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.

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 triple 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. For 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. Its action on a basic tensor $\psi \otimes v \otimes w$ is defined by $g \cdot (\psi \otimes v \otimes w) = (g_1 \psi) \otimes (g_3 v) \otimes (g_2 w)$, where $g_3 \in SU(3)$ and $g_2 \in SU(2)$ act on their respective factors via fundamental representations, and $g_1 \in U(1)$ acts as complex scalar multiplication.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. For any element $g \in \mathcal{G}$, let $g_3 \in SU(3)$, $g_2 \in SU(2)$, and $g_1 \in U(1)$ be its components under the projection maps. Given a left-handed Weyl spinor $\psi \in S_L$, a color vector $v \in \mathbb{C}^3$, and a weak isospin vector $w \in \mathbb{C}^2$, the action of the gauge group representation $\rho$ on a pure tensor $\psi \otimes v \otimes w$ in the quark doublet space is given by: $$\rho(g)(\psi \otimes v \otimes w) = (g_1 \cdot \psi) \otimes (g_3 v) \otimes (g_2 w)$$ where $g_1$ acts on $\psi$ via scalar multiplication, and $g_3$ and $g_2$ act on $v$ and $w$ respectively via their fundamental representations as linear transformations.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. Let the space of quark doublets be represented by the tensor product $S_L \otimes \mathbb{C}^3 \otimes \mathbb{C}^2$, 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. Let $\{\psi_k\}_{k \in \{0,1\}}$, $\{e_i\}_{i \in \{0,1,2\}}$, and $\{f_j\}_{j \in \{0,1\}}$ be the standard bases for $S_L$, $\mathbb{C}^3$, and $\mathbb{C}^2$, respectively. For any element $g = (g_3, g_2, g_1) \in \mathcal{G}$, the action of the gauge group representation $\rho(g)$ on the basis element $\psi_k \otimes e_i \otimes f_j$ is given by the sum: $$\rho(g)(\psi_k \otimes e_i \otimes f_j) = \sum_{i'=0}^2 \sum_{j'=0}^1 (g_1 \cdot (g_3)_{i'i} \cdot (g_2)_{j'j}) (\psi_k \otimes e_{i'} \otimes f_{j'})$$ where $g_1 \in \mathbb{C}$ is the $U(1)$ component, and $(g_3)_{i'i}$ and $(g_2)_{j'j}$ denote the matrix entries of the $SU(3)$ and $SU(2)$ components of $g$.

Let $g_1, g_2 \in SU(3) \times SU(2) \times U(1)$ be elements of the Standard Model gauge group. Let $\rho$ be the representation of this group on the space of quark doublets. The linear endomorphisms $\rho(g_1)$ and $\rho(g_2)$ are equal if and only if for all indices $i, i' \in \{1, 2, 3\}$ and $j, j' \in \{1, 2\}$, the following equality holds: $$(g_1)_{U(1)} \cdot ((g_1)_{SU(3)})_{i'i} \cdot ((g_1)_{SU(2)})_{j'j} = (g_2)_{U(1)} \cdot ((g_2)_{SU(3)})_{i'i} \cdot ((g_2)_{SU(2)})_{j'j}$$ where $(g)_{U(1)} \in \mathbb{C}$ is the component of $g$ in $U(1)$, and $((g)_{SU(3)})_{i'i}$ and $((g)_{SU(2)})_{j'j}$ denote the matrix elements of the $SU(3)$ and $SU(2)$ components of $g$, respectively.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group and let $\rho$ be its representation on the space of quark doublets. An element $g \in \mathcal{G}$ belongs to the kernel of $\rho$ if and only if there exist complex scalars $a, b \in \mathbb{C}$ such that: 1. The $SU(2)$ component of $g$ is a scalar matrix $a I_2$; 2. The $SU(3)$ component of $g$ is a scalar matrix $b I_3$; 3. The product of these scalars and the $U(1)$ component of $g$ satisfies $a \cdot b \cdot g_{U(1)} = 1$.

Let $G_I = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group (without discrete quotients), and let $\rho$ be the representation of $G_I$ on the space of quark doublets. Let $H_{\mathbb{Z}_6}$ be the central subgroup of $G_I$ associated with the choice of the discrete quotient $\mathbb{Z}_6$. Then $H_{\mathbb{Z}_6}$ is a subgroup of the kernel of $\rho$ ($H_{\mathbb{Z}_6} \subseteq \ker \rho$).

For any choice of a discrete quotient $Q$ of the Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$, the central subgroup $Z_Q$ associated with $Q$ is contained within the kernel of the representation of $G_I$ on the space of quark doublets.