Spinor Structure and Matter Spectrum

TL;DR

This paper classifies relativistic wave equations using Lorentz group representations, defining a matter spectrum of elementary particles with specific energy and spin properties, revealing non-degenerate and degenerate spectra for different fields.

Contribution

It introduces a unified classification framework for relativistic fields and their matter spectrum based on interlocking Lorentz group representations, including stability levels.

Findings

Energy spectrum is non-degenerate for $(l,0)(0,l)$ fields.

Degenerate energy spectra occur in arbitrary spin chains.

Stability levels exhibit fractal threshold structures.

Abstract

Classification of relativistic wave equations is given on the ground of interlocking representations of the Lorentz group. A system of interlocking representations is associated with a system of eigenvector subspaces of the energy operator. Such a correspondence allows one to define matter spectrum, where the each level of this spectrum presents a some state of elementary particle. An elementary particle is understood as a superposition of state vectors in nonseparable Hilbert space. Classification of indecomposable systems of relativistic wave equations is produced for bosonic and fermionic fields on an equal footing (including Dirac and Maxwell equations). All these fields are equivalent levels of matter spectrum, which differ from each other by the value of mass and spin. It is shown that a spectrum of the energy operator, corresponding to a given matter level, is non-degenerate for the fields of type $(l,0)\oplus(0,l)$, where $l$ is a spin value, whereas for arbitrary spin chains we have degenerate spectrum. Energy spectra of the stability levels (electron and proton states) of the matter spectrum are studied in detail. It is shown that these stability levels have a nature of threshold scales of the fractal structure associated with the system of interlocking representations of the Lorentz group.