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.