Abstract
This work presents new round of the author’s pursuit for consistent description of the finite sized objects in classical and quantum field theory. Current paper lays out an adequate mathematical background for this quest. A novel framework of the matter-induced physical affine geometry is developed. Within this framework, (1) an intrinsic nonlinearity of the Dirac equation becomes self-explanatory; (2) the spherical symmetry of an isolated localized object is of dynamic origin; (3) the auto-localization is a trivial consequence of nonlinearity and wave nature of the Dirac field; (4) localized objects are split into two major categories that are clearly associated with the positive and negative charges; (5) of these, only the former can be stable as isolated objects, which explains the global charge asymmetry of the matter observed in Nature. In the second paper, the nonlinear Dirac equation is written down explicitly. It is solved in one-body approximation (in absence of external fields). Its two analytic solutions unequivocally are positive (stable) and negative (unstable) isolated charges. From the author’s current perspective, the so for obtained results must be developed further and applied to various practical and fundamental problems in particle and nuclear physics, and also in cosmology.
Highlights
This work addresses the long-standing puzzle of how the physical Dirac field of real matter becomes a finitesized particle
( ) on, we look at the ψσ (P) as the physical Dirac field over four-dimensional manifold
Thesurfaces emerging from the Dirac equation and differential identities for the Dirac currents point to a fairly simple geometric structure of the lines and surfaces of the admissible coordinate net
Summary
This work addresses the long-standing puzzle of how the physical Dirac field of real matter becomes a finitesized particle. At any point in spacetime continuum (the principal differentiable manifold ), there exist four fields of quadruples of these forms (the Dirac currents), which are linearly independent and Lorentzorthogonal, and can serve as local algebraic basis for any four-dimensional vector space, including the infinitesimal displacements in coordinate space 4. The components of the tetrad vectors ha (P) with respect to the basis eA (P) must have invariant values (2.10) These equations together with normalization conditions (2.5) and unitarity, det VAa = 1 , allow one to interpret VAa (P) as the matrix of a local Lorentz rotation between the bases eA (P) and ha (P) with parameters that are determined by the Dirac field ψ (P) 4. We consider the Dirac field as a known function of coordinates and do not employ its equation of motion
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