Neutrinos and TGD

In what follows, the problem of missing right-handed neutrinos and the problem created by apparently contradictory findings of Mini-Boone and Micro-Boone about neutrino mixing are discussed. Also the topological model for neutrino and D-quark CKM mixing is briefly considered.

Why only left-handed neutrinos are observed?

A basic theoretical motivation for the sterile neutrinos is the difficulty posed by the fact that the neutrinos behave like massive particles. This is not consistent with their left-handedness, which is an experimental fact.

As a matter of fact, the sterile neutrinos would be analogous to the covariantly constant right-handed neutrinos in TGD if only J(cP₂) would be present.

Remark: As already stated, in the sequel it is assumed that leptons as bound states of 3 antiquarks can be described using spinors of H with chirality opposite to that for quarks. They have colored modes and the action of super-symplectic algebra is assumed to neutralize the color and also give rise to a massless state getting its small mass by p-adic thermodynamics.

How could one understand the fact that only left-handed neutrinos are observed although neutrinos are massive? One can consider two approaches leading to the same conclusion.

Is it possible to have time evolution respecting M⁴ chirality and neutrinos with fixed chirality possible despite their mass?

1. All spinor modes in CP₂ are of the form Φ_L or D(CP₂)Φ_L and therefore generated from left-handed spinors Φ_L.

   If one assumes D(H)Ψ=0, the spinor modes of H are of the form D(M⁴)Ψ_R× Φ_L + Ψ_R× D(CP₂)Φ_L. The modes of form D(M⁴)Ψ_L× Φ_R + Ψ_L× D(CP₂)Φ_R are therefore of the form D(M⁴)Ψ_L× DΦ_L + Ψ_L× D²(CP₂)Φ_L. The mixing of chiralities is unavoidable.

2. However, if one assumes only the condition D²(H)Ψ=0, one can obtain both left- and right-handed modes without mixing of M⁴ chiralities and M⁴ Kähler structure could make the lowest mass second right-handed neutrino (covariantly constant in CP₂) tachyonic. The time evolution generated by the exponent of L₀ would respect M⁴ chirality.

   This does not prevent superpositions of right- and left-handed fermions if their masses are the same. If only charged leptons can satisfy this condition, one can understand why right-handed neutrinos are not observed.

An alternative approach would rely on quantum measurement theory but leads to the same conclusion.

1. Suppose that neutrinos can appear as superpositions of both right- and left-handed components. To detect a right-handed neutrino, one must have a measurement interaction, which entangles both length and right-handed components of the neutrino with the states of the measuring system. Measurement would project out the right-handed neutrino. If only the J(CP₂) form is present, the right-handed neutrino has only gravitational interactions, and this kind of measurement interaction does not seem to be realizable.
2. Putting it more explicitly, the reduction probability should be determined by a matrix element of a neutral (charged) weak current between a massive neutrino (charged lepton) spinor with a massless right-handed neutrino spinor. This matrix element should have the form Ψbar_R OΨ_L, where O transforms like a Dirac operator. If it is proportional to D(H), the matrix element vanishes by the properties of the massless right-handed neutrino.
3. There is however a loophole: the transformation of left- to right-handed neutrinos analogous to the transformation to sterile neutrino in the neutrino beam experiments could demonstrate the existence of ν_R just like it was thought to demonstrate the existence of the inert neutrino in Mini-Boone experiment. Time evolution should thus respect M⁴ chirality.

If J(M⁴) is present, one might understand why right- and left-handed neutrinos have different masses.

1. Also the right-handed neutrino interacts with Kähler gaug potential A(M⁴) and one can consider an entanglement distinguishing between right- and left-handed components and the measurement would project out the right-handed component. How could this proposal fail?

   Could it be that right- and left-handed neutrinos cannot have modes with the same mass so that these superpositions are not possible as mass eigen states? Why charged modes could have the same mass squared but not the neutral ones?

2. The modes with right-handed CP₂ chirality are constructed from the left-handed ones by applying the CP₂ Dirac operator to them and they have the same CP₂ contribution to mass squared. However, for the right-handed modes the J^kl(M⁴)Σ_kl term splits the masses. Could it be that for right- and left-handed charged leptons the same value of mass is possible.

   The presence of J(M⁴) breaks the Poincare symmetry to that for M² which corresponds to a Lagrangian manifold. This suggests that the physical mass is actually M² mass and the QCD picture is consistent with this. Also the p-adic mass calculations strongly support this view. The E² degrees of freedom would be analogous to Kac-Moody vibrational degrees of freedom of string. This would allow right- and left-handed modes to have different values of "cyclotron" quantum numbers n₁ and n₂ analogous to conformal weights. This could allow identical masses for left- and right-handed modes. For a Lagrangian manifold M², one would have n₁=n₂=0, which could correspond to ground states of super-symplectic representation.

3. Why identical masses would be impossible for right- and left-handed neutrinos? Something distinguishing between right- and left-handed neutrinos should explain this. Could the reason be that Z⁰ couples to left-handed neutrinos only? Could the fact that charged leptons and neutrinos correspond to different representations of color group explain why only charged states can have right and left chiralities with the same mass?

   Perhaps it is of interest to notice that the presence of J^kl(M⁴)Σ_kl for right-handed modes makes possible the existence of a mode for which mass can vanish for a suitable selection of B.

Mini-Boone and Micro-Boone anomalies and TGD

After these preliminaries we are ready to tackle the anomalies associated with the neutrino mixing experiments. The incoming beam consists of muonic neutrinos mixing with electron neutrinos. The neutrinos are detected as they transform to electrons by an exchange of W boson with nuclei of the target and the photon shower generated by the electron serves as the experimental signature.

The basic findings are as follows.

1. Mini-Boone collaboration reported 2018 (see this) an anomalously large number of electrons generated in the charged weak interaction assumed to occur between neutrino and a nucleus in the detector. "Anomalous" meant that the fit of the analog of the CKM matrix of neutrinos could not explain the finding. Various explanations including also inert neutrinos were proposed. Muonic inert neutrino would transform to inert neutrino and then to electron neutrino increasing the electro neutrino excess in the beam.
2. The recently published findings of Micro-Boone experiment (see this) studied several channels denoted by 1eNpM\pi where N=0,1 is the number of protons and M=0,1 is the number of pions. Also the channel 1eX, where "X" denotes all possible final states was studied.

   It turned out that the rate for the production of electrons is below or consistent with the predictions for channels 1e1p, 1eNp0\pi and 1eX. Only one channel was an exception and corresponds to 1e0p0\pi.

   If one takes the finding seriously, it seems that a neutrino might be able to transform to an electron by exchanging the W boson with a nucleus or hadron, which does not belong to the target.

In TGD, the only imaginable candidate for this interaction could be charged current interaction with a dark nucleus or with a nucleon with h_eff>h. This could explain the absence of ordinary hadrons in the final state for 1e events.

1. Dark particles are identified as h_eff>h phases of the ordinary matter because they are relatively dark with respect to phases with a different value of h_eff. Dark protons and ions play a key role in the TGD inspired quantum biology (see For a summary of earlier postings (see this) and even in the chemistry of valence bonds (seethis). Dark nuclei play a key role in the model for "cold fusion" (see this) and this) and also in the description of nuclear reactions with nuclear tunnelling interpreted as a formation of dark intermediate state (see this).
2. I have proposed that dark protons are also involved with the lifetime anomaly of the neutron (see this). The explanation relies on the transformation of some protons produced in the decay of neutrons to dark protons so that the measured life time would appear to be longer than real lifetime. In this case, roughly 1 percent of protons from the decay of n had to transform to dark protons.

3. If dark protons have a high enough value of h_eff and weak bosons interacting with them have also the same value of h_eff, their Compton length is scaled up and dark W bosons behave effectively like massless particles below this length scale. The minimum scale seems to be nuclear or atomic scale. This would dramatically enhance the dark rate for ν p→ e+n so that it would have the same order of magnitude as the rates for electromagnetic interactions. Even a small fraction of dark nucleons or nuclei could explain the effect.

CKM mixing as topological mixing and unitary time evolution as a scaling

The scaling generator L₀ describes basically the unitary time evolution between SSFRs (see this) involving also the deterministic time evolutions of space-time surfaces as analogs of Bohr orbits appearing in the superposition defining the zero energy state. How can one understand the neutrino mixing and more generally quark and lepton mixing in this picture?

1. In the TGD framework, quarks are associated with partonic 2-surfaces as boundaries of wormhole contacts, which connect two Minkowskian space-time sheets and have an Euclidean signature of induced metric and light-like projection to M⁴ (see this) and this).

2. For some space-time surfaces in their superposition defining a zero energy state, the topology of the partonic 2-surfaces can change in these time evolutions. The mixing of boundary topologies would explain the mixing of quarks and leptons. The CKM matrix would describe the difference of the mixings for U and D type quarks and for charged and neutral leptons. The topology of a partonic 2-surface is characterized by the genus g as the number of handles attached to a sphere to obtain the topology.

   The 3 lowest genera with g≤ 2 have the special property that they always allow Z₂ as a conformal symmetry. The proposal is that handles behave like particles and thanks to Z₂ symmetry g=2 the handles form a bound state. For g>2 one expects a quasi-continuous spectrum of mass eigenvalues. These states could correspond to so-called unparticles introduced by Howard Georgi (\url{https://cutt.ly/sRZKSFm}).

3. The time evolution operator defined by L₀ induces mixing of the partonic topologies and in a reasonable idealization one can say that L₀ has matrix elements between different genera. The dependence of the time evolution operator on mass squared differences is natural in this framework. In standard description it follows from the approximation of relativistic energies as p₀\simeq p+ m²/2p. Also the model of hadronic CKM relies on mass squared as a basic notion and involves therefore L₀ rather than Hamiltonian.

See the article Neutrinos and TGD.

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

Articles and other material related to TGD. 


Quantum hydrodynamics in nuclear physics and hadron physics

The field equations of TGD defining the space-time surfaces have interpretation as conservation laws for isometry charges and therefore have a hydrodynamics character. The hydrodynamic character is actually characterized in quite concrete ways (see this, this, and this).

Also nuclear and hadron physics suggest applications for Quantum Hydrodynamics (QHD). The basic vision about what happens in high energy nuclear and hadron collisions is that two BSFRs take place. The first BSFR creates the intermediate state with h_eff>h: the entire system formed by colliding systems need not be in this state. In nuclear physics this state corresponds to a dark nucleus which decays in the next BSFR to ordinary nuclei.

The basic notions are the notion of dark matter at MB and ZEO, in particular the change of the arrow of time in BSFR.

1. Cold fusion, nuclear tunnelling, ℏ_eff, and BSFRs

This model allows us to understand "cold fusion" in an elegant manner (see this, this, and this). The dark protons at flux tubes associated with water and created by the Pollack effect have much smaller nuclear binding energy than ordinary nucleons. This energy is compensated to a high degree by the positive Coulomb binding energy which corresponds roughly to distance given by electron Compton length.

Dark nuclear reactions between these kinds of objects do not require large collision energy to increase the value of h_eff and can take place at room temperature. After the reaction the dark nuclei can transform to ordinary nuclei and liberate the ordinary nuclear binding energy. One can say that in ordinary nuclear reactions one must get to the top of the energy hill and in "cold fusion" one already is at the top of the hill.

Quite generally, the mechanism creating intermediate dark regions in the system of colliding nuclei in BSFR, would be the TGD counterpart of quantum tunnelling in the description of nuclear reactions based on Schrödinger equation. This mechanism could be involved with all tunnelling phenomena.

2. QHD and hadron physics

Hadron physics suggests applications of QHD.

2.1 Quark gluon plasma and QHD

In hadron physics quark gluon plasma (see this) has turned out to be what it was thought to be originally. Instead of being like a gas of quarks and gluons with a relatively large dissipation, it has turned out to behave like almost perfect fluid. This means that the ratio η/s of viscosity and entropy is near to its minimal value proposed by string model based arguments to be η/s=ℏ/m.

To be a fluid means that the system has long range correlations whereas in gas the particles move randomly and one cannot assign to the system any velocity field or more general currents. In the TGD framework, the existence of a velocity field means at the level of the space-time geometry generalized Beltrami flow allowing to define a global coordinate varying along the flow lines (see this and this). This would be a geometric property of space-time surfaces and the finite size of the space-time surface would serve as a limitation.

In the TGD framework the replacement ℏ→ ℏ_eff requires that s increases in the same proportion. If the fluid flow is realized in terms of vortices controlled by pairs of monopole flux tubes defining their cores and Lagrangian flux tubes with gradient flow defining the exteriors of the cores, this situation is achieved.

In this picture entropy could but need not be associated with the monopole flux tubes with non-Beltrami flow and with non-vanishing entropy since the number of the geometric degrees of freedom is infinite which implies limiting temperature known has Hagedorn temperature T_H which is about 175 MeV for hadrons, and slightly higher than pion mass. In fact, the Beltrami property holds for the flux tubes with 2-D CP₂ projection, which is a complex manifold for monopole flux tubes. The fluid flow associated with (controlled by) the monopole flux tubes would have non-vanishing vorticity for monopole fluxes and could dissipate.

The monopole flux tube at the core of the vortex could therefore serve as the source of entropy. One expects that η/s as minimal value is not affected by h→ h_eff. One expects that s → (ℏ_eff/ℏ)s= ns since the dimension of the extension of rationals multiplies the Galois degrees of freedom by n.

Almost perfect fluids are known to allow almost non-interacting vortices. For a perfect fluid, the creation of vortices is impossible due to the absence of friction at the walls. This suggests that the ordinary viscosity is not the reason for the creation of vortices, and in the TGD picture the situation is indeed this. The striking prediction is that the masses of Sun and Earth appear as basic parameters in the gravitational Compton lengths Λ_gr determining ν_gr= Λ_grc.

2.2 The phase transition creating quark gluon plasma

The phase transition creating what has been called quark gluon plasma is now what it was expected to be. That the outcome behaves like almost perfect fluid was the first example. TGD leads however to a proposal that since quantum criticality is involved, phases with ℏ_eff>h must be present.

p-Adic length scale hypothesis led to the proposal (see this and this) that this transition could allow production of so called M₈₉ hadrons characterized by Mersenne prime M₈₉=2⁸⁹-1 whereas ordinary hadrons would correspond to M₁₀₇. The mass scale of M⁸⁹ hadrons would be by a factor 512 higher than that of ordinary hadrons and there are indications for the existence of scaled versions of mesons.

How M₈₉ hadrons could be created. The temperature T_H= 175 MeV is by a factor 1/512 lower than the mass scale of M₈₉ pion. Somehow the colliding nuclei or hadrons must provide the needed energy from their kinetic energy. What certainly happens is that this energy is materialized in the ordinary nuclear reaction to ordinary pions and other mesons. The mesons should correspond to closed flux tubes assignable to circular vortices of the highly turbulent hydrodynamics flow created in the collision.

Could roughly 512 mesonic flux tubes reconnect to circular but flattened long flux tubes having length of M₈₉ meson, which is 512 times that of ordinary pions? I have proposed this kind of process, analogous to BEC, to be fundamental in both biology (see this, this, and this) and also to explain the strange findings of Eric Reiter challenging some basic assumptions of nuclear physics if taken at face value (see this).

The process generating an analog of BEC would create in the first BSFR M₈₉ mesons having ℏ_eff/ℏ=512. In the second BSFR the transition ℏ_eff→ ℏ would take place and yield M⁸⁹ mesons. It would seem that part of the matter of the composite system ends up to n M₈₉ hadronic phase with 512 times higher T_H. In the number theoretic picture, these BEC like states would be Galois confined states (see this and this).

2.3 Can the size of a quark be larger than the size of a hadron?

The Compton wavelength Λ_c= ℏ/m is inversely proportional to mass. This implies that the Compton length of the quark as part of the hadron is longer than the Compton length of the hadron. If one assigns to Compton length a geometric interpretation as one does in M⁸-H duality mapping mass shell to CD with radius given by Compton length, this sounds paradoxical. How can a part be larger than the whole? One can think of many approaches to what might look like a paradox.

One could of course argue that being a part in the sense of tensor product has nothing to with being a part in geometric sense. However, if one requires quantum classical correspondence (QCC), one could argue that a hadron is a small region to which much larger quark 3-surfaces are attached.

One could also say that Compton length characterizes the size of the MB assignable to a particle which itself has size of order CP₂ length scale. In this case the strange looking situation would appear only at the level of MBs and the magnetic bodies could have sizes which increase when the particle mass decreases.

What if one takes QCC completely seriously? One can look at the situation in ZEO.

1. The size of the CD corresponds to Compton length and CDs for different particle masses have a common center and form a Russian doll-like hierarchy. One can continue the geodesic line defining point of CD associated with the hadron mass so that it intersects the CDs associated with quarks, in particular that for the lightest quark.
2. The distances between the quarks would define the size scale of the system in this largest CD and in the case of light hadrons containing U and D quarks it would be of the order of the Compton length of the lightest quark involved having mass about 5 MeV: this makes about .2 × 10^-13 m. There are indeed indications that the MB of proton has this size scale.

One could also require that there must be a common CD based on such an identification of h_eff for each particle that its size does not depend on the mass of the particles.

1. Here ℏ_gr= GMm/β₀ provides a possible solution. The size of the CD would correspond to Λ_gr =GM/v₀ for all particles involved. One could call this size the quantum gravitational size of the particle.

2. There is an intriguing observation related to this. To be in gravitational interaction could mean ℏ_eff=ℏ_gr=GMm/v₀ so that the size of the common CD would be given by Λ_gr= GMm/v₀. The minimum mass M given ℏ_gr>ℏ would be M=β₀ M_Pl²/m. For protons this gives M ≥ 1.5 × 10³⁸ m_p. Assuming density ρ ≈ 10³⁰A/m³, A the atomic number, the length L for the side cube with minimal mass M is L×β₀× 10²/A^1/3. For β₀= 2^-11 assignable to the Sun-Earth system, this gives L∼ 5/A^1/3 mm. The value of Λ_gr for Earth is 4.35 mm for β₀=1. The orders of magnitude are the same. Is this a mere accident?

One solution to the problem is that the ratio ℏ_eff(H)/ℏ_eff(q) is so large that the problem disappears.

1. If ℏ_eff(1)=ℏ, the value of ℏ_eff for hadron should be so large that the geometric intuitions are respected: this would require h_eff/h;≥ m_H/m_q. The hadrons containing u, d, and c quarks are very special.
2. Second option is that the value of h_eff for quarks is smaller than h to guarantee that the Compton length is not larger than ℏ. The perturbation theory for states consisting of free quarks would not converge since Kähler coupling strength α_K ∝ 1/ℏ_eff would be too large. This would conform with the QCD view and provide a reason for color confinement. Quarks would be dark matter in a well-defined sense.
3. The condition would be ℏ_eff(H)/ℏ_eff(q)≥ m(H)/m_q, where q is the lightest quark in the hadron. For heavy hadrons containing heavy quarks this condition would be rather mild. For light hadrons containing u,d, and c quarks it would be non-trivial. Ξ gives the condition ℏ/ℏ_eff≥ 262. The condition could not be satisfied for too small masses of the value of ℏ= 7!ℏ₀=5040ℏ₀ identifiable as the ratio of dark CP₂ deduced from p-adic mass calculations and Planck length.

See the article TGD and Quantum Hydrodynamics or the chapter chapter with the same title.

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

Articles and other material related to TGD.

The emergence of twistor spaces from M8-H duality

It has become clear that M⁸-H duality generalizes and there is a connection with the twistorialization at the level of H.

Space-time surfaces as images of associative surfaces in M⁸

M⁸-H duality would provide an explicit construction of space-time surfaces as algebraic surfaces with an associative normal space (see this, this, and this). M⁸ picture codes space-time surface by a real polynomial with rational coefficients. One cannot exclude coefficients in an extension of rationals and also analytic functions with rational or algebraic coefficients can be considered as well as polynomials of infinite degree obtained by repeated iteration giving rise algebraic numbers as extension and continuum or roots as limits of roots.

M⁸-H duality maps these solutions to H and one can consider several forms of this map. The weak form of the duality relies on holography mapping only 3-D or even 2-D data to H and the strongest form maps entire space-time surfaces to H. The twistor lift of TGD allows to identify the space-time surfaces in H as base spaces of 6-D surfaces representing the twistor space of space-time surface as an S² bundle in the product of twistor spaces of M⁴ and CP₂. These twistor spaces must have Kähler structure and only the twistor spaces of M⁴ and CP₂ have it so that TGD is unique also mathematically.

An interesting question relates to the possibility that also 6-D commutative space-time surfaces could be allowed. The normal space of the space-time surface would be a commutative subspace of M⁸_c and therefore 2-D. Commutative space-time would be a 6-D surface X⁶ in M⁸.

This raises the following question: Could the inverse image of the 6-D twistor-space of 4-D space-time surface X⁴ so that X⁶ would be M⁸ analog of twistor lift? This requires that X⁶⊂ M⁸_c has the structure of an S² bundle and there exists a bundle projection X⁶→ X⁴.

The normal space of an associative space-time surface actually contains this kind of commutative normal space! Its existence guarantees that the normal space of X⁴ corresponds to a point of CP₂. Could one obtain the M⁸_c analog of the twistor space and the bundle bundle projection X⁶→ X⁴ just by dropping the condition of associativity. Space-time surface would be a 4-surface obtained by adding the associativity condition.

One can go even further and consider 7-D surfaces of M⁸ with real and therefore well-ordered normal space. This would suggest dimensional hierarchy: 7→ 6→ 4.

This leads to a possible interpretation of twistor lift of TGD at the level of M⁸ and also about generalization of M⁸-H correspondence to the level of twistor lift. Also the generalization of twistor space to a 7-D space is suggestive. The following arguments representa vision about "how it must be" that emerged during the writing of this article and there are a lot of details to be checked.

Commutative 6-surfaces and twistorial generalization of M⁸-H correspondence

Consider first the twistorial generalization of M⁸-H correspondence.

1. The complex 6-D surface X⁶_c⊂ M⁸_c has commutative normal space and thus corresponds to complexified octonionic complex numbers (z₁+z₂I). X⁶_c has real dimension 12 just as the product T(M⁴)× T(CP₂) of 6-D twistor spaces of M⁴ and CP₂. It has a bundle structure with a complex 4-D base space which is mapped M⁴× CP₂ by M⁸H duality. The fiber has complex dimension 2 and corresponds to the dimension for the product of twistor spheres of the twistor spaces of M⁴ and CP₂.
2. This suggests that M⁸-H duality generalizes so that it maps X⁶_c ⊂ M⁸_c to T(M⁴)× T(CP₂) . It would map the point of X⁶_c to its real projection identified as a point of T(M⁴). "Real" means here that the complex continuation of the number theoretical norm squared for octonions is real so that the components of M⁸ point are either real or imaginary with respect to the commuting imaginary unit i. The complex 6-D tangent space of X⁶_c would be mapped to a point of T(CP₂).

   The beauty of this picture would be that the entire complex 6-D surface would carry physical information mapped directly to the twistor space.

One can try to guess the form of the map of X⁶_c to the product T(M⁴)× T(CP₂).

The surfaces X⁶ have local normal space basis 1⊕ e₇ . The problem is that this space is invariant under SU(3) for M⁸-H for CP₂. Could one choose the 2-D normal space to be something else without losing the duality. If e₇ and e₁ are permuted, the tangent space basis vector transforms by a phase phase factor under U(1)× U(1). The 4-D sub-basis of normal space would be now (1,e₁,e₇,e₂). This does not affect the M⁸-H-duality map to CP₂. The 6-D space of normal spaces would be the flag manifold SU(2)/U(1)times U(1), which is nothing but the twistor space T(CP₂).

What about the twistorial counterpart for the map of M⁴⊂ M⁸→ M⁴⊂ M⁸? One can consider several options.

1. At the level of M⁸, M⁴ is replaced by M⁶ at least locally in the sense that one can use M⁶ coordinates for the point of X⁶. Can one identify the M⁶ image of this space as the projective space C⁴/C_× obtained from C⁴ by dividing with complex scalings? This would give the twistor space CP₃= SU(4)/U(3) of M⁴. This is not obvious since one has (complexified) octonions rather than C⁴ or its hypercomplex analog. This would be analogous to using several (4) coordinate charts glued together as in the case of sphere CP₁.
2. If M⁸-H duality generalizes as such, the points of M⁶ could be mapped to the 6-D analog of cd₄ such that the image point is defined as the intersection of a geodesic line with direction given by the 6-D momentum with the 5-D light-like boundary of 6-D counterpart cd₆ of cd? Does the slicing of M⁶ by 5-D light-boundaries of cd₆ for various values of 6-D mass squared have interpretation as CP₃? Note that the boundary of cd₆ does not contain origin and the same applies to CP₃= C⁴/C_×.
3. Or could one identify the octonionic analog of the projective space CP₃=C⁴/C_×? Could the octonionic M⁸ momenta be scaled down by dividing with the momentum projection in the commutative normal space so that one obtains an analog of projective space? Could one use these as coordinates for M⁶?

   The scaled 8-momenta would correspond to the points of the octonionic analog of CP₃. The scaled down 8-D mass squared would have a constant value.

   A possible problem is that one must divide either from left or right and results are different in the general case. Could one require that the physical states are invariant under the automorphisms generated o→ gog^-1, where g is an element of the commutative subalgebra in question?

What about the physical interpretation at the level of M⁸_c.

1. The first thing to notice is that in the twistor Grassmann approach twistor space provides an elegant description of spin. Partial waves in the fiber S² of twistor space representation of spin as a partial wave. All spin values allow a unified treatment.

   The problem is that this requires massless particles. In the TGD framework 4-D masslessness is replaced with its 8-D variant so that this difficulty is circumvented. This kind of description in terms of partial waves is expected to have a counterpart at the level of the twistor space T(M(⁴)× T(CP₂). At level of M⁸ the description is expected to be in terms of discrete points of M⁸_c.

2. Consider first the real part of X⁶_c⊂ M⁸_c. At the level of M⁸ the points of X⁴ correspond to points. The same must be true also at the level of X⁶. Single point in the fiber space S² would be selected. The interpretation could be in terms of the selection of the spin quantization axis.

   Spin quantization axis corresponds to 2 diametrically opposite points of S². Could the choice of the point also fix the spin direction? There would be two spin directions and in the general case of a massive particle they must correspond to the values S_z= +/- 1/2 of fermion spin. For massless particles in the 4-D sense two helicities are possible and higher spins cannot be excluded. The allowance of only spin 1/2 particles conforms with the idea that all elementary particles are constructed from quarks and antiquarks. Fermionic statistics would mean that for fixed momentum one or both of the diametrically opposite points of S² defining the same and therefore unique spin quantization axis can be populated by quarks having opposite spins.

3. For the 6-D tangent space of X⁶_c or rather, its real projection, an analogous argument applies. The tangent space would be parametrized by a point of T(CP₂) and mapped to this point. The selection of a point in the fiber S² of T(CP₂) would correspond to the choice of the quantization axis of electroweak spin and diametrically opposite points would correspond to opposite values of electroweak spin 1/2 and unique quantization axis allows only single point or pair of diametrically opposite points to be populated.

   Spin 1/2 property would hold true for both ordinary and electroweak spins and this conforms with the properties of M⁴× CP₂ spinors.

4. The points of X⁶_c⊂ M⁸_c would represent geometrically the modes of H-spinor fields with fixed momentum. What about the orbital degrees of freedom associated with CP₂?

   M⁴ momenta represent orbital degrees of M⁴ spinors so that E⁴ parts of E⁸ momenta should represent the CP₂ momenta. The eigenvalue of CP₂ Laplacian defining mass squared eigenvalue in H should correspond to the mass squared value in E⁴ and to the square of the radius of sphere S³ ⊂ E⁴.

   This would be a concrete realization for the SO(4)=SU(2)_L× SU(2)_R↔ SU(3) duality between hadronic and quark descriptions of strong interaction physics. Proton as skyrmion would correspond to a map S³ with radius identified as proton mass. The skyrmion picture would generalize to the level of quarks and also to the level of bound states of quarks allowed by the number theoretical hierarchy with Galois confinement. This also includes bosons as Galois confined many quark states.

5. The bound states with higher spin formed by Galois confinement should have the same quantization axis in order that one can say that the spin in the direction of the quantization axis is well-defined. This freezes the S² degrees of freedom for the quarks of the composite.

7-surfaces with real normal space and generalization of the notion of twistor space

It would seem that twistorialization could correspond to the introduction of 6-surfaces of M⁸, which have commutative normal space. The next step is to ask whether it makes sense to consider 7-surfaces with a real norma space allowing well-ordering? This would give a hierarchy of surfaces of M⁸ with dimensions 7, 6, and 4. The 7-D space would have bundle projection to 6-D space having bundle projection to 4-D space.

What could be the physical interpretation of 7-D surfaces of M⁸ with real normal space in the octonionic sense and of their H images?

1. The first guess is that the images in H correspond to 7-D surfaces as generalizations of 6-D twistor space in the product of similar 7-D generalization of twistor spaces of M⁴ and CP₂. One would have a bundle projection to the twistor space and to the 4-D space-time.
2. SU(3)/U(1)× U(1) is the twistor space of CP₂. SU(3)/SU(2)× U(1) is the twistor space of M⁴? Could 7-D SU(3)/U(1) resp. SU(4)/SU(3) correspond to a generalization of the twistor spaces of M⁴ resp. CP₂? What could be the interpretation of the fiber added to the twistor spaces of M⁴, CP₂ and X⁴? S³ isomorphic to SU(2) and having SO(4) as isometries is the obvious candidate.
3. The analog of M⁸-H duality in Minkowskian sector in this case could be to use coordinates for M⁷ obtained by dividing M⁸ coordinates by the real part of the octonion. Is it possible to identify RP₇= M⁸/R_× with SU(4)/SU(3) or at least relate these spaces in a natural manner. It should be easy to answer these questions with some knowhow in practical topology.

   A possible source of problems or of understanding is the presence of a commuting imaginary unit implying that complexification is involved in Minkowskian degrees of freedom whereas in CP₂ degrees of freedom it has no effect. RP₇ is complexified to CP₇ and the octonionic analog of CP₃ is replaced with its complexification.

What could be the physical interpretation of the extended twistor space?

1. Twistorialization takes care of spin and electroweak spin. The remaining standard model quantum numbers are Kähler magnetic charges for M⁴ and CP₂ and quark number. Could the additional dimension allow their geometrization as partial waves in the 3-D fiber?

   The first thing to notice is that it is not possible to speak about the choice of quantization axis for U(1) charge. It is however possible to generalize the momentum space picture also to the 7-D branes X⁷ of M⁸ with real normal space and select only discrete points of cognitive representation carrying quarks. The coordinate of 7-D generalized momentum in the 1-D fiber would correspond to some charge interpreted as a U(1) momentum in the fiber of 7-D generalization of the twistor space.

2. One can start from the level of the 7-D surface with a real normal space. For both M⁴ and CP₂, a plausible guess for the identification of 3-D fiber space is as 3-sphere S³ having Hopf fibration S³→ S² with U(1) as a fiber.

   At H side one would have a wave exp(iQ φ/2π) in U(1) with charge Q and at M⁸ side a point of X⁷ representing Q as 7:th component of 7-D momentum.

   Note that for X⁶ as a counterpart of twistor space the 5:th and 6:th components of the generalized momentum would represent spin quantization axis and sign of quark spin as a point of S². Even the length of angular momentum might allow this kind representation.

3. Since both M⁴ and CP₂ allow induced Kähler field, a possible identification of Q would be as a Kähler magnetic charge. These charges are not conserved but in ZEO the non-conservation allows a description in terms of different values of the magnetic charge at opposite halfs of the light-cone of M⁸ or CD.

   Instanton number representing a change of magnetic charge would not be a charge in strict sense and drops from consideration.

One expects that the action in the 7-D situation is analogous to Chern-Simons action associated with 8-D Kahler action, perhaps identifiable as a complexified 4-D Kähler action.

1. At M⁴ side, the 7-D bundle would be SU(4)/SU(3)→ SU(4)/SU(3)× U(1). At CP₂ side the bundle would be SU(3)/U(1)→ SU(3)/U(1)× U(1).
2. For the induced bundle as 7-D surface in the SU(4)/SU(3)× SU(3)/U(1), the two U(1):s are identified. This would correspond to an identification φ(M⁴)= φ(CP₂) but also a more general correspondence φ(M⁴)= (n/m)φ(CP₂) can be considered. m/n can be seen as a fractional U(1) winding number or as a pair of winding numbers characterizing a closed curve on torus.
3. At M⁸ level, one would have Kähler magnetic charges Q_K(M⁴), Q_K(CP₂) represented associated with U(1) waves at twistor space level and as points of X⁷ at M⁸ level involving quark. The same wave would represent both M⁴ and CP₂ waves that would correlate the values of Kähler magnetic charges by Q_K,m(M⁴)/Q_K,m(CP₂)= m/n if both are non-vanishing. The value of the ratio m/n affects the dynamics of the 4-surfaces in M⁸ and via twistor lift the space-time surfaces in H.

How could the Grassmannians of standard twistor approach emerge number theoretically?

One can identify the TGD counterparts for various Grassmann manifolds appearing in the standard twistor approach.

Consider first, the various Grassmannians involved with the standard twistor approach (this) can be regarded as flag-manifolds of 4-complex dimensional space T.

1. Projective space is FP_n-1 the Grasmannian F₁(Fⁿ) formed by the k-D planes of Vⁿ where F corresponds to the field of real, complex or quaternionic numbers, are the simplest spaces of this kind. The F-dimension is d_F=n-1. In the complex case, this space can be identified as U(n)/U(n-1)× U(1)= CP_n-1.
2. More general flag manifolds carry at each point a flag, which carries a flag which carries ... so that one has a hierarchy of flag dimensions d₀=0<d₁<d₂...d_k=n. Defining integers n_i= d_i-d_i-1, this space can in the complex case be expressed as U(n)/U(n₁)×.....U(n_k). The real dimension of this space is d_R=n²-∑_in_i².
3. For n=4 and F=C, one has the following important Grassmannians.
   1. The twistor space CP₃ is projective is of complex planes in T=C⁴ and given by CP₃=U(4)/U(3)× U(1) and has real dimension d_R=6.
   2. M=F₂ as the space of complex 2-flags corresponds to U(4)/U(2)× U(2) and has d_R=16-8= 8. This space is identified as a complexified Minkowski space with D_C= 4.
   3. The space F_1,2 consisting of 2-D complex flags carrying 1-D complex flags has representation U(4)/U(2)× U(1)× U(1) and has dimension D_R=10.

      F_1,2 has natural projection ν to the twistor space CP₃ resulting from the symmetry breaking U(3)→ U(2)× U(1) when one assigns to 2-flag a 1-flag defining a preferred direction. F_1,2 also has a natural projection μ to the complexified and compactified Minkowski space M=F₂ resulting in the similar manner and is assignable to the symmetry breaking U(2)× U(2)→ U(1)× U(1) caused by the selection of 1-flag.

      These projections give rise to two correspondences known as Penrose transform. The correspondence μ ∘ ν_-1 assigns to a point of twistor space CP₃ a point of complexified Minkowski space. The correspondence ν ∘ μ_-1 assigns to the point of complexified Minkowski space a point of twistor space CP₃. These maps are obviously not unique without further conditions.

This picture generalizes to TGD and actually generalizes so that also the real Minkowski space is obtained naturally. Also the complexified Minkowski space has a natural interpretation in terms of extensions of rationals forcing complex algebraic integers as momenta. Galois confinement would guarantee that physical states as bound states have real momenta.

1. The basic space is Q_c=Q² identifiable as a complexified Minkowski space. The idea is that number theoretically preferred flags correspond to fields R,C,Q with real dimensions 1,2,4. One can interpret Q_c as Q² and Q as C² corresponding to the decomposition of quaternion to 2 complex numbers. C in turn decomposes to R× R.
2. The interpretation C²= C⁴ gives the above described standard spaces. Note that the complexified and compactified Minkowski space is not same as Q_c=Q² and it seems that in TGD framework Q_c is more natural and the quark momenta in M⁴_c indeed are complex numbers as algebraic integers of the extension.

Number theoretic hierarchy R→ C→ Q brings in some new elements.

1. It is natural to define also the quaternionic projective space Q_c/Q=Q²/Q (see this), which corresponds to real Minkowski space. By non-commutativity this space has two variants corresponding to left and right division by quaternionic scales factor. A natural condition is that the physical states are invariant under automorphisms q→ hqh^-1 and depend only on the class of the group element. For the rotation group this space is characterized by the direction of the rotation axis and by the rotation angle around it and is therefore 2-D.

   This space is projective space QP₁, quaternionic analog of Riemann sphere CP₁ and also the quaternionic analog of twistor space CP₃ as projective space. Therefore the analog of real Minkowski space emerges naturally in this framework. More generally, quaternionic projective spaces Qⁿ have dimension d=4n and are representable as coset spaces of symplectic groups defining the analogs of unitary/orthogonal groups for quaternions as Sp(n+1)/Sp(n)× Sp(1) as one can guess on basis of complex and real cases. M⁴_R would therefore correspond to Sp(2)/Sp(1)× SP(1).

   QP₁ is homeomorphic to 4-sphere S⁴ appearing in the construction of instanton solutions in E⁴ effectively compactified to S⁴ by the boundary conditions at infinity. An interesting question is whether the self-dual Kähler forms in E⁴ could give rise to M⁴ Kähler structure and could correspond to this kind of self-dual instantons and therefore what I have called Hamilton-Jacobi structures.

2. The complex flags can also contain real flags. For the counterparts of twistor spaces this means the replacement of U(1) with a trivial group in the decompositions.

   The twistor space CP₃ would be replaced U(4)/U(3) and has real dimension d_R=7. It has a natural projection to CP₃. The space F_1,2 is replaced with representation U(4)/U(2) and has dimension D_R=12.

To sum up, the Grassmannians associated with M⁴ as 6-D twistor space and its 7-D extension correspond to a complexification by a commutative imaginary unit i - that is "vertical direction". The Grassmannians associated with CP₂ correspond to "horizontal ", octonionic directions and to associative, commutative and well-ordered normal spaces of the space-time surface and its 6-D and 7-D extensions. Geometrization of the basic quantum states/numbers - not only momentum - representing them as points of these spaces is in question.

See the article Summary of TGD as it is towards end of 2021 or the chapter chapter with the same title.

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

Articles and other material related to TGD.

Braids, anyons, and Galois groups

Braids and anyons in the TGD framework are discussed here. Braid statistics has an interpretation in terms of rotations as homotopies at a 2-D plane of the space-time surfaces instead of rotations in M⁴. One can use M⁴ coordinates for the M⁴ projection of the space-time surface.

As a matter of fact, arbitrary isometry induced flows of H can be lifted to rotations as flows along the lifted curve at the space-time surface and for many-sheeted space-time the flows, which correspond to identity in H can lead to a different space-time sheet so that the braid groups structure emerges naturally (see here).

The representations of H isometries at the level of WCW act on the entire 3-surface identifiable as a generalized point-like particle and by holography on the entire space-time surface. The braid representations of isometries act inside the space-time surface. This suggests a generalization of the notions of gravitational and inertial masses so that they apply to all conserved charges. Generalization of Equivalence Principle would state that gravitational and inertial charges are identical.

The condition that the Dirac operator at the level of H has tangential part equivalent to the Dirac operator for induced spinors, implies that the conserved isometry currents of H are conserved along the flow lines of corresponding Killing vector fields and proportional to the Killing vectors lifted/projected to the space-time surface. This has an interpretation as a local hydrodynamics conservation law analogous to the conservation of ρ v²/2+p along a flow line.

One can ask whether the 2-dimensionality, which makes possible non-trivial and non-Abelian homotopy groups, is really necessary for the notion of the braid group in the TGD framework. As a matter of fact, the conditions are not expected to be possible for all conserved charges, and the intuitive guess that they hold true only for Cartan algebra representing maximal set of commuting observables would provide a space-time correlate of the Uncertainty Principle. If so, the space-time surface would depend on the choice of quantization axes. This conforms with quantum classical correspondence. For instance, the Cartan algebra of rotation group would act on a plane so that the effective 2-dimensionality of braid group and quantum group representations would hold true.

This view has some nice consequences.

1. If the space-time surface is n-sheeted, the rotation of 2π can take the particle to a different space-time sheet, and only n fold-rotation brings it back to its original position. The formula for fractional Hall conductivity is the same as in the case of integer Hall effect except that the 1/ℏ-proportionality is replaced with 1/ℏ_eff-proportionality in TGD framework (see this).
2. Degeneracy of fermion states also makes non-Abelian braid statistics possible. Since the Galois group acts as a symmetry group, the degeneracy would be naturally associated with the representations of the Galois group. Galois singletness of the many-anyon states guarantees reduces braid statistics to ordinary statistics for these. Galois confinement is proposed to be a central element of quantum biology (see this and this).

Braid statistics could also relate to the problem created by Bose-Einstein and Fermi statistics.

1. The problem is that many-boson and many-fermion states are maximally entangled so that state function reduction is in the QFT framework possible only for the entanglement between fermions and bosons.

   In the TGD framework the situation is even more difficult since all elementary particles can be constructed from quarks. The replacement of point-like particles with 3-surfaces however forces us to re-consider the notion of particle identity. Number theoretic definition of identity applying to cognitive representations is attractive.

2. The intuitive proposal is that Galois representations can entangle and that the reduction of entanglement is possible. In particular, the decomposition of extension to a hierarchy of extensions with Galois groups forming a hierarchy of normal subgroups allows the notions of cognitive measurement cascade \cite{btart/SSFRGalois}.
3. A more rigorous basis for the intuition emerges from the TGD view about braiding. The Galois group can be always represented as a subgroup of a suitable symmetric group S_n. S_n allows braidings and therefore induces a braiding of the Galois group. The discrete subgroups of symmetry groups of TGD could allow representation as a Galois group of the space-time surface. They could also allow braiding defined by the lift of the continuous isometry flow to the space-time surface. This suggests that the notion of a quantum group could allow a geometric interpretation in terms of the braiding based on the many-sheeted sub-manifold geometry.
4. The Galois group is in general non-Abelian and the braided Galois group would define braid statistics allowing higher-D representations. This would also make possible a non-maximal entanglement and the reduction of entanglement for the tensor products would be possible..

See the article TGD and condensed matter or the chapter with the same title.

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

Articles and other material related to TGD.

Adelic physics and quantum measurement theory

Adelic physics forces us to reconsider the notion of entanglement and what happens in state function reductions (SFRs). Let us leave the question whether the SFR can correspond to SSFR or BSFR or both open for a moment.

1. The natural assumption is that entanglement is a number-theoretically universal concept and therefore makes sense in both real and various p-adic senses. This is guaranteed if the entanglement coefficients are in an extension E of rationals associated with the polynomial Q defining the space-time surface in M⁸ and having rational coefficients.

   In the general case, the diagonalized density matrix ρ produced in a state function reduction (SFR) has eigenvalues in an extension E₁ of E. E₁ is defined by the characteristic polynomial P of ρ.

2. Is the selection of one of the eigenstates in SFR possible if E₁ is non-trivial? If not, then one would have a number-theoretic entanglement protection.
3. On the other hand, if the SFR can occur, does it require a phase transition replacing E with its extension by E₁ required by the diagonalization?

Let us consider the option in which E is replaced by an extension coding for the measured entanglement matrix so that something also happens to the space-time surface.

1. Suppose that the observer and measured system correspond to 4-surfaces defined by the polynomials O and S somehow composed to define the composite system and reflecting the asymmetric relationship between O and S. The simplest option is Q=O∘ S but one can also consider as representations of the measurement action deformations of the polynomial O× P making it irreducible. Composition conforms with the properties of tensor product since the dimension of extension of rationals for the composite is a product of dimensions for factors.
2. The loss of correlations would suggest that a classical correlate for the outcome is a union of uncorrelated surfaces defined by O and S or equivalently by the reducible polynomial defined by the O× S (see this). Information would be lost and the dimension for the resulting extension is the sum of dimensions for the composites. O however gains information and quantum classical correspondence (QCC) suggests that the polynomial O is replaced with a new one to realize this.
3. QCC suggests the replacement of the polynomial O the polynomial P∘ O, where P is the characteristic polynomial associated with the diagonalization of the density matrix ρ. The final state would be a union of surfaces represented by P∘ O and S: the information about the measured observable would correspond to the increase of complexity of the space-time surface associated with the observer. Information would be transferred from entangled Galois degrees of freedom including also fermionic ones to the geometric degrees of freedom P∘ O. The information about the outcome of the measurement would in turn be coded by the Galois groups and fermionic state.
4. This would give a direct quantum classical correspondence between entanglement matrices and polynomials defining space-time surfaces in M⁸. The space-time surface of O would store the measurement history as kinds of Akashic records. If the density matrix corresponds to a polynomial P which is a composite of polynomials, the measurement can add several new layers to the Galois hierarchy and gradually increase its height.

   The sequence of SFRs could correspond to a sequence of extensions of extensions of..... This would lead to the space-time analog of chaos as the outcome of iteration if the density matrices associated with entanglement coefficients correspond to a hierarchy of powers P^k.

Does this information transfer take place for both BSFRs and SSFRs? Concerning BSFRs the situation is not quite clear. For SSFRs it would occur naturally and there would be a connection with SSFRs to which I have associated cognitive measurement cascades (see this and this).

1. Consider an extension, which is a sequence of extensions E₁→ ..E_k → E_k+1..→ E_n defined by the composite polynomial P_n∘ ....∘ P₁. The lowest level corresponds to a simple Galois group having no non-trivial normal subgroups.
2. The state in the group algebra of Galois group G= G_n having G_n-1 as a normal subgroup can be expressed as an entangled state associated with the factor groups G_n/G_n-1 and subgroup G_n-1 and the first cognitive measurement in the cascade would reduce this entanglement. After that the process could but need not to continue down to G₁. Cognitive measurements considerably generalize the usual view about the pair formed by the observer and measured system and it is not clear whether O-S pair can be always represented in this manner as assumed above: also small deformations of the polynomial O× S can be considered.

   These considerations inspire the proposal the space-time surface assigned to the outcome of cognitive measurement G_k,G_k-1 corresponds to polynomial the Q_k,k-1∘ P_n, where Q_k,k-1 is the characteristic polynomial of the entanglement matrix in question.

See the article TGD Towards the End of 2021.

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

Articles and other material related to TGD. 


Uncertainty Principle and M8-H duality

The detailed realization of M⁸-H duality (for what this duality means, see this) involves still uncertainties. The quaternionic normal spaces containing fixed 2-space M² (or an integrable distribution of M²) are parametrized by points of CP₂. One can map the normal space to a point of CP₂.

The tough problem has been the precise correspondence between M⁴ points in M⁴× E⁴ and M⁴× CP₂ and the identification of the sizes of causal diamonds (CDs) in M⁸ and H. The identification is naturally linear if M⁸ is analog of space-time but if M⁸ is interpreted as momentum space, the situation changes. The ealier proposal maps mass hyperboloids to light-cone proper time =constant hyperboloids and it has turned out that this correspondence does not correspond to the classical picture suggesting that a given momentum in M⁸ corresponds in H to a geodesic line emanating from the tip of CD.

M⁸-H duality in M⁴ degrees of freedom

The following proposal for M⁸-H duality in M⁴ degrees of freedom relies on the intuition provided by UP and to the idea that a particle with momentum p^k corresponds to a geodesic line with this direction emanating from the tip of CD.

1. The first constraint comes from the requirement that the identification of the point p^k∈ X⁴⊂ M⁸ should classically correspond to a geodesic line m^k= p^kτ/m (p²=m²) in M⁸ which in Big Bang analogy should go through the tip of the CD in H. This geodesic line intersects the opposite boundary of CD at a unique point.

   Therefore the mass hyperboloid H³ is mapped to the 3-D opposite boundary of cd⊂ M⁴⊂ H. This does not fix the size nor position of the CD (=cd× CP₂) in H. If CD does not depend on m, the opposite light-cone boundary of CD would be covered an infinite number of times.

2. The condition that the map is 1-to-1 requires that the size of the CD in H is determined by the mass hyperboloid M⁸. Uncertainty Principle (UP) suggests that one should choose the distance T between the tips of the CD associated with m to be T= ℏ_eff/m.

   The image point m^k of p^k at the boundary of CD(m,h_eff) is given as the intersection of the geodesic line m^k= p^kτ from the origin of CD(m,h_eff) with the opposite boundary of CD(m,h_eff):

   m^k=ℏ_effX× (p^k/m²),

   X= 1/(1+ p₃/p₀) .

   Here p₃ is the length of 3-momentum.

   The map is non-linear. At the non-relativistic limit (X\rightarrow 1), one obtains a linear map for a given mass and also a consistency with the naive view about UP. m^k is on the proper time constant mass shell so the analog of the Fermi ball in H³ ⊂ M⁸ is mapped to the light-like boundary of cd⊂ M⁴⊂ H.

3. What about massless particles? The duality map is well defined for an arbitrary size of CD. If one defines the size of the CD as the Compton length ℏ_eff/m of the massless particle, the size of the CD is infinite. How to identify the CD? UP suggests a CD with temporal distance T= 2ℏ_eff/p₀ between its tips so that the geometric definition gives p^k= ℏ_effp^k/p₀² as the point at the 2-sphere defining the corner of CD. p-Adic thermodynamics strongly suggests that also massless particles generate very small p-adic mass, which is however proportional to 1/p rather than 1/p^{1/2. The map is well defined also for massless states as a limit and takes massless momenta to the 3-ball at which upper and lower half-cones meet.}
4. What about the position of the CD associated with the mass hyperboloid? It should be possible to map all momenta to geodesic lines going through the 3-ball dividing the largest CD involved with T determined by the smallest mass involved to two half-cones. This is because this 3-ball defines the geometric "Now" in TGD inspired theory of consciousness. Therefore all CDs in H should have a common center and have the same geometric "Now".

   M⁸-H duality maps the slicing of momentum space with positive/negative energy to a Russian doll-like slicing of t≥0 by the boundaries of half-cones, where t has origin at the bottom of the double-cone. The height of the CD(m,h_eff) is given by the Compton length L(m,h_eff) = ℏ_eff/m of quark. Each value of h_eff corresponds its own scaled map and for h_gr=GMm/v₀, the size of CD(m,h_eff)=GM/v₀ does not depend on m and is macroscopic for macroscopic systems such as Sun.

5. The points of cognitive representation at quark level must have momenta with components, which are algebraic integers for the extension of rationals considered. A natural momentum unit is m_Pl=ℏ₀/R, h₀ is the minimal value of h_eff=h₀ and R is CP₂ radius. Only "active" points of X⁴⊂ M⁸ containing quark are included in the cognitive representation. Active points give rise to active CD:s CD(m,h_eff) with size L(m,h_eff).

   It is possible to assign CD(m,h_eff) also to the composites of quarks with given mass. Galois confinement suggest a general mechanism for their formation: bound states as Galois singlets must have a rational total momentum. This gives a hierarchy of bound states of bound states of ..... realized as a hierarchy of CDs containing several CDs.

6. This picture fits nicely with the general properties of the space-time surfaces as associative "roots" of the octonionic continuation of a real polynomial. A second nice feature is that the notion of CD at the level H is forced by this correspondence. "Why CDs?" at the level of H has indeed been a longstanding puzzle. A further nice feature is that the size of the largest CD would be determined by the smallest momentum involved.
7. Positive and negative energy parts of zero energy states would correspond to opposite boundaries of CDs and at the level of M⁸ they would correspond to mass hyperboloids with opposite energies.
8. What could be the meaning of the occupied points of M⁸ containing fermion (quark)? Could the image of the mass hyperboloid containing occupied points correspond to sub-CD at the level of H containing corresponding points at its light-like boundary? If so, M⁸-H correspondence would also fix the hierarchy of CDs at the level of H.

It is enough to realize the analogs of plane waves only for the actualized momenta corresponding to quarks of the zero energy state. One can assign to CD as total momentum and passive resp. active half-cones give total momenta P_tot,P resp. P_tot,A, which at the limit of infinite size for CD should have the same magnitude and opposite sign in ZEO.

The above description of M⁸-H duality maps quarks at points of X⁴ ⊂ M⁸ to states of induced spinor field localized at the 3-D boundaries of CD but necessarily delocalized into the interior of the space-time surface X⁴ ⊂ H. This is analogous to a dispersion of a wave packet. One would obtain a wave picture in the interior.

Does Uncertainty Principle require delocalization in H or in X⁴?

One can argue that Uncertainty Principle (UP) requires more than the naive condition T=ℏ_eff/m on the size of sub-CD. I have already mentioned two approaches to the problem: they could be called inertial and gravitational representations.

1. The inertial representations assigns to the particle as a space-time surface (holography) an analog of plane wave as a superposition of space-time surfaces: this is natural at the level of WCW. This requires delocalization space-time surfaces and CD in H.
2. The gravitational representation relies on the analog of the braid representation of isometries in terms of the projections of their flows to the space-time surface. This does not require delocalization in H since it occurs in X⁴.

Consider first the inertial representation. The intuitive idea that a single point in M⁸ corresponds to a discretized plane wave in H in a spatial resolution defined by the total mass at the passive boundary of CD. UP requires that this plane wave should be realized at the level of H and also WCW as a superposition of shifted space-time surfaces defined by the above correspondence.

1. The basic observation leading to TGD is that in the TGD framework a particle as a point is replaced with a particle as a 3-surface, which by holography corresponds to 4-surface.

   Momentum eigenstate corresponds to a plane wave. Now planewave could correspond to a delocalized state of 3-surface - and by holography that of 4-surface - associated with a particle.

   A generalized plane wave would be a quantum superposition of shifted space-time surfaces inside a larger CD with a phase factor determined by the 4-momentum. M⁸-H duality would map the point of M⁸ containing an object with momentum p to a generalized plane wave in H. Periodic boundary conditions are natural and would force the quantization of momenta as multiples of momentum defined by the larger CD. Number theoretic vision requires that the superposition is discrete such that the values of the phase factor are roots of unity belonging to the extension of rationals associated with the space-time sheet. If momentum is conserved, the time evolutions for massive particles are scalings of CD between SSFRs are integer scalings. Also iterated integer scalings, say by 2 are possible.

2. This would also provide WCW description. Recent physics relies on the assumption about single background space-time: WCW is effectively replaced with M⁴ since 3-surface is replaced with point and CP₂ is forgotten so that one must introduce gauge fields and metric as primary field variables.

As already discussed, the gravitational representation would rely on the lift/projection of the flows defined by the isometry generators to the space-time surface and could be regarded as a "subjective" representation of the symmetries. The gravitational representation would generalize braid group and quantum group representations.

The condition that the "projection" of the Dirac operator in H is equal to the modified Dirac operator, implies a hydrodynamic picture. In particular, the projections of isometry generators are conserved along the lifted flow lines of isometries and are proportional to the projections of Killing vectors. QCC suggests that only Cartan algebra isometries allow this lift so that each choice of quantization axis would also select a space-time surface and would be a higher level quantum measurement.

See the article TGD as it is towards the end of 2021 or the chapter with the same title.

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

Articles and other material related to TGD.

Galois confinement

The notion of Galois confinement emerged in TGD inspired biology. Galois group for the extension of rationals determined by the polynomial defining the space-time surface X⁴⊂ M⁸ acts as a number theoretical symmetry group and therefore also as a physical symmetry group.

1. The idea that physical states are Galois singlets transforming trivially under the Galois group emerged first in quantum biology. TGD suggests that ordinary genetic code is accompanied by dark realizations at the level of magnetic body (MB) realized in terms of dark proton triplets at flux tubes parallel to DNA strands and as dark photon triplets ideal for communication and control. Galois confinement is analogous to color confinement and would guarantee that dark codons and even genes, and gene pairs of the DNA double strand behave as quantum coherent units.
2. The idea generalizes also to nuclear physics and suggests an interpretation for the findings claimed by Eric Reiter in terms of dark N-gamma rays analogous to BECs and forming Galois singlets. They would be emitted by N-nuclei - also Galois singlets - quantum coherently. Note that the findings of Reiter are not taken seriously because he makes certain unrealistic claims concerning quantum theory.

Galois confinement as a number theoretically universal manner to form bound states?

It seems that Galois confinement might define a notion much more general than thought originally. To understand what is involved, it is best to proceed by making questions.

1. Why not also hadrons could be Galois singlets so that the somewhat mysterious color confinement would reduce to Galois confinement? This would require the reduction of the color group to its discrete subgroup acting as Galois group in cognitive representations. Could also nuclei be regarded as Galois confined states? I have indeed proposed that the protons of dark proton triplets are connected by color bonds.
2. Could all bound states be Galois singlets? The formation of bound states is a poorly understood phenomenon in QFTs. Could number theoretical physics provide a universal mechanism for the formation of bound states. The elegance of this notion is that it makes the notion of bound state number theoretically universal, making sense also in the p-adic sectors of the adele.
3. Which symmetry groups could/should reduce to their discrete counterparts? TGD differs from standard in that Poincare symmetries and color symmetries are isometries of H and their action inside the space-time surface is not well-defined. At the level of M⁸ octonionic automorphism group G₂ containing as its subgroup SU(3) and quaternionic automorphism group SO(3) acts in this way. Also super-symplectic transformations of δ M⁴_+/-× CP₂ act at the level of H. In contrast to this, weak gauge transformations acting as holonomies act in the tangent space of H.

   One can argue that the symmetries of H and even of WCW should/could have a reduction to a discrete subgroup acting at the level of X⁴. The natural guess is that the group in question is Galois group acting on cognitive representation consisting of points (momenta) of M⁸_c with coordinates, which are algebraic integers for the extension.

   Momenta as points of M⁸_c would provide the fundamental representation of the Galois group. Galois singlet property would state that the sum of (in general complex) momenta is a rational integer invariant under Galois group. If it is a more general rational number, one would have fractionation of momentum and more generally charge fractionation. Hadrons, nuclei, atoms, molecules, Cooper pairs, etc.. would consist of particles with momenta, whose components are algebraic, possibly complex, integers. Also other quantum numbers, in particular color, would correspond to representations of the Galois group. In the case of angular moment Galois confinement would allow algebraic half-integer valued angular momenta summing up to the usual half-odd integer valued spin.

4. Why Galois confinement would be needed? For particles in a box of size L the momenta are integer valued as multiples of the basic unit p₀= ℏ n× 2π/L. Group transformations for the Cartan group are typically represented as exponential factors which must be roots of unity for discrete groups. For rational valued momenta this fixes the allowed values of group parameters. In the case of plane waves, momentum quantization is implied by periodic boundary conditions.

   For algebraic integers the conditions satisfied by rational momenta in general fail. Galois confinement for the momenta would however guarantee that they are integer valued and boundary conditions can be satisfied for the bound states.

Two further remarks are in order.

1. Besides the simplest realization also a higher level realization is possible: Galois singlets are not realized in the space of momenta but in the space of wavefunctions of momenta. States of an electron in an atom serve as an analogy. Origin is invariant under the rotation group and electron at origin would be the classical analog of a rotationally invariant state. In quantum theory, this state is replaced with an s-wave invariant under rotations although its argument is not.

   In the recent situation, one would have a wave function in the space of algebraic integers representing momenta, which are not Galois invariants but if one has Galois singlet, the average momentum as Galois invariant is ordinary integer. Also single-quark states could be Galois invariant in this sense.

2. The proposal inspired by TGD inspired quantum biology is that the polynomials defining 4-surface in M⁸ vanish at origin: P(0)=0. One can form increasingly complex 4-surfaces in M⁸ by forming composite polynomials P_n∘ P_n-1∘ ...∘ P₁ and these polynomials have roots of P₁....and P_n-1 as their roots. These roots are like conserved genes: also the momentum spectra of Galois singlets are analogous to conserved genes. This construction applies to Galois singlets in both classical and quantal sense.

   At the highest level one can construct states as singlets under the entire Galois group. One can use non-singlets of previous level as building bricks of these singlets.

See the articles TGD as it is towards the end of 2021 and TGD and condensed matter physics

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

Bird's Eye of View about the Topics of the Book "TGD and Condensed Matter"

This book tries to provide a view about the applications of TGD to condensed matter physics. Quantum TGD in its recent form. Quantum TGD relies on two different views about physics: physics as an infinite-dimensional spinor geometry based on the notion of "World of Classical Worlds" (WCW) and physics as a generalized number theory.

WCW picture generalizes Einstein's geometrization program to a geometrization of the entire quantum physics. Number theoretic vision states that so-called adelic physics provides a dual view about physics.

M⁸-H duality realizes these dual views in terms of space-time surfaces X⁴⊂ H and X⁴⊂ M⁸ mapped to each other byM⁸-H duality. This duality turns out to be a generalization of momentum-position duality of wave mechanics. Also the duality of number theory and geometry suggested by Langlands correspondence pops up into mind.

The view about physics at the level of H

An important guiding principle in the development of TGD has been quantum classical correspondence (QCC), whose most profound implications follow almost trivially from the basic structure of the classical theory forming an exact part of quantum theory (here TGD differs from quantum field theories (QFTs)).

1. 4-D General Coordinate Invariance (GCI) forces holography: the space-time surface associated with a given 3-surface is almost unique as an analog of Bohr orbit. X⁴ is therefore a preferred extremal of an action principle. This realizes QCC at space-time level and leads to zero energy ontology (ZEO) generalizing the ontology of standard physics.
2. The new view about space-time as 4-surface X⁴⊂ H= M⁴× CP₂ is central for applications. One manner to formulate this is that X⁴ is simultaneously minimal surface and extremal of Kähler action S_K analogous to Maxwell action. The twistor lift of TGD forces the presence of both S_K and the volume term in the action.
3. Especially important minimal surfaces are CP₂ type extremals representing building bricks of elementary particles, cosmic strings and magnetic flux tubes as their deformations so that their M⁴ and CP₂ projections have dimension larger than 2, and so called massless extremals (MEs). Magnetic flux tubes appear as two variants depending on whether they carry monopole flux or not. Monopole flux tubes require no current to create the magnetic field and are not possible in Maxwellian theory. Both are in a crucial role also in condensed matter applications.
4. The new view about space-time differs dramatically from that of GRT. The space-time surface is topologically non-trivial in all scales and many-sheeted in the sense that CP₂ coordinates as function of M⁴ coordinates and vice versa are many-valued. The space-time of GRT is obtained from the many-sheeted one in long length scale limit by replacing the sheets with a single region of M⁴ and by deforming its metric. The gauge potentials are defined as sums of induced gauge potentials for sheets. The deviation of the metric is the sum of the deviations of the induced metric from the M⁴ metric.

   The new physics related to the many-sheetedness is not describable in terms of the QFT approach.

5. The classical field equations reduce to conservation laws for the conserved charges defined by the isometries of H. Therefore they are essentially hydrodynamical and this together with QCC is essential for TGD inspired quantum hydrodynamics (QHD). The conjecture that the extremals allow generalized Beltrami property, which implies the existence of a global coordinate varying along the flow lines of flow. For instance, Beltrami property provides purely classical geometric correlates for supra flows and supracurrents. Global coordinates allow identification of order parameters having interpretation in terms of quantum coherence.
6. The requirement that modified Dirac operator at the level of space-time surface is in a well-defined sense a projection of the Dirac operator of H implies that for preferred extremals the isometry currents are proportional to projections if the corresponding Killing vectors with proportionality factor constant along the projections of their flow lines. This implies as generalization of the energy conservation along flow lines of hydrodynamical flow (ρ v²/2+p=constant).

   This also leads to a braiding type representations for isometry flows of H in theirs of their projections to the space-time surface and it seems that quantum groups emerge from these representations. Physical intuition suggests that only the Cartan algebra corresponding to commuting observables allows this representation so that the selection of quantization axes would select also space-time surface as a higher level state function reduction.

   One also ends up to a generalization of Equivalence Principle stating that the charges assignable to "inertial" or "objective" representations of H isometries in WCW affecting space-time surfaces as analogs of particles are identical with the charges of "gravitational" or subjective representations which act inside space-time surfaces. This has also implications for M⁸-H duality.

The view about physics at the level of M⁸

Over the years, the number theoretical vision has evolved to what I call adelic physics. M⁸-H duality as analog of momentum-position duality and of Complementarity Principle crystallizes number theoretical vision.

1. Complexified octonions M⁸_c have interpretation as an analog of momentum space. There are 4-surfaces in both M⁸ and H and they are related by M⁸-H duality. 4-surfaces X⁴ ⊂ M⁸ have associative normal spaces and are are defined as "roots" of an octonionic polynomial defined as an algebraic continuation of a real polynomial P with rational coefficients.

   A real polynomial (its roots) therefore defines the entire 4-surface, which means holography taken to an extreme. This also motivates the preferred extremal property at the H side, where one has partial differential equations and variational principle instead of algebraic equations. The analog of Bohr orbit property is one manner to formulate the restrictions.

2. The notion of cognitive representation is motivated by adelic physics, and corresponds to a subset of points of X⁴ ⊂ M⁸ in M⁸_c such that the coordinates of the points are in the extension of rationals defined by P. They define a unique discretization of the 4-surface. Cognitive representation contains common points of the real 4-surface and its p-adic variants and makes possible the number-theoretical universality of adelic physics. It is not completely clear whether the cognitive representations are needed only on the M⁸ side or at both sides of the duality.
3. The interpretation of the points of M⁸ as momenta leads to the proposal that the points of cognitive representation are algebraic integers. Contrary to the naive expectations, all algebraic numbers of extension belong to the cognitive representation for the roots of octonionic polynomials. A natural restriction is that 4-momenta correspond to algebraic integers. A further restriction is that the "active" points of the cognitive representation are occupied by quarks (in TGD leptons can correspond to bound states of quarks). Therefore the 4-surface in H is analogous to the Fermi ball containing discrete quark momenta as an analog of cognitive representation. Condensed matter physics relies strongly on the use of momentum space so that M⁸ picture is central for the applications in condensed matter.
4. Galois group of the polynomial defining X⁴ ⊂ M⁸ acts as symmetries for the roots of the polynomials. The order of Galois group of P is identified as effective Planck constant h_eff/h₀=n, where the ordinary Planck constant is a multiple of h₀. n is in general not the same as the dimension of extension defined by the degree of P. The original identification was as the dimension of extension counting the number of roots. One must keep an open mind here.

   The identification of as h_eff as the order of the Galois group (rather than dimension of extension) finds support from the following. The order gives the number of regions at the orbit of Galois group (in the case that the isotropy group of the point is trivial!) and at the level of H, the action is sum of identical contributions over these regions so that Planck constant would be nh₀.

   The phases of matter with different values of h_eff behave like dark matter relative to each other (this does not of course imply that galactic dark matter and energy correspond to h_eff>h phases). Various quantum scales, typically proportional to h_eff, can be arbitrarily long and quantum coherence is possible in all scales and assigned with the magnetic body (MB).

   Galois confinement generalizes the notion of periodic boundary conditions and could serve as a universal mechanism for the formation of bound states. One implication is that the total momentum of a bound state formed by quarks has total momentum, which is a rational integer. This serves as an extremely powerful constraint.

   The evolutionary hierarchies formed by the extensions of rationals defined by functional composition of polynomials are characterized by root inheritance if the condition P(0)=0 is satisfied. This gives rise to an evolutionary hierarchy of bound states such that the new level contains all bound states of the previous level (conserved genes serve as an analogy). What is nice is that the states of Galois representations which are not Galois singlets can serve as composites of Galois singlets at the next level.

   For these reasons Galois confinement is expected to play a fundamental role in TGD and in the TGD based view about condensed matter.

5. Nottale introduced the notion of gravitational Planck constant ℏ_eff=ℏ_gr = GMm/v₀, which allows us to understand the planetary planets as Bohr orbits. The form of ℏ_gr is dictated by the Equivalence Principle. In the TGD framework, the hierarchy of Planck constants, the infinite range of gravitational interaction, and the absence of screening motivate this notion.

   The gravitational Compton length for a particle with mass m is Λ_gr = GM/v₀=r_sc/2v₀ and thus expressible in terms of Schwarzschild radius and has r_s/2 as lower bound. For Earth of order TGD view about living matter involves ℏ_gr in an essential manner.

   Amazingly, there are indications that Λ_gr for Earth could play a key role in super-conductivity, superfluidity, and in quantum analogies of hydrodynamical systems. Also the role of Λ_gr for Sun is suggestive. Note that the original motivation for the large value of h_eff was the TGD based model for effects of em radiation at ELF frequencies on the vertebrate brain.

M⁸-H duality

The proposal is that the description of physics in terms of geometry and number theory are dual to each other. There are several observations motivating M⁸-H duality.

1. There are four classical number fields: reals, complex numbers, quaternions, and octonions with dimensions 1,2,4,8. The dimension of the embedding space is D(H)= 8, the dimension of octonions. Spacetime surface has dimension D(X⁴)=4 of quaternions. String world sheet and partonic 2-surface have dimension D(X²) =2 of: complex numbers. The dimension D(string)=1 of string is that of reals.
2. Isometry group of octonions is a subgroup of automorphism group G₂ of octonions containing SU(3) as a subgroup. CP₂=SU(3)/U(2) parametrizes quaternionic 4-surfaces containing a fixed complex plane.

Could M⁸ and H= M⁴× CP₂ provide alternative dual descriptions of physics?

1. Actually a complexification M⁸_c== E⁸_c by adding an imaginary unit i commuting with octonion units is needed in order to obtain sub-spaces with real number theoretic norm squared. M⁸_c fails to be a field since 1/o does not exist if the complex valued octonionic norm squared ∑ o_i² vanishes.
2. The four-surfaces X⁴ ⊂ M⁸ are identified as "real" parts of 8-D complexified 4-surfaces X⁴_c by requiring that M⁴⊂ M⁸ coordinates are either imaginary or real so that the number theoretic metric defined by octonionic norm is real. Note that the imaginary unit defining the complexification commutes with octonionic imaginary units and number theoretical norm squared is given by ∑_i z_i² which in the general case is complex.
3. The space H would provide a geometric description, classical physics based on Riemann metric, differential geometric structures and partial differential equations deduced from an action principle. M⁸_c would provide a number theoretic description: no partial differential equations, no Riemannian metric, no connections...

   M⁸_c has only the number theoretic norm squared and bilinear form, which are real only if M⁸_c coordinates are real or imaginary. This would define "physicality". One open question is whether all signatures for the number theoretic metric of X⁴ should be allowed? Similar problem is encountered in the twistor Grassmannian approach.

4. The basic objection is that the number of algebraic surfaces is very small and they are extremely simple as compared to extremals of action principle. Second problem is that there are no coupling constants at the level of M⁸ defined by action.

   Preferred extremal property realizes quantum criticality with universal dynamics with no dependence on coupling constants. This conforms with the disappearance of the coupling constants from the field equations for preferred extremals in H except at singularities, with the Bohr orbitology, holography and ZEO. X⁴⊂ H is analogous to a soap film spanned by frame representing singularities and implying a failure of complete universality.

5. In M⁸, the dynamics determined by an action principle is replaced with the condition that the normal space of X⁴ in M⁸ is associative/quaternionic. The distribution of normal spaces is always integrable to a 4-surface.

   One cannot exclude the possibility that the normal space is complex 2-space, this would give a 6-D surface. Also this kind of surfaces are obtained and even 7-D with a real normal space. They are interpreted as analogs of branes and are in central role in TGD inspired biology.

   Could the twistor space of the space-time surface at the level of H have this kind of 6-surface as M⁸ counterpart? Could M⁸-H duality relate these spaces in 16-D M⁸_c to the twistor spaces of the space-time surface as 6-surfaces in 12-D T(M⁴)× T(CP₂)?

6. Symmetries in M⁸ number theoretic: octonionic automorphism group G₂ which is complexified and contains SO(1,3). G₂ contains SU(3) as M⁸ counterpart of color SU(3) in H. Contains also SO(3) as automorphisms of quaternionic subspaces. Could this group appear as an (approximate) dynamical gauge group?

   M⁸=M⁴× E⁴ as SO(4) as a subgroup. It is not an automorphism group of octonions but leaves the octonion norm squared invariant. Could it be analogous to the holonomy group U(2) of CP₂, which is not an isometry group and indeed is a spontaneously broken symmetry.

   A connection with hadron physics is highly suggestive. SO(4)=SU(2)_L× SU(2)_R acts as the symmetry group of skyrmions identified as maps from a ball of M⁴ to the sphere S³⊂ E⁴. Could hadron physics ↔ quark physics duality correspond to M⁸-H duality. The radius of S³ is proton mass: this would suggest that M⁸ has an interpretation as an analog of momentum space.

   One implication of M⁸-H duality is that the image of a generic point of X⁴⊂ M⁸ is a single point of CP₂. There is complete localization in color degrees of freedom in Einsteinian regions of space-time having 4-D M⁴ projection and one cannot even speak of color. This would solve in a trivial manner the problem of color confinement.

   CP₂ type extremals however correspond to singularities for which a single point of line has ill-defined normal space and normal spaces correspond to a 3-D surface in CP₂. In this case, one can assign representations of color group SU(3) to the image of the line which is essentially CP₂. Color therefore makes sense inside the Euclidean wormhole contacts.

   For the string-like entities, the singularity at a given point of string world sheets corresponds to a 2-D surface of CP₂, which is a complex manifold or Lagrangian manifold. For the two geodesic spheres the color group reduces to U(2) or SO(3), and one can speak about spontaneous breaking of the color symmetry. Also S¹ singularity is possible for objects with 3-D M⁴ projection. In this case the color symmetry reduces to U(1).

7. What is the interpretation of M⁸? Massless Dirac equation in M⁸ for the octonionic spinors must be algebraic. This would be analogous to the momentum space Dirac equation. Solutions would be discrete points having interpretation as quark momenta! Quarks pick up discrete points of X⁴⊂ M⁸.

   States turn out to be massive in the M⁴ sense: this solves the basic problem of 4-D twistor approach (it works y for massless states only). Fermi ball is replaced with a region of a mass shell (hyperbolic space H³).

   M⁸ duality would generalize the momentum-position duality of the wave mechanics. QFT does not generalize this duality since momenta and position are not anymore operators.

See the book TGD and Condensed Matter.

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

Articles and other material related to TGD.