^{4}⊂ M

^{8}acts as a number theoretical symmetry group and therefore also as a physical symmetry group.

- 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.
- 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.

- 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.
- 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.
- 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 M
^{8}octonionic automorphism group G_{2}containing as its subgroup SU(3) and quaternionic automorphism group SO(3) acts in this way. Also super-symplectic transformations of δ M^{4}_{+/-}× CP_{2}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 X

^{4}. The natural guess is that the group in question is Galois group acting on cognitive representation consisting of points (momenta) of M^{8}_{c}with coordinates, which are algebraic integers for the extension.Momenta as points of M

^{8}_{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. - 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 p
_{0}= 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.

- 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.

- The proposal inspired by TGD inspired quantum biology is that the polynomials defining 4-surface in M
^{8}vanish at origin: P(0)=0. One can form increasingly complex 4-surfaces in M^{8}by forming composite polynomials P_{n}∘ P_{n-1}∘ ...∘ P_{1}and these polynomials have roots of P_{1}....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.

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

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