Differentials of the basis in Clifford Geometric Algebra

Yingqiu Gu

doi:10.59400/jam.v2i4.1700

Differentials of the basis in Clifford Geometric Algebra

Yingqiu Gu School of Mathematical Science, Fudan University, Shanghai 200433, China

Ariticle ID: 1700

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DOI: https://doi.org/10.59400/jam.v2i4.1700

Keywords: Clifford algebra; connection; Dirac-γ; moving frame; variation

Abstract

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

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Published

2024-08-18

How to Cite

Gu, Y. (2024). Differentials of the basis in Clifford Geometric Algebra. Journal of AppliedMath, 2(4), 1700. https://doi.org/10.59400/jam.v2i4.1700

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