Physlib.Particles.StandardModel.Fermions.LeptonDoublet

The equivalence $\text{valEquiv}$ provides an isomorphism between the space of lepton doublets and the tensor product of left-handed Weyl spinors and the 2-dimensional complex weak isospin space. Mathematically, it identifies the type `LeptonDoublet` with the complex tensor product $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors carrying the Lorentz index, and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the space carrying the weak index.

The space of lepton doublets, denoted by the type `LeptonDoublet`, is equipped with the structure of an additive commutative group. This structure is induced via the equivalence `valEquiv` from the tensor product $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the 2-dimensional complex space carrying the weak isospin index. This allows for the addition of lepton doublets, the existence of a zero doublet, and the definition of additive inverses.

The space of lepton doublets $\text{LeptonDoublet}$ is equipped with the structure of a module over the complex numbers $\mathbb{C}$, making it a complex vector space. This structure is inherited from the tensor product $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$ via the equivalence $\text{valEquiv}$, where $W_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors carrying the Lorentz index and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the space carrying the weak isospin index.

The space of lepton doublets $\text{LeptonDoublet}$ is $\mathbb{C}$-linearly isomorphic to the tensor product $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors carrying the Lorentz index and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the complex vector space carrying the weak isospin index. This isomorphism provides the formal linear identification between the lepton doublet wrapper and its underlying tensor-product representation.

For any lepton doublet $l$ in the space $\text{LeptonDoublet}$, applying the linear isomorphism $\text{valLinEquiv}$ to $l$ yields its underlying value $l.\text{val}$. This value resides in the tensor product space $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ (represented by `Fermion.LeftHandedWeyl`) is the space of left-handed Weyl spinors and $\mathbb{C}^2$ (represented by `EuclideanSpace ℂ (Fin 2)`) is the space carrying the weak isospin index.

Let $W_L$ be the space of left-handed Weyl spinors (represented by `Fermion.LeftHandedWeyl`) and $\mathbb{C}^2$ be the complex vector space carrying the weak isospin index (represented by `EuclideanSpace ℂ (Fin 2)`). For any element $m$ in the tensor product space $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, the inverse of the linear isomorphism $\text{valLinEquiv}$ maps $m$ to the lepton doublet $\langle m \rangle$.

For any two lepton doublets $l_1$ and $l_2$ in the space $\text{LeptonDoublet}$, the underlying value of their sum is equal to the sum of their individual values: $(l_1 + l_2).\text{val} = l_1.\text{val} + l_2.\text{val}$. Here, the addition on the left-hand side is the addition operation defined for lepton doublets, and the addition on the right-hand side is the addition in the underlying tensor product space $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$.

For any complex scalar $r \in \mathbb{C}$ and any lepton doublet $l$ in the space $\text{LeptonDoublet}$, the underlying value of the scalar multiplication $r \cdot l$ is equal to the scalar multiplication of $r$ with the underlying value of $l$: $(r \cdot l).\text{val} = r \cdot l.\text{val}$. Here, the left-hand side uses the scalar multiplication defined for lepton doublets, while the right-hand side uses the scalar multiplication in the underlying tensor product space $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$.

This definition establishes the complex linear representation of the group $\text{SL}(2, \mathbb{C})$ (the double cover of the restricted Lorentz group) on the space of lepton doublets. The space of lepton doublets is modeled as the tensor product $W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ is the vector space of left-handed Weyl spinors and $\mathbb{C}^2$ represents the two-dimensional space of weak isospin. For any Lorentz transformation $\Lambda \in \text{SL}(2, \mathbb{C})$, the representation acts by applying the left-handed Weyl representation to the first factor of the tensor product while acting trivially (as the identity) on the second factor.

The representation of the unquotiented Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ on the space of lepton doublets $\text{LeptonDoublet} \cong W_L \otimes_{\mathbb{C}} \mathbb{C}^2$, where $W_L$ is the space of left-handed Weyl spinors. For an element $g = (g_3, g_2, g_1) \in \mathcal{G}$, the action on a tensor product $v \otimes w$ (where $v \in W_L$ and $w \in \mathbb{C}^2$) is given by: $$g \cdot (v \otimes w) = (g_1^*)^3 v \otimes (g_2 w)$$ In this representation, the $SU(3)$ component $g_3$ acts trivially (as the lepton doublet is a singlet), the $SU(2)$ component $g_2$ acts on the weak isospin index $w$, and the $U(1)$ component $g_1$ acts via the hypercharge $Y = -1/2$ (normalized as $6Y = -3$), represented by the scalar multiplication by $(g_1^*)^3$.

Let $g$ be an element of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ with components $(g_3, g_2, g_1)$. For any pure tensor $v \otimes w$ in the lepton doublet space, where $v$ is a left-handed Weyl spinor and $w \in \mathbb{C}^2$ is a vector in weak isospin space, the gauge action of $g$ is given by: $$g \cdot (v \otimes w) = (g_1^*)^3 v \otimes (g_2 w)$$ where $g_1^* \in U(1)$ is the complex conjugate of $g_1$ and $g_2 \in SU(2)$ acts linearly on $w$.

Let $g = (g_3, g_2, g_1)$ be an element of the unquotiented Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$. Let $\{e_k\}_{k \in \{0, 1\}}$ be the basis for the space of left-handed Weyl spinors $W_L$ and $\{f_j\}_{j \in \{0, 1\}}$ be the standard basis for the weak isospin space $\mathbb{C}^2$. The gauge representation of $g$ acting on the basis element $e_k \otimes f_j$ of the lepton doublet space is given by the expansion: $$g \cdot (e_k \otimes f_j) = \sum_{j' \in \{0, 1\}} (\bar{g}_1^3 (g_2)_{j'j}) (e_k \otimes f_{j'})$$ where $\bar{g}_1$ denotes the complex conjugate of the $U(1)$ component $g_1$, and $(g_2)_{j'j}$ denotes the entry in the $j'$-th row and $j$-th column of the $SU(2)$ matrix component of $g$.

Let $g_1, g_2$ be elements of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$. Let $\rho$ be the representation of $\mathcal{G}$ on the space of lepton doublets. The actions of these two elements are identical, $\rho(g_1) = \rho(g_2)$, if and only if for all indices $j, j' \in \{1, 2\}$, the following equality of coefficients holds: $$(g_{1, U(1)}^*)^3 (g_{1, SU(2)})_{j'j} = (g_{2, U(1)}^*)^3 (g_{2, SU(2)})_{j'j}$$ where $g_{i, U(1)} \in U(1) \subset \mathbb{C}$ denotes the $U(1)$ component, $(g_{i, SU(2)})_{j'j}$ denotes the $(j', j)$-th entry of the $SU(2)$ matrix component, and $^*$ denotes complex conjugation.

An element $g = (g_3, g_2, g_1)$ of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ acts trivially on the lepton doublet space if and only if there exists a complex scalar $a \in \mathbb{C}$ such that the $SU(2)$ component $g_2$ is a scalar matrix $a \mathbb{I}$ (where $\mathbb{I}$ is the $2 \times 2$ identity matrix) and the $U(1)$ component $g_1$ satisfies $a (g_1^*)^3 = 1$.

For the representation $\rho$ of the unquotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ on the space of lepton doublets $L \cong (1, 2)_{-3}$, the central subgroup $\mathbb{Z}_6 \subset G_I$ is contained within the kernel of the representation $\rho$. This implies that the central $\mathbb{Z}_6$ subgroup acts trivially on the lepton doublet space.

For any valid discrete quotient $Q$ of the Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$, the associated central subgroup $K_Q \subseteq G_I$ is contained in the kernel of the representation $\rho$ of $G_I$ on the space of lepton doublets. That is, every element in the subgroup used to define the quotient acts trivially on the lepton doublet space.

Given a quotient parameter $Q$ of type `GaugeGroupQuot`, this definition provides the representation of the corresponding global Standard Model gauge group $\text{GaugeGroup}(Q)$ on the complex vector space of lepton doublets $\text{LeptonDoublet}$. For the unquotiented case $Q = I$, the representation is the standard action $\rho_I$ of $SU(3) \times SU(2) \times U(1)$ where the lepton doublet transforms in the $(1, 2)_{-3}$ representation. For the quotient cases $Q \in \{\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_6\}$, the representation is defined by lifting the unquotiented representation $\rho_I$ to the quotient group $G_I / K_Q$, utilizing the fact that the central subgroup $K_Q$ lies within the kernel of $\rho_I$.