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S. V. Rumyantseva

Publications and source records attributed to S. V. Rumyantseva.

2 recordsLinked to original sources
arXiv2025

Lorentz Invariance of the Multidimensional Dirac-Hestenes Equation

This paper investigates the Lorentz invariance of the multidimensional Dirac-Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac-Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1,n) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action.

math-ph↗
arXiv2025

Introducing Multidimensional Dirac-Hestenes Equation

It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac-Hestenes equation instead of a complex solution to the Dirac equation. The current research presents a formulation of the multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra $\mathbb{C}\otimes{C \kern -0.1em \ell}_{1,n}$ depends on the parity of $n$, we examine even and odd cases separately. In the geometric algebra ${C \kern -0.1em \ell}_{1,3}$, there is a lemma on a unique decomposition of an element of the minimal left ideal into the product of the idempotent and an element of the real even subalgebra. The lemma is used to construct the four-dimensional Dirac-Hestenes equation. The analogous lemma is not valid in the multidimensional case, since the dimension of the real even subalgebra of ${C \kern -0.1em \ell}_{1,n}$ is bigger than the dimension of the minimal left ideal for $n>4$. Hence, we consider the auxiliary real subalgebra of ${C \kern -0.1em \ell}_{1,n}$ to prove a similar statement. We present the multidimensional Dirac-Hestenes equation in ${C \kern -0.1em \ell}_{1,n}$. We prove that one might obtain a solution to the multidimensional Dirac-Hestenes equation using a solution to the multidimensional Dirac equation and vice versa. We also show that the multidimensional Dirac-Hestenes equation has gauge invariance.

math-ph↗