Physlib.Particles.StandardModel.Basic

The global gauge group of the Standard Model (without discrete quotients) is defined as the product group $SU(3) \times SU(2) \times U(1)$. Here, $SU(3)$ denotes the special unitary group of $3 \times 3$ complex matrices, $SU(2)$ denotes the special unitary group of $2 \times 2$ complex matrices, and $U(1)$ denotes the unitary group of $1 \times 1$ complex matrices (the group of complex numbers with absolute value 1).

This group homomorphism maps an element $g$ of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ to its component in the special unitary group $SU(3)$. It is defined as the projection onto the first factor of the product group.

Let the Standard Model gauge group be denoted by $\mathcal{G} = SU(3) \times SU(2) \times U(1)$. The function is the group homomorphism that maps an element $g = (g_3, g_2, g_1) \in \mathcal{G}$ to its component in $SU(2)$, where $SU(2)$ is the special unitary group of $2 \times 2$ complex matrices.

This group homomorphism maps an element of the Standard Model gauge group $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ to its component in the unitary group $U(1)$. The group $U(1)$ is represented as the set of complex numbers with absolute value $1$ under multiplication.

Let $\mathcal{G}$ denote the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$. For any two elements $g, g' \in \mathcal{G}$, if their projections onto each of the three factor groups are equal—specifically, $\text{toSU3}(g) = \text{toSU3}(g')$, $\text{toSU2}(g) = \text{toSU2}(g')$, and $\text{toU1}(g) = \text{toU1}(g')$—then $g = g'$.

The gauge group $G = SU(3) \times SU(2) \times U(1)$ is equipped with a star operation (involution). For an element $g = (g_1, g_2, g_3) \in G$, where $g_1 \in SU(3)$, $g_2 \in SU(2)$, and $g_3 \in U(1)$, the star operation is defined component-wise as $g^* = (g_1^*, g_2^*, g_3^*)$. Here, the operation on each component corresponds to the adjoint (conjugate transpose) within the respective unitary groups.

Let $G = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. For any element $g = (g_1, g_2, g_3) \in G$, where $g_1 \in SU(3)$, $g_2 \in SU(2)$, and $g_3 \in U(1)$, the involution (star operation) $g^*$ is defined component-wise such that $g^* = (g_1^*, g_2^*, g_3^*)$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. For any element $g \in \mathcal{G}$, the projection of the involution (star operation) $g^*$ onto the $SU(3)$ component is equal to the involution of the projection of $g$ onto $SU(3)$. That is, $\text{toSU3}(g^*) = (\text{toSU3}(g))^*$.

Let $\mathcal{G} = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group. For any element $g \in \mathcal{G}$, the projection of the involution (star operation) $g^*$ onto the $SU(2)$ component is equal to the involution of the projection of $g$ onto $SU(2)$. That is, $\text{toSU2}(g^*) = (\text{toSU2}(g))^*$.

Let $\mathcal{G}$ be the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$. For any element $g \in \mathcal{G}$, the projection of the involution (star operation) $g^*$ onto the $U(1)$ component is equal to the involution of the projection of $g$ onto $U(1)$. That is, $\text{toU1}(g^*) = (\text{toU1}(g))^*$.

The star operation (involution) defined on the Standard Model gauge group $G = SU(3) \times SU(2) \times U(1)$ is involutive. That is, for any element $g \in G$, applying the star operation twice returns the original element: $(g^*)^* = g$.

The function maps an element $u \in U(1)$ (represented as a unitary complex number) to an element in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$. The resulting triple is $(I_3, \begin{pmatrix} \bar{u}^3 & 0 \\ 0 & u^3 \end{pmatrix}, u)$, where $I_3$ is the identity matrix of $SU(3)$ and $\bar{u}$ denotes the complex conjugate of $u$.

For any unitary complex number $u \in U(1)$, the projection onto the $SU(3)$ factor of the element in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined by the $U(1)$ subgroup inclusion map (which maps $u$ to $(I_3, \begin{pmatrix} \bar{u}^3 & 0 \\ 0 & u^3 \end{pmatrix}, u)$) is equal to the $3 \times 3$ identity matrix $I_3$.

For any unitary complex number $u \in U(1)$, the projection onto the $SU(2)$ factor of the element in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined by the map `ofU1Subgroup` is equal to the $2 \times 2$ matrix $$\begin{pmatrix} \bar{u}^3 & 0 \\ 0 & u^3 \end{pmatrix}$$ where $\bar{u}$ denotes the complex conjugate of $u$.

For any unitary complex number $u \in U(1)$, the projection onto the $U(1)$ factor of the element in the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined by the subgroup inclusion map (which sends $u$ to the triple $(I_3, \begin{pmatrix} \bar{u}^3 & 0 \\ 0 & u^3 \end{pmatrix}, u)$) is equal to $u$.

The subgroup $\mathbb{Z}_6 \subset SU(3) \times SU(2) \times U(1)$ is defined as the set of elements of the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$, where $\alpha$ is a sixth root of unity in $\mathbb{C}$ (satisfying $\alpha^6 = 1$), and $I_n$ denotes the $n \times n$ identity matrix. This subgroup corresponds to the elements of the un-quotiented gauge group $G_I$ that act trivially on all particle fields in the Standard Model.

The type represents the smallest possible gauge group of the Standard Model, defined as the quotient group $G_I / \mathbb{Z}_6$, where $G_I$ is the gauge group `GaugeGroupI` and $\mathbb{Z}_6$ is the subgroup `gaugeGroupℤ₆SubGroup`.

The $\mathbb{Z}_2$ subgroup of the un-quotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ is defined as the unique subgroup of order 2 within the $\mathbb{Z}_6$ kernel (the `gaugeGroupℤ₆SubGroup`). It consists of elements of the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$, where $\alpha \in \mathbb{C}$ satisfies $\alpha^2 = 1$, and $I_n$ denotes the $n \times n$ identity matrix. This subgroup contains the identity and the element $(I_3, -I_2, -1)$, and it acts trivially on all particle fields in the Standard Model.

The type represents the gauge group of the Standard Model defined as the quotient group $G_I / \mathbb{Z}_2$, where $G_I$ is the gauge group `GaugeGroupI` and $\mathbb{Z}_2$ is the subgroup `gaugeGroupℤ₂SubGroup`.

The $\mathbb{Z}_3$ subgroup of the un-quotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ is defined as the unique subgroup of order 3 within the $\mathbb{Z}_6$ kernel (the `gaugeGroupℤ₆SubGroup`). It consists of elements of the form $(\alpha^2 I_3, I_2, \alpha)$, where $\alpha \in \mathbb{C}$ satisfies $\alpha^3 = 1$, and $I_n$ denotes the $n \times n$ identity matrix. This subgroup acts trivially on all particle fields in the Standard Model.

The type represents the gauge group of the Standard Model defined as the quotient group $G_I / \mathbb{Z}_3$, where $G_I$ is the gauge group `GaugeGroupI` and $\mathbb{Z}_3$ is the subgroup `gaugeGroupℤ₃SubGroup`.

The type `StandardModel.GaugeGroupQuot` represents the set of valid discrete quotients of the product group $SU(3) \times SU(2) \times U(1)$ that can serve as the gauge group for the Standard Model. In particle physics, while the local symmetry is defined by the Lie algebra of $SU(3) \times SU(2) \times U(1)$, the global structure of the gauge group can be a quotient by a discrete subgroup of its center.

Given a quotient parameter $q$ of type `GaugeGroupQuot`, this function returns the corresponding global gauge group of the Standard Model. This group is defined as the quotient of the internal product group $G_I = SU(3) \times SU(2) \times U(1)$ by a discrete subgroup of its center as specified by the choice $q$.

The internal gauge group of the Standard Model, denoted as `GaugeGroupI` and defined as the product group $SU(3) \times SU(2) \times U(1)$, is a Lie group.

For every quotient parameter $q \in \text{GaugeGroupQuot}$, the associated gauge group $\text{GaugeGroup}(q)$ of the Standard Model is a Lie group.

The definition `gaugeBundleI` represents the trivial principal bundle over the spacetime manifold $M$ (denoted as `SpaceTime`) with the structure group $G = SU(3) \times SU(2) \times U(1)$ (the internal gauge group of the Standard Model, denoted as `GaugeGroupI`).

A gauge transformation is defined as a global section of the gauge bundle $\text{gaugeBundleI}$. In the context of the Standard Model, this corresponds to a field of gauge group elements acting on the theory's physical fields.