There seems to be two different views about fractons as one learns by going to Wikipedia. Fracton can be regarded as a as self-similar particle-like entity (see this or as "sub-dimensional" particle unable to move in isolation (see this). I do not understand the motivation for "sub-dimensional". It is also unclear whether the two notions are related. The popular article assigns to the fractons both the fractal character and the inability to move in isolation.
The basic idea is however that discrete translational symmetry is replaced with a discrete scaling invariance. The analog of lattice which is invariant under discrete translations is fractal invariant under discrete scalings.
One can also consider the possibility that the time evolution operator would act as scaling rather than translation. This is something totally new from quantum field theory (QFT) point of view. In QFTs energy corresponds to time translational symmetry and Hamiltonian generates infinitesimal translations. In string models the analog of stringy Hamiltonian is the infinitesimal scaling operator, Virasoro generator L0.
In TGD the extension of physics to adelic physics provides number theoretic and geometric descriptions as dual descriptions of physics (see for instance this, this, and this). This approach also provides insights about fractons as scale invariant entities and.
- In TGD the analog of time evolution between "small" state function reductions is the exponent of the infinitesimal scaling operator, Virasoro generator L0. One could imagine fractals as states invariant under discrete scalings defined by the exponential of L0. They would be counterparts of lattices but realized at the level of space-time surfaces having quite concrete fractal structure.
- In p-adic mass calculations the p-adic analog of thermodynamics for L0 proportional to mass squared operator M2 replaces energy. This approach is the counterpart of the Higgs mechanism which allows only to reproduce masses but does not predict them. I carried out the calculations already around 1995 and the predictions were amazingly successful and eventually led to what I call adelic physics fusing real and various p-adic physics (see this).
- Long range coherence and absence of thermal equilibrium are also mentioned as properties of fractons (at least those of the first kind). Long range coherence could be due to the predicted hierarchy of Planck constants heff=n×h0 assigned with dark matter and predicting quantum coherence in arbitrarily long scales and associated with what I called magnetic bodies.
If translations are replaced by discrete scalings, the analogs of thermodynam equilibria would be possible for L0 rather than energy. Fractals would be the analogs of thermodynamic equilibria. In p-adic thermodynamic elementary particles are thermodynamic equilibria for L0 but it is not clear whether the analogy with fractal analog of a plane wave in lattice makes sense.
In TGD number theoretical vision leads to the notion of Galois confinement analogous to color confinement. The Galois group of a given extension of rationals indeed acts as a symmetry at space-time level. In TGD inspired biology Galois groups would play a fundamental role. For instance, dark analogs of genetic codons, codon pairs, and genes would be singlets (invariant) under an appropriate Galois group and therefore behave as a single quantum coherent dynamical and informational unit. See (see this and this) .
Suppose that one has a system - say a fractal analog of a lattice consisting of Galois singlets. Could fracton be identified as a state which is analogous to quark or gluon and therefore not invariant under the Galois group. The physical states could be formed from these as Galois singlets and are like hadrons.
For a summary of earlier postings see Latest progress in TGD.