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With conjecture of fractional charge quantization (quantum dipole/multiple moments), Fourier transform stretching, twisting and twigging of an electron quanta and waver strings of electron quanta, the mathematical expressions for mesoscopic fractional electron fields in a cavity of viscous medium and the associated quantum dielectric susceptibility are developed. Agreement of this approach is experimentally evidenced on barite and Fanja site molecular sieves. These findings are in conformity with experimental results of 2012 Physics Nobel prize winning scientists, Serge Haroche and David J. Wineland especially for cavity quantum electro-dynamics electron and its associated mesoscopic electric fields. The mover electron quanta strings lead to warping of space and time following the behaviour of quantum electron dynamics.

Jonscher [

results. The orientation polarizability, i.e.,

and the deformation polarizibility, i.e.,

When a dielectric material at a fixed voltage is subjected to varying frequencies, polarization occurs with different magnitude of energies [

where

where

where in Equation (4), h is a plank’s constant (quantum action),

Due to fractional charge quantization or sub quanta or twigs of an electron string, electric field is also fractionally quantized. Thus Equation (3) changed into

putting Equation (5) in Equation (2), we have

Equation (6) can be transformed into frequency domain by taking its Fourier transform [

The imaginary part of dielectric susceptibility

For our case, the above equation is written as follows

where

The time dependent eigenfunction would lead to fractional quantization of electric fields within molecules or atoms. Using

One would have

where

The fractional quantum electric fields have already been obtained [

where

where

where

where r equivalent to displacement due to polarization

The relaxation time in classical wave mechanics is defined by

mechanical system. With quantum action, energies oscillate between two arbitrary fixed points at the atomic or molecular level. Atoms or molecules when polarized and fractionally quantized behave like micro wave cavity resonators (mesoscopic fluid resonators) with their sub-quanta or twigs on an electron quanta string or many electron woven quanta string. So, a new definition of relaxation time for quantum behaviour is introduced, i.e.,

Equation (17) shows relaxation time for quantum action of atoms, molecules or ions in the viscous medium and is inversely proportional to applied frequencies. Using Equation (17) in Equation (16), i.e.,

The term

Equation (19) is obtained by using Fourier transformed with new definition of relaxation time in a mesoscopic cavity resonator with woven electron string and their corresponding twigs (sub-quanta) for fractional charge quantization.

The Equation (18) can be re written in this form

the negative sign for eigenfunction

where

Taking cube of both sides

Equation (21) is the expression for mesoscopic electric fields due to fractional charge quantization of atoms, molecules or ions only due to woven electron quanta string. This observation is consistent [

The negative sign shows quantum wells with woven electron strings of varying lengths in a cavity for mesoscopic electric fields and follows periodicity of

Rewriting Equation (21) for molecular field

Now rewriting Equation (21) for

The term

Taking the cube root of Equation (26) we have

Equation (27) shows that the giant magnate resistance (GMR) [

Hence, the imaginary dielectric susceptibility in a viscous medium is directly proportional to the cube root of real permittivity and square cube root of imaginary permittivity

where

The mathematical result for electrical susceptibility is obtained in terms of giant magneto resistance (GMR), real permittivity and the imaginary permittivity and the imaginary permittivity, i.e.,

GMR is given in Equation (28), the polarization at zero frequency

and

SaleemIqbal,SalmaJabeen,FarhanaSarwar,Syed MohsinRaza, (2016) Quantum Theory of Mesoscopic Fractional Electric Fields in a Cavity of Viscous Medium. World Journal of Condensed Matter Physics,06,39-44. doi: 10.4236/wjcmp.2016.61006