Public
Edited
May 29, 2022
g-numbers
From herehttps://arxiv.org/pdf/1905.08102.pdf
${tex`𝐚^2 = 𝐛^2 = 0`}
${tex`2𝐚⋅𝐛 ≡ 𝐛𝐚 + 𝐚𝐛 = 1`}
${tex`𝐚𝐛𝐚 = (1 - 𝐛𝐚) 𝐚 = 𝐚 - 𝐛𝐚^2 = 𝐚`}
${tex`(𝐚𝐛)(𝐛𝐚) = 0 = (𝐛𝐚)(𝐚𝐛)`}
---
Also from herehttps://www.garretstar.com/hyprevisited12-17-2017.pdf (in slightly different order, mostly re: idempotents).
And some craziness from herehttps://www.garretstar.com/sobczyk09-mar-2020.pdf, largely around the Clifford clock, and building up any G_p,q from standard basis.
Multiplication table, reordered for bidiagonal blocks:
× 𝐛𝐚 𝐚 𝐛 𝐚𝐛
𝐛𝐚 𝐛𝐚 0 𝐛 0
𝐛 0 𝐛𝐚 0 𝐛
𝐚 𝐚 0 𝐚𝐛 0
𝐚𝐛 0 𝐚 0 𝐚𝐛
Another alternative, reordered as a direct sum:
× 𝐚𝐛 𝐚 𝐛 𝐛𝐚
𝐚𝐛 𝐚𝐛 𝐚 0 0
𝐛 𝐛 𝐛𝐚 0 0
𝐚 0 0 𝐚𝐛 𝐚
𝐛𝐚 0 0 𝐛 𝐛𝐚
lift_scalar = (p, q) => (q ? Math.sign(q) * ((p || 0) % Math.abs(q)) : (p || 0))
reversion = (g_) => {
const g = lift(g_);
return lift({
ba: g.ab,
b: g.b,
a: g.a,
ab: g.ba
});
}
inversion = (g_) => {
const g = lift(g_);
return lift({
ba: g.ba,
b: -g.b,
a: -g.a,
ab: g.ab
});
}
star = (g) => inversion(reversion(g))
tr = (g_) => {
const g = lift(g_);
return lift_scalar(g.ba + g.ab);
}
det = (g_) => {
const g = lift(g_);
return lift_scalar(g.ba * g.ab - g.b * g.a);
}
Arithmetic
add = (f_, g_) => {
const f = lift(f_);
const g = lift(g_);
return lift({
ba: f.ba + g.ba,
b: f.b + g.b,
a: f.a + g.a,
ab: f.ab + g.ab
});
}
multiply = (f_,g_) => {
const f = lift(f_)
const g = lift(g_)
return lift({
ba: f.ba * g.ba + f.b * g.a,
b: f.ba * g.b + f.b * g.ab,
a: f.a * g.ba + f.ab * g.a,
ab: f.a * g.b + f.ab * g.ab,
})
}
reversion([-3, -4, 2, 11])
controls
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