How to construct scattering amplitudes?

Lubos Motl ) told about two new hep-th papers, by Pate, Raclariu, and Strominger (see this) and by Nandan, Schreiber, Volovich, Zlotnikov (see this) related to a new approach to scattering amplitudes based on the replacement of the quantum numbers associated with Poincare group labelling particles appearing in the scattering amplitudes with quantum numbers associated with the representations of Lorentz group.

Why I got interested was that in zero energy ontology (ZEO) the key object is causal diamond (CD) defined as intersection of future and past directed M⁴ light-cones with points replaced with CP₂. Space-time surfaces are inside CD and have ends at its light-like boundaries. The Lorentz symmetries associated with the boundaries of CD could be more natural than Poincare symmetry, which would emerge in the integration over the positions of CDs of external particles arriving to the opposite light-like boundaries of the big CD defining the scattering region where preferred extremal describing the scattering event resides.

I did my best to understand the articles and - of course relate these ideas to TGD, where the construction of scattering amplitudes is the basic challenge. My technical skills are too limited for to meet this challenge at the level of explicit formulas but I can try to understand the physics and mathematics brought in by TGD.

While playing with more or less crazy and short-lived ideas inspired by the reading of the articles I finally realized that there is perhaps no point in starting from quantum field theories. TGD is not quantum field theory and I must start from TGD itself.

In TGD framework the picture inspired by adelic physics is roughly following.

1. Cognitive representations realizing number theoretic universality of adelic physics consist of points of imbedding space with coordinates in the extension of rationals. The number of points is typically finite. Cognitive representation should contain as subset the points associated with n-point functions, which are essentially correlation functions.
   

   Fundamental fermions are building bricks of elementary particles, and a good guess is that fundamental fermions correspond to singular points for which the action of subgroup of Galois group of extension is trivial so that several points collapse together.
   
   

2. One must sum over the orbits of a subgroup S_D of symplectic group of light-cone boundary acting as isometries of both boundaries of CD. S_D consists of isometries with parameters in the extension of rationals defining the adele. All orbits needed to represent the pairs of initial and final 3-surfaces at the boundaries of CD allowed by the action principle must be realized so that single orbit very probably is not enough.
   
   
3. Correlations code for the quantum dynamics. In quantum field theories quantum fluctuations of fields at distinct points of space-time correlate and give rise to n-point functions expressible in terms of propagators and vertices: massless fields and conformal fields define the basic example. Operator algebra or path integral describes them mathematically.
   

   In TGD correlations between imbedding space points belonging to the space-time surface result from classical deterministic dynamics: the points of 3-surface at opposite boundaries of CD are not independent.
   

   This dynamics is non-linear geometric analog for the dynamics of massless fields: space-time sheets as preferred extremals are indeed minimal surfaces with string world sheets appearing as singularities. Minimal surface property is forced by the volume action implied by the twistor lift and having interpretation in terms of cosmological constant. The correlation between points at the same boundary of CD are expected to be independent since these 3-surfaces chosen rather freely as analogs of boundary values for fields.
   

   Fermionic dynamics governed by modified Dirac action is dictated completely by super-symplectic and super-conformal symmetries. Second quantization of fermions at space-time level is necessary to realized WCW spinor structure: WCW gamma matrices are linear combinations of fermionic oscillator operators.
   
   

4. This suggests that the attempts to guess the conformal field theory producing the correlation functions makes things much more complex than they actually are. It should be possible to understand how these correlations emerge from the classical dynamics of space-time surfaces.
   
   

As the first brave guess one could try to calculate directly the correlations of spinor harmonics of imbedding space assigned with these points.

1. Sum over the symplectic orbits of cognitive representations must be involved as also vacuum expectation values in the fermionic sector for fermionic fields which must appear in vertices for external particles. At the level of cognitive representations anti-commutators for oscillator operators involve Kronecker deltas so that one has discretized variant of second quantization.
   
   
2. This could be achieved by expanding the restriction Ψ^A_|X³ of the imbedding space harmonic Ψ^A restricted to 3-surface at end of space-time surface as sum of modes Ψ_n of the induced spinor field. This would be counterpart for the induction procedure. One can assign to singular points bilinear of type Ψbar^A_|X³ D^↔Ψ. D denotes modified Dirac operator. This gives to propagators connecting fermions vertices at the opposite ends of space-time surface.
   
   
3. A more concrete picture must rely on a concrete model for elementary particles. Elementary particles have as building bricks pair of wormhole contacts with fermion lines at the light-like orbits of the throats at which the signature of the metric changes from Minkowskian to Euclidian. Particle is necessarily a pair of two wormhole contacts and flux tube connects them at both space-time sheets and forms a closed flux tube carrying monopole flux.
   

   All particles consist of fundamental fermions and anti-fermions: for instance gauge bosons involve fermion and anti-fermion responsible for the quantum numbers at the opposite throats of second wormhole contact. Second wormhole contact involves neutrino pair neutralizing electroweak isospin in scales longer than the size of the flux tube structure.
   
   

4. The topological counterpart of 3-vertex appearing in Feynman diagram corresponds to a replication of this kind of 3-surface highly analogous to bio-replication. In replication vertex, there is no singularity of 3-surface analogous to that appearing in the vertices of stringy diagrams but space-time surface is singular just like 1-D manifold is singular for at vertex of Feynman diagram.
   

   These singular replicating 3-surfaces and the partonic 2-surfaces give rise to the counterparts of interaction vertices. Fermionic 4-vertex is impossible and fermion lines can only be re-shared between outgoing partonic orbits. This is however not enough as will be found. It will be found that also the creation of fermion pair as effective turning of fermion lines entering along "upper" wormhole throat and turning back at Euclidian wormhole throat and continuing along the orbit of "lower" wormhole throat must be possible.
   
   

To see how this conclusion emerges consider the following problem. One should obtain also emission of bosons identified as fermion pairs from fermion line. One has incoming fermion and outgoing fermion and fermion pair describing boson which represents gauge boson or graviton with vanishing B and L. Fermionic 4-vertex is not allowed since this would bring in divergences.

1. The appearance of a sub-CD assignable to the partonic 2-surface is possible but does not solve the problem considered. There would be incoming fermion line at lower boundary and 1 fermion line and fermion and anti-fermion line associated with the boson at the "upper" boundary of CD. There would be non-locality in the scale of the partonic 2-surface and sub-CD meaning that the lines can end to vacuum. Now one would encounter the same difficulty but only in shorter scale.
   
   
2. Could one say that fermion line turns backwards in time? A line turning back could be described as an annihilation of fermion pair to vacuum carrying classical gauge field, which is standard process. In QFT picture this would be achieved if a bilinear Ψ DΨ is allowed in the vertex where annihilation takes place. Not in TGD: fermionic action vanishes identically by field equations expressing essentially the conservation of fermion current and various super currents obtained as contractions fermion field with modes.
   

   Could fermion-antifermion pair creation occur at singular points associated with partonic surfaces representing the turning of fermion line backwards in time. This looks still too singular.
   

   Rather, the turning backwards in time should mean that a fermion line arriving from future along the orbit of "upper" throat (say) goes through Euclidian wormhole throat and continues along the orbit of "lower" throat back to future than making discontinuous turn-around. Euclidian regions of space-time surface representing one key distinction between GRT and TGD would thus be absolutely essential for the generalized scattering diagrams. An exchange of momentum with classical field would be Feynman diagrammatic manner to say this.
   

   New oscillator operator pairs emerge at the partonic vertices and would correspond to the above described turn-around for fermion line at wormhole contact. Fermion pairs present at the "lower" boundary of CD could also disappear.
   
   

3. The anti-commutation relations fermions are modified due to the presence of vacuum gauge fields so that the anti-commutator of fermionic creation operators a^†_m and anti-fermionic creation operators b^†_n is non-vanishing. A proper formulation of the fermionic anti-commutation relations at the ends of space-time surface is needed. One can imagine that although standard anti-commutation relations at the lower end of space-time surface hold true, the time evolution of Ψ in the presence of vacuum gauge potentials implies that the vacuum expectations < vac| a^†_mb^†_n|vac > are non-vanishing. This would require that for instance b^†_n and a_n are mixed.
   
   

There are still questions to be answered.

1. Is the first guess enough? It is not as becomes clear after a thought about the continuum limit. The WCW degrees of freedom are described at continuum limit in terms of super-symplectic algebra (SSA) acting on ground state are neglected. Imbedding space spinor modes characterizee only the ground staes of these representations. These degrees of freedom are essential already in elementary particle physics.
   

   Sub-algebra SSA_m of SSA with conformal weights coming as m-multiples of those of SSA and its commutator with SSA annihilate the physical states, and one obtains a hierarchy. How to describe these states in the discretization? The natural possibility are the representations of S_D such that (S_D)_m and the subgroup generated by the commutator algebra are represented trivially. One has non-trivial S_Dm representations at both ends of WCW such that the action of S_D on the tensor product acts trivially.
   
   There are also fermionic degrees of freedom. The challenge is to identify among other things WCW gamma matrices as fermionic super charges and it would be nice if all charges were Noether charges. The simplest guess is that the algebra generated by fermionic Noether charges Q^A for symplectic transformations h^k→ h^k+j^Ak assumed to induce isometries of WCW and Noether supercharges Q_n and their conjugates for the shifts Ψ→ Ψ+ε u_n, where u_n is a solution of the modified Dirac equation, is enough.
   

   The commutators Γ^A_n=[Q^A,Q_n] are super-charges labelled by A and n. One would like to identify them as gamma matrices of WCW. The problem is that they are labelled by (A,n) and isometry generators are labelled by A only. There should be one-one correspondence. Do all supercharges Γ^A_n except Γ^A₀ corresponding to u₀=constant annihilate the physical state so that one would have 1-1 correspondence. This would correspond to what happens quite generally in super-conformal algebras.
   

   The generators of this fermionic algebra could be used to generate more general states. One should also construct the discretized versions of the generators as sums over points of the cognitive representation at the ends of space-time surface. Note that this requires tangent space data.
   
   

2. What about the conservation of four-momentum and other conservation laws? This can be handled by quantum classical correspondence (QCC). The momentum and color labels defined by fermionic quantum numbers in Cartan algebra can be assumed to be equal to the corresponding classical Noether charges for particle-like space-time surfaces entering to CD. The technical problem is that if one knows only the discretization - even with tangent space data - one does not know the values of these charges! It might be that M⁸-H correspondence in which M⁸side fixes space-time surfaces as roots for real or imaginary parts of octonionic polynomials from the data at discrete set of points is needed.
   
   
3. ZEO means deviations from ordinary description. S_D invariance of zero energy state forces sum over the 4-surfaces of the orbit with identical coefficients. Symplectic invariance implies time-like entanglement. One can describe this in terms of hermitian square root Ψ of density matrix satisfying Ψ †Ψ =ρ. The coefficients of different orbits need not be same and allows description in terms of dynamical density matrix. If there is Yangian symmetry also this entanglement is analogous to the entanglement due to statistics.
   

   Surprisingly - and somewhat disappointingly after decades of attempts to understand unitarity in TGD - unitarity is trivial in ZEO since state basis is defined essentially by the rows of matrices and orthogonality conditions their orthogonality and therefore unitarity. More concretely, for single state at the passive end state function normalization to unity defined by inner product as sum over 3-surfaces at active end would give conservation of probability. Orthogonality of the state basis with inner product as sum over surfaces passive boundary gives orthogonality for the coefficients defining rows of a matrix and therefore unitarity. In the case that single orbit or even several of them defines the states one obtains the same result.
   

   What then guarantees the orthogonality of zero energy states? In ordinary quantum mechanics the property of being eigenstates of some hermitian operator guarantees orthogonality. In TGD zero energy states would be solutions of the analog of massless Dirac equation in WCW consisting of pairs of 3-surfaces with members at the ends of preferred extremals inside CD. This generalizes Super Virosoro conditions of superconformal theories and would provide the orthonormal state basis.
   
   

The outcome would be amazingly simple. There would be no propagators, no vertices, just spinor harmonics of imbedding assigned with these n=n₁+n₂ points at the boundaries of CD, and summation over the orbits of the symplectic group. All these mathematical objects would emerge from classical dynamics. The sum over the orbits for chosen spinor harmonics would produce n-point functions, vertices and propagators. It is difficult to imagine anything simpler and quantum classical correspondence would be complete.

See the article Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

About list of unsolved problems in physics

Thanks to Savyasanchi Ghose for posting to FB a long list about unsolved problems of physics . I descided to write about TGD based answers to the questions. I have done some re-organization to the questions since they contain so many sub-questions. Also some re-ordering.

A. Questions related to time, entropy, causality, and T violation

A1. Arrow of time (e.g. entropy's arrow of time): Why does time have a direction?

Understanding of the arrow of time requires a new ontology. I have talked a lot about zero energy ontology (ZEO) which also solves the basic problem of quantum measurement theory and allows free will without conflict with the deterministic laws of physics such as field equations: in TGD dynamics of space-time surfaces and modified Dirac equation.

ZEO allows to distinguish between geometric time and experienced time and to understand their correlation. One could speak about quantum arrow inducing thermodynamical arrow of time and arrow of psychological time. Thermodynamical arrow basically reduces to a generation of entanglement entropy accompanied by entanglement negentropy associated with p-adic degrees of freedom assignable to cognition. Cognitive evolution as a "positive" aspect of the arrow of time.

That there is only single arrow of time is only a belief. There are many examples, where this is not the case: the spontaneous build up of molecules from their their building bricks and phase conjugate laser beams for which second law holds true in reverse time direction.

ZEO predicts both arrows of time and in living system the arrow of time at control level (magnetic body carrying dark matter in TGD sense) can change. TGD based view about memory and motor actions relies on this. Re-incarnation in very general sense is one quite radical prediction.

A2. Questions about entropy

Why did the universe have such low entropy in the past, and time correlates with the universal (but not local) increase in entropy, from the past and to the future, according to the second law of thermodynamics.

In TGD cosmology radiation dominated era is preceded by a period during which normal space-time with 4-D M⁴ projection did not exist. Space-time surfaces were cosmic strings with 2-D M⁴ projection. A phase transition occurred. The cosmic strings are unstable against developing 4-D M⁴ projection. The projection started to increase and produced flux tubes and the magnetic and volume energy of flux tubes decayed to ordinary particles (possibly dark in TGD sense). This is TGD counterpart of inflation. This process still continues and especially interesting are phase transitions reducing length scale dependent cosmological constant.

In primordial phase particles were absent and their contribution to entropy was not present. This might explain why the entropy was low. How much entropy did string degrees of freedom carry? If this contribution was small then one can argue that the very early Universe had low entropy. The fact that the amount of string energy per volume went to zero like cosmic time a suggests that also the entropy per unit volume went to zero.

A3. Questions about causality

Are there exceptions to the principle of causality? Is there a single possible past? Is the present moment physically distinct from the past and future, or is it merely an emergent property of consciousness? What links the quantum arrow of time to the thermodynamic arrow?

There are two causations: causality of field equations which is not violate and causality of free will associated with state
function reduction in ZEO. The two causalities are consistent since act of free will does not affect single time evolution but replaces superposition of deterministic time evolutions with a new one. "Now" is associated with experienced time, not with geometric time With respect to geometric time states are superpositions of classical time evolutions- preferred extremals - inside causal diamond defining what one might call perceptive field of self.

A4. Questions about CP and T violation

Why are CP violations observed in certain weak force decays, but not elsewhere? Are CP violations somehow a product of the Second Law of Thermodynamics, or are they a separate arrow of time?

The violation of T symmetry implied by CP violation must be distinguished from the generation of arrow of time and I would not speak about this as quantum arrow. T violation means that the for conscious entities with opposite arrows of time is slightly different already at the geometric level- dynamics of space-time surfaces in TGD. The analog of Kahler action in M⁴ degrees of freedom forced by twistor lift of TGD implies this. CP/T violation induces also matter antimatter asymmetry, which is not a small effect at all. Twistor lift of TGD predicts small CP breaking and the mechanism generating matter antimatter asymmetry. The decay of cosmic string to ordinary matter produces slightly more matter than antimatter outside strings and after annihilation only matter remains.

B. Interpretation of quantum mechanics

How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse or quantum decoherence, give rise to the reality we perceive?

Another way of stating this question regards the measurement problem: What constitutes a "measurement" which apparently causes the wave function to collapse into a definite state? Unlike classical physical processes, some quantum mechanical processes (such as quantum teleportation arising from quantum entanglement) cannot be simultaneously "local", "causal", and "real", but it is not obvious which of these properties must be sacrificed,[5] or if an attempt to describe quantum mechanical processes in these senses is a category error such that a proper understanding of quantum mechanics would render the question meaningless.

Answer: No "interpretation" of QM is needed in ZEO. What is new that quantum measurement theory extends to a theory of consciousness describing also observer as "self" and as part of physical system rather than outsider. State function reduction is completely universal process rather than something performed only by human observer.

We perceive the world as classical because the very feature of state function reduction as a basic building of conscious experience makes the world classical in degrees of freedom about which it give information. Quantum measurement is always an answer to a question and state function reduction selects one option.

State function reduction in the sense of ZEO involves two new elements.

1. The poorly undestood weak measurements assumes also in standard QM corresponds to "small" state function (SSR) essentially as time localization (measurement of clock time): the time evolution of self is sequence of SSRs. The members of state pairs defining zero energy state that are associated at active, shifting boundary of CD changes and gives rise to sensory input and all induced by it. The members of state pairs at passive boundary does not change and this corresponds to generalize Zeno effect. The states remain unentangled and give rise to permanent part of self.
   
   
2. "Big" reduction (BSR) corresponds to ordinary state function reduction and changes the arrow of time for self. At elementary particle level the duration of single lifetime for self is very short. At our level situation is different. The larger the size scale of quantum system, the longer the duration of life cycle. Many-sheeted space-time and hierarchy of CDs is required to understand self hierarchy geometrically.
   

   Space-time locality in TGD is replaced with locality in the world of classical worlds (WCW). There is no spooky action. Entangled states have flux tubes connecting them as correlates of entanglement: they form single space-time surface essentially, single point of WCW. Apart from state fuction reduction quantum TGD is purely classical: WCW spinor fields obey formally single particle Dirac equation and there is no second quantization at this level.
   
   

3. In ZEO the quantum universe is re-created in every SR and this occurs in all scales corresponding scale hierarchy of CDs. The is no unique past and geometric past is also re-created to some degree.
   
   

C. Questions about Grand Unification Theory/Theory of everything

C.1 Is there a theory which explains the values of all fundamental physical constants?

Answer: In TGD there the fundamental constants are CP₂ size scale which is not coupling parameter but provides universal unit of length. Kaehler couplings strength analogous to critical temperature is fundamental dimensionless constant. Its values are fixed as analogs of critical temperature. Quantum criticality determines the values of all coupling parameters. Basically the coupling constants should be expressible in terms number theoretical information. For instance, the value of effective Planck constant h_eff=nh₀ would correspond to the dimension of extension of rationals characterizing adele. I have made a concrete proposal for the discrete coupling constant evolution.

C2. Is there a theory which explains why the gauge groups of the standard model are as they are

Answer: GUT approach is given up in TGD as too naive and "too easy". It leads to proton instability, predicts desert, and is untestable and l there is not a single piece of evidence for a larger gauge group.

To me the correct question is "Why standard model gauge group is so special and how it emerges". There are 3 manners for this in TGD.

1. The construction of WCW geometry requires maximal isometry group since WCW is infinite-D. Already in the case of loops spaces this is required. Hence one expects that TGD is highly unique from the condition that it exists.
   
   
2. CP₂ codes for the standard model symmetries.
   
   
3. M⁴ and CP₂ are also the only 4-D spaces having twistor space with Kahler structure so that in TGD M⁴xCP₂ is completely unique.
   
   
4. M⁴ and CP₂ are also number theoretically unique. SU(3) as isometry group is also automorphism group of octonions. In M⁸-H duality at the octonionic M⁸ level provides purely algebraic description of space-time surfaces. Algebraic equations instead of partial differential equations for space-time surface as in M⁴xCP₂.
   

   M⁸-H duality maps quaternionic algebraic surfaces in octonionic M⁸ to preferred extremals in H. Dynamics at M⁸ level states that space-time surfaces are associative and thus quaternionic and therefore 4-D. 2-D string world sheets and 1-D strings appear also as complex (commutative) and real surfaces.
   
   

C3. Is there a theory explaining why observed spacetime has 3 spatial dimensions and 1 temporal dimension?

1+3 signature is completely unique in TGD framework. For 4-D space surfaces light-like 3-surfaces identified as boundaries between Euclidian and Minkowski regions of space-time surface have extended KAc-Moody symmetry since light-like 3-surface is metrically 2-D. In other dimensions one does not obtain this. Also the light-cone boundary of M⁴ (boundaries of CD) are light-like and this gives rise to infinite-D maximal isometry group of WCW as consisting of symplectic transformations.

C4. Questions about dimensionless physical constants

At the present time, the values of the dimensionless physical constants cannot be calculated; they are determined only by physical measurement.[8][9] What is the minimum number of dimensionless physical constants from which all other dimensionless physical constants can be derived? Are dimensional physical constants necessary at all?

Answer: Basic dimensionless coupling constant in TGD is Kaehler coupling strength analogous to critical temperature. Its possible values are determined by quatum criticality condition but I cannot really calculated yet. Other dimensionless coupling strengths are derived from this. I have proposed general scenario for the evolution of Kaehler coupling constant having length scale dependent cosmological constant as evolution parameter.

CP₂ length R is the only dimensional parameter. It is not coupling constant, however. It defines universal unit for length measurement and p-adic length scales are derived units. Note that in string models dimensional coupling has no geometric interpretation. Cosmological constant is generated in twistor lift dynamically and is scale dependent. In short scales approaching CP₂ length it is very large but in long scales very small since it decreases like the inverse of p-adic length scale squared. Cosmological constant replaces cuffoff length in discrete coupling constant evolution.

CP₂ length takes the role of Planck length and is about 10³.5 times longer. Newton' constant is identified as G= R²/ℏ₁. ℏ₁ =n₁ℏ. This assumes that the n= ℏ_eff/ℏ₀ decomposes to n= n₁× n₂ meaning that space-time surface corresponds to n₁-fold covering of M⁴ (n₁ valuedness of CP₂ coordinates) and n₂-fold covering over CP₂ n₂ copies of n₁ valued regions which could look like n₁ flux tubes in parallel. G is predicted to have spectrum and its measurements indeed give values, which differ "too much". I wrote about this a blog posting and article: see this.

C5. Do "fundamental physical constants" vary over time?

Answer: In TGD fundamental constants do not vary over time. They vary as a function of the scale of the space-time surface. p-Adic length scales are fundamental and the hierarchy of Planck constants assigns a scale hierarchy to each p-adic scale as its multiples n= h_eff/h₀.

D. Questions about particles

Are any of the fundamental particles in the standard model of particle physics actually composite particles too tightly bound to observe as such at current experimental energies? Are there fundamental particles that have not yet been observed, and, if so, which ones are they and what are their properties? Are there unobserved fundamental forces?

Answer: Many-sheeted space-time is crucial element. This is completely new as compated to GUTS and allows to avoid the fine tuning problems and understand particle mass scales number theoretically. For instance, electron corresponds to Mersenne prime M₁27 which is the smallest Mersenne not completely super-astrophysical: therefore electron is the lightest stable elementary particle.

In TGD all particles - also bosons - are bound states of fundamental fermions that correspond to spinor fields of H=M⁴xCP₂. They have electroweak quantum numbers and spin of ordinary elementary fermions. Color is not spinlike quantum number but CP₂ partial wave and baryon and lepton number correpond to different H-chiralities.

p-Adic fractality of TGD Universe predicts entire scaled variants of hadronic and electroweak physics and the are pieces of evidence for M₈9 hadron physics for which hadrons would have mass scale 512 times that of ordinary hadron physics: two handfull of ordinary mesons appear as bumps with predicted mass. These findings have been forgotten since they do not fit to SUSY predictions. Lubos Motl has been enthusiastic about them as SUSY Higges but has suffered disappoint every time! My role in the game has been took particle data tables and identify the meson!

TGD predicts only standard model forces but there could be scaled up versions of them.

E. Questions about physical information

E1. Are there physical phenomena, such as wave function collapse or black holes, which irrevocably destroy information about their prior states?

TGD State function is definitely real process. The self is a system hopping around in Platonia - state space formed by WCW spinor modes defining fermionic Fock states depending on space-time surface - and learning about it by coding information as memories to its quantum state. Adelic physics makes this concrete by providing entanglement negentropies as a measure for conscious information and by predicting evolution as increase of dimension of extension of rationals identied as =h_eff/h₀.

How is quantum information stored as a state of a quantum system? Does state function reduction destroy information? Depends on what kind of information one is talking about. State function reduction answers question and therefore produces conscious information: observer knows the state of the system. On the other hand, when performed for ensemble it generates entropy but this is loss of information about ensemble.

E2. How is quantum information stored as a state of a quantum system?

Conscious information has negentropic entanglement assignable to p-adic degrees of freedom describing cognition as a correlate. It is measured by p-adic negentropy which is just the analog of Shannon entropy with logarithms of propabilities replaced with logarithms of their p-adic norms. The unavoidable increase of dimenaion of algebraic extension of rationals means increase of maximal negentropy and therefore evolution. The difference for the sum of p-adic entanglement negentropies and real entanglement entropy tends to grow and is zero for rational number based physics at the lowest level of evolutionary hierarchy. This also explains why evolution tends to produce entropy.

G. Questions about fine-tuned Universe

The values of the fundamental physical constants are in a narrow range necessary to support carbon-based life.[10][11][12] Is this because there exist other universes with different constants, or are our universe’s constants the result of chance, or some other factor or process?

Answer: In TGD the Universe fine-tunes itself: evolution is the manifestation of this process. In adelic physics predicts the dimension n of extension of rationals is bound to increase in statistical sense since there is infinite number of extensions larger than given value n and only finite number them with smaller dimension. The increase of n=h_eff/h₀ implies increase of scale of quantum coherence and algebraic complexity increases cognitive resources and the maximal entanglement negentropy increases.

In ZEO state functions replace the superposition of space-time surfaces with a new one so that this process became possible.

H. Questions about quantum field theory

H1. Is it possible to construct, in the mathematically rigorous framework of algebraic QFT, a theory in 4-dimensional spacetime that includes interactions and does not resort to perturbative methods?

Answer: I do not believe that QFT is enough. Pointlike particles must be replaced with extended particles and the replacement with 3-surfaces means unification of the notions of particle and 3-space.

In TGD framework number theoretical universality more or less forces the conclusion that loop corrections must vanish since scattering amplitudes must be rational functions of momenta. Therefore scattering amplitudes must reduce to sums over finite number of terms. For twistor lift this implies that scattering amplitudes are sums over resonances: string models emerged from dual resonance models assuming this. Discrete mass spectrum is implied and would be due to the gravitational biding.

Coupling constant evolution discretizes and coupling constants are analogous to critical temperatures. TGD can be indeed regarded as a square root of thermodynamics.

H2. Given an arbitrary compact gauge group, does a non-trivial quantum Yang-Mills theory with a finite mass gap exist?

This problem is also listed as one of the Millennium Prize Problems in mathematics.

Answer: I would not have added this to a list of millenium problems. First of all, I do not believe that gauge theory can be more than effective description in long length scales. There is a simple argument for thow standard model emerges from the many-sheeted space-time of TGD.

YM theory suggests continuum mass spectrum contrary to experimental facts. The understanding of mass requires a scale and YM theories have no scale. Scales emerges by the replacement of point-like particle with extended objects so that YM theories are very probably not enough to explain mass gap. In TGD this object is 3-D. The notions of particle and space are unified. p-Adic length scale hypothesis allows to understand the emergence of scales quantitatively. This indeed leads to p-adic mass calculations which are very successful. This however requires bringing cognition to the realm of described phenomena.

In TGD variant of twistor approach masslessness in 4-D sense is replaced with masslessness in 8-D sense. Since particle massless in 8-D sense can be massive in 4-D sense this allows to overcome the basic problems of gauge theories - also twistor Grassmann approach - related to mass gap. But this requires space-time as 4-surface and particles are small space-times.

Fluctuations of Newton's constant in sub-millimeter scales as evidence for TGD

Sabine Hossenfelder had a posting about rather interesting article related to possible new gravitational physics appearing in sub-millimeter length scales. Washington group has reported evidence for the variation of the effective value of Newton's constant in sub-millimeter scales. The proposed interpretation in terms of spatially oscillating variation of the effective value of G discussed in the article "Hints of Modified Gravity in Cosmos and in the Lab?" does not look convincing. The reader is encouraged to look for the figure 19 of this article about the observed anomalous torque claimed to have interpretation in terms of spatial oscillations of the gravitational potential. The fit is however far from convincing and several wavelength minimizing the deviation of the fit are possible. A more convincing intepretation is in terms of fluctuations of Newton's constant not predicted by standard modifications of General Relativity. These fluctuations have been observed allready earlier seem to be larger than allowed by measurement resolution.

TGD suggest an interpretation in terms of fluctuations in the value of G. The fluctuations are predicted to be make themselves visible in p-adic lengths scales assignable to the density of vacuum energy. The correspondence between these relatively short p-adic length scales and the hierarchy of cosmological p-adic length scales defined by length scale dependent cosmological constant Λ suggests that gravitational physics at short and long length scales reflect each other. Biologically interesting p-adic length scales would be of special interest quantum gravitationally.

See the article Fluctuations of Newton's constant in sub-millimeter scales as evidence for TGD or the chapter About the Nottale's formula for h_gr and the possibility that Planck length l_P and CP₂ length R are identical of "Physics in Many-sheeted space-time".

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

Articles and other material related to TGD.

Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals

Number theorist Minhyong Kim has speculated about very interesting general connection between number theory and physics (see this). The reading of a popular article about Kim's work revealed that number theoretic vision about physics provided by TGD has led to a very similar ideas and suggests a concrete realization of Kim's ideas (see this). The identification of points of algebraic surface with coordinates, which are rational or in extension of rationals, gives rise to what one can call identification problem. In TGD framework the imbedding space coordinates for points of space-time surface belonging to the extension of rationals defining the adelic physics in question are common to reals and all extensions of p-adics induced by the extension. These points define what I call cognitive representation, whose construction means solving of the identification problem.

Cognitive representation defines discretized coordinates for a point of "world of classical worlds" (WCW) taking the role of the space of spaces in Kim's approach. The symmetries of this space are proposed by Kim to help to solve the identification problem. The maximal isometries of WCW necessary for the existence of its K\"ahler geometry provide symmetries identifiable as symplectic symmetries. The discrete subgroup respecting extension of rationals acts as symmetries of cognitive representations of space-time surfaces in WCW, and one can identify symplectic invariants characterizing the space-time surfaces at the orbits of the symplectic group.

This picture could be applied to the construction of scattering amplitudes with finite cognitive precision in terms of cognitive representations and their orbits under subgroup S_D of symplectic group respecting the extension of rationals defining the adele. One could pose to S_D the additional condition that it leaves the value of action invariant: call this group S_D,S: this would define what I have called micro-canonical ensemble (MCE).

The obvious question is whether the simplest zero energy states could correspond to single orbit of S_D or whether several orbits are required. For the more complex option zero energy states would be superposition of states corresponding to several orbits of S_D with coefficients constructed of symplectic invariants. The following arguments lead to the conclusion that MCE and single orbit orbit option are non-realistic, and raise the question whether the orbits of S_D could combine to an orbit of its Yangian analog.

See the article Scattering amplitudes and orbits of cognitive representations under subgroup of symplectic group respecting the extension of rationals or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

Secret Link Uncovered Between Pure Math and Physics

I learned about a possible existence of a very interesting link between pure mathematics and physics (see this). The article told about ideas of number theorist Minhyong Kim working at the University of Oxford. As I read the popular article, I realized it is somethin g very familiar to me but from totally different view point.

Number theoretician encounters the problem of finding rational points of an algebraic curve defined as real or complex variant in which case the curve is 2-D surface and 1-D in complex sense. The curve is defined as root of polynomials polynomials or several of them. The polynomial have typically rational coefficients but also coefficients in extension of rationals are possible.

For instance, Fermat's theorem is about whether xⁿ+yⁿ=1, n=1,2,3,... has rational solutions for n≥1. For n=1, and n= 2 it has and these solutions can be found. It is now known that for n>2 no solutions do exist. Quite generally, it is known that the number is finite rather than infinite in the generic case.

A more general problem is that of finding points in some algebraic extension of rationals. Also the coefficients of polynomials can be numbers in the extension of rationals.

Connection with TGD and physics of cognition

The problem is extremely difficult even for mathematicians - to say nothing about humble physicist like me with hopelessly limited mathematical skills. It is however just this problem which I encounter in TGD inspired vision about adelic physics. Recall that in TGD space-times are 4-surfaces in H=M⁴×CP₂, preferred extremals of the variational principle defining the theory.

1. In this approach p-adic physics for various primes p provide the correlates for cognition: there are several motivations for this vision. Ordinary physics describing sensory experience and the new p-adic physics describing cognition for various primes p are fused to what I called adelic physics. The adelic physics is characterized by extension of rationals inducing extensions of various p-adic number fields. The dimension n of extension characterizes kind of intelligence quotient and evolutionary level since algebraic complexity is the larger, the larger the value of n is. The connection with quantum physics comes from the conjecture that n is essentially effective Planck constant h_eff/h_0=n characterizing a hierarchy of dark matters. The larger the value of n the longer the scale of quantum coherence and the higher the evolutionary level, the more refined the cognition.
   
   
2. An essential notion is that of cognitive representation. It has several realizations. One of them is the representation as a set of points common to reals and extensions of various p-adic number fields induced by the extension of rationals. These space-time points have points in the extension of rationals considered defining the adele. The coordinates are the imbedding space coordinates of a point of the space-time surface.
   
   
3. The gigantic challenge is to find these points common to real number field and extensions of various p-adic number fields appearing in the adele.
   
   
4. If this were not enough, one must solve an even tougher problem. In TGD the notion of "world of classical worlds" (WCW) is also a central notion. It consists of space-time surfaces in imbedding space H =M⁴× CP₂, which are so called preferred extremals of the action principle of theory. Quantum physics would reduce to geometrization of WCW and construction of classical spinor fields in WCW and representing basically many-fermion states: only the quantum jump would be genuinely quantal in quantum theory.
   

   There are good reasons to expect that space-time surfaces are minimal surfaces with 2-D singularities, which are string world sheets - also minimal surfaces. This gives nice geometrization of gauge theories since minimal surfaces equations are counterparts for massless field equations.
   

   One must find the algebraic points, the cognitive representation, for all these preferred extremals representing points of WCW (one must have preferred coordinates for H - the symmetries of imbedding space crucial for TGD and making it unique, provide the preferred coordinates)!
   
   

5. What is so beautiful is that in given cognitive resolution defined by the extension of rationals inducing the discretization of space-time surface, the cognitive representation defines the coordinates of the space-time surfaces as a point of WCW. This huge infinite-dimensional space WCW discretizes and the situation can be handled using finite mathematics.
   

Connection with Kim's work

So: what is then the connection with the work and ideas of Kim. Also he is interested in the above problem of finding rational points of given surface. There has been a lot of progress in understanding the problem: here I an only refer to the popular article.

1. One step of progress has been the realization that if one uses the fact that the solutions are common to both reals and various p-adic number fields helps a lot. The reason is that for rational points the rationality implies that the solution of equation representable as infinite power series of p contains only finite number powers of p. If one manages to prove the this happens for even single prime, a rational solution has been found.
   

   The use of reals and all p-adic numbers fields is nothing but adelic physics. Real surfaces and all its p-adic variants form pages of a book like structure with infinite number of pages. The rational points or points in extension of rationals are the cognitive representation and are points common to all pages in the back of the book.
   

   This generalizes also to algebraic extensions of rationals. Solving the number theoretic problem is in TGD framework nothing but finding the points of the cognitive representation. The surprise for me was that this viewpoint helps in the problem rather than making it more complex. There are however problematic situations in some cases the hypothesis about fintie set of algebraic points need not make sense. A good example is Fermat for x+y=1. All rational points and also algebraic points are solutions. For x²+ y²=1 the set of Pythagorean triangles characterizing the solutions is infinite. How to cope with these situations in which one has accidental symmetries as one might say.
   
   

2. Kim however argues that one can make even further progress by considering the situation from even wider perspective by making the problem even bigger. Introduce what popular article calls the space of spaces. The space of string world sheets is what string models suggests. The "world of classical worlds", WCW is what TGD suggests. One can get a wider perspective of the problem of finding algebraic points of a surface by considering the problem in the space of surfaces and at this level it might be possible to gain much more understanding. The notion of WCW would not mean horrible complication of a horribly complex problem but possible manner to understand the problem!
   
   
3. A further TGD based simplification would be M⁸-H (H=M⁴×CP₂) duality in which space-time surfaces at the level of M⁸ are algebraic surfaces which are mapped to surfaces in H identified as preferred extremals of action principle by the M⁸-H duality. Algebraic surfaces satisfying algebraic equations are very simple as compared to preferred extremals satisfying partial differential equations but "preferred" is what makes possible the duality. This huge simplification of the solution space of field equations guarantees holography necessitated by general coordinate invariance implying that space-time surfaces are analogous to Bohr orbits. It would also guarantee the huge symmetries of WCW making it possible to have Kähler geometry.
   

   This suggests in TGD framework that one finds the cognitive representation at the level of M⁸ using methods of algebraic geometry and maps the points to H by using the M⁸-H duality. TGD and octonionic variant of algebraic geometry would meet each other.
   

   It must be made clear that now solutions are not points but 4-D surfaces and this probably means also that points in extension of rationals are replaced with surfaces with imbedding space coordinates defining function in extensions of rational functions rather than rationals. This would bring in algebraic functions. This might provide also a simplification by providing a more general perspective. Also octonionic analyticity is extremely powerful constraint that might help.
   
   

Can one make Kim's idea about the role of symmetries more concrete in TGD framework?

The crux of the Kim's idea is that somehow symmetries of space of spaces could come in rescue in the attempts to understand the rational points of surface. The notion of WCW suggest in TGD framework rather concrete realization of this idea that I have discussed from the point of view of construction of quantum states.

1. A little bit more of zero energy ontology (ZEO) is needed to follow the argument. In ZEO causal diamonds (CDs) defined as intersections of future and past directed light-cones with points replaced with CP₂ and forming a scale hierarchy are central. Space-time surfaces are preferred extremals with ends at the opposite boundaries of CD indeed looking like diamond. Symplectic group for the boundaries of causal diamond (CD) is the group of isometries of WCW. Its Lie algebra has structure of Kac-Moody algebra with respect to the light-like radial coordinate of the light-like boundary of CD, which is piece of light-cone boundary. This infinite-D group plays central role in quantum TGD: it acts as WCW isometries and zero energy states are invariant under its action at opposite boundaries.
   
   
2. As one replaces space-time surface with a cognitive representation associated with an extension of rationals, WCW isometries are replaced with their infinite discrete subgroup acting in the number field define by the extension of rationals defining the adele. These discrete isometries do not leave the cognitive representation invariant but replace with it new one having the same number of points and one obtains entire orbit of cognitive representations. This is what the emergence of symmetries in wider conceptual framework would mean.
   
   
3. One can in fact construct invariants of the symplectic group. Symplectic transformations leave invariance the Kaehler magnetic fluxes associated with geodesic polygons with edges identified as geodesic lines of H. The simplest polygons are geodesic triangles. The symplectic fluxes associated with the geodesic triangles define symplectic invariants characterizing the cognitive representation. For the twistor lift one must allow also M⁴ to have analog of Kähler form and it would be responsible for CP violation and matter antimatter asymmetry. Also this defines symplectic invariants so that one obtains them for both M⁴ and CP₂ projections and can characterize the cognitive representations in terms of these invariants.
   

   More complex cognitive representations in an extension containing the given extension are obtained by adding points with coordinates in the larger extension and this gives rise to new geodesic triangles and new invariants.
   
   

4. Also in this framework one can have accidental symmetries. For instance, M⁴ with CP₂ coordinates taken to be constant is a minimal surface, and all rational and algebraic points for given extension belong to the cognitive representation so that they ar infinite. Could this has something to do with the fact that we understand M⁴ so well and have even identified space-time with Minkowski space! Linear structure would be cognitively easy for the same reason and this could explain why we must linearize.
   

   CP₂ type extremals with light-like M⁴ geodesic as M⁴ projection is second example of accidental symmetries. The number of rational or algebraic points with rational M⁴ coordinates at light-like curve is infinite - the situation is very similar to x+y=1 for Fermat. Simplest cosmic strings are geodesic sub-manifolds, that is products of plane M² ⊂ M⁴ and CP₂ geodesic sphere. Also they have exceptional symmetries.
   

   What is interesting from the point of view of proposed model of cognition is that these cognitively easy objects play a central role in TGD: their deformations represent more complex dynamical situations. For instance, replacing planar string with string world sheet replaces cognitive representation with a discrete or perhaps even finite one in M⁴ degrees of freedom.
   
   

See the article Secret Link Uncovered Between Pure Math and Physics or the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry?.

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

Articles and other material related to TGD.

About the physical interpretation of ramified primes in TGD framework

Adelic physics corresponds to a hierarchy of extensions of rationals inducing extensions of p-adic number fields and the proposal is that ramified primes of extension correspond to preferred p-adic primes.

1. Adelic physics suggests that prime p and quite generally, all preferred p-adic primes, could correspond to ramified primes for the extension of rationals defining the adele. Ramified prime divides discriminant D(P) of the irreducible polynomial P (monic polynomial with rational coefficients) defining the extension (see this).
   

   Discriminant D(P) of polynomial whose, roots give rise to extension of rationals, is essentially the resultant Res(P,P') for P and its derivative P' defined as the determinant of so called Sylvester polynomial (see this). D(P) is proportional to the product of differences r_i-r_j, i≠ j the roots of p and vanishes if there are two identical roots.
   

   Remark: For second order polynomials P(x)=x²+bx+c one has D= b²-4c.
   
   

2. Ramified primes divide D. Since the matrix defining Res(P,P') is a polynomial of coefficients of p of order 2n-1, the size of ramified primes is bounded and their number is finite. The larger coefficients P(x) has, the larger the value of ramified prime can be. Small discriminant means small ramified primes so that polynomials having nearly degenerate roots have also small ramifying primes. Galois ramification is of special interest: for them all primes of extension in the decomposition of p appear as same power. For instance, the polynomial P(x)=x²+p has discriminant D=-4p so that primes 2 and p are ramified primes.
   
   
3. What does ramification mean algebraically? The ring O(K)/(p) of integers of the extension K modulo p=π_i^e_i can be written as product ∏_i O/π_i^e_i (see this). If p is ramified, one has e_i>1 for at least one i. Therefore there is at least one nilpotent element in O(K)/(p).
   
   

Could one interpret nilpotency quantum physically?

1. For Galois extensions one has e_i=e>1 for ramified primes. e divides the dimension of extension. For the quadratic extensions ramified primes have e=2. Quadratic extensions are fundamental extensions - kind of conserved genes -, whose further extensions give rise to physically relevant extensions.
   

   On the other hand, fermionic oscillator operators and Grassmann number used to describe fermions "classically" are nilpotent. Could they correspond to nilpotent elements of order e_i=e=2 in O(K)/(p)? Fermions are building bricks of all elementary particles in TGD. Could this number theoretic analogy for the fermionic statistics have a deeper meaning?
   
   

2. What about ramified primes with e_i=e>2? Could they correspond to para-statistics (see this) or braid statistics (see this)?
   

   Both parabosonic and parafermionic fields of order n have the representation Ψ=∑_i=1ⁿ Ψ_i. For parafermion field one has {Ψ_i(x),Ψ_i(y)}=0 and [Ψ_i(x),Ψ_j(y)]=0, i≠ j, when x and y have space-like separation. For parabosons the roles of commutator and anti-commutator are changed.
   

   The states containing N identical parafermions are described by a representation of symmetric group S_N with N rows with at most e columns (anti-symmetrization). For states containing N identical parabosons one has N columns and at most e rows. For parafermions the wave function is symmetric in horizontal direction but the modes are different so that Bose-Einstein condensation is not possible.
   

   For parafermion of order n operator ∑_i=1ⁿ Ψ_i one has (∑_i=1ⁿ Ψ_i)ⁿ= ∏ Ψ₁ Ψ₂...Ψ_n and higher powers vanish so that one would have e-nilpotency. Therefore the interpretation for the nilpotent elements of order e in O(K)/(p)$ in terms of parafermion of order n=e-1 might make sense.
   

   It seems impossible to build a nilpotent operator from parabosonic field Ψ= ∑_iΨ_i: the reason is that the powers Ψ_iⁿ are non-vanishing for arbitrarily high values of n.
   
   

3. Braid statistics differs from para-statistics and is assigned with quantum groups. It would naturally correspond to quantum phase exp(iπ/p) assignable to the exchange of particles by braid operation regarded as a homotopy permuting braid strands. Could ramified prime p would correspond to braid statistics and the index e_i=e characterizing it to para-statistics of order e-1? This possibility cannot be excluded since this exotic physics would be associated in TGD framework to dark matter assigned to algebraic extensions of rationals whose dimension n equals to h_eff/h₀.
   
   

Why the primes, which do not split maximally in given extension - the ramified ones - would be physically special?

1. Do ramified primes possess exceptional evolutionary fitness and are ramified primes present for lower-dimensional extensions present also for higher-dimensional extensions? If higher extensions are formed as extensions of already existing extensions, this is the case. Hierarchy of polynomials of polynomials would to this kind of hierarchy with ramified primes of starting point polynomials analogous to conserved genes.
   
   
2. Quadratic extensions are the simplest ones and could serve as starting point extensions. Polynomials of form x²-c are the simplest among them. Discriminant is now D= -4c.
   
   
3. Why c= M_n=2ⁿ-1 allowing p=2 and Mersenne prime p=M_n as ramified primes would be favored? Extension of rationals defined by x=2ⁿ is non-trivial for odd n and is equivalent with extension containing 2^1/2. c=M_n=2ⁿ-1 as a small deformation of c=2ⁿ gives an extension having both 2 as M_n as ramified primes.
   

   For c=M_n the number of ramified primes is smallest possible and equal to 2: why minimal number of ramified primes would give rise to a fittest extension? Why smallest number of fermionic p-adic mass scales assignable to the ramified primes would be the fittest option?
   

   The p-adic length scale corresponding ro M_n would be maximal and mass scale minimal. Could one think that other quadratic extension are unstable against transforming to Mersenne extensions with smallest p-adic mass scale?
   
   

See the appendix of the article Shnoll effect decade later or of the chapter A possible explanation of Shnoll effect of "Hyper-finite Factors and Dark Matter Hierarchy: part II" .

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

Articles and other material related to TGD.

Shnoll effect decade later

As I wrote the first version of this chapter about Shnoll effect for about decade ago I did not yet have the recent vision about adelic physicsas a unification of real physics and various p-adics and real number based physics to describe the correlates of both sensory experience and cognition.

The recent view is that the hierarchy of extensions of rational numbers induces a hierarchy of extensions of p-adic number fields in turn defining adele. This hierarchy gives rise to dark matter hierarchy labelled by a hierarchy of Planck constants and also evolutionary hierarchy. The hierarchy of Planck constants h_eff=n× h₀ is and essential element of quantum TGD and adelic physics suggests the identification of n as the dimension of extension of rationals. n could be seen as a kind of IQ for the system.

What is also new is the proposal that preferred p-adic primes labelling physical systems could correspond to so called ramified primes, call them p, of extension of rationals for which the expression of the rational p-adic prime as product of primes of extension contains less factors than that the dimension n of extension so that some primes of extension appear as higher powers. This is analog for criticality as the appearance of multiple roots of a polynomial so that the derivative vanishes at the root besides the polynomial itself.

Before continuing it is good to make some confessions. Already in the earlier approach I considered two options for explaining the Shnoll effect: either in terms of p-adic fractality or in terms of quantum phase q of both. I however too hastily concluded that the p-adic option fails and choose the quantum phase option.

In the following both options are seen as parts of the story relying on a principle: the approximate scaling invariance of probability distribution P(n) for fluctuations under scalings by powers of p-adic prime p implying that P(n) is approximately identical for the divisions for which the interval Δ defining division differs by a power of p.

Second new idea is the lift of P(n) to wave function Ψ(n) in the space of counts. For quantum phase q_m, m=p, Ψ_m would have quantum factor proportional to a wave function in finite field F_p, and the notion of counting modulo p suggests that the wave function corresponds to particle in box - standing wave - giving rise to P(n) representing diffraction pattern.

Basic facts about Shnoll effect

Usually one is not interested in detailed patterns of the fluctuations of physical variables, and assumes that possible deviations from the predicted spectrum are due to the random character of the phenomena studied. Shnoll and his collaborators have however studied during last four decades the patterns associated with random fluctuations and have discovered a strange effect described in detail in the articles of Shnoll (references can be found in
A Possible Explanation of Shnoll Effect).

1. Some examples studied by Shnoll and collaborators are fluctuations of chemical and nuclear decay rates, of particle velocity in external electric field, of discharge time delay in a neon lamp RC oscillator, of relaxation time of water protons using the spin echo technique, of amplitude of concentration fluctuations in the Belousov-Zhabotinsky reaction. Shnoll effect appears also in financial time series, which gives additional support for its universality. Often the measurement reduces to a measurement of a number of events in a given time interval τ. More generally, it is plausible that in all measurement situations one divides the value range of the studied observable observable to intervals of fixed length and counts the number of events in each interval to get a histogram representing the distribution N(n), where n is the number of events in a given interval and N(n) is the number of intervals with n events. These histograms allow to estimate the probability distribution P(n), which can be compared with theoretical predictions for the spectrum of fluctuations of n. Typical theoretical expectations for the fluctuation spectrum are characterized by Gaussian and Poisson distributions.
   
   
2. Contrary to the expectations, the histograms describing the distribution of N(n) has a distribution having several maxima and minima (see the figures in the article of Shnoll and collaborators (see this). Typically -say for Poisson distribution - one expects single peak. As the duration of the measurement period increases, this structure becomes gets more pronounced: standard intuition would suggest just the opposite to take place. The peaks also tend to be located periodically. According to Shnoll the smoothed out distribution is consistent with the expected distribution in the case that it can be predicted reliably.
   
   
3. There are also other strange features involved with the effect. The anomalous distribution for the number n of events per fixed time interval (or more general value interval of measured observable) seems to be universal as the experiments carried out with biological, chemical, and nuclear physics systems demonstrate. The distribution seems also to be same at laboratories located far away from each other. The comparison of consecutive histograms shows that the histogram shape is likely to be similar to the shape of its nearest temporal neighbors. The shapes of histograms tend to recur with periods of 24 hours, 27 days, or 365 days. The regular time variation of consecutive histograms, the similarity of histograms for simultaneous independent processes of different nature and occurring in different geographical positions, and the above mentioned periods, suggest a common reason for the phenomenon possibility related to gravitational interactions in Sun-Earth and Earth-Moon system.
   
   

In the case that the observable is number n of events per given time interval, theoretical considerations predict a distribution characterized by some parameters. For instance, for Poisson distribution the probabilities P(n) are given by the expression

P(n|λ)= exp(-λ) λⁿ/n! .

The mean value of n is λ>0 and also variance equals to λ. The replacement of distribution with a many-peaked one means that the probabilities P(n| λ) are modified so that several maxima and minima result. This can occur of course by the randomness of the events but for large enough samples the effect should disappear.

The universality and position independence of the patterns suggest that the modification changes slowly as a function of geographic position and time. The interpretation of the periodicities as periods assignable to gravitational interactions in Sun-Earth system is highly suggestive. It is however very difficult to imagine any concrete physical models for the effect since distributions look the same even for processes of different nature. It would seem that the very notion of probability somehow differs from the ordinary probability based on real numbers and that this deformation of the notion of probability concept somehow relates to gravitation.

Quantum group inspired model for Shnoll effect

Usually quantum groups are assigned with exotic phenomena in Planck length scale. In TGD they are assignable to a finite measurement resolution. TGD inspired quantum measurement theory describes finite measurement resolution finite measurement resolution in terms of inclusions of hyper-finite factors of type II₁ (HFFs) and quantum groups related closely to the inclusions and appear also in the models of topological quantum computation quantum computation based on topological quantum field theories.

Consider first the original version of the proposed model. If I would rewrite it now correcting also the small errors, the summary would be as follows. This slightly revised model can be included as such to the new model.

1. The possibility that direct p-adic variants of real distribution functions such as Poisson distribution might allow to understand the findings was discussed also in the original version. The erratic conclusion was that this cannot the case. In fact, for λ=1/p^k the sum of probabilities P(n) without normalization factor is finite, and the appriximate scaling symmetry P(n)≈ P(p^rn) emerges for k=1. p-Adicity predicts approximate p-periodicity corresponding to the periodic variation of n_R with the lowest pinary digit of n.
   

2. It was argued that one should replace the integer n! in P(n) with quantum integer (n!)_qm, q=exp(iπ/m), identified as the product of quantum integers r_qm=(q^r-q^-r)/(q-q^-1), r<n.
   

   This however leads to problems since r_qm can be negative. The problem can be circumvented by interpreting n! as p-adic number and expanding it in powers of p with pinary digits x_k<p. For m=p the replacement of x_k with quantum integer yields positive pinary digits.
   

   The resulting quantum variant of p-adic integer can be mapped to its real counterpart by a generalization of canonical identification x= ∑ x_npⁿ→ ∑ x_np^-n. Whatever the detailed definition, quantum integers are non-zero and positive. The quantum replacement r→ r_qm of the integers appearing in rational parameters in P(n|λ_i) might therefore make sense. It however does not make sense in the exponents like λⁿ and λ=p^k, k>1,2,.., is forced by convergence condition.
   
   

3. I proposed also another modification of quantum integers x_qm, x<p=m appearing in as pinary digits by decomposing x into a product of primes s<p and replacing s with quantum primes s_qp so that also the notion of quantum prime would make sense: one might talk about quantum arithmetics. This is possible but is not necessary.
   
   

Adelic model for Shnoll effect

At the first re-reading the original model looked rather tricky, and this led to a revised model feeding in the adelic wisdom. One implication hierarchy of Planck constants h_eff/h₀=n with n identified as the dimension of Galois extension.

One also ends up to the proposal that preferred p-adic primes p correspond to so called ramified primes of the extension of rationals inducing the extensions of p-adic number fields defining the adele. This kind of prime would naturally define a small-p p-adicity associated with Shnoll effect, which would thus serve as a direct signature of adelic physics.

1. The first observation in conflict with the original belief is that one can actually define purely p-adic variant of the Poisson distribution P(n| λ) by replacing 1/n! with its image (n!)_R under canonical identification. For instance, for Poisson distribution one must have λ= p^-k, k=1,2,.. for both real and p-adic distributions to nake sense. The sum of the probabilities P(n) is finite. Poisson distribution with trivial quantum part is determined uniquely.
   
   
2. One can also consider quantization P(n)=|Ψ(n)|², suggested by the vision about quantum TGD
   as complex square root of thermodynamics and hierarchy of Planck constants making possible macroscopic quantum coherence in arbitrarily long scales. The complexity of Ψ(n) could genuine quantum interpretation. Quantum factor of Ψ(n) allows interpretation as a wave function in finite field F_p representing the space of counts modulo p. The existence of quantum p-adics requires m=p. Scaling by p is not a symmetry but multiplication by 0<k<p and shift by 0≤ k<p act as symmetries analogous to rotations and translations acting on waves functions in Euclidian 3-space.
   
   
3. The objections against Shnoll effect lead to an additional condition - or should one say principle - stating that the P(n) is approximately invariant under scalings n→ p^kn. This could be seen as a manifestation of p-adic fractality in turn reflecting quantum criticality of TGD Universe.
   
   
4. Taking n as the observable simplifies p-adicization crucially since the highly non-unique p-adicization of classical observables is avoided. One could speak of quantum measurement in the space of counts n defining universal observables. In quantum measurements the results are typically expressed as numbers of counts in given bin so that this kind of p-adicization is physically natural. The division of measurement interval would define an ensemble and n would be measured in each interval. State function reduction would localize Ψ(n) to n in each interval.
   
   

This picture leads to an alternative and simpler view about Shnoll effect. The scaling invariance is an essential additional condition now.

1. The factorials n! appearing in P(n)=(dⁿf/dxⁿ)/n! identified as coefficients of Taylor series of its generating function developed in pinary expansion for p=m. In this expansion one must invert powers of p in (n!)_R and could also replace the coefficients of powers of p with quantum integers or replace even primes in their prime composition with quantum primes. For given norm (n!)_R has period p approximately.
   
   
2. The n:th derivative X(n)= dⁿf/dxⁿ appearing as coefficient of 1/n! is replaced with X(n)_R/X(n)_p giving approximate periodicity and scaling invariance n→ pn.
   
   
3. Quantum phase is associated with the ansatz stating P(n)= | Ψ(n)|². In the "diffractive" situation quantum counterpart corresponds to | (kn)_qm|², 0<k<p-1. This gives rise to periodicity with period m=p.
   
   

The universal modifications of the probability distributions P(n|λ_i) considered predict patterns analogous to the ones observed by Shnoll. The p-adic prime p=m characterizes the deformation of the probability distribution and implies approximate p-periodicity, which could explain the periodically occurring peaks of the histograms for N(n) as function of n.

One can imagine several explanations for the dependence of the time series distribution P(n) on the direction of the momentum of alpha particle and on the dependence of P(n) on time.

1. The change of ramified prime p induced by the change of the extension of rationals would affect the periods. An interesting question is whether the effects understood in terms of the effect of the measurement apparatus on many-sheeted space-time many-sheeted space-time topology and geometry on p. Can one speak about measurement of p and of extension of rationals?
   
   
2. The extension of rationals (and thus p) need not change. The "quantum factor of Ψ in P(n)=| Ψ(n)|² has part depending on q_p. The dependence on q_p could change without change in p so that the extension of rationals need not change. One could speak about measurement of an observable related to the quantum factor of Ψ. A more concrete model relies on wave function proportional to (kn)_qp ∝ q_m^kn+q_m^-kn - analog to a superposition of plane waves with momenta k propagating to opposite directions in the space of counts and producing in P(n) diffraction pattern proportional to (qn)_qp². Change of momentum k by scaling or shift induced by variation of the gravitational parameters or time evolution could be in question.
   
   

The p-adic primes p in question are rather small, not much larger than 100 and the periods of P(n) provide a stringent test for the proposal. If p corresponds to ramified prime as adelic physics suggests, it can be indeed small.

To sum up, I cannot avoid the thought that fluctuations regarded usually as a mere nuisance could be actually a treasure trove of new physics. While we have been busily building bigger and bigger particle accelerators, the truth would have been staring directly at our face and even winking eye to us.

See the updated article Shnoll Effect decade later.

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

Articles and other material related to TGD.

Number theoretical view about unitarity conditions for twistor lift

Twistorialization leads to the proposal that cuts in the scattering amplitudes are replaced with sums over poles, and that also many-particle states have discrete momentum and mass squared spectrum having interpretation in terms of bound states. Gravitation would be the natural physical reason for the discreteness of the mass spectrum and in string models it indeed emerges as "stringy" mass spectrum. The situation is very similar to that in dual resonance models, which were predecessors of super string theories.

Number theoretical discretization based on the hierarchy of extensions of rationals defining extensions of p-adic number fields gives rise to cognitive representatations as discrete sets of space-time surface and discretization of 4-momenta and S-matrix with discrete momentum labels. In number theoretic discretization cuts reduce automatically to sequences of poles. Whether this discretization is an approximation reflecting finite cognitive resolution or whether finite cognitive representation is a property of physical states reflecting itself as a condition that various parameters characterizing them belong to the extension considered, remains an open question.

One can approach the unitarity conditions also number theoretically. In the discretization forced by the extension of rationals the amplitudes are defined between states having a discrete spectrum of 4-momenta. Unitarity condition reduces to a purely algebraic condition involving only sums. In these conditions the Dirac delta functions associated with the mass squared of the resonances are replaced with Kronecker deltas.

1. For given extension of rationals the unitary conditions are purely algebraic equations
   

   i(T_mn+T*_nm)= ∑_r T_mrT*_nr = T_mnT*_nn +T_mmT_mn + ∑_{r≠ m,n} T_mrT*_nr .
   

   where T_mn belongs the extension. Complex imaginary unit i corresponds to that appearing in the extension of octonions in M⁸-H duality (see this).
   
   

2. In the forward direction m=n one obtains
   

   2Im(T_mm)= Re(T_mm)² + Im(T_mm)²+ P_m , P_m= ∑_{r≠ m} T_mrT*_mr .
   

   P_m represents total probability for non-forward scattering.
   
   

3. One can think of solving Im(T_mm) algebraically from this second order polynomial in the lowest order approximation in which T_mn=0 for m ≠ n. This gives
   

   2Im(T_mm)= 1+ (1-P_m-Re(T_mm)²)^1/2 .
   

   Reality requires 1-Re(T_mm)²-P_m≥ 0 giving
   

   Re(T_mm)²+P_m≤ 1 .
   

   This condition is identically true by unitarity since the probability for scattering cannot be larger than 1.
   

   Besides this the real root must belong to the original extension of rationals. For instance, if the extension of rationals is trivial, the quantity 1-P_m-Re(T_mm)² must be a square of rational y giving 1-P_m= y²+Re(T_mm)². In the case of extension y is replaced with a number in the extension. I am not enough of number theorist to guess how powerful this kind of number theoretical conditions might be. In any case, the general ansatz for S is a unitary matrix in extension of rationals and this kind of matrices form a group so that there is no hope about unique solution.
   
   

4. One could think of iterative solution of the conditions by assuming in the zeroth order approximation T_mn=0 for m≠ n giving Re(T_mm)² +Im(T_mm)²= 1 reducing to cos²(θ)+sin²(θ)=1. For trivial extension of rationals θ would correspond to Pythagorean triangle.
   

   For non-diagonal elements of T_mn one would obtain at the next step the conditions
   

   i(T_mn+T*_nm)= T_mnT*_nn + T_mmT*_nm .
   

   This gives a 2 linear equations for T_mn.
   
   

5. These conditions are not enough to give unique solution. Time reversal invariance gives additional conditions and might help in this respect. T invariance is slightly broken but CPT symmetry could replace T symmetry in the general situation.
   

   Time reversal operator T (to be not confused with T_mn above) is anti-unitary operator and one has S^†= T(S). In wave mechanics one can show that T-invariant S-matrix and thus also T-matrix is symmetric. The matrices of this kind do not form a group so that the conditions can be very powerful.
   

   Combined with the above equations symmetry gives
   

   2Im(T_mn)= T_mnT*_nn + T_mmT*_mn .
   

   The two conditions for T_mn in principle fix it completely in this order.
   

   One obtains from the real part of the equation
   

   2Im(T_mn)= Re[T_mnT*_nn + T_mmT*_mn] .
   

   The vanishing of the imaginary part gives
   

   Im[T_mnT*_nn + T_mmT*_mn]=0 .
   

   giving a linear relation between the real and imaginary parts of T_mn. No new number theoretical conditions emerge. This relation requires that real and imaginary parts belong to the extension.
   
   

6. At higher orders one must feed the resulting ansatz to the unitarity conditions for the diagonal elements T_nn. One can hope that the lowest order ansatz leads to rather unique solution by iteration of the unitarity conditions. In higher order conditions the higher order corrections appear linearly so that no new number theoretic conditions emerge at higher orders.
   

   Physical picture suggests that the S-matrices could be obtained by an iterative procedure. Since infinitely long procedure very probably leads out of the extension, one can ask whether the procedure should stop after finite steps. This property would pose an additional conditions to the S-matrix.
   
   

7. Diagonal matrices are solutions to the conditions and for then the diagonal elements are roots of unity in the extension of rationals considered. The automorphisms S_d→ US_dU^-1 produce new S-matrices and if the unitary matrix U is orthogonal real matrix in algebraic extension satisfying therefore UU^T=1, the condition S=S^T is satisfied. There are therefore a large number of solutions.
   

   S-matrices diagonalizable in the extension are not the only solutions. The diagonalization of a unitary matrix S=S^T in general gives a diagonal S-matrix, for which the roots of unity in general do not belong to the extension. Also the diagonalizating matrix fails to be in the extension. This non-diagonalizability might have deep physics content and explain why the physically natural state basis is not the one in which S-matrix is diagonal. In the case of density matrix it would guarantee stability of entanglement.
   
   

To sum up, number theoretic conditions could give rise to highly unique discrete S-matrices, when CPT symmetry can be formulated purely algebraically and be combined with unitarity. CPT symmetry might not however allow formulation in terms of automorphisms of diagonal unitary matrices analogous to orthogonal transformations.

See the article More about the construction of scattering amplitudes in TGD framework or the chapter The Recent View about Twistorialization in TGD Framework.

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

Articles and other material related to TGD.

Icosa-tetrahedral and icosa-dodecahedral bioharmonies as candidates for genetic code

Both the icosa-tetrahedral (see this) and icosa-dodecahedral harmony to be discussed below can be considered as candidates for bio-harmony as also the harmony involving fusion of 2 icosahedral harmonies and toric harmony (see this). The basic reason is that the third harmony corresponds to doublets. One cannot exclude the possibility of several equivalent representations of the code.

Icosa-tetrahedral harmony

Icosahedral harmonies can be characterized by a subgroup of icosahedral isometries A₅ having 60 elements. If reflections are included the isometry group, oneas A₅× Z₂ with 120 elements. The group of symmetries is Z₆,Z₄, or Z₂. There are two choices for Z₂ and the interpretation has been that Z₂ correspond to either reflection or rotation by π. A₅ however allows also Z₂× Z2 as subgroup. Amino-acids (AAs) correspond to orbits of the symmetry group and DNA codons coding for the AA correspond to triangles (3-chords) at the orbit. In purely icosahedral model on obtains 20+20+20 codons. A fusion with tetrahedral harmony gives 64 codons.

1. Z₆ gives rise to 3 AAs coded by 6 codons each (leu,se,arg) and 2 AAs coded by 2 codons: the choice of the doublet would require additional conditions. One option is ile doublet.
   
   
2. Depending on whether one includes reflection or not, one can have either Z₄⊂ A₅ (60=4× 15) or Z₄=Z_2,rot× Z₂ ⊂ A₅× Z₂. I have assumed that Z₄=Z_2,rot× Z₂ but the recent argument suggests the first option. This does not have any implications for the earlier model. Icosahedral Z₄ gives rise to 5 AAs coded by 4 codons each (5× 4=20). This leaves 11 AAs and 3 "empty" AA formally coded by stop codons.
   
   
3. Icosahedral Z₂ gives rise to 10 dublets. These 4-plets would correspond to (phe, tyr, his, gln, asn, lys, asp, glu, cys, stop-doublet) This leaves (stop,trp) double and (ile,met) doublet with broken Z₂ symmetry.
   

   The fusion with tetrahedral code with 4- codons and 4 AAs should explain these 4 AAs. Tetrahedral isometries form group S₃ and reduce to group Z₃ for tetrahedral cycle.
   
   

   1. One could argue that ile-triplet and and met correspond to 3-element orbits with 1-element orbit. (stop,trp) would be formed by Z₂ symmetry breaking from trp doublet and there is no obvious mechanism for this.
      
      
   2. If one tetrahedral face is fixed as a face shared with icosahedron, the symmetry group of tetrahedral cycle reduces to Z₁. This would give 4 singlets identifiable as (ile,met) and (stop,trp) symmetry broken doubles. Since ile appears also in doublet, tetrahedral 1-orbit and icosahedral 2-orbit must have a common doubled triangle identifiable as the common face of icosahedron and tetrahedron. The doubling of the common triangle replaces ile-doublet with ile-triplet. This option looks rather reasonable.
      
      

Dodecadedral harmony

Dodecahedral harmony correspond to the unique Hamilton cycle at dodecahedron. Dodecahedral harmony as 20 notes and and 12 5-chords. If one assumes that the octave divides to 20 notes, this brings in mind "eastern" view about harmony.

The obvious objection against dodecahedral harmony is that dodecahedral faces are pentagons so that dodecahedral chords would be 5- rather than 3-chords so that the correspondence between chords and DNA codons would be lost. The situation changes if 3 notes - 3-chord - determine the 5-chord completely and one can assign a unique 3-chord to each pentagon. This is indeed the case!

1. 3-edges meet in every dodecahedral vertex (this makes the dodecahedral cycle unique apart from rotations) and each edge pair in the vertex belongs to same pentagon (in the case of icosahedron there are 5 edges per vertex so that this is not true). Therefore each pentagon must contain at least 2 edges of Hamilton's cycle.
   

   The cycle must visit all vertices of pentagon, and the visit to the vertex means that the cycle shares at least one edge with pentagon. Since all vertices of the pentagon must be visited, there are two options. For option a) given pentagon shares with the cycle disjoint 2-edge with 3 vertices and 1-edge with two vertices. For option b) the pentagon shares with the cycle 4-edge with 5 vertices.
   
   

2. The numbers n_a of pentagons with 4-edges and n_b=12-n_a 2-edge+ 1-edge (making 3 edges) can be deduced easily. Cycle has 20 edges. Pentagon of type a) shares 3 edges with the cycle and the edge is shared by 2 pentagons. This gives 3n_a/2 edges. Pentagon of type b) shares 4 edges with the cycle. This gives 2n_b= 2(12-n_a) edges. The total number of edges is 3n_a/2+2n_b= 20, which gives n_a=8 and n_b=4. Dodecahedral Hamilton's cycle can be found from web (see this). The structure is as deduced here.
   

   For case a) the 3-chords correspond naturally to the 3 vertices of the 2-edge shared with the cycle. Therefore it is possible to assign unique 3-chords to the dodecahedral harmony and to obtain connection with codons in this case. One however obtains also 12 2-chords: could they have some genetic counterpart?
   

   What about 5-chords for pentagons of type b)? Hamiltonian cycle can be oriented and this is induces orientation of the pentagons. One can say that the first vertex in the 4-edge is the vertex at which cycle arrives to the pentagon and identify the 3-chord as the first three vertices. It turns out that for the replacement of quint cycle this is not actually necessary.
   
   

Is icosa-dodecahedral harmony consistent with the genetic code?

One must check whether icosa-dodecahedral harmony is consistent with the degeneracies of the genetic code.

1. A fusion of 2 icosahedral harmonies and 2 copies of dodecahedral harmony would be in question. As in the case of icosahedral harmony already discussed, the two icosahedral harmonies would have symmetry groups Z₆ and Z₄ and give the codons coding for 3 6-plets and 1 doublet+ 5 4-plets + two copies of dodecahedral harmony.
   
   
2. Can the model predict correctly the numbers of codons coding for AAs? It is known that dodecahedral Hamilton cycle divides dodecahedron to two congruent pieces related by Z₂ symmetry (see this). Also the Hamiltonian cycle defining the common boundary has Z₂ symmetry. A good guess is that these Z₂:s corresponds to reflection symmetry and rotation by π but I am not able to exclude Z₄⊂ G₀, where G₀ consists of 60 orientation preserving isometries. In this case some orbits - presumably all 3 of them - could contain 4 pentagons. This is not consistent with the condition that one has doublets and singlets.
   

   If the second symmetry corresponds to reflection, it can be excluded by simply assuming that reflections change the orientation of the cycle.
   
   

3. Rotation by π has two fixed points corresponding to opposite poles so that one has 5 2-orbits and 2 1-orbits giving 12 triangles for each copy. Two copies of dodecahedral harmony would give 5+5=10 doublets and 2+2=4 singlets. A possible interpretation would be as (ile,met) and (stop,trp).
   
   

Consider now objections against dodecahedral harmony.

1. Why two copies of dodecahedral code? What distinguishes between them? If imirror symmetry leaves the cycle invariant apart from orientation the copies could be mirror images and consist of same faces. The second option is that they related by a rotation?
   
   
2. The number of dodecahedral AAs is 24 rather than 20. Could the additional 4 AAs as orbits have interpretation as AAs in some sense. Could the "empty" AAs coded by stop codons be counted as AAs exceptional in some sense. In TGD framework one can consider the possibility that although AA is "empty", there is analog of AA as physical signature for the end of protein telling what stopping codon it corresponds. The magnetic body of protein is a good candidate.
   

   Genetic code has several slightly differing variants. Could the 2 additional exotic AAs Pyl and Sec correspond in some situations to the additional AAs?
   
   

3. Essential for the bio-harmony as a fusion of harmonies is that one can select from each orbit single face as a representative of the AA it codes - kind of gauge choice is in question - and that the orbits corresponding to different AAs can be chosen to be disjoint. Otherwise codons belonging to the orbits of different Hamilton cycles can code for the same AA if the AA can be chosen to be in intersection. If not, the same codon can code for 2 different AAs - this can indeed occur in reality (see this)!
   

   The condition that orbits of different cycles do not interesect seems quite stringent but has not been proven. But what if it is actually broken? Indeed, in the case of icosahedral harmony with Z₁ symmetry tetrahedron and icosahedron could have common a doubled face the breaking of this condition would geometrically explain why ile belongs to both icosahedral and tetrahedral orbit.
   

   Ile is the problem also in the case if icosa-dodecahedral harmony. Dodecahedral singlet codes for ile as also icosahedral doublet. Could one talk about doubling of ile face so that it corresponds to a pair of triangle and pentagon (in 1-1 correspondence with triangle as chord).
   
   

4. The two copies of the dodecahedral code should correspond to 5 doublets and 2 singlets each. One expects that together they give rise to 10+2 +10+2 =24 faces. Do they? Mirror symmetry and rotation by π act as symmetries of the cycle so that neither can map the two cycles to each other. Dodecahedral (equivalently icosahedral) rotations give rise to new equivalent cycles. The action on pentagons corresponds to the action on vertices of icosahedron so that it easy to understand what happens.
   

   Each symmetry corresponds to a rotation around some axis and has opposite icosahedral vertices at this axis as fixed points. Hence any two cycles obtained in this manner have 2 common pentagons. This means reduction 24→ 22 unless one interprets the situation in terms doubled faces? Could the disappearing doublet correspond to stop-doublet? What about the remaining stop of the vertebrate code pairing with trp? Why does second singlet correspond to empty AA and not something else such as exotic AA.
   
   

5. There is also further problem. Suppose that an intersection of orbits takes place at single triangle. Suppose that one cannot choose this triangle to be "AA" triangle for both orbits. In this case it is not clear to which AA the codon codes. This kind of phenomenon actually takes place in some cases and is known as homonymy. It is associated with the deviations of the code from the vertebrate code and involves exotic AAs Pyl and Sec. Codons can serve as a stop codon or code for an exotic AA.
   
   

Clearly, the notion of bio-harmony involves many unclear aspects but my strong feeling is that there is very beautiful mathematics involved.

See the article Hachimoji DNA from TGD perspective.

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

Articles and other material related to TGD.