Spherical Symmetry in Anisotropic Non-Integer Dimensional Space: Application to Fractal Charge Distributions
- 1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia
Abstract
Fractal charge distributions (FCD) can be described by models with non-integer dimensional spaces (NIDS) and a special form of density of states. The NID linear space and integration over the NID space have been suggested in quantum field theory by Wilson in 1973. The integration in NID space is applied in statistical physics, in quantum field theory and in fractal physics. Mathematically, the description of the electric fields of FCD can be realized by the self-consistent calculus in NID spaces and the vector calculus in these spaces is proposed in 2025. with nearly 200 pages. In the previous article of 2026, a Lorentz-invariant and gauge-invariant theory of the electromagnetic fields in NID space was constructed. In the proposed article, a possibility of spherical symmetry of electric fields in anisotropic NID space is investigated and the conditions under which these symmetries exist are described. Previously, due to the lack of self-consistent calculus in NID space, errors arose even in the application of the generalized Gauss law in spherically symmetric cases. In contrast to the standard case, in a space with non-integer dimensions and thus in FCD, only one of the three quantities (electric charge density, scalar potential, and electric field strength) can be spherically symmetric. In this work we obtain the conditions under which physical quantities can be spherically symmetric in an anisotropic NID space. The conditions of spherical symmetry in NID space are derived for the following quantities: the density of charge distribution, scalar potential, electric field strength vector, projection of the electric field strength vector onto the normal, and the separation of radial and angular variables for the modulus of the electric field strength vector and for the density of electrical energy. The results obtained in this paper may be useful for describing fractal distribution of charges in plasma. Note that the proposed results can also be used for anisotropic fractal distributions of mass and particles in classical gravitation theory, statistical mechanics and its NIDS generalizations.
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BibTeX
@article{tarasov2026spherical,
title = {{Spherical Symmetry in Anisotropic Non-Integer Dimensional Space: Application to Fractal Charge Distributions}},
author = {Vasily E. Tarasov},
journal = {Fractal and Fractional},
year = {2026},
month = sep,
doi = {10.3390/fractalfract10100685},
url = {https://www.mdpi.com/2504-3110/10/10/685},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Spherical Symmetry in Anisotropic Non-Integer Dimensional Space: Application to Fractal Charge Distributions
AU - Vasily E. Tarasov
PY - 2026
DA - 2026/09/30
JO - Fractal and Fractional
DO - 10.3390/fractalfract10100685
UR - https://www.mdpi.com/2504-3110/10/10/685
AB - Fractal charge distributions (FCD) can be described by models with non-integer dimensional spaces (NIDS) and a special form of density of states. The NID linear space and integration over the NID space have been suggested in quantum field theory by Wilson in 1973. The integration in NID space is applied in statistical physics, in quantum field theory and in fractal physics. Mathematically, the description of the electric fields of FCD can be realized by the self-consistent calculus in NID spaces and the vector calculus in these spaces is proposed in 2025. with nearly 200 pages. In the previous article of 2026, a Lorentz-invariant and gauge-invariant theory of the electromagnetic fields in NID space was constructed. In the proposed article, a possibility of spherical symmetry of electric fields in anisotropic NID space is investigated and the conditions under which these symmetries exist are described. Previously, due to the lack of self-consistent calculus in NID space, errors arose even in the application of the generalized Gauss law in spherically symmetric cases. In contrast to the standard case, in a space with non-integer dimensions and thus in FCD, only one of the three quantities (electric charge density, scalar potential, and electric field strength) can be spherically symmetric. In this work we obtain the conditions under which physical quantities can be spherically symmetric in an anisotropic NID space. The conditions of spherical symmetry in NID space are derived for the following quantities: the density of charge distribution, scalar potential, electric field strength vector, projection of the electric field strength vector onto the normal, and the separation of radial and angular variables for the modulus of the electric field strength vector and for the density of electrical energy. The results obtained in this paper may be useful for describing fractal distribution of charges in plasma. Note that the proposed results can also be used for anisotropic fractal distributions of mass and particles in classical gravitation theory, statistical mechanics and its NIDS generalizations.
SN - 2504-3110
ER -