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The Periodic Table and the Group SO(4,4)

V. V. Varlamov

TL;DR

This work builds a four-parameter, spin-inclusive mathematical framework for the periodic table by mapping chemical elements to the weight diagram of the Lie algebra $ rak{so}(4,4)$, the four-dimensional extension of the conformal group's $ rak{so}(4,2)$. Central to the construction is the hydrogen-based realization of $ rak{so}(4,2)$ (Barut representation), with a Cartan subalgebra of rank 3 that is augmented by a fourth generator to encode spin, yielding a 4D weight space whose nodes correspond to elements and their masses via explicit formulas. The approach integrates twofold coverings ($ m SU(2,2)$) and Yao bases, dissects the $ rak{so}(4,2)$ subalgebras ($ rak{so}(3,1)$, $ rak{so}(4)$, $ rak{so}(2,2)$), and culminates in a $ rak{so}(4,4)$ weight diagram that naturally includes antimatter and produces matter/antimatter pyramids akin to Haenzel/Finke graphical schemes. Mass formulas attached to weight nodes and the projection structure (three-dimensional $ rak{so}(4,2)$-towers into four-dimensional $ rak{so}(4,4)$ towers) provide a quantitative bridge between group theory and element periodicity. Overall, the paper offers a conceptually unified, group-theoretic depiction of the periodic system, extends classical 3D spin representations to a four-dimensional framework, and suggests a formal mechanism for antimatter incorporation within the periodic scheme.

Abstract

The periodic system of chemical elements is represented within the framework of the weight diagram of the Lie algebra of the fourth rank of the rotation group of an eight-dimensional pseudo-Euclidean space. The hydrogen realization of the Cartan subalgebra and Weyl generators of the group algebra is studied. The root structure of the subalgebras of the group algebra of a conformal group in the framework of a twofold covering is analyzed. Based on the analysis, the Cartan-Weyl basis of the group algebra is determined. The root and weight diagrams are constructed. A mass formula associated with each node of the weight diagram is introduced. Spin is interpreted as the fourth generator of the Cartan subalgebra, whose two eigenvalues correspond to two three-dimensional projections of the weight diagram containing elements of the periodic system from hydrogen to moscovium (the first projection) and from helium to oganesson (the second projection). One of the main advantages of the proposed group-theoretic construction of the periodic system is the natural inclusion of antimatter in the general scheme.

The Periodic Table and the Group SO(4,4)

TL;DR

This work builds a four-parameter, spin-inclusive mathematical framework for the periodic table by mapping chemical elements to the weight diagram of the Lie algebra , the four-dimensional extension of the conformal group's . Central to the construction is the hydrogen-based realization of (Barut representation), with a Cartan subalgebra of rank 3 that is augmented by a fourth generator to encode spin, yielding a 4D weight space whose nodes correspond to elements and their masses via explicit formulas. The approach integrates twofold coverings () and Yao bases, dissects the subalgebras (, , ), and culminates in a weight diagram that naturally includes antimatter and produces matter/antimatter pyramids akin to Haenzel/Finke graphical schemes. Mass formulas attached to weight nodes and the projection structure (three-dimensional -towers into four-dimensional towers) provide a quantitative bridge between group theory and element periodicity. Overall, the paper offers a conceptually unified, group-theoretic depiction of the periodic system, extends classical 3D spin representations to a four-dimensional framework, and suggests a formal mechanism for antimatter incorporation within the periodic scheme.

Abstract

The periodic system of chemical elements is represented within the framework of the weight diagram of the Lie algebra of the fourth rank of the rotation group of an eight-dimensional pseudo-Euclidean space. The hydrogen realization of the Cartan subalgebra and Weyl generators of the group algebra is studied. The root structure of the subalgebras of the group algebra of a conformal group in the framework of a twofold covering is analyzed. Based on the analysis, the Cartan-Weyl basis of the group algebra is determined. The root and weight diagrams are constructed. A mass formula associated with each node of the weight diagram is introduced. Spin is interpreted as the fourth generator of the Cartan subalgebra, whose two eigenvalues correspond to two three-dimensional projections of the weight diagram containing elements of the periodic system from hydrogen to moscovium (the first projection) and from helium to oganesson (the second projection). One of the main advantages of the proposed group-theoretic construction of the periodic system is the natural inclusion of antimatter in the general scheme.

Paper Structure

This paper contains 18 sections, 1 theorem, 155 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Cartan subalgebra and Weyl generators
  3. The conformal group $\mathop{\mathrm{SO}}\nolimits(4,2)$
  4. Hydrogen implementation of the algebra $\mathfrak{so}(4,2)$
  5. Cartan subalgebra
  6. The group $\mathop{\mathrm{SU}}\nolimits(2,2)$ and the Yao basis
  7. Subalgebra $\mathfrak{so}(3,1)$
  8. Subalgebra $\mathfrak{so}(4)$
  9. Subalgebra $\mathfrak{so}(2,2)$
  10. The root diagram of the algebra $\mathfrak{so}(4,2)$
  11. The weight diagram of the Lie algebra $\mathfrak{so}(4,2)$
  12. Haenzel Polygonfläche
  13. The three-dimensional Finke system
  14. A general note on the Haenzel and Finke systems
  15. Group $\mathop{\mathrm{SO}}\nolimits(4,4)$
  16. ...and 3 more sections

Key Result

Theorem 1

(Racah theorem) For every $n$-dimensional Lie algebra $\mathfrak{g}$ of rank $m$, there are a total of $m$ Casimir operators ${\boldsymbol{\sf C}}_\mu$ ($\mu=1\rightarrow m$) that commute with generators ${\boldsymbol{\sf X}}_i$ ($i=1\rightarrow n$) of the algebra $\mathfrak{g}$.

Figures (16)

  • Figure 1: The root diagram of the Lie algebra $\mathfrak{sl}(2,\bb C)$. The action of each Weyl generator is shown in the (${\sf X}_3,{\sf Y}_3$)-plane.
  • Figure 2: The first three weight diagrams (Weyl diagrams) of the Lie algebra $\mathfrak{sl}(2,\bb C)$: a) $(\tfrac{1}{2},\tfrac{1}{2})$-multiplet; b) $(1,1)$-multiplet; c) $(\tfrac{3}{2},\tfrac{3}{2})$- multiplet.
  • Figure 3: Extended Weyl diagram of the algebra $\mathfrak{sl}(2,\bb C)$. A state is associated with each node $(l,\dot{l})$ of the diagram, the mass of which is determined by the formula (\ref{['Mass2']}).
  • Figure 4: The root diagram of the Lie algebra $\mathfrak{so}(4)$. The action of each Weyl generator is shown in the (${\boldsymbol{\sf K}}_3,{\boldsymbol{\sf J}}_3$)-plane.
  • Figure 5: The root diagram of the Lie algebra $\mathfrak{so}(4,2)$.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1