A fractal-like structure for the fractional quantum Hall effect

Abstract

We have pursued in the literature a fractal-like structure for the fractional quantum Halll effect-FQHE which consider the Hausdorff dimension associated with the quantum mechanics paths and the spin of the particles or quasiparticles termed fractons. These objects carry rational or irrational values of spin and satisfy a fractal distribution function associated with a fractal von Neumann entropy. We show that our approach offers a rationale for all FQHE data including possible filling factors suggested by some authors. Our formulation is free of any empirical formula and this characteristic appears as a foundational insight for this FQHE-phenomenon. The connection between a geometrical parameter, the Hausdorff dimension h, associated with the quantum paths and the spin s of particles, h = 2 − 2s, 0 < s < 1 2 , is a physical analogous to the fractal dimension formula, ∆(Γ) = 2− H, of the graph of the functions in the context of the fractal geometry, where H is known as Hölder exponent, with 0 < H < 1. We emphasize that our fractal approach to the fractional spin particles gives us a new perspective for charge-flux systems defined in two-dimensional multiply connected space.

Max Planck Half Quanta as a Natural Explanation for Ordinary and Dark Energy of the Cosmos

“…The innocent conclusion of the above half quanta is that our postulate gained mainly from experiments that quanta are indivisible cannot be as straight forward as one could naively have thought and who knows, it may open the door to unsuspected connections related to fractional-Hall effects and similar things [119]. Historically speaking this (1/2) which ought to be quite famous because it gives a clear justification for the Casimir effect, goes back to the pioneering efforts of Max Planck to make sense out of his own discovery of the quantization of energy [21]- [23].…”

“…Either way we see that E = mc 2 consists of two quasi quantum components well hidden inside the deceptively simple Einstein's beauty E = mc 2 [119]. We could touch upon trisecting E = mc 2 not only into two parts E(O) and E(D) but into three parts making a distinction between dark matter energy E(DM) and pure dark energy…”

“…[1]- [76] are the main readings while Refs. [77]- [119] are additional readings which we think deepen and enhance understanding of the subject.…”