Abstract
The Riemann curvature tensor is constructed using the Clifford-Dirac geometric algebra and division-algebraic operator structure.
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Notes
Ref. [28], p. 89.
Ref. [1], Ch. 3 & Sec. 3.3.
Ref. [17], Secs. 11.5 & 24.6.
See Ref. [23], Sec. III, for the step-by-step derivation, which results in Eq. (8) herein.
See, e.g., Ref. [3], Eqs. (9.29)–(9.30).
Ref. [2], p. 246.
See, e.g., Ref. [7], p. 404.
Ref. [13], p. 224.
Ref. [2], pp. 403–405.
Ref. [20], Sec. 6.2, p. 155: Any symmetric matrix can be transformed into a diagonal matrix with main diagonal \(\pm 1\) entries.
In like manner the 8-component strong interaction gauge field: \(G_{\mu }^{a}\mid a:1\rightarrow 8\), requires an analogous 8-component operator, which has been constructed (Ref. [26], Sec. 3.3).
See, e.g., Ref. [7], p. 404, for the component expression of \(\textbf{R}\) in the \(\{\partial _{\mu }\}\) basis.
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Wolk, B.J. A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure. Adv. Appl. Clifford Algebras 33, 42 (2023). https://doi.org/10.1007/s00006-023-01286-8
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DOI: https://doi.org/10.1007/s00006-023-01286-8