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A Note on the Representation of Clifford Algebra
Version 1
: Received: 28 February 2020 / Approved: 29 February 2020 / Online: 29 February 2020 (12:52:32 CET)
How to cite: Gu, Y. A Note on the Representation of Clifford Algebra. Preprints 2020, 2020020466. https://doi.org/10.20944/preprints202002.0466.v1 Gu, Y. A Note on the Representation of Clifford Algebra. Preprints 2020, 2020020466. https://doi.org/10.20944/preprints202002.0466.v1
Abstract
In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.
Keywords
Clifford algebra; Pauli matrix; gamma matrix; matrix representation
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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