Ulla gave in the comment section of previous posting a link to an article by Bilson-Thompson , Hackett , Kauffman, and Wan with title "Emergent Braided Matter of Quantum Geometry". The article summarizes the recent state of an attempt to replace space-time continuum with discrete structure involving braids. The article satisfies high technical standards - note that one of the authors is Louis Kauffman, a leading knot theorist. The mathematician inside me is however skeptic for several reasons.
- Continuous symmetries - in particular Lorentz and Poincare invariance are lost. One must however get continuous space-time, which we use to organize our observational data and this requires ad hoc assumptions to get "long wave length limit" of the theory.
The notion of space-time dimension becomes questionable. In algebraic topology continuous space is replaced with a web of simplexes of various dimensions embedded into the space. The simplexes with maximum number of vertices define the dimension of manifold. Genuine discretization would not allow the imbedding and one would lose all information about manifold. Everything would reduce to combinatorics and trying to get continuous space-time from mere combinatorics is like squaring of circle.
The notion of distance realized in terms of Riemann geometry is fundamental for (quantum) physics. Without it one has just topological quantum field theory. Braids indeed appear naturally in topological quantum theory. The notion of metric must be introduced in ad hoc manner if space-time is discretized.
- The proposed approach identifies particles as 3-braids (this does not work and sin network is essential for the proposed generalization). The introduction of braiding is in conflict with the original idea about discreteness at fundamental level. Braiding requires the imbedding of the spin network into some continuous space. Continuous imbedding space would of course make possible to introduce also the notion of length and also "long wave length limit" could be more than a trick of magician. The continuous space-time seems to pop up irresistibly even in these noble attempts to get rid of it!
- The identification of braid invariants with standard model quantum numbers might look like an innocent operation. Braid invariants are however discrete topological invariants whereas standard model quantum numbers are group theoretical invariants. The latter ones are much more refined requiring continuum topology, differential structure, and Riemann metric. These quantum numbers are always with respect to some choice of quantization axes unlike topological quantum numbers. This makes the idea of assigning gauge interactions to topological invariants highly implausible.
Finite measurement resolution is seen as a property of state rather than a limitation preventing to know everything about it. The solutions of the modified Dirac equation indeed lead to this notion automatically: by conservation of electric charge they are localized to 2-D surfaces (string orbits) of space-time surface defining orbits of space-like braids and partonic 2-surfaces. Their ends at 3-D light-like wormhole throats define light-like braids.
At quantum level inclusion of hyper-finite factor is part of definition of state and leads to "quantum quantum theory" with non-commutative WCW ("world of classical worlds") spinors. Infinite-dimensional space of quantum states is replaced with space of "quantum quantum states" with finite fractional dimension. More concretely:
- Finite measurement resolution (and finite resolution of cognition) replaces partonic 2-surfaces with the collections of ends of light-like orbits of wormhole throats. They correspond to external particles of generalized Feynman diagrams and define also their vertices as surfaces at which the wormhole throat orbits are glued to together along their ends like lines of Feynman diagram. Braid strands carry quantum numbers of fermion or antifermion (quark or lepton). String world sheets result when space-like braid strands connecting braid ends at different wormhole throats move.
The orbits of the ends of space-like braid strands move along 3-D light-like orbit of wormhole throat defining what might be called light-like braid strands. Space-like and light-like braidings become part of fundamental physics. This physics is new physics: there is no attempt to interpret it as standard model physics.
This new physics becomes interesting only when the number of fermions and antifermions at partonic 2-surface is at least three as also in the original variant of Bilson-Thompson model. Standard model particles in TGD framework contain only single fermion or antifermion per wormhole throat and thus correspond to 1-braids. The experimental braid physics at elementary particle level there would belong to - I cannot say how distant - future in TGD Universe;-).
- Partonic 2-surfaces could however have macroscopic size and one can ask whether anyonic systems could correspond to large partonic 2-surfaces containing many-electron states and having non-standard value of Planck constant scaling up the size of the wormhole contact. The electrons would move with parallel momenta and it is not clear whether this can make sense. What is interesting that the fermionic oscillator operator algebra at partonic 2-surface defines SUSY algebra with large value of N and therefore anyonic system would represent SUSY with large N.
To my opinion the basic problematic assumption of "Emergent Braided Matter of Quantum Geometry" is the assumption that discretization is something fundamental. Mathematician would see spin networks without imbedding as something analogous to the attempt to square the circle. The irony is that by introducing braiding the authors are forced to give up the motivating assumption! Why not try to deduce braids from a theory starting directly from space-time?