Charge

The physical notion of elementary electric charge follows from a mathematical theory of the constitution of real physical space. Set theory, topology and fractal geometry allow us to construct space, as a mathematical lattice of topological balls - the tessel-lattice, which possesses fractal properties.

A fractal volumetric deformation of a topological ball is associated with the notion of mass. A fractal surface deformation of a topological ball is associated with notion of elementary electric charge. In the degenerate tessel-lattice one can distinguish a middle radius of cells such that amplitudes of oscillations of the cells' surfaces (surface wavelets) cross the surface both out and in. Then the quant of surface deformation, when all amplitudes, i.e. needles, of the surface oscillations are directed outward of the cell can be associated with a positive electric charge. When all surface amplitudes, i.e. needles, are directed inward of the cell, the form can be called a negative electric charge.

Figure 1: Completely free topological ball outside of the tessel-lattice (left figure); topological ball as part of the tessel-lattice, which can be referred to here as a superparticle (central figure); the formation of particles from the topological ball (right figures).

How many such amplitudes, or needles, cover the surface of a topological ball in the tessel-lattice? Obviously as many as the number of harmonics in the tessel-lattice. This number $N$ is defined by the quantity of balls that forms the tessel-lattice. If the middle size of a cell of the tessel-lattice equals the Planck's fundamental length $r_{\rm cell} \sim l_{\rm f} =\sqrt{\hbar G/c^3}$ $\approx 10^{-35}$ m and the radius of the visible universe $R_{\rm univ}\approx 10^{26}$, we can easily estimate the number of needles that cover an elementary electric charge: $N \sim R_{\rm univ}^3 / r_{\rm cell}^3 = 10^{183}$.

In the macroscopic world the most descriptive analogy of the positive charge is a chestnut; ramified roots of plants are also look like a positive charge. Hence the negative charge must be an inverse spatial topological structure such as the stomachs of animals. The motion of a charged particle in the tessel-lattice can be described by means of the appropriate Lagrangian 

(1)
\begin{align} {\cal L}_n=C \Bigl\{{1\over 2}{\kern 1pt}{\dot \Phi}^2_n + \frac 12 {\kern 1pt} {\dot{\vec {\cal A}}}^2_n + {1\over 2}{\kern 1pt}{\dot \phi}^2_n +\frac 12 {\kern 1pt}{\dot {\vec \alpha}}^2_{{\kern 1pt} n} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \qquad \qquad \qquad \qquad \qquad \qquad \\ - v_0 \Bigl({\dot \Phi}_n \nabla {\vec \alpha}_{{\kern 1pt} n} +{\dot \phi}_n \nabla {\vec {\cal A}}_n \Bigr) - v_0^2 \Bigl( \nabla \times {\vec {\cal A}}_n \Bigr) \Bigl( \nabla \times {\vec {\alpha}}_{{\kern 1pt}n} \Bigr) \ \Bigr\} \end{align}

Here, $\Phi$ and ${\vec {\cal A }}$ are scalar and vector fields determined on the surface of the particle; $\phi$ and $\vec \alpha$ are the appropriate scalar and vector fields located on the particle's inertons; in other words, these polarized inertons represent a cloud of photons around the charged particle. $\upsilon_0$ is the velocity of the particle.

The equations of motion, i.e. the Euler-Lagrange equations derived on the basis of the Lagrangian (1) result in the Maxwell equations written in the d'Alambert form. The motion of the charge is associated with the ejection and reabsorption of photons, which are electromagnetically polarized inertons - because photons represent the same inertons whose surface is additionally covered with fractal polarization. Running each odd section $\lambda/2$ of the charge's path, where $\lambda$ is the de Broglie wavelength of the corresponding particle, step-by-step, the charge on the surface transforms its electric state to the magnetic state, i.e. the sate of a magnetic monopole.

Figure 2: Motion of the charged particle accompanied by the cloud of its photons

The magnetic state, or magnetic charge, looks like a combed electric charge, i.e. all needles are stowed, or bended to the surface of the particulate ball. Then emitted photons (or rather 'inerton-photon' cloud) gradually comes back to the particle in even sections $\lambda/2$ and restores the particle's initial pure electric surface state.

### Bibliography

 V. Krasnoholovets, On the nature of the electric charge, Hadronic Journal Supplement 18, no. 4, pp. 425-456 (2003) (also http://arXiv.org/abs/physics/0501132).

page revision: 29, last edited: 17 Jun 2012 15:42