_{eff}is so large gauge coupling strength proportional to 1/ℏ

_{eff}becomes so small that perturbation series converges. In the sequel a generalization of this connection and also the notion of quantum holography.

It is convenient to call the pair of a fermion and antifermion with vanishing total quantum numbers (apart from momentum) a "glue particle" . Galois singlet property would be a natural additional property of the glue particles formed by fermion antifermion pairs. One can also imagine a generalization of the proposed equivalence between "Mother Nature who likes her theorists" principle and holography principle.

**Could "glue particles" be also Galois singlets**

For hadrons, and perhaps quite generally, they would be color entangled color singlets with vanishing total quantum numbers (momentum forms an exception) but without any other kind of entanglement.

Galois confinement implies that the components of momentum are integers in the scale determined by the causal diamond (CD). Without this condition, the momentum components would be in general complex algebraic numbers. The 4-momenta can be however tachyonic so that analogs of virtual particles with quantized 4-momenta and negative mass squared value (integer) would be in question. The virtual masses of the glue particles could be tachyonic suggesting and interpretation as an analog of Coulomb potential.

This suggests that color singlet property could be strengthened with the Galois singlet property.

**Hierarchy of pairings associated with a hierarchy of MBs**

Number theoretic view (see this, this, this, this) of TGD predicts hierarchies of magnetic bodies (MBs) with levels labelled increasing value of h_{eff}. Galois confinement as a candidate for a universal mechanism for the formation of bound states predicts a hierarchy of Galois singlets as physical states.

- One could take Galois singlets at a given level of the hierarchy with h
_{eff}≥ h and deform them to Galois non-siglets, and form their bound states as Galois singlets. This would give an entire hierarchy of bound states formed by the proposed mechanism of quantum holography and assignable to the slaving hierarchy of MBs. - The holographic pairing would be only between the fundamental fermions and antifermions assignable to the MBs which are nearest neighbours in the hierarchy. The pairs, "glue particles", would have vanishing net quantum numbers other than four-momenta.
The total energy would be sum over contributions from various levels in the magnetic hierarchy. The masses of the fundamental fermions are very small as compared to the magnetic energies, and the color magnetic energies for the nucleons would give a dominant contribution. Higher hierarchy levels would give only a small contribution.

- At least in the case of hadrons, the holography would be by a formation of glue particles as meson-like pairs of a quark at with h
_{eff,1}and dark quark with h_{eff,2}>h_{eff,1}, having vanishing electroweak quantum numbers and spin and being color entangled color singlets. Also Galois singlet property looks very natural. - For example, U-shaped radial gravitational flux tube loops mediating gravitational interaction and also other interacting flux tubes could realize the holography. The fermion and antifermion at flux tube would be located at strings connecting wormhole contacts so that one would have direct analogy with the AdS/CFT holography but AdS interior replaced by the interior of the space-time surface.

**Physical interpretation of the glue particles**

What could be the physical interpretation of the pairing of quarks and antiquarks to glue particles. In the case of leptons the simplest scenario would be that leptons are bound states of quarks in CP_{2} scale so that the pairing would reduce to quark-antiquark pairing also in this case.

- Could the glue particles defining the holography correspond to an interaction potential energy in the classical description? In accordance with the string model picture, the pairs would reside at strings inside monopole flux tubes. Glue particles could also be seen as analogs of virtual boson-like particles with vanishing quantum numbers (total momenta could be non-vanishing) responsible for the binding between fermions and antifermions.
- If gluons and even electroweak bosons appear as partons also their pairs are formed. It has been proposed that gravitons can be expressed as pairs of gauge bosons (gravitation is "square" of gauge theory). Could these pairs have interpretation as virtual (possibly "strong") gravitons with a vanishing spin. This is analogous to AdS/CFT correspondence.
- Black hole evaporation can be formally regarded as a generation of pairs with the members of pair going to opposite sides of the horizon. Could one regard the glue particles as analogs of virtual pairs of this kind.

**Symmetry breaking is necessary**

At least at the hadron level, quarks and antiquarks and perhaps also gluons are involved, but pair into color singlets by quantum entanglement in color degrees of freedom. Other forms of entanglement are not allowed by the proposed form of holography.

- The glue particles are entangled only in color degrees of freedom and differ from gauge bosons and Higgs, which are in TGD framework superpositions of fermion pairs and are quantum entangled with respect to spin and weak isospin.
- The total quantum numbers of glue particles vanish but symmetry breaking SO(3) → SO(2) takes place. SO(2) would naturally correspond to the direction of the magnetic field in the flux tube. The same happens also in the case of weak interactions and could correspond to electroweak symmetry breaking.
- Could the Bose-Einstein condensate for glue particles made of gauge bosons serve an analogue of the sigma meson condensate in hadron physics. The sigma analogy would be a scalar only with respect to the SO(2) ⊂ SO(3). Could sigma mesons be associated with the pairing of hadrons and its magnetic body?

**How could one understand masses?**

A test for the proposal is whether one can understand the masses of macroscopic systems.

- If the paired fundamental fermions are each other's antiparticles, they must be fundamental fermions or bosons such as gluons (which also reduce to fermion-antifermion pairs). Sensible values of mass are expected if one has a hierarchy in levels such that the energies are sums of the magnetic energies and fermionic energies from various levels. Given level would give only the magnetic contribution and fermion contribution of fermions at it. Its scale would be determined by the p-adic scale assignable to the level.
- Virtual dark quarks at the strands and their ordinary counterparts at the ends of the strands, have very low-mass compared to the contribution of Kähler magnetic energy to the mass. The color magnetic energy at the hadron level would practically give almost the entire mass. This could hold true also at higher levels of the hierarchy of layers of MB with decreasing magnetic energies.
- The hierarchy of magnetic bodies would give a dominant contribution to the mass at the lowest level and the contribution of the few lowest levels could dominate the mass because the energy/strand tension of the magnetic flux tube quickly approaches zero as the strand thickens.

**Earth as an example**

It is instructive to consider the Earth as an example.

- The mass of the Earth's MB in the exterior of Earth is negligible when compared to the mass of the Earth as a simple order of magnitude estimate shows. The assumption that the monopole flux tubes with magnetic field strength of order of Earth's magnetic field carry quantized monopole flux implies that their radii are at least of the of order magnetic length of order cell size and fixes the string tension as the density of magnetic energy per unit length. The mass of the flux tube of length L is proportional to L/S, where S the transverse area of the flux tube. Assume that the flux tubes have length L of order of the size of the magnetosphere. Assume that the flux tubes fill the entire volume with scale given by the radius of the magnetosphere.
With these assumptions the total magnetic assignable to the monopole flux tubes is a negligible fraction of the mass of Earth determined by the lowest, nucleonic level of the hierarchy.

- In the interior of the Earth one would have a flux tube spaghetti and flux tubes within flux tubes corresponding to the magnetic slaving hierarchy. The color magnetic energy associated with nucleonic monopole flux tubes would give a dominating contribution to the Earth's mass. There would be atomic nuclei with mass number A with nucleon flux tubes with radius of order nucleon size. The flux tubes with a thickness of the order of the size of an atom would give a much smaller contribution to the magnetic energy. Fractality would therefore reduce the situation to nucleon level as far as masses are considered.
This idea is actually already old: also the interior of a star would be like this. In condensed matter for a region with size of an atom, the number of nucleon flux tubes equals the atomic weight A of the nucleus.

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