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.

For any sixth root of unity $\alpha \in \mathbb{C}$, the projection of the homomorphism $\Phi : \mu_6 \to SU(3) \times SU(2) \times U(1)$ (defined as `gaugeGroupℤ₆Hom`) onto the $SU(2)$ factor of the Standard Model gauge group is equal to the element $\Phi_{SU(2)}(\alpha)$ (defined as `gaugeGroupℤ₆SU2OfRoot α`).

For any sixth root of unity $\alpha \in \mathbb{C}$ (where $\alpha^6 = 1$), the projection of the element $\text{gaugeGroup}\mathbb{Z}_6\text{Hom}(\alpha)$ onto the $U(1)$ factor of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ is equal to $\text{gaugeGroup}\mathbb{Z}_6\text{UnitaryOfRoot}(\alpha)$.

Let $\alpha \in \mathbb{C}$ be a sixth root of unity (such that $\alpha^6 = 1$). Then the element of the Standard Model gauge group $SU(3) \times SU(2) \times U(1)$ defined by $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$ is a member of the $\mathbb{Z}_6$ subgroup.

Let $G_I = SU(3) \times SU(2) \times U(1)$ be the global gauge group of the Standard Model without discrete quotients. An element $g \in G_I$ belongs to the subgroup $\mathbb{Z}_6$ if and only if there exists a sixth root of unity $\alpha \in \mathbb{C}$ (satisfying $\alpha^6 = 1$) such that $g = (\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$, where $I_n$ denotes the $n \times n$ identity matrix.

Let $G_I = SU(3) \times SU(2) \times U(1)$ be the global gauge group of the Standard Model without discrete quotients. Let $\mathbb{Z}_6$ be the subgroup of $G_I$ consisting of elements of the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$, where $\alpha \in \mathbb{C}$ is a sixth root of unity (i.e., $\alpha^6 = 1$) and $I_n$ denotes the $n \times n$ identity matrix. Then $\mathbb{Z}_6$ is a subgroup of the center of $G_I$.

The subgroup $\mathbb{Z}_6$ of the Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$—defined as the set of elements of the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$ where $\alpha \in \mathbb{C}$ is a sixth root of unity—is a normal subgroup.

The type `GaugeGroupℤ₆`, representing the Standard Model gauge group $G_I / \mathbb{Z}_6$, is equipped with a group structure. This defines the multiplication, identity, and inverse operations inherited from the quotient of the group $G_I$ by the normal subgroup $\mathbb{Z}_6$.

The function is the canonical quotient group homomorphism $\pi: G_I \to G_I / \mathbb{Z}_6$ from the un-quotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ to the quotient gauge group $G_I / \mathbb{Z}_6$, where $\mathbb{Z}_6$ is the specific normal subgroup of $G_I$ that acts trivially on all particle fields.

Let $\mu_6 = \{ \alpha \in \mathbb{C} \mid \alpha^6 = 1 \}$ be the group of sixth roots of unity. For any $\alpha \in \mu_6$, let $g_\alpha \in G_I$ be the element in the global Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ constructed from $\alpha$ via the function `gaugeGroupℤ₆OfRoot`. Let $\pi : G_I \to G_I / \mathbb{Z}_6$ be the canonical quotient homomorphism. Then for every $\alpha \in \mu_6$, $\pi(g_\alpha) = 1$ in the quotient group $G_I / \mathbb{Z}_6$.

This is the group homomorphism mapping the second roots of unity, $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, into the sixth roots of unity, $\mu_6 = \{z \in \mathbb{C} \mid z^6 = 1\}$. This map represents the inclusion $\mu_2 \hookrightarrow \mu_6$ based on the fact that any element satisfying $z^2 = 1$ also satisfies $z^6 = 1$.

Given a second root of unity $\alpha \in \mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, this definition returns the corresponding element in the global Standard Model gauge group $SU(3) \times SU(2) \times U(1)$.

For any second root of unity $\alpha \in \mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, the projection of the Standard Model gauge group element associated with $\alpha$ onto its $SU(3)$ component is equal to the $SU(3)$ element associated with the image of $\alpha$ under the natural inclusion $\mu_2 \hookrightarrow \mu_6$.

For any second root of unity $\alpha \in \mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, the projection onto the $SU(2)$ factor of the Standard Model gauge group element (in $SU(3) \times SU(2) \times U(1)$) associated with $\alpha$ is equal to the $SU(2)$ element associated with the image of $\alpha$ under the inclusion map $\mu_2 \hookrightarrow \mu_6$.

Let $\mu_n = \{z \in \mathbb{C} \mid z^n = 1\}$ denote the group of $n$-th roots of unity. For any second root of unity $\alpha \in \mu_2$, let $g(\alpha) \in SU(3) \times SU(2) \times U(1)$ be the element of the Standard Model gauge group constructed from $\alpha$. Let $\pi_{U(1)} : SU(3) \times SU(2) \times U(1) \to U(1)$ be the group homomorphism that projects an element onto its $U(1)$ component. Then $$\pi_{U(1)}(g(\alpha)) = h(\iota(\alpha))$$ where $\iota : \mu_2 \hookrightarrow \mu_6$ is the natural inclusion map and $h : \mu_6 \to U(1)$ is the map identifying a sixth root of unity with an element of the unitary group $U(1)$.

Let $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ be the set of second roots of unity, and let $G = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group (without discrete quotients). For any $\alpha \in \mu_2$, the element $\text{gaugeGroupℤ₂OfRoot}(\alpha) \in G$ belongs to the center $Z(G)$ of the gauge group.

The definition represents the group homomorphism from the second roots of unity, $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, to the un-quotiented Standard Model gauge group $SU(3) \times SU(2) \times U(1)$. This map embeds the discrete group $\mu_2 \cong \mathbb{Z}_2$ into the product of the unitary groups.

Let $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ be the group of second roots of unity and let $G = SU(3) \times SU(2) \times U(1)$ be the global gauge group of the Standard Model (without discrete quotients). For any element $\alpha \in \mu_2$, the value of the group homomorphism $\text{gaugeGroup}\mathbb{Z}_2\text{Hom}: \mu_2 \to G$ applied to $\alpha$ is equal to the element $\text{gaugeGroup}\mathbb{Z}_2\text{OfRoot}(\alpha)$.

Let $\mu_n = \{z \in \mathbb{C} \mid z^n = 1\}$ be the group of $n$-th roots of unity in the complex numbers. Let $G = SU(3) \times SU(2) \times U(1)$ be the un-quotiented gauge group of the Standard Model. Consider the group homomorphism $\phi: \mu_2 \to G$ that embeds the second roots of unity into $G$. For any $\alpha \in \mu_2$, the projection of the element $\phi(\alpha)$ onto the $SU(3)$ factor is equal to the image of $\alpha$ under the map from $\mu_6$ to $SU(3)$, where $\alpha$ is treated as an element of $\mu_6$ via the inclusion $\mu_2 \hookrightarrow \mu_6$.

Let $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ be the group of second roots of unity. Let $\Phi : \mu_2 \to SU(3) \times SU(2) \times U(1)$ be the group homomorphism that embeds $\mu_2$ into the product of the gauge groups of the Standard Model. For any $\alpha \in \mu_2$, the projection of $\Phi(\alpha)$ onto the $SU(2)$ component is equal to the value of the $SU(2)$-mapping for sixth roots of unity applied to $\alpha$ (treated as an element of $\mu_6$).

Let $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ and $\mu_6 = \{z \in \mathbb{C} \mid z^6 = 1\}$ be the groups of second and sixth roots of unity, respectively. For any $\alpha \in \mu_2$, the projection onto the $U(1)$ factor of the image of $\alpha$ under the homomorphism $\mu_2 \to SU(3) \times SU(2) \times U(1)$ is equal to the image of $\alpha$ (viewed as an element of $\mu_6$) under the map defining the $U(1)$ component for sixth roots of unity.

In the context of the Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$, let $\mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ be the set of second roots of unity. For any $\alpha \in \mu_2$, the gauge group element $g(\alpha)$ (defined as $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$) is an element of the $\mathbb{Z}_2$ subgroup of $G_I$.

Let $G = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group (without discrete quotients). An element $g \in G$ belongs to the $\mathbb{Z}_2$ subgroup (the kernel of the gauge group, denoted as `gaugeGroupℤ₂SubGroup`) if and only if there exists a second root of unity $\alpha \in \mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$ such that $g$ is the element corresponding to $\alpha$ (given by the map `gaugeGroupℤ₂OfRoot`). Elements of this subgroup take the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$, specifically consisting of the identity and the element $(I_3, -I_2, -1)$.

The $\mathbb{Z}_2$ subgroup of the un-quotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ is a subgroup of the $\mathbb{Z}_6$ kernel. Specifically, if $G_{\mathbb{Z}_2}$ is the subgroup consisting of elements of the form $(\alpha^2 I_3, \alpha^{-3} I_2, \alpha)$ where $\alpha^2 = 1$, and $G_{\mathbb{Z}_6}$ is the subgroup consisting of elements of the same form where $\alpha^6 = 1$, then $G_{\mathbb{Z}_2} \le G_{\mathbb{Z}_6}$.

Let $G = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group (without discrete quotients). The $\mathbb{Z}_2$ subgroup of $G$ is a subgroup of the center of $G$.

In the un-quotiented Standard Model gauge group $G = SU(3) \times SU(2) \times U(1)$, the $\mathbb{Z}_2$ subgroup (the unique subgroup of order $2$ within the $\mathbb{Z}_6$ kernel, consisting of the elements $(I_3, I_2, 1)$ and $(I_3, -I_2, -1)$) is a normal subgroup of $G$.

The gauge group of the Standard Model, defined as the quotient $G_I / \mathbb{Z}_2$, is equipped with a group structure. Here $G_I$ denotes the internal gauge group `GaugeGroupI` and $\mathbb{Z}_2$ is the normal subgroup `gaugeGroupℤ₂SubGroup`.

The canonical group homomorphism $\pi: G_I \to G_I / \mathbb{Z}_2$ which maps an element of the un-quotiented Standard Model gauge group $G_I = SU(3) \times SU(2) \times U(1)$ to its corresponding element in the quotient gauge group $G_I / \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the specific normal subgroup `gaugeGroupℤ₂SubGroup`.

Let $G_I = SU(3) \times SU(2) \times U(1)$ be the Standard Model gauge group without discrete quotients, and let $\pi: G_I \to G_I / \mathbb{Z}_2$ be the canonical quotient homomorphism. For any second root of unity $\alpha \in \mu_2 = \{z \in \mathbb{C} \mid z^2 = 1\}$, the image of the element $\text{gaugeGroupℤ₂OfRoot}(\alpha) \in G_I$ under the quotient map $\pi$ is the identity element of $G_I / \mathbb{Z}_2$.

The group homomorphism representing the natural inclusion of the third roots of unity $\mu_3 = \{ \alpha \in \mathbb{C} \mid \alpha^3 = 1 \}$ into the sixth roots of unity $\mu_6 = \{ \alpha \in \mathbb{C} \mid \alpha^6 = 1 \}$.

For a given third root of unity $\alpha \in \mu_3(\mathbb{C}) = \{ z \in \mathbb{C} \mid z^3 = 1 \}$, this function defines a corresponding element in the Standard Model gauge group $G_{SM} = SU(3) \times SU(2) \times U(1)$.

For any third root of unity $\alpha \in \mu_3(\mathbb{C})$, the projection onto the $SU(3)$ factor of the Standard Model gauge group element associated with $\alpha$ is equal to the $SU(3)$ element associated with the image of $\alpha$ under the natural inclusion $\mu_3 \hookrightarrow \mu_6$. Mathematically, $\pi_{SU(3)}(g_{\mu_3}(\alpha)) = g_{SU(3), \mu_6}(\iota(\alpha))$, where $\pi_{SU(3)}$ is the projection from $SU(3) \times SU(2) \times U(1)$ to $SU(3)$, $g_{\mu_3}$ maps a third root of unity to the gauge group, $\iota$ is the inclusion of $\mu_3$ into $\mu_6$, and $g_{SU(3), \mu_6}$ maps a sixth root of unity to an element of $SU(3)$.