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which I found from this page at Tony Smith's website, thanks to Jere Northrop.
Michael Gibbs's Bit Representation of Clifford Algebras
| Signature {$P-N$} | 8k | 8k+1 | 8k+2 | 8k+3 | 8k+4 | 8k+5 | 8k+6 | 8k+7 |
| Type | {$\mathbb{R}$} | {$\mathbb{R}\oplus\mathbb{R}$} | {$\mathbb{R}$} | {$\mathbb{C}$} | {$\mathbb{H}$} | {$\mathbb{H}\oplus\mathbb{H}$} | {$\mathbb{H}$} | {$\mathbb{C}$} |
| Bits(B) | {$\frac{1}{2}(P+N)$} | {$\frac{1}{2}(P+N-1)$} | {$\frac{1}{2}(P+N)$} | {$\frac{1}{2}(P+N-1)$} | {$\frac{1}{2}(P+N-2)$} | {$\frac{1}{2}(P+N-3)$} | {$\frac{1}{2}(P+N-2)$} | {$\frac{1}{2}(P+N-1)$} |
| Bits(B) P=0 | {$0$} | {$3$} | {$3$} | {$2$} | {$1$} | {$0$} | {$0$} | {$0$} |
Bits B indicates that there are 2B rows and columns in the nonzero blocks of the matrix.
From Bits to Matrix
The number of basis elements is a power of 2, thus the basis elements are representable by a string of P+N bits.
Gibbs provides a reasonably simple encoding that translates the bit string into the full matrix which represents the Clifford algebra.
Links
- Michael Gibbs
- M. Gibbs, Spacetime as a Quantum Graph, Ph.D. thesis of J. Michael Gibbs, with advisor David Finkelstein, Georgia Tech (1994).
- David Ritz Finkelstein (1929-2016)
- David Finkelstein, J. Michael Gibbs. Quantum relativity. 1993. We propose a unified description of the known forces. We formulate a quantum relativistic spacetime as a (directed) graph of causal arrows with indefinite Hilbert metric, whose physical meaning is given.
- Frank D. (Tony) Smith, Jr. Gravity and the Standard Model with 130 GeV Truth Quark from D4 − D5 − E6 Model using 3 × 3 Octonion Matrices.
- Michael Gibbs's Bit Representation of Clifford Algebras and their Generators, including a Constructive Proof of Clifford Periodicity 8