Physlib.Particles.StandardModel.Fermions.QuarkDoublet

There is a linear equivalence between the type `QuarkDoublet` and the tensor product space $S_L \otimes \mathbb{C}^3 \otimes \mathbb{C}^2$. In this context, $S_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors, $\mathbb{C}^3$ (represented by `EuclideanSpace ℂ (Fin 3)`) is the complex vector space corresponding to color degrees of freedom, and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the complex vector space corresponding to weak isospin degrees of freedom.

The type `QuarkDoublet` is endowed with the structure of an additive commutative group. This group structure is inherited from the tensor product space $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$ via the equivalence `valEquiv`, where $S_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors, $\mathbb{C}^3$ represents color degrees of freedom, and $\mathbb{C}^2$ represents weak isospin degrees of freedom.

The type `QuarkDoublet` is endowed with the structure of a module over the complex numbers $\mathbb{C}$. This module structure is inherited from the tensor product space $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$ via the equivalence `valEquiv`, where $S_L$ (the space of left-handed Weyl spinors), $\mathbb{C}^3$ (the space of color degrees of freedom), and $\mathbb{C}^2$ (the space of weak isospin degrees of freedom) are complex vector spaces.

The linear equivalence over $\mathbb{C}$ between the space of quark doublets $\text{QuarkDoublet}$ and the triple tensor product space $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$. Here, $S_L$ (represented by `Fermion.LeftHandedWeyl`) denotes the space of left-handed Weyl spinors, $\mathbb{C}^3$ represents the color degrees of freedom, and $\mathbb{C}^2$ represents the weak isospin degrees of freedom. The map identifies a quark doublet with its representation in this underlying tensor product space.

For any quark doublet $q$, applying the linear equivalence $\text{valLinEquiv}$ to $q$ results in its underlying value $q.\text{val}$ in the tensor product space $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$. Here, $S_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors, $\mathbb{C}^3$ is the space of color degrees of freedom, and $\mathbb{C}^2$ is the space of weak isospin degrees of freedom.

Let $S_L$ be the space of left-handed Weyl spinors, $\mathbb{C}^3$ represent the color degrees of freedom, and $\mathbb{C}^2$ represent the weak isospin degrees of freedom. Given the linear equivalence $\text{valLinEquiv} : \text{QuarkDoublet} \simeq_{\mathbb{C}} S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$, then for any element $m$ in the triple tensor product space $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$, its image under the inverse map $\text{valLinEquiv}^{-1}(m)$ is the quark doublet $\langle m \rangle$ constructed from $m$.

The representation of the Lorentz group $SL(2, \mathbb{C})$ on the space of quark doublets. Given the linear equivalence $\text{QuarkDoublet} \cong S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$, where $S_L$ is the space of left-handed Weyl spinors, $\mathbb{C}^3$ represents color degrees of freedom, and $\mathbb{C}^2$ represents weak isospin degrees of freedom, the action of an element $\Lambda \in SL(2, \mathbb{C})$ is defined as the tensor product of the left-handed Weyl representation $\rho_L(\Lambda)$ acting on $S_L$ and the trivial representation (identity map) acting on both the color and weak isospin spaces.

The representation of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the space of quark doublets. The space of quark doublets is linearly isomorphic to the tensor product $S_L \otimes_{\mathbb{C}} \mathbb{C}^3 \otimes_{\mathbb{C}} \mathbb{C}^2$, where $S_L$ is the space of left-handed Weyl spinors, $\mathbb{C}^3$ is the color space, and $\mathbb{C}^2$ is the weak isospin space. Given an element $g = (g_3, g_2, g_1) \in \mathcal{G}$, the representation maps $g$ to a linear endomorphism of the quark doublet space that acts as: 1. The identity on the spinor factor $S_L$; 2. The fundamental representation of $g_3 \in SU(3)$ on the color factor $\mathbb{C}^3$; 3. The fundamental representation of $g_2 \in SU(2)$ on the weak isospin factor $\mathbb{C}^2$; 4. Scalar multiplication by $g_1 \in U(1)$ on the resulting tensor.