Fractional Charge in Quantum Hall effect

In 1976 Jackiw and Rebbi found 1/2 of a fermion number by using Dirac equation. Schrieffer, in several proposals made an effort to suggest that there is a fractional charge. The calculations of Peierls distortion, Berry’s phase and classical action were presented to accomodate the fractional charge in non-relativistic theory. Laughlin used the antisymmetry to define the charge density per unit area in a two dimensional system. In order to elliminate the area, Laughlin introduced the incompressibility which fixed the area and the odd number which determines the antisymmetry of the electron wave function, gave the charge. The antisymmetry relies on the odd number which can be equal to 3 so that the charge became 1/3. We have used the orbital angular momentum and the spin of the electron to define the effective charge through the Bohr magneton which already has the charge of the electron. When both signs of the spin are used, j = l ± s, we generate a series of charges which are in agreement with those measured by Stormer in the quantum Hall effect experiments.