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Introducing Multidimensional Dirac–Hestenes Equation
Advances in Applied Clifford Algebras. 2025. Vol. 35. Article 24.
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 present research outlines the multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra ℂ⊗Cℓ1,n depends on a parity of n, we explore even and odd cases separately. In the geometric algebra Cℓ1,3, 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 Cℓ1,n is bigger than the dimension of the minimal left ideal for n>4. Hence, we consider the auxiliary real subalgebra of Cℓ1,n to prove a similar statement. We present the multidimensional Dirac-Hestenes equation in Cℓ1,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.
Publication based on the results of:
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Filimoshina E., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 16
This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that these Lie groups can be equivalently defined using norm functions of multivectors applied in the theory of spin groups. We also study the ...
Added: January 12, 2026
Sofia Rumyantseva, Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 Article 5
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 ...
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Filimoshina E., Shirokov D., , in: Volume 267: International Conference on Machine Learning, 13-19 July 2025, Vancouver Convention Center, Vancouver, CanadaVol. 267.: [б.и.], 2025. P. 17153–17188.
We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization ...
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