(depr.) Standard Model 🪩 Symmetries ❡

Update ❡

Sadly, again ... these matrizes are just tricks to have them act like GA does.

So see here σᵧ f.e.:

⎡  i  0  ⎤
⎣  0 -i  ⎦ 

now this format just does the same as the anti-commutative property of GA with that +/- stuff and the fact that matrices are not commutative (non commutativity is not the same as anti-commutativity!)

You can learn this, but again - its just a hack. I'm still looking for ways to have Cohl Fureys lectures in a GA language though but ill disregard matrizes from here on.

Update End ❡

U(1) SU(2) SU(3)

complex numbers C are written as z = a + bi

The conjugate is z̃ = a - bi

Translation from GA ❡

"Lie Algebra" - vector space of Bivectors

"Lie Group" (G) - manifold of Rotors

So these technically can all be handled from just GA but it's smart to keep some understanding about these groups still as most of literature will reference them.

U(1) ❡

Electromagnetism / 2D Phase ❡

[ C ]

So thats 2 values in 1 cell. To get this as a real number matrix imagine a simple circle rotation / sin/cos

⎡  cos, -sin  ⎤
⎣  sin,  cos  ⎦

Thats akin to

⎡  0, -1 ⎤
⎣  1,  0 ⎦

And the reason for that is that this rotates a point [x, y] with matmul

x' =  cos(θ)  +  sin(θ)
y' = -sin(θ)  +  cos(θ)

and if you square [0, -1 | 1, 0 ] you get

⎡  0, -1  ⎤ × ⎡  0, -1  ⎤ = ⎡ -1,  0  ⎤
⎣  1,  0  ⎦   ⎣  1,  0  ⎦   ⎣  0, -1  ⎦

which is the the -1 identity matrix

SU(2) ❡

Quantum Spin and the Weak Force ❡

I never encountered these until now (i mean technically i do since birth i guess)

That's now 8 values in 4 cells

⎡ C, C ⎤
⎣ C, C ⎦

But ! These are used with quaternions like

1 = ⎡  1  0  ⎤
    ⎣  0  1  ⎦

i = ⎡  i  0  ⎤      -kj = i:    ⎡  0 -i  ⎤ × ⎡  0  1  ⎤ = ⎡  i  0  ⎤
    ⎣  0 -i  ⎦                  ⎣ -i  0  ⎦   ⎣ -1  0  ⎦   ⎣  0 -i  ⎦

j = ⎡  0  1  ⎤      -ik = j     ⎡ -i  0  ⎤ × ⎡  0  i  ⎤ = ⎡  0  1  ⎤
    ⎣ -1  0  ⎦                  ⎣  0  i  ⎦   ⎣  i  0  ⎦   ⎣ -1  0  ⎦

k = ⎡  0  i  ⎤      -ji = k:    ⎡  0 -1  ⎤ × ⎡  i  0  ⎤ = ⎡  0  i  ⎤
    ⎣  i  0  ⎦                  ⎣  1  0  ⎦   ⎣  0 -i  ⎦   ⎣  i  0  ⎦

so every matrix is

A = ⎡  x  y  ⎤
    ⎣ -ỹ  x̃  ⎦

in quaternions the coefficients are sometimes noted w,x,y,z but thats highly misleading so lets add s for the scalar and keep x,y,z

H = ⎡  s + iz ,  x + iy  ⎤
    ⎣ -x + iy ,  s - iz  ⎦

Determinant is 1:

    (s + iz)(s - iz) - (x + iy)(-x + iy)
=   s² -(i²) z² - (-x² + (i² y²))
=   s² + z²     + x² + y²

=   1

in GA these are ❡

i = -e₂₃ j = -e₃₁ k = -e₁₂

-e₂₃ * -e₃₁ = e₂₃₃₁ = -e₁₂
-e₃₁ * -e₁₂ = e₃₁₁₂ = -e₂₃
-e₁₂ * -e₂₃ = e₁₂₂₃ = -e₃₁

Works conjugated too though

e₃₁ * e₂₃ = e₃₁₂₃ = e₁₂
e₂₃ * e₁₂ = e₂₃₁₂ = e₃₁
e₁₂ * e₃₁ = e₁₂₃₁ = e₂₃

So how do you build these?

Unit vectors are e₁ e₂ e₃

e₁ = ⎡ 0  1 ⎤       e₂ = ⎡ 0 -i ⎤       e₃ = ⎡ 1  0 ⎤
     ⎣ 1  0 ⎦            ⎣ i  0 ⎦            ⎣ 0 -1 ⎦