Publication: Geometry of the symmetries in dimension 4 = (1+[1)+“2”] ,
and general Time-Space-Spin vectors (matrices).
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2023
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Abstract
Escribimos los vectores en dimensión tres en términos de matrices cuadradas, que diagonalizamos. Proponemos una parametrización para estos vectores, con variables de tipo ángulo (2ϕ,φ), diferente de la usual. Añadimos una dimensión tipo tiempo, definimos una segunda parametrización para sus 4-vectores, y escribimos los vectores (las matrices tipo spin) como funciones de estas nuevas variables angulares (ϕ,φ). Estudiamos las simetrías para algunos valores especiales relacionados de una de las nuevas variables angulares (ϕ), y también varios valores específicos de esta variable. Definimos un producto intermediado y anticonmutadores. Incluimos una primera aproximación breve al spin, al spin-vector y a la quiralidad con sus posibles implicaciones al final.
We write the vectors in dimension three in terms of square matrices, which we diagonalize. We propose a parametrization for these vectors, with angle variables (2ϕ,φ), different to the usual one. We append a time type dimension, we define a second parametrization for their 4-vectors, and we write the vectors (the spin type matrices) as functions of these new angle variables (ϕ,φ). We study the symmetries for some specially related values of one of the angle variables (ϕ), and we also consider various specific values of this variable. We define an intermediated product and anticommutators. We include a first brief approximation to the spin, the vector-spin and the chirality with their possible implications at the end.
We write the vectors in dimension three in terms of square matrices, which we diagonalize. We propose a parametrization for these vectors, with angle variables (2ϕ,φ), different to the usual one. We append a time type dimension, we define a second parametrization for their 4-vectors, and we write the vectors (the spin type matrices) as functions of these new angle variables (ϕ,φ). We study the symmetries for some specially related values of one of the angle variables (ϕ), and we also consider various specific values of this variable. We define an intermediated product and anticommutators. We include a first brief approximation to the spin, the vector-spin and the chirality with their possible implications at the end.