The Periodic Table and the Group SO(4,4): II. Double SO(4,2)-tower

Abstract

A group-theoretic interpretation of the periodic system of elements is given within the framework of the weight diagram of the Lie algebra $\mathfrak{so}(4,4)$ of the fourth rank, where the four quantum numbers $n$, $l$, $m$, $s$ correspond to the eigenvalues (weights) of the Cartan generators of the maximal Abelian subalgebra (the maximal torus of the group SO(4,4)). It is shown that the root system of the algebra $\mathfrak{so}(4,4)$ forms a regular four-dimensional self-dual polyhedron (24-cell). The action of the fourth Cartan generator associated with spin leads to a splitting of the Cartan-Weyl basis of the algebra $\mathfrak{so}(4,4)$ into two structurally identical bases, each of which is isomorphic to the Yao basis of the subalgebra $\mathfrak{so}(4,2)$ (the Lie algebra of the conformal group). At this point, a four-dimensional 24-cell is projected onto two three-dimensional cuboctahedra, each of which defines the root system of the subalgebra $\mathfrak{so}(4,2)$. This splitting physically corresponds to spin doubling (two-valuedness). The structure of the energy levels of a periodic system is studied, the states of which (chemical elements) are represented as nodes of the weight diagram of the group algebra $\mathfrak{so}(4,4)$. The structure of the double SO(4,2)-towers of Mendeleev, Seaborg, and 10-periodic extension is examined in detail. The period doubling associated with the sequence of period lengths 2, 8, 8, 18, 18, 32, 32, $\ldots$ of the periodic system of elements is explained by the action of the fourth Cartan generator. It is shown that antimatter (Mendeleev anti-table consisting of antihydrogen, antihelium, antilitium, $\ldots$) is naturally included in the general group-theoretic scheme of description of the periodic table.