We discuss a procedure to construct multi-resolution analyses (MRA) of L2(R) starting from a given seed function h(s) which should satisfy some conditions. Our method, originally related to the quantum mechanical hamiltonian of the fractional quantum Hall effect (FQHE), is shown to be model independent. The role of a canonical map between certain canonically conjugate operators is discussed. Th...

We study the fractional quantum Hall effect in a bilayer with charge-distribution imbalance induced, for instance, by a bias gate voltage. The bilayer can either be intrinsic or it can be formed spontaneously in wide quantum wells, due to the Coulomb repulsion between electrons. We focus on fractional quantum Hall effect in asymmetric bilayer systems at filling factor ν = 4/11 and show that an ...

The second-quantized form of the Laughlin states for the fractional quantum Hall effect is discussed by decomposing the Laughlin wavefunctions into the N -particle Slater basis. A general formula is given for the expansion coefficients in terms of the characters of the symmetric group, and the expansion coefficients are shown to possess numerous interesting symmetries. For expectation values of...

One of the most memorable lessons of an undergraduate course in quantum mechanics is that identical particles can have two types of “statistics.” An exchange of two identical bosons leaves the many-particle wave function unchanged, while an exchange of two identical fermions introduces a minus sign. The consequences of this minus sign are profound in all areas of physics. Moreover, it has long ...

We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1 < h < 2, a fractal distribution function associated with a fractal von Newmann entropy. Fractons are charge-flux systems defined in two-dimensional multiply connected space and they carry rational or irrational ...

Quasielectrons and quasiholes in the fractional quantum Hall liquids obey fractional (including nontrivial mutual) exclusion statistics. Their statistics matrix can be determined from several possible state-counting scheme, involving different assumptions on statistical correlations. Thermal activation of quasiparticle pairs and thermodynamic properties of the fractional quantum Hall liquids ne...

We propose a new hierarchy scheme for the filling factor, a parameter which characterizes the occurrence of the Fractional Quantum Hall Effect ( FQHE ). We consider the Hausdorff dimension, h, as a parameter for classifying fractional spin particles, such that, it is written in terms of the statistics of the collective excitations. The number h classifies these excitations with different statis...

The Fermi-hypernetted-chain ~FHNC! theory and the effective hypernetted-chain method are applied to study the composite-fermion ~CF! states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for n51...

We extend the noncommutative geometry model of the fractional quantum Hall effect, previously developed by Mathai and the first author, to orbifold symmetric products. It retains the same properties of quantization of the Hall conductance at integer multiples of the fractional Satake orbifold Euler characteristics. We show that it also allows for interesting composite fermions and anyon represe...

A small number of M-theory branes as giant gravitons in the M-theory sector of LLM geometry is studied as a probe. The abelian way shows that the low energy effective action for M-theory brane is exactly the 2d electron subject to a vertical magnetic field. We also briefly discuss the microscopic description of M2-brane giant graviton in this geometry, in the language of a combination of D0-bra...