Derivation of the Elementary Particles Total Energy by Quantum Statistical Physics and Generalized Special Relativity

Einstein’s theory of special relativity is used to refine the description of Fermi energy within a quantum statistical framework. Starting from the relativistic energy–momentum relation, the authors derive general expressions for the Fermi momentum and Fermi energy of an ideal, highly degenerate gas, showing explicitly that the Fermi energy depends on the particle rest mass and system size (modeled via an atomic or mass radius). They then introduce a generalized, potential-dependent form of special relativity to account for the influence of external fields, particularly a weak gravitational field, and derive an expression for the self-energy of gravitational mass under equilibrium conditions. For a uniformly dense spherical mass distribution, the total Fermi energy includes rest-mass energy, kinetic energy from quantum degeneracy, and gravitational potential energy. The analysis demonstrates that both the Fermi momentum and Fermi energy depend on the rest mass, radius, and the gravitational potential, yielding an explicit formula for the total energy of elementary particles in this framework.