On algebraic structure of matter spectrum

TL;DR

This paper explores the algebraic structure of the matter spectrum, proposing a two-level Hilbert space framework that unifies various particle sectors and classifies symmetries into fundamental, dynamical, and gauge types.

Contribution

It introduces a novel two-level Hilbert space model for matter spectrum, incorporating both operator algebras and physical state spaces, unifying different particle sectors.

Findings

Decomposition of physical Hilbert space into coherent subspaces

Unified description of lepton, meson, and baryon spectra

Classification of matter spectrum symmetries into three types

Abstract

An algebraic structure of matter spectrum is studied. It is shown that a base mathematical construction, lying in the ground of matter spectrum (introduced by Heisenberg) , is a two-level Hilbert space. Two-level structure of the Hilbert space is defined by the following pair: 1) a separable Hilbert space with operator algebras and fundamental symmetries; 2) a nonseparable (physical) Hilbert space with dynamical and gauge symmetries, that is, a space of states (energy levels) of the matter spectrum. The each state of matter spectrum is defined by a cyclic representation within Gel'fand-Naimark-Segal construction. A decomposition of the physical Hilbert space onto coherent subspaces is given. This decomposition allows one to describe the all observed spectrum of states on an equal footing, including lepton, meson and baryon sectors of the matter spectrum. Following to Heisenberg, we assume that there exist no fundamental particles, there exist fundamental symmetries. It is shown also that all the symmetries of matter spectrum are divided onto three kinds: fundamental, dynamical and gauge symmetries.