Association between SM doublets and coefficients of Aff(3)

We describe here a correspondence between the coefficients \(a^i_\mu\) of the affine group and electroweak doublet of the Standard Model of particle physics.

red quarks green quarks blue quarks leptonic sector
1. generation a11 ⇔ (d,u)r a12 ⇔ (d,u)g a13 ⇔ (d,u)b a10 ⇔ (e,νe)
2. generation a21 ⇔ (s, c)r a22 ⇔ (s, c)g a23 ⇔ (s, c)b a20 ⇔ (μ,νμ)
3. generation a31 ⇔ (b,t)r a32 ⇔ (b,t)g a33 ⇔ (b,t)b a30 ⇔ (τ,ντ)

As a consequence, the configuration space corresponding to a single electroweak doublet is a single, real-valued lattice function \(a^i_\mu(n_1,n_2,n_3)\), where the upper index i denotes the generation, and the lower index μ denotes, for \(\mu>0\), the color of the quark doublet, while \(\mu=0\) describes the lepton doublet.

Leptons have no color charge, which may be characterized by giving them the "color" black (or white). With this identification, we can simply name the lower index μ the "color index".

The corresponding phase space consists of a pair of such lattice functions \(a^i_\mu(n_1,n_2,n_3),\pi^i_\mu(n_1,n_2,n_3))\). We can define a complex structure by \[ z^i_\mu(n_1,n_2,n_3) = a^i_\mu(n_1,n_2,n_3) + i \pi^i_\mu(n_1,n_2,n_3)\]

This complex structure has a remarkable property: It is preserved by all SM gauge fields.

A consequence of the identification is that we can identify the action of Euclidean symmetry on the electroweak doublets. This symmetry has a remarkable property: It commutes with all SM gauge fields.


Rotations \((\omega^i_j)\) are also affine transformations. Therefore they can act on affine transformations using the composition rule for affine transformations. \begin{eqnarray} a^1_\mu &\to& \omega^1_1 a^1_\mu + \omega^1_2 a^2_\mu + \omega^1_3 a^3_\mu\\ a^2_\mu &\to& \omega^2_1 a^1_\mu + \omega^2_2 a^2_\mu + \omega^2_3 a^3_\mu\\ a^3_\mu &\to& \omega^3_1 a^1_\mu + \omega^3_2 a^2_\mu + \omega^3_3 a^3_\mu \end{eqnarray}

These rotations preserve the lower index \(\mu\): The expression for all components \(a^i_\mu\) depends only on \(a^j_\mu\) with the same lower index \(\mu\). Thus, the color of the quarks remains unchanged, and quarks depend only on quarks, and leptons only on leptons. What changes is the upper index i, which denotes the generation.


Translations, characterized by a shift vector (ti), act on the elements of the affine group in a simple way: \[ a^i_0 \to a^i_0 + t^i, \]

while the quark doublets \(a^i_\mu\) with \(\mu > 0\) remain unchanged. As a consequence, translations act only on the leptonic sector.