Algebra of hypersymmetry (extended version) applied to state transformations in strongly relativistic interactions illustrated on an extended form of the Dirac equation

TL;DR

This paper introduces the algebra of hypersymmetry (HySy), an invariance group that transforms scalar and vector quantities into each other, and demonstrates its application to extended Dirac equations in high-energy physics.

Contribution

It defines the tau algebra and hypersymmetry group, showing their properties and invariance in fundamental equations, extending the understanding of state transformations in relativistic interactions.

Findings

Hypersymmetry group is isomorphic with SU(2).

The algebra applies to extended Dirac equations at high energies.

Invariance under combined Lorentz and hypersymmetry transformations.

Abstract

There are several 3+1 parameter quantities in physics (like vector + scalar potentials, 4-currents, space-time, 4-momentum). In most cases (but space-time), the 3- and the 1-parameter characterised elements of these quantities differ in the field-sources (e.g., inertial and gravitational masses, Lorentz- and Coulomb-type electric charges) associated with them. The members of the field-source pairs appear in the vector- and the scalar potentials, respectively. Sec. 1 and 2 present an algebra what demonstrates that the members of the field-source siblings are subjects of an invariance group that can transform them into each other. (This includes, the conservation of the isotopic field-charge spin, proven in previous publications.) The paper identifies the algebra of that transformation and characterises the group of the invariance, it discusses the properties of this group, shows how they can be classified in the known nomenclature, and why is this pseudo-unitary group isomorphic with the SU(2) group. This algebra is denoted by tau. The invariance group generated by the tau algebra is called hypersymmetry (HySy). The group of HySy had not been described. The defined symmetry group is able to make correspondence between scalars and vector components that appear often coupled in the characterisation of physical states. In accordance with conclusions in previous papers, the second part (Sec. 3 and 4) shows that the equations describing the individual fundamental physical interacions are invariant under the combined application of the Lorentz transformation and the here explored invariance group at high energy approximation (while they are left intact at lower energies). As illustration, the paper presents a simple form for an extended Dirac equation and a set of matrices to describe the combined transformation in QED. Sec. 2.2 shows applicability of this algebra for genetic matrices.