Fractional Quantum Hall Effect as Consequence of Symmetry Invariance

The non-relativistic Hamiltonian for an electron in the presence of an electromagnetic field, described using Landau's gauge, is solved analytically based on the conserved operators of the system: one for the canonical momentum in the x-axis, a second one for the canonical momentum in the y-axis, and the final one for the energy operator. It is shown that the Lorentz force can be recovered only if both of the conserved momenta are considered; otherwise, the system cannot be fully described. The wave function is calculated by solving an eigenvalue equation for the momentum operators, and a ground state of this function is then constructed. Based on the conserved properties of the system, a set of unitary operators defining the symmetries of the Hamiltonian is established. However, in Schr\"odinger's scheme, the necessary conditions for the invariance of the wave function after a unitary transformation give rise to a couple of quantized identities: one for the electric field and the second one for the magnetic field. Using the electric current expression defined by the continuity equation, the Hall and longitudinal resistivity were calculated, showing that the former is proportional to von Klitzing's constant and the latter vanishes when the time increment is t<<m₂q{ E} x. Finally, if the invariance condition is satisfied, then the Hall resistivity is quantized in integer multiples proportional to von Klitzing's constant. This implies that the fractional quantum Hall effect is a manifestation of symmetry invariance.