 requiring P(x)= xP1(x), the roots of P are inherited. In this case fixed points correspond to the points with P(x)=1. Assume also that the coefficients are rational. Monic polynomials are an especially interesting option.

For a k:th iterate of P, the mass squared spectrum is obtained as a union of spectra obtained as images of the spectrum of P under iterates P-r, r< k, for the inverse of P, which is an n-valued algebraic function if P has degree n. This set is a subset of Fatou set (see this) and for polynomials a subset of filled Julia set.

At the limit of large k, the limiting contributions to the spectrum approach a subset of Julia set defined as a P-invariant set which for polynomials is the boundary of the set for which the iteration divergences. The iteration of all roots except x=0 (massless particles) leads to the Julia set asymptotically.

All inverse iterates of the roots of P are algebraic numbers. The Julia set itself is expected to contain transcendental complex numbers. It is not clear whether the inverse iterates at the limit are algebraic numbers or transcendentals. For instance, one can ask whether they could consist of n-cycles for various values of n consisting of algebraic points and forming a dense subset of the Julia set. The fact that the number of roots is infinite at this limit, suggests that a dense subset is in question.

The basis properties of Julia set

The basic properties of Julia set deserve to be listed.

1. At the real axis , the fixed points satisfying P(x)=x with |dP/dx|>1 are repellers and belong to the Julia set. In the complex plane, the definition of points of the Julia set is |P(w)-P(z)|> |w-z| for point w near to z.
2. Julia set is the complement of the Fatou set consisting of domains. Each Fatou domain contains at least one critical point with dP/dz=0. At the real axis, this means that P has maximum or minimum. The iteration of P inside Fatou domain leads to a fixed point inside the Fatou set and inverse iteration to its boundary. The boundaries of Fatou domains combine to form the Julia set. In the case of polynomials, Fatou domains are labeled by the n-1 solutions of dP/dz= P1 +zdP1/dz=0.
3. Julia set is a closure of infinitely many periodic repelling orbits. The limit of inverse iteration leads towards these orbits. These points are fixed points for powers Pn of P.
4. For rational functions Julia set is the boundary of a set consisting of points whose iteration diverges to infinity. For polynomials Julia set is the boundary of the so-called filled Julia set consisting of points for which the iterate remains finite.
Chaos theory also studies the dependence of Julia set on the parameters of the polynomials. Mandelbrot fractal is associated to the polynomial Q(z)= a+z2 for which origin is an stable critical point and corresponds to the boundary of the region in a-plane containing origin such that outside the boundary the iteration leads to infinity and in the interior to origin.

The critical points of P with dP/dz=0 for z= zcr located inside Fatou domains are analogous to point z=0 for Q(z) associated with Fatou domains and quadratic polynomial a+b(z-zcr)2, b>0, would serve as an approximation. The variation of a is determined by the variation of the coefficients of P required to leave zcr invariant.

Feigenbaum studied iteration of a polynomial a-x2 for which origin is unstable critical point and found that the variation of a leads to a period doubling sequence in which a sequence of 2n-cycles is generated (see this). Origin would correspond to an unstable critical point dP(z)/dz=0 belonging to a Julia set.

The physical implications of this picture are highly interesting.

1. For a large number of interacting quarks, the mass squared spectrum of quarks as roots of the iterate of P in the interaction region would approach the Julia set as infinite inverse iterates of the roots of P. This conforms with the idea that the complexity increases with the particle number.

Galois confinement forces the mass squared spectrum to be integer valued when one uses as a unit the p-adic mass scale defined by the larger ramified prime for the iterate. The complexity manifests itself only as the increase of the microscopic states in interaction regions.

2. Julia set contains a dense set consisting of repulsive n-cycles, which are fixed points of P and the natural expectation is that the mass spectrum decomposes into n-multiplets. Whether all values of n are allowed, is not clear to me. The limit of a large quark number would also mean an approach to (quantum) criticality.
Two objections

There is a useful objection against this picture. M8-H duality requires the same momentum and mass spectrum for quarks in M8 and H. However, the mass spectrum of color partial waves for quark spinors in H is very simple and characterized by 2 integers labeling triality t=1 representations of SU(3) (see this). How can these pictures be consistent with each other? Do the quarks in M4 and H differ from each other and what does this mean?

To answer this question, one must ask what one means with quark in H and in M8.

1. There are good reasons to assume that the quark spinor modes in H are annihilated only by the H d'Alembertian but not by the H Dirac operator (see this). This allows different M4 chiralities to propagate separately and solves problems related to the notion of right-handed neutrino νR and also conforms with the right and left-handed character of standard model couplings.
2. Apart from νR all quark partial harmonics have CP2 mass scale and also the correlation between color and electroweak quantum numbers is wrong (see this). Therefore the physical quarks cannot correspond to the solutions of H spinor d'Alembertian.

The M4 Kähler structure forced by the twistor lift of TGD (see this) is part of the solution. It predicts that νR, if modeled as a mode of H spinor with Kähler coupling giving correct leptonic charges, has a tachyonic mass. The first guess is that the physical states contain an appropriate number of right handed neutrinos to build a tachyonic ground state from which one can construct a massless state. A more general approach allows roots of P with a negative real part as tachyonic virtual quarks. The virtual particles of standard QFT would correspond to quarks with masses coming as roots of P and they can also be tachyonic. Galois singlets would be analogous to on-mass shell particles.

3. How to construct quark states which are physical in the sense that they are massless and color-electroweak correlation is correct? The reduction to quark masses to zero requires a tachyonic ground state in p-adic mass calculations (see this) . Also colored operators are required to make all quarks state color triplets.

The solution of the problem is provided by the identification of physical quarks as states of super-symplectic representations. Also the generalized Kac-Moody algebra assignable to the light-like partonic orbits or both of these representations can be considered. These representations could correspond to inertial and gravitational representations realized at "objective" embedding space level and "subjective" space-time level.

Supersymplectic generators are characterized by a conformal weight h completely analogous to mass squared. The conformal weights naturally correspond to algebraic integers associated with P. The mass squared values for the "physical" quarks are algebraic integers and Galois confinement forces integer-valued conformal weights for the physical states consisting of quarks. This conforms with the earlier picture about conformal confinement.

4. These "physical" quarks constructed as states of super-symplectic representation, as opposed to modes of H spinor field, would correspond to quarks in M8. The complex momenta and mass squared values would be generated by supersymplectic generators with conformal weights h coming as algebraic integers associated with P. Most importantly, the modes of H-spinors would have integer-valued momenta and mass spectrum of the spinor d'Alembertian.
Second useful objection relates to unitarity. Complex energies and mass squared values are not allowed in the standard picture based on unitary time evolution.
1. Here several new concepts lend a hand. Galois confinement could solve the problems if one considers only Galois singlets as physical particles. ZEO replaces quantum states with entangled pairs of positive and negative energy states at the boundaries of CD and entanglement coefficients define transition amplitudes.

The notion of the unitary time evolution is replaced with the Kähler metric in quark degrees of freedom and its components correspond to transition amplitudes. The analog of the time evolution operator assignable to SSFRs corresponds naturally to a scaling rather than time translation and mass squared operator corresponds to an infinitesimal scaling.

2. The complex eigenvalues of mass squared as roots of P be allowed when unitarity at quark level is not required to achieve probability conservation. For complex mass squared values, the entanglement coefficients for quarks would be proportional to mass squared exponents exp(im2λ), λ the scaling parameter analogous to the duration of time evolution. For Galois singlets these exponentials would sum up to imaginary ones so that probability conservation would hold true.
To sum up, it would seem that chaos (or rather complexity-) theory could be an essential part of the fundamental physics of many-quark systems rather than a mere source of pleasures of mathematical aesthetics.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Wednesday, January 26, 2022

### Quantum Statistical Brain

The following considerations were inspired by a popular article (see this) telling about findings (see this) of Li et al supporting the view that neural noise carries information in the sense that it represents the uncertainty of visual short term memories so that both the content of memory and its uncertainty are represesented. Thanks for the link to Jouko Alanko.

Does neural noise carry information about the uncertainty of visual short term memories?

The highlights of Li et al are following:

• Humans know the uncertainty of their working memory and use it to make decisions.
• The content and the uncertainty of working memory can be decoded from so called BOLD signals.
• Decoding errors predict memory errors at the single-trial level.
• Decoded uncertainty correlates with behavioral reports of working memory uncertainty.
It is not surprising that the states of feature detector neurons should obey a statistical distribution. It is however not obvious that the reliability of the memory should correlate with the width of this distribution and that even the subjective estimate for the reliability should reflect this width.

Does the distribution in the feature space reflect quantum non-determinism?

Could the distribution in the feature space reflect quantum non-determinism rather than uncertainty of sensosry perceptions and somehow also the uncertainty of memories.

1. If features as states of feature detector neurons or groups of them correspond to the outcomes of quantum measurements, they have a probability distribution. The real input to these neutron would have produced this distribution and could be estimated from the probability distribution.

The outcomes are eigenstates of density matrix determined by the entanglement and determined apart from phase factors. For instance, in the measurement of spin of spin 1/2 particle the probabilities of spin 1/2 and spin -1/2 states can be deduced for an ensemble of identical particles but the relative phase of the spin 1/2 and spin -1/2 state cannot be deduced.

2. The interpretation of quantum measurement would differ from the classical one. Classically, and according to recent neuroscience, sensory perception means that brain, system A, detects the state of system B in the external world. Quantum mechanically, the entanglement between A and B is reduced in the measurement and entangled state becomes a tensor product of are eigenstates of the density matrix. The relationship between A and B is what is "measured". For an ensemble of outcomes, the probabilities of outcomes allow to deduce information about the entanglement before measurement.
3. If the reduction of the entanglement between sensory organ and external world can be measured repeatedly, it gives rise to a distribution of outcomes coding also the uncertainty caused by the quantum measurement. This however requires that the entanglement is regenerated between these measurements. Is this possible?
The distribution of features would not reflect uncertainty of memories but the non-determinism of the outcome in the reduction of entanglement. Interestingly, in quantum computation this kind of ensemble is produced and from the distribution of outcomes of the measurement halting the quantum computation, the outcome of the quantum computation is deduced. The method is essentually statistical.

In TGD framework the notion of magnetic body (MB) using biological body as sensory receptor and motor instrument emerges as a new notion. The entanglement between magnetic body and sensory organs could be reduced in sensory perception. There is a hierarchy of levels and entanglements at them and SFR is replaed with a cascade of SFRs proceeding from long to short scales.

Is the feature distribution realized as a temporal ensemble?

In sensory perception, the distribution of features should correspond to a distribution of states of feature detector neurons or their groups. How is this distribution realized? How does this distribution relate to the distribution of memories?

Let us consider the questions about sensory perceptions.

1. The neuroscience based answer to question in the case of sensory perceptions would be "As a spatial ensemble consisting of feature neurons". But how does this distribution relate to the distribution of memories?
2. In TGD framework, the answer would be "As a temporal ensemble". Zero energy ontology (ZEO) leads to a new view about quantum states as superpositions of deterministic time evolutions and modifies the view about quantum measurements allowing to circumvent the basic paradox of quantum measurement theory leading to various interpretations.

The outcome is the notion of 4-D brain, which suggests a temporal ansemble formed by memory mental images of the feature. In ZEO, the sequences of "small" state function reductions (SSFRs) as counterparts of so called weak measurements would form temporal ensembles of memory mental images so that the connection with short term memory would be direct. The spatial ensemble would be replaed by temporal ensemble experienced consciously as memories.

TGD based view about sensations and short term memories

To develop a more detailed model based on the proposed ideas, one must answer several questions in the TGD framework. What sensory experiences, perceptions, and features are in TGD Universe? What could the phrase "statistical ensemble of features" mean? What does sensory perception as a quantum measurement and quantum measurement itself correspond to?

The notions of sensation, perception, and feature

Sensation as the core of sensory experience must be distinguished from perception. Sensation is just the sensory awareness with nothing added. Perception involves a cognitive representation providing an interpretation of perception and consists of objects and the associations and memories associated with them.

Brain is believed to analyze the sensory input from the sensory organs to features. Features are just those aspects of the input that are relevant to survival or target of attention. Neutrons serve as feature detectors (see this).

This deconstruction process is followed by reconstruction which proceeds upwards from features to objects of the perceptive field so that the perceptive field decomposes to standardized mental images representing objects with various attributes, orientation and motion are such attributes. This is basically pattern recognition. Features are basic building bricks of the sensory mental images and not necessarily conscious to us.

The reconstruction process is analogous to first drawing a simple drawing consisting of lines and then gradually filling the picture by adding colors with varying intensities. Something analogous happens also when the sound-scape of a movie is constructed. One starts from the actual sound-scape but the outcome is quite different and very far from the original. One could say that sensory perception is essentially an artwork.

In the mathematical modeling, one can speak of a feature space. Features have attributes and the claim of the article discussed is that one can assign to features a probability distribution. Brain would not only build features but also represent this probability distribution making it possible to estimate the reliability of the visual short memory. It is however not clear how the distribution gives rise to a conscious experience about reliability and how the short term memory relates to the sensory perception.

Ensemble of features as temporal ensemble of memory mental images?

The probability distribution for features should be realized somehow as a statistical ensemble. One can consider two alternative options.

1. In the standard physics framework spatial ensemble seems to be the only possible realization. The perception would be represented as a large number of copies. The fact that the inputs in the retina are mapped in a topographic manner to various parts of the visual cortex poses strong constraints on the number and location of the copies. If there is a spatial ensemble its neurons should form groups of neary neurons. The problem is how the distribution of features in this ensemble can code for the reliability of sensory or memory mental images and this requires a theory of consciousness.
2. In the TGD framework, the brain is 4-D and it makes sense to speak of a temporal ensemble of memory mental images. These temporal ensembles would correspond to temporal sequences of memory mental images and the distribution aspect would be automatically realized. The variance of this distribution would provide conscious experience about the reliability of the mental images. The natural interpretation would be in terms of short term memory.
For the TGD option, the sensory input to the sensory organ, say retina, would generate a temporal ensemble of visual mental images making possible short term memory. This ensemble would be characterized by a probability distribution. The probability distribution for the states of feature neurons would be a neuronal level example of this kind of distribution. Variance would be one characteristic of this distribution and characterize the reliability of short term memory. Sensory perceptions would give rise to short term memories.

Many questions remain to be answered. How are these memory mental images generated in quantum measurements? How does the memory recall of long term memory generate a short term memory represented as a temporal ensemble of visual mental images?

1. For instance, in the memory recall of a phone number, long term memory is involved. Somehow the memory recall creates "almost" sensory, that is virtual, perception, which suggests that a virtual sensory input from MB is involved and creates a virtual sensory perception giving rise to a visual short term memory.
2. In the TGD framework, these virtual sensory perceptions would also make possible imagination. The virtual sensory input would come from MB to cortex and proceed to the lower levels of the brain but would not reach sensory organs except during dreams, hallucinatory states, and sensory memories (memory feats of idiot savants).
3. The sensation associated with the sensory experience would correspond to a state function reduction (SFR) occurring in quantum measurement. But what does SFR correspond to in TGD?

In the zero energy ontology (ZEO), the notion of SFR generalizes. The are two kinds of SFRs: "big" SFRS (BSFRs) as analogs of ordinary quantum measurements in which a large change is possible and "small" SFRs (SSFRs) as analogs of so called weak measurements, which are assumed in quantum optics but are not very-well defined in the standard quantum theory and do not appear in the text books.

SSFRs relate closely to the Zeno effect which states that the state of the physical system remains unaffected if the same measurement is repeated. In reality this is not quite true, and the sequence of SSFRs represents a generalization of a repeated quantum measurement allowing us to understand what really happens.

Sensory perception would be repetition of SSFRs following analogs of unitary time evolutions and would produce an temporal ensemble of sensory mental images giving rise to short term memory. The system would be measured, it would return back to almost its original state and would be measured again. SSFR is almost a classical measurement.

See the article Quantum Statistical Brain or the chapter with the same title.

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

## Tuesday, January 25, 2022

### The honeycomb of large voids as a tesselation of hyperbolic space by membrane like structures?

Large voids of size scale or order 108 light years forming honeycomb like structures are rather mysterious objects, or rather non-objects. The GRT based proposal is that the formation of gravitational bound states leads to these kinds of structures in general relativity but I do now know how convincing these arguments really are.

One should answer two questions: what are these voids and why do they form these lattice-like structures? Even new physics is allowed!

One explanation of large voids is based on the TGD based view about space-time as a 4-surface in M4×CP2.

1. Space-time surfaces have M4 projection, which is 4-D for what I call Einsteinian space-times. At this limit general relativity is expected to be a good approximation for the field theory limit of TGD.

However, the M4 projection can be also 3-D , 2-D or 1-D. In these cases one has what looks like a membrane, string, or point-like particle. All these options are realized.

2. The simplest membranes would look like M1×S2×S1, S1 a geodesic circle of CP2 which depends on a point of M1×S2 defining M4 projection. Only this assumption allows us to have a minimal surface. I discovered this quite recently although the existence of membrane like entities was almost obvious from the beginning (see this).
3. Small perturbations tend to thicken the dimension of M4 projection to 4 but the deformed objects are in an excellent approximation still 3-D, 2-D or 1-D.
Membranes are everywhere: large voids, blackhole horizons, Fermi bubbles, cell membranes, soap bubbles, bubbles in water, shock wave fronts, etc.... Large voids could be really voids in a good idealization!! Even 4-D space-time would be absent!! The void would be the true vacuum!

It should be noticed that matter as smaller objects, say cosmic strings thickened to flux tubes, would in turn have galaxies as tangles, which in turn would have stars as tangles. The TGD counterparts of blackholes would be dense flux tube spaghettis filling the entire volume.

What could then give rise to the lattice like structures formed from voids? Here TGD suggests a rather obvious solution.

1. The lattices could correspond to tessellations of the 3-D hyperbolic space H3 for which cosmic time coordinate identified as light-cone proper time is constant. H3 allows an infinite number of tessellations whereas Euclidean 3-space allows a relatively small number of lattices.
2. There is even empirical evidence for these tessellations. Along the same line of sight there are several sources of light and the redshifts are quantized. One speaks of God's fingers. This is what any tessellation of cosmic voids would predict: cosmic redshift would define effective distance. Of course also tessellations in smaller scales can be considered.
3. Also ordinary atomic lattices could involve this kind of tessellations with atomic nuclei at the centers of the unit cells as voids. The space between nucleus and atom would literally be empty, even the 4-D space-time would be absent!
4. Also the TGD inspired model for genetic code involves a particular tessellation of H3 realized at the magnetic body (MB) of a biological system and realizing genetic code. This leads to the conjecture that genetic code is universal and does not characterize only living matter. It would be induced to the space-time surface in the sense that part of tessellation would define a tessellation at the space-time surface. At the level of dark matter at MB, 1-D DNA could also have 2-D and even 3-D analogs, even in ordinary living matter (see this)!
See the article What could 2-D minimal surfaces teach about TGD? or the chapter ZEO, Adelic Physics, and Genes.

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

## Sunday, January 23, 2022

### A possible generalization of number theoretic approach to analytic functions

A possible generalization of number theoretic approach to analytic functions

M8-H duality also allows the possibility that space-time surfaces in M8 are defined as roots of real analytic functions. This option will be considered in this section. One of the open problems of the number-theoretic vision is whether the space-time surfaces associated with rational or even monic polynomials are an approximation or not.

1. One could argue that the cognitive representations are only a universal discretization obtained by approximating the 4-surface in M8 by a polynomial. This discretization relies on an extension of rationals and more general than rational discretizations, which however appear via Galois confinement for the momenta of Galois singlets.

One objection against space-time surfaces as being determined by polynomials in M8 was that the resulting 4-surfaces in M8 would bre algebraic surfaces. There seems to be no hope about Fourier analysis. The problem disappeared with the realization that polynomials determine only the 3-surfaces as mass-shells of M4 and that M8-H duality is realized by an explicit formula subject to I(D)= exp(-K) condition.

2. Galois confinement provides a universal mechanism for the formation of bound states. Could evolution be a development of real states for which cognitive representations in terms of quarks become increasingly precise.

That the quarks defining the active points of the representation are at 3-D mass shells would correspond to holography at the level of M8. At the level of H they would be at the boundaries of CD. This would explain why we experience the world as 3-dimensional.

Also the 4-surfaces containing quark mass shells defined in terms of roots of arbitrary real analytic functions are possible.

1. Analytic functions could be defined in terms of Taylor or Laurent series. In fact, any representation can be considered. Also now one can consider representation involving only integers, rationals, algebraic numbers, and even reals as parameters playing a role of Taylor coefficients.

2. Does the notion of algebraic integers generalize? The roots of the holomorphic functions defining the meromorphic functions as their ratios define an extension of rationals, which is in the general transcendental. It is plausible that the notion of algebraic integers generalizes and one can assume that quarks have momentum components, which are transcendental integers. One can also define the transcendental analog of Galois confinement.

3. One can form functional composites to construct scattering amplitudes and this would make possible particle reactions between particles characterized by analytic functions. Iteration of analytic functions and approach to chaos would emerge as the functions involved appear very many times as one expects in case of IR photons and gravitons.

What about p-adicization requiring the definition discriminant D and identification of the ramified primes and maximal ramified prime? Under what conditions do these notions generalize?

1. One can start from rational functions. In the case of rational functions R, one can generalize the notion of discriminant and define it as a ratio D= D1/D2 of discriminants D1 and D2 for the polynomials appearing as a numerator and denominator of R. The value of D is finite irrespective of the values of the degrees of polynomials.

2. Analytic functions define function fields. Could a generalization of discriminant exist. If the analytic function is holomorphic, it has no poles so that D could be defined as the product of squares of root differences.

If the roots appear as complex conjugate pairs, D is real. This is guaranteed if one has f(z*) = f(z)*. The real analyticity of f guarantees this and is necessary in the case of polynomials. A stronger condition would be that the parameters such as Taylor coefficients are rational.

If the roots are rationals, the discriminant is a rational number and one can identify ramified primes and p-adic prime if the number of zeros is finite.

3. Meromorphic functions are ratios of two holomorphic functions. If the numbers of zeros are finite, the ratio of the discriminants associated with the numerator and denominator is finite and rational under the same assumptions as for holomorphic functions.

4. M8-H duality leads to the proposal that the discriminant interpreted as a p-adic number for p-adic prime defined by the largest ramified prime, is equal to the exponent of exp(-K) of Kähler function for the space-time surface in H.

If one can assign ramified primes to D, it is possible to interpret D as a p-adic number having a finite real counterpart in canonical identification. For instance, if the roots of zeta are rationals, this could make sense.

2. Questions related to the emergence of mathematical consciousness

These considerations inspire further questions about the emergence of mathematical consciousness.

1. Could some mathematical entities such as analytic functions have a direct realization in terms of space-time surfaces? Could cognitive processes be identified as a formation of functional composites of analytic functions? They would be analogs of particle reactions in which the incoming particles consist of quarks, which are associated with mass-shells defined by the roots of analytic function.

These composites would decay to products of polynomials in cognitive measurements involving a cascade of SSFRs reducing the entanglement between a relative Galois group and corresponding normal group acting as Galois group of rationals (see this).

2. Could the basic restriction to cognition come from the Galois confinement: momenta of composite states must be integers using p-adic mass scale as a unit.

Or could one think that the normal sub-group hierarchies formed by Galois groups actually give rise to hierarchies of states, which are Galois confined for an extension of the Galois group.

Could these higher levels relate to the emergence of consciousness about algebraic numbers. Could one extend computationalism allow also extensions of rationals and algebraic integers as discussed here).

Galois confinement for an extension of rationals would be analogous to the replacement of a description in terms of hadrons with that in terms of quarks and mean increase of cognitive resolution. Also Galois confinement could be generalized to its quantum version. One could have many quark states for which wave function in the space of total momenta is Galois singlet whereas total momenta are algebraic integers. S-wave states of a hydrogen atom define an obvious analog.

3. During the last centuries the evolution of mathematical consciousness has made huge steps due to the discovery of various mathematical concepts. Essentially a transformation of rational arithmetics with real analysis and calculus has taken place since the times of Newton. Could these evolutionary explosions correspond to the emergence of space-time surfaces defined by analytic functions or is it that only a conscious awareness about their existence has emerged?
3. Space-time surfaces defined by zeta functions and elliptic functions

Several physical interpretations of Riemann zeta have been proposed. Zeta has been associated with chaotic systems, and the interpretation of the imaginary parts of the roots of zeta as energies has been considered. Also an interpretation as a formal analog of a partition function has been considered. The interpretation as a scattering amplitude was considered by Grant Remmen (see this).

3.1. Conformal confinement as Galois confinement for polynomials?

TGD suggests a totally different kind of approach in the attempts to understand Riemann Zeta. The basic notion is conformal confinement.

1. The proposal is that the zeros of zeta correspond to complex conformal weights sn=1/2+iyn. Physical states should be conformally confined meaning that the total conformal weight as the sum of conformal weights for a composite particle is real so that the state would have integer value conformal weight n, which is indeed natural. Also the trivial roots of zeta with s=-2n, n>0, could be considered.
2. In M8-H duality, the 4-surfaces X4⊂ M8 correspond to roots of polynomials P. M8 has an interpretation as an analog of momentum space. The 4-surface involves mass shells m2= rn, where rn is the root of the polynomial P, algebraic complex number in general.

The 4-surface goes through these 3-D mass-shells having M4 ⊂ M8 as a common real projection. The 4-surface is fixed from the condition that it defines M8-H duality mapping it to M4× CP2. One can think X4 as a deformation of M4 by a local SU(3) element such that the image points are U(2) invariant and therefore define a point of CP2. SU(3) has an interpretation as octonionic automorphism.

3. Galois confinement states that physical states as many-quark states with quark momenta as algebraic integers in the extension defined by the polynomial have integer valued momentum components in the scale defined by the causal diamond also fixed by the p-adic prime identified as the largest ramified prime associated with the discriminant D of P.

Mass squared in the stringy picture corresponds to conformal weight so that the mass squared values for quarks are analogous to conformal weights and the total conformal weight is integer by Galois confinement.

3.2. Conformal confinement for zeta functions

At least formally, TGD also allows a generalization of real polynomials to analytic functions. For a generic analytic function it is not possible to find superpositions of roots that would be integers and this could select Riemann Zeta and possible other analytic functions are those with infinite number of roots since they might allow a large number of bound states and be therefore winners in the number theoretic selection.

Riemann zeta is a highly interesting analytic function in this respect.

1. Actually an infinite hierarchy of zeta functions, one for any extension of rationals and conjectured to have zeros at the critical line, can be considered. Could one regard these zetas as analogous to polynomials with an infinite degree so that the allowed mass squared values for quarks would correspond to the roots of zeta?
2. Conformal confinement requires integer valued momentum components and total conformal weights as mass squared values. The fact that the roots of zetas appear as complex conjugates allows for a very large number of states with real conformal weights. This is however not enough. The fact that the roots are of the form zn= 1/2+iyn or z=-2n implies that the conformal weights of Galois/conformal singlets are integer-valued and the spectrum is the same as in conformal field theories.
3. Riemann zeta has only a single pole at s=1. Discriminant would be the product ∏m≠ n (ym-yn2) ∏m≠ n 4(m-n)2m,n(4m2+yn2) since the pole gives D=1. D would be infinite.
4. Fermionic zeta ζF(s)= ζ(s)/ζ(2s) is analogous to the partition function for fermionic statistics and looks more appropriate in the case of quarks. In this case, the zeros are zn resp. zn/2 and the ratio of determinants would reduce to an infinite power of 2. The ramified prime would be the smallest possible: p=2! D= D1/D2 would be infinite power of 2 and 2-adically zero so that exp(-K) should vanish and Kähler function would diverge. 3-adically it would be infinite power of -1. If one can say that the number of roots is even, one has D=1 3-adically. Kähler function would be equal to zero, which is in principle possible.

For Mersenne primes Mn=2n-1, 2n would be equal to 1+ Mn=1 mod Mn and one would obtain an infinite power 1+Mn, which is equal to 1 mod Mn. Could this relate to the special role of Mersenne primes?

5. What about Riemann Hypothesis? By ζ(s*)= ζ*(s), the zeros of zeta appear in complex conjugate pairs. By functional equation, also s and 1-s are zeros. Suppose that there is a zero s+= s0+iyn with s0≠ 1/2 in the interval (0,1). This is accompanied by zeros s*+, 1-s+, s-= 1-s*+. The sum of these four zeros is equal to s=2. Therefore Galois singlet property does not allow us to say anything about the Riemann hypothesis.
3.3 Conformal confinement for elliptic functions

Elliptic functions (see this) provide examples of analytic functions with infinite number of roots forming a doubly periodic lattice and are therefore cood candidates for analogs of polynomials with infinite degree.

1. Elliptic functions are doubly periodic and characterized by the ratio τ of complex periods ω1 and ω2. One can assume the convention ω1=1 giving ω2=τ. The roots of the elliptic function for an infinite lattice and complex rational roots are of obvious interest concerning the generalization of Galois/conformal confinement.
2. The fundamental set of zeros is associated with a cell of this lattice. The finite number of zeros (with zero with multiplicity m counted as m zeros) in the cell is the same as the number poles and characterizes partially the elliptic function besides τ.
3. Weierstrass P-function and its derivative dP/dz are the building blocks of elliptic functions. A general elliptic function is a rational function of P and dP/dz. In even elliptic functions only the even funktion P appears.
4. The roots of Weierstrass P-function P(z)= ∑λ 1/(z-λ)2 appear in pairs +/- z whereas the double poles at at the points of the modular lattice (see the article "The zeros of the Weierstrass P-function and hypergeometric series" of Duke and Imamoglu (see this).

The roots are given by Eichler-Zagier formula z+/-(m,n) =1/2+ m+nτ +/- z1, where z1 contains an imaginary transcendental part log(5+2×61/2)/2π) plus second part, which depends on τ (see formula 6) of the article.

5. Conformally confined states with conformal weights h= 1+(m1+m2)+(n1+n2)τ can be realized as pairs with conformal weights (z+(m1,n1),z-(m2,n2). The condition n1=-n2 guarantees integer-valued conformal weights and conformal confinement for a general value of τ.
6. A possible problem is that the total conformal weights can be also negative, which means tachyonicity. This is not a problem also in the case of Riemann zeta if trivial zeros are included.

As a matter of fact, already at the level of M8, M4 Kähler structure implies that right-handed neutrino νR is a tachyon (see this). However, νR provides the tachyon needed to construct massless super-symplectic ground states and also allows us to understand why neutrinos can be massive although right-handed neutrinos are not detected. The point is that only the square of Dirac equation in H is satisfied so that different M4 chiralities can propagate independently.

In M8-H duality, non-tachyonicity makes it possible to map the momenta at mass shell to the boundary of CD in H. Hence the natural condition would be that the total conformal weight of a physical state is non-negative.

What about the notion of discriminant and ramified prime? One can assign to the algebraic extensions primes as prime ideals for algebraic integers and this suggests that the generalization of p-adicity and p-adic prime is possible. If this is the case also for transcendental extensions, it would be possible to define transcendental p-adicity.

One can however ask whether the discriminant is rational under some conditions. D could also allow factorization to the primes of the transcendental extension.

1. Elliptic functions are meromorphic and have the same number of poles and zeros in the basic cell so that there are some hopes that the ratio of discriminants is finite and even rational or integer for a suitable choice of the modular parameter τ as the ratio of the periods and the other parameters. Discriminant D as the ratio D1/D2 of the discriminants defined by the products of differences of roots and poles could be finite although they diverge separately.
2. For the Weierstrass P-function, the zeros appear as pairs +/- z0 and also as complex conjugate pairs. Complex pairs are required by real analyticity essential for the number theoretical vision. It might be possible to define the notion of ramified prime under some assumptions.

For z+(m,n) or z-(m,n), D1 in D1/D2 would reduce to a product ∏m,n Δm,n)2m,n +2z1) (Δm,n -2z1), Δm,n =Δ m+Δ nτ, which is a complex integer valued if τ has integer components. D1 would be a product of Gaussian integers.

3. The number of poles and zeros for the basic cell is the same so that D2 as a product of the pole differences would have an identical general form. For large values of m,n, the factors in the product approach Δm,n for both zeros and poles so that the corresponding factors combine to a factor approaching unity.

The double poles of P(z)= ∑λ 1/(z-λ)2 are at points of the lattice. One has D2=∏m,n Δm,n)4. This gives

D= D1/D2=∏m,n(1 +2z0m,n) (1 -2z0m,n)= ∏m,n(1-8z0m,n2) .

Therefore D is finite. z0 contains a transcendental constant term plus a term depending on τ (see this). The existence of values of τ for which D is rational, seems plausible.

In the number theoretic vision, the construction of many-particle states corresponds to the formation of functional composites of polynomials P. If the condition P(0)=0 is satisfied, the n-fold composite inherits the roots of n-1-fold composites and the roots are like conserved genes. If one multiplies zeta functions and elliptic functions by z, one obtains similar families and the formation of composites gives rise to iteration sequences and approach to chaos (see this).

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Friday, January 21, 2022

### Bubble-like structures in TGD Universe

Membrane-like structures appear in all length scales from soap bubbles to large cosmic voids and it would be nice if they were fundamental objects in the TGD Universe. The Fermi bubble in the galactic center is an especially interesting membrane-like structure also from the TGD point of view as also the membrane-like structure presumably associated with the analog of horizon of the TGD counterpart of blackhole. Cell membrane is an example of a biological structure of this kind. I have however failed to identify candidates for the membrane-like structures.

In M8-H duality surface with M4 projections smaller than four appear as singularities of algebraic surfaces in M^8. The dimension of M4 projection varies and known extremals can be interpreted in terms of singularities.

An especially interesting singularity would be a static 3-D singularity M1× X2 with a geodesic circle S1 ⊂ CP2 as a local blow-up.

1. The simplest guess is as product M1× S2× S1. The problem is that a soap bubble is not a minimal surface: a pressure difference between interior and exterior of the bubble is required so that the trace of the second fundamental form is constant. Quite generally, closed 2-D surfaces cannot be minimal surfaces in a flat 3-space since the vanishing curvature of the minimal surface forces the local saddle structure.
2. A correlation between M4 and CP2 degrees of freedom is required. In order to obtain a minimal surface, one must achieve a situation in which the S2 part of the second fundamental form contains a contribution from a geodesic circle S1 ⊂ CP2 so that its trace vanishes. A simple example would correspond to a soap bubble-like minimal surface with M4 projection M1× X2, which has having geodesic circle S1 as a local CP2 projection, which depends on the point of M1× X2.
3. The simplest candidate for the minimal surface M1× S2⊂ M4. One could assign a geodesic circle S1⊂ CP2 to each point of S2 in such a manner that the orientation of S1⊂ CP2 depends on the point of S2.

4. A natural simplifying assumption is that one has S1⊂ S21⊂ CP2, where S21 is a geodesic sphere of CP2 which can be either homologically trivial or non-trivial. One would have a map S2→ S21 such that the image point of point of S2 defines the position of the North pole of S21 defining the corresponding geodesic circle as the equatorial circle.

The maps S2→ S21 are characterized by a winding number. The map could also depend on the time coordinate for M1 so that the circle S1 associated with a given point of S1 would rotate in S21. North pole of S21 defining the corresponding geodesic circle as an equatorial circle. These maps are characterized by a winding number. The map could also depend on the time coordinate for M1 so that the circle S1 associated with a given point of S1 would rotate in S21.

The minimal surface property might be realized for maximally symmetric maps. Isometric identification using map with winding number n=+/- 1 is certainly the simplest imaginable possibility.

See What could 2-D minimal surfaces teach about TGD? or the chapter Zero Energy Ontology.

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

### About the relationship of Kaehler approach to the standard picture

The replacement of the notion of unitary S-matrix with Kähler metric of fermionic state space generalizes the notion of unitarity. The rows of the matrix defined by the contravariant metric are orthogonal to the columns of the covariant metric in the inner product (T○ U)ABbar = TACbar ηCbarDUDBbar, where ηCbarD is flat contravariant Kähler metric of state space. Although the probabilities are complex, their real parts sum up to 1 and the sum of the imaginary parts vanishes.

The counterpart of the optical theorem in TGD framework

The optical theorem generalizes. In the standard form of the optical theorem i(T-T)mm=2Im(T) = TTmm states that the imaginary part of the forward scattering amplitude is proportional to the total scattering rate. Both quantities are physical observables.

In the TGD framework the corresponding statement

TABbarηBbar CABbarTBbar C +TABbarTBbar C=0 .

Note that here one has G= η+ T, where G and T are hermitian matrices. The correspondence with the standard situation would require the definition G= η +iU. The replacement T→ T= iU, where U is antihermitian matrix, gives

One has

i[UABbarηBbarC + ηABbarIBbarC] = UABbarUBbar C .

This statement does not reduce to single condition but gives two separate conditions.

1. The first condition is analogous to Optical Theorem:

Im[ηABbarUCBbar+UABbarηBbarC]=- Re[UABbarUBbar C] = Re[UABbarUCBbar] .

2. Second condition is new and reflects the fact that the probabilities are complex. It is necessary to guarantee that the sum of the probabilities reduces to the sum of their real parts.

Re[ηABbarUCBbar+UABbarηBbarC]= -Im[UABbarUCBbar] .

The challenge would be to find a physical meaning for the imaginary parts of scattering probabilities. This is discussed in (see this). The idea is that the imaginary parts could make themselves visible in a Markov process involving a power of the complex probability matrix.

In the applications of the optical theorem, the analytic properties of the scattering matrix T make it possible to construct the amplitude as a function of mass shell momenta using its discontinuity at the real axis. Indeed, 2Im(T) for the forward scattering amplitude can be identified as the discontinuity Disc(T).

In the recent case, this identification would suggests the generalization

Disc[TABbarηBbarC]= TABbarηBbar CABbarTCBbar .

Therefore covariant and contravariant Kähler metric could be limits of the same analytic function from different sides of the real axis. One assigns the hermitian conjugate of S-matrix to the time reflection. Are covariant and contravariant forms of Kähler metric related by time reversal? Does this mean that T symmetry is violated for a non-flat Kähler metric.

The emergence of QFT type scattering amplitudes at long length scale limit

The basic objection against the proposal for the scattering amplitudes is that they are non-vanishing only at mass shells with m2=n. A detailed analysis of this objection improves the understanding about how the QFT limit of TGD emerges.

1. The restriction to the mass shells replaces cuts of QFT approach with a discrete set of masses. The TGD counterpart of unitarity and optical theorem holds true at the discrete mass shells.
2. The p-adic mass scale for the reaction region is determined by the largest ramified prime RP for the functional composite of polynomials characterizing the Galois singlets participating in the reaction. For large values of ramified prime RP for the reaction region, the p-adic mass scale increases and therefore the momentum resolution improves.
3. For large enough RP below measurement resolution, one cannot distinguish the discrete sequence of poles from a continuum, and it is a good approximation to replace the discrete set of mass shells with a cut. The physical analogy for the discrete set of masses along the real axis is as a set of discrete charges forming a linear structure. When their density becomes high enough, the description as a line charge is appropriate and in 2-D electrostatistics this replaces the discrete set of poles with a cut.
This picture suggests that the QFT type description emerges at the limit when RP becomes very large. This kind of limit is discussed in the article considering the question whether a notion of a polynomial of infinite degree as an iterate of a polynomial makes sense (see this). It was found that the number of the roots is expected to become dense in some region of the real line so that effectively the QFT limit is approached.
1. If the polynomial characterizing the scattering region corresponds to a composite of polynomials participating in the reaction, its degree increases with the number of external particles. At the limit of an infinite number of incoming particles, the polynomial approaches a polynomial of infinite degree. This limit also means an approach to a chaos as a functional iteration of the polynomial defining the space-time surface (see this). In the recent picture, the iteration would correspond to an addition of particles of a given type characterized by a fixed polynomial. Could the characteristic features for the approach of chaos by iteration, say period doubling, be visible in scattering in some situations. Could p-adic length scale hypothesis stating that p-adic primes near powers of two are favored, relate to this.
2. For a large number of identical external particles, the functional composite defining RG involves iteration of polynomials associated with particles of a particular kind, if their number is very large. For instance, the radiation of IR photons and IR gravitons in the reaction increases the degree of RP by adding to P very high iterates of a photonic or gravitonic polynomial.

Gravitons could have a large value of ramified prime as the approximately infinite range of gravitational interaction and the notion of gravitational Planck constant (see this) originally proposed by Nottale suggest. If this is the case, graviton corresponds to a polynomial of very high degree, which increases the p-adic length scale of the reaction region and improves the momentum resolution. If the number of gravitons is large, this large RP appears at very many steps of the SFR cascade.

A connection with dual resonance models

There is an intriguing connection with the dual resonances models discussed already in (see this).

1. The basic idea behind the original Veneziano amplitudes (see this) was Veneziano duality. The 4-particle amplitude of Veneziano was generalized by Yoshiro Nambu, Holber-Beck Nielsen, and Leonard Susskind to N-particle amplitude (see this) based on string picture, and the resulting model was called dual resonance model. The model was forgotten as QCD emerged.
2. Recall that Veneziano duality (1968) was deduced by assuming that scattering amplitude can be described as sum over s-channel resonances or t-channel Regge exchanges and Veneziano duality stated that hadronic scattering amplitudes have a representation as sums over s- or t-channel resonance poles identified as excitations of strings. The sum over exchanges defined by t-channel resonances indeed reduces at larger values of s to Regge form.
3. The resonances have zero width and the imaginary part of the amplitude has a discontinuity only at the resonance poles, which is not consistent with unitarity so that one must force unitarity by hand by an iterative procedure. Further, there were no counterparts for the sum of s-, t-, and u-channel diagrams with continuous cuts in the kinematical regions encountered in QFT approach. What puts bells ringing is the u-channel diagrams would be non-planar and non-planarity is the problem of the twistor Grassmann approach.
It is interesting to compare this picture with the twistor Grassman approach and TGD picture.
1. In the TGD framework, one just picks up the residue of what would be analogous to stringy scattering amplitude at mass shells. In the dual resonance models, one keeps the entire amplitude and encounters problems with the unitarity outside the poles. In the twistor Grassmann approach, one assumes that the amplitudes are completely determined by the singularities whereas in TGD they are the residues at singularities. At the limit of an infinite-fold iterate the amplitudes approach analogs of QFT amplitudes.
2. In the dual resonance model, the sums over s- and t-channel resonances are the same. This guarantees crossing symmetry. An open question is whether this can be the case also in the TGD framework. If this is the case, the continuum limit of the scattering amplitudes should have a close resemblance with string model scattering amplitudes as the M4× CP2 picture having magnetic flux tubes in a crucial role indeed suggests.
3. In dual resonance models, only the cyclic permutations of the external particles are allowed. As found, the same applies in TGD if the scattering event is a cognitive measurement (see this), only the cyclic permutations of the factors of a fixed functional composite are allowed. Non-cyclic permutations would produce the counterparts of non-planar diagrams and the cascade of cognitive state function reductions could not make sense for all polynomials in the superposition simultaneously. Remarkably, in the twistor Grassmann approach just the non-planar diagrams are the problem.
See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Sunday, January 16, 2022

### Bosonic strange metals and oscillatory behavior of magnetoresistance

Strange metals are difficult to understand in the standard paradigm. Linear dependence with respect to temperature instead of quadratic in fermionic liquids is one problem. In the TGD framework, dark particles as ordinary particles with effective Planck constant heff=nh0>h at magnetic flux tubes, explains this: see the blog posting or the article TGD and condensed matter.

In the article "Signatures of a strange metal in a bosonic system" by Yang et al published in Nature, bosonic strange metals are studied instead of fermionic ones. The system can also be superconducting and this seems to be essential.

The linear dependence on magnetoresistance in an external magnetic field B is the second interesting phenomenon.

1. Below the onset of temperature Tc1>Tc, the low-field magneto-resistance varies with a periodic dictated by superconducting flux quantum suggesting that the density of charge carriers varies with this period.
2. What comes to mind is the De Haas-Van Alphen effect in field B (see this).

The magnetic susceptibility of the system varies periodically with the inverse of the magnetic flux Φ = e ∫ BdS defined by extremal orbit of electrons at the Fermi surface in field B. Φ is measured in units defined by elementary flux quantum h/2e.

3. Could spin=0 Cooper pairs be formed from the electrons at the Fermi surface and lead to the De Haas-Van Alphen effect. They would go to the flux tubes of the external magnetic field B with a rate determined by the magnetic flux.

The rate for this highest, when the extremal orbit at the Fermi surface corresponds to a quantized flux. Otherwise, energy is needed to kick the electrons from the Fermi surface to a larger orbit in order to satisfy the flux quantization condition.

Now one considers magnetoresistance rather than susceptibility. The linearity in magnetoresistance suggests that the resistance in the external field is mostly due to magnetoresistance.
1. Could the analog of the De Haas-Van Alphen effect be present so that the density of Cooper pairs as current carriers at "endogenous" magnetic flux tubes has an oscillatory behavior as a function of the external magnetic field B? Could there be a competition for the Cooper pairs between the magnetic fields of flux tubes and external magnetic field B?
2. When the flux Φ for the external B is near the multiple of the elementary flux quantum at extremal orbits at Fermi surface, the formation of spin=0 Cooper pair and transgder to the flux tubes of B would become probable by De-Haas-van Alphen effect. The number of Cooper pairs at "endogenous" flux tubes is therefore reduced and the current therefore reduced.
See the article TGD and condensed matter or the chapter with the same title.

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

## Friday, January 14, 2022

### Why the resistance of strange metals is linear in temperature?

The so-called strange metals are a real troublemaker for condensed matter theorists. This posting was inspired by the most recent findings about strange metals (see this).

Could one understand the somewhat mysterious looking linear high T dependence of the resistivity of strange metals in TGD the framework?

In the TGD based model of high T superconductivity (see this) charge carriers are dark electrons, or rather Cooper pairs of them, at magnetic flux tubes which are effectively 1-D systems. Magnetic flux tubes are much more general aspect of TGD based model of condensed matter (see this).

Could magnetic flux tubes carrying dark matter with heff=nh0> h also explain the resistance of strange metals. I have actually asked this question earlier.

More precisely: Could the effective 1-dimensionality of flux tubes, darkness of charge carriers, and isolation from the rest of condensed matter together explain the finding?

Isolation would mean that only the collisions of dark electrons with each other cause resistance.

One can make a dimensional estimate.

1. Assume that the resistance ρ can be written in the form

ρ= (4π/ω2)/τ= (me/ne e2)/τ .

Here ω is the plasma frequency

ω2= 4πne e2/me.

ne is 3-D electron density.

What happens for 3-D ne in the case of 1-D flux tube? It would seem that ne must be replaced with linear density divided by the transversal area S of the flux tube: ne= (dne/dl)×(1/S).

2. τ is the time spent by the charge carrier in free motion between collisions. Charge carrier is in thermal motion with thermal velocity vth= kT/m . The length Lf of the free path is determined non-thermally. Hence one has

τ= Lf/vth= mLf/kT .

This gives 1/τ= kT/mLf.

3. For the resistivity ρ one obtains

ρ= (me/nee2)kT/mLf,

which indeed depends linearly on T as it does for strange metals.

For m=me, one would have

ρ= kT/(nee2Lf) .

For the TGD based view of condensed matter, see for instance this.

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

## Wednesday, January 12, 2022

### Critical questions related to the number theoretical view about fundamental dynamics

One can pose several critical questions helping to further develop the proposed number theoretic picture.

Is mere recombinatorics enough as fundamental dynamics?

Fundamental dynamics as mere re-combination of free quarks to Galois singlets is attractive in its simplicity but might be an over-simplification. Can quarks really continue with the same momenta in each SSFR and even BSFR?

1. For a given polynomial P, there are several Galois singlets with the same incoming integer-valued total momentum pi. Also quantum superpositions of different Galois singlets with the same incoming momenta pi but fixed quark and antiquark numbers are in principle possible. One must also remember Galois singlet property in spin degrees of freedom.
2. WCW integration corresponds to a summation over polynomials P with a common ramified prime (RP) defining the p-adic prime. For each P of the Galois singlets have different decomposition to quark momenta. One can even consider the possibility that only the total quark number as the difference of quark and antiquark numbers is fixed so that polynomials P in the superposition could correspond to different numbers of quark-antiquark pairs.
3. One can also consider a generalization of Galois confinement by replacing classical Galois singlet property with a Galois-singlet wave function in the product of quark momentum spaces allowing classical Galois non-singlets in the superposition.

Hydrogen atom serves as an illustration: electron at origin would correspond to classical ground state and s-wave correspond to a state invariant under rotations such that the position of electron is not anymore invariant under rotations. The proposal for transition amplitudes remains as such otherwise.

Note however that the basic dynamics at the level of a single polynomial would be recombinatorics for all these options.

General number theoretic picture of scattering

Only the interaction region has been considered hitherto. One must however understand how the interaction region is determined by the 4-surfaces and polynomials associated with incoming Galois singlets. Also the details of the map of p-adic scatting amplitude to a real one must be understood.

The intuitive picture about scattering is as follows.

1. The incoming particle "i" is characterized by p-adic prime pi, which is RP for the corresponding 4-surface in M8. Also the "adelic" option that all RPs characterize the particle, is considered below.
2. The interaction region corresponds to a polynomial P. The integration over WCW corresponds to a sum over several P:s with at least one common RP allowing to map the superposition of amplitudes to real amplitude by canonical identification I: ∑ xnpn→ ∑ xnp-n.

If one gives up the assumption about a shared RP, the real amplitude is obtained by applying I to the amplitudes in the superposition such that RP varies. Mathematically, this is an ugly option.

3. If there are several shared RPs, in the superposition over polynomials P, one can consider an adelic picture. The amplitude would be mapped by I to a product of the real amplitudes associated with the shared RP:s. This brings in mind the adelic theorem stating that rational number is expressible as a product of the inverses of its p-adic norms. The map I indeed generalizes the p-adic norm as a map of p-adics to reals. Could one say that the real scattering amplitude is a product of canonical images of the p-adic amplitudes for the shared RP:s? Witten has proposed this kind of adelic representation of real string vacuum amplitude.

Whether the adelization of the scattering amplitudes in this manner makes sense physically is far from clear. In p-adic thermodynamics one must choose a single p-adic prime p as RP. Sum over ramified primes for mass squared values would give CP2 mass scale if there are small p-adic primes present.

The incoming polynomials Pi should determine a unique polynomial P assignable to the interaction regions as CD to which particles arrive. How?
1. The natural requirement would be that P possess the RPs associated with Pi:s. This can be realized if the condition Pi=0 is satisfied and P is a functional composite of polynomials Pi. All permutations π of 1,...,n are allowed: P= Pi1○ Pi2○ ....Pin with (i1,...in)=(π(1),...,π(n)). P possesses the roots of Pi.

Different permutations π could correspond to different permutations of the incoming particles in the proposal for scattering amplitudes so that the formation of area momenta xi+1= ∑k=1ipk in various orders would corresponds to different orders of functional compositions.

2. Number theoretically, interaction would mean composition of polynomials. I have proposed that so-called cognitive measurements as a model for analysis could be assigned with this kind of interaction (see this and this). The preferred extremal property realized as a simultaneous extremal property for both K\"ahler action and volume action suggests that the classical non-determinism due to singularities as analogs of frames for soap films serves as a classical correlate for quantum non-determinism (see this).
3. If each incoming state "i" corresponds to a superposition of Pi:s with some common RPs, only the RP:s shared by all compositions P from these would appear in the adelic image. If all polynomials Pi are unique (no integration over WCW for incoming particles), the canonical image of the amplitude could be the product over images associated with common RPs.

The simplest option is that a complete localization in WCW occurs for each external state, perhaps as a result of cognitive state preparation and reduction, so that P has the RP:s of Pis as RP:s and adelization could be maximal.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Tuesday, January 11, 2022

### Fundamental quark dynamics as recombinatorics for Galois singlets

The proposal that unitary S-matrix should be replaced with the Kähler metric for the fermionic sector of the state space is extremely attractive. In the sequel the explicit expressions for the scattering probabilities are deduced and are of the same form as in the case of unitary S-matrix. The natural question is whether the notions of virtual and real particles can be geometrized.

Virtual states correspond to Galois non-singlets with momentum which is algebraic integer and real states to Galois singlets with momentum which is ordinary integer. The scattering amplitudes are shown to have basic properties as in QFT. Scattering amplitudes associated with mere re-combinations of quark states to different Galois singlets as in the initial states. Quarks move as free particles.This corresponds to OZI rule and conforms with the assumption that all particles are Galois composites of quarks

One can also ask whether the counterpart of S-matrix has on mass shell virtual states as singularities. This turns out to be the case. Also the analogs of non-planar amplitudes are allowed.

Explicit expressions for scattering probabilities

The proposed identification of scattering probabilities as P(A→B)= gABbargABbar in terms of components of the Kähler metric of the fermionic state space.

Contravariant component gABbar of the metric is obtained as a geometric series ∑n&ge 0 Tn from from the deviation TABbar= gABbarABbar of the covariant metric gABbar from δABbar.

g this is not a diagonal matrix. It is convenient to introduce the notation ZA, A=1,...,n, ZAbar=Zn+k, k=n+1,...,2n. So that the gBbarC corresponds to gk+n,l= δk,l+Tk,l.

and one has

gABbar to gk,l+n= δk,l+T1k,l.

The condition gABbargBbarC= δAC gives

gk,l+ngl+n,m= δkm .

giving

lk,l+T1k,l)(δl,m+Tl,m)

= δk,m + (T1+ T + T1T)km = δk,m ,

which resembles the corresponding condition guaranteeing unitarity. The condition gives

T1= -T/(1+T)>=- ∑n>1 ((-1)nTn .

The expression for PA→B reads as

P(A→B)=gABbargBbarA

=[1-T/(1+T)+T -(T/(1+T))ABT]AB .

It is instructive to compare the situation with unitary S-matrix S=1+T. Unitarity condition SS= 1 gives

T=-T/(1+T) ,

and

P(A→B)=δAB+ TAB+TAB+ TABTAB= [δAB-(T/(1+T))AB+TAB -(T/(1+T))ABTAB .

The formula is the same as in the case of Kähler metric.

Do the notions of virtual state, singularity and resonance have counterparts?

Is the proposal physically acceptable? Does the approach allow to formulate the notions of virtual state, singularity and resonance, which are central for the standard approach?

1. The notion of virtual state plays a key role in the standard approach. On-mass-shell internal lines correspond to singularities of S-matrix and in a twistor approach for N=4 SUSY, they seem to be enough to generate the full scattering amplitudes.

If only off-mass-shell scattering amplitudes between on- mass-shell states are allowed, one can argue that only the singularities are allowed, which is not enough.

2. Resonance should correspond to the factorization of S-matrix at resonance, when the intermediate virtual state reduces to an on-mass-shell state. Can the approach based on Kähler metric allow this kind of factorization if the building brick of the scattering amplitudes as the deviation of the covariant Kähler metric from the unit matrix δABbar is the basic building bricks and defined between on mass shell states?

Note that in the dual resonance model, the scattering amplitude is some over contribution of resonances and I have proposed that a proper generalization of this picture could make sense in the TGD framework.

The basic question concerns the number theoretical identification of on-mass-shell and off-mass-shell states.
1. Galois singlets with integer valued momentum components are the natural identification for on-mass-shell states. Galois non-singlet would be off-mass-shell state naturally having complex quark masses and momentum components as algebraic integers.

Virtual states could be arbitrary states without any restriction to the components of quark momentum except that they are in the extension of rationals and the condition coming from momentum conservation, which forces intermediate states to be Galois singlets or products of them.

Therefore momentum conservation allows virtual states as on mass shell states, that is intermediate states, which are Galois singlets but consist of Galois non-singlets identified as off-mass-shell lines. The construction of bound states formed from Galois non-singlets would indeed take place in this way.

2. The expansion of the contravariant part of the scattering matrix T1 = T/(1+T) appearing in the probability

PA→B=gABbargABbar

=[1-T/(1+T)]AB+TAB -[T/(1+T)]ABT]AB .

would give a series of analogs of diagrams in which Galois singlets of intermediate states are deformed to non-singlets states.

3. Singularities and resonances would correspond to the reduction of an intermediate state to a product of Galois singlets.
What about the planarity condition in TGD?

The simplest proposal inspired by the experience with the twistor amplitudes is that only planar polygon diagrams are possible since otherwise the area momenta are not well-defined. In the TGD framework, there is no obvious reason for not allowing diagrams involving permutations of external momenta with positive energies resp. negative energies since the area momenta xi+1= ∑k=1i pk are well-defined irrespective of the order. The only manner to uniquely order the Galois singlets as incoming states is with respect to their mass squared values given by integers.

Generalized OZI rule

In TGD, only quarks are fundamental particles and all elementary particles and actually all physical states in the fermionic sector are composites of them. This implies that quark and antiquark numbers are separately conserved in the scattering diagrams and the particle reaction only means the-arrangement of the quarks to a new set of Galois singlets.

At the level of quarks, the scattering would be completely trivial, which looks strange. One would obtain a product of quark propagators connecting the points at mass shells with opposite energies plus entanglement coefficients arranging them at positive and negative energy light-cones to groups which are Galois singlets.

This is completely analogous to the OZI role. In QCD it is of course violated by generation of gluons decaying to quark pairs. In TGD, gauge bosons are also quark pairs so that there is no problem of principle.

See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

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

## Friday, January 07, 2022

### Horizontal Gene Transfer by Remote Replication?

This chapter was inspired by the discovery that a horizontal gene transfer (HGT) between eukaryotes is possible. The belief has been that HGT is possible only from prokaryotes to prokaryotes or eukaryotes. The basic obstacles are that the host DNA is within the cell nucleus and that DNA is tightly bound to chromosomes. The transfer should also occur to germ cells in order to have a lasting effect.

The case considered is HGT of antifreezing gene (AFG) from herring to smelt, which could have occurred during simultaneous spawning of herring and smelt in the same area. The AFT of herring associated with a transposon could have somehow attached to the sperm cell of the smelt and carried by it to the egg of the smelt. Vector carrying AFT to the sperm cell of smelt is needed and there are only guesses about what it might be.

That HGT however occurs, justifies a heretical question. Could it be only the genetic information, which is transferred and used to construct DNA in the host as a kind of remote replication analogous to quantum transportation? The findings of Gariaev and Montagnier indeed suggest remote replication and TGD provides a new physics model for it.

See the article Horizontal Gene Transfer by Remote Replication? or the chapter with the same title.

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