Physlib.Relativity.Tensors.ComplexTensor.Weyl.Basic

This definition characterizes the fundamental representation $\rho$ of the special linear group $SL(2, \mathbb{C})$ on the complex vector space $V \cong \mathbb{C}^2$. Given a matrix $M \in SL(2, \mathbb{C})$ and a spinor $\psi \in V$, the representation acts via matrix-vector multiplication $\rho(M)\psi = M\psi$. In the context of particle physics, this representation describes the transformation properties of a left-handed Weyl fermion, typically denoted in index notation as $\psi^a$.

The standard basis for the complex vector space associated with left-handed Weyl fermions. This basis, indexed by $\{0, 1\}$, identifies the representation space $V_L \cong \mathbb{C}^2$ such that the fundamental representation of $SL(2, \mathbb{C})$ acts via standard matrix multiplication. Specifically, each basis element $e_i$ corresponds to the standard unit vector in $\mathbb{C}^2$ where the $i$-th component is $1$ and all other components are $0$.

Let $M \in SL(2, \mathbb{C})$ and let $\rho(M)$ be the linear map corresponding to the fundamental representation of $SL(2, \mathbb{C})$ on the space of left-handed Weyl fermions. Let $\mathcal{B}$ be the standard basis for this vector space. Then the $(i, j)$-th entry of the matrix of $\rho(M)$ with respect to $\mathcal{B}$ is equal to the $(i, j)$-th entry of the matrix $M$.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the complex vector space of left-handed Weyl fermions. Let $\Phi$ be the canonical identification map that sends a spinor in this space to its representation in $\mathbb{C}^2$. Then for each $i \in \{0, 1\}$, the image of the $i$-th basis element under this identification is the standard unit vector in $\mathbb{C}^2$ (the vector whose $i$-th component is $1$ and all other components are $0$).

This definition describes a representation of the special linear group $SL(2, \mathbb{C})$ on the complex vector space $\mathbb{C}^2$. Under this representation, an element $M \in SL(2, \mathbb{C})$ acts on a vector $\psi$ (representing a Weyl fermion) by the matrix transformation $\psi \mapsto (M^{-1})^T \psi$. In the language of physics, this corresponds to the transformation law for a left-handed Weyl fermion with a lower index, denoted as $\psi_a$.

The standard basis for the vector space of alt-left-handed Weyl fermions, which carries the representation of $SL(2, \mathbb{C})$ where an element $M$ acts via the matrix transformation $(M^{-1})^T$. This basis $\{e_0, e_1\}$ corresponds to the standard unit vectors of $\mathbb{C}^2$ and represents the components of a left-handed Weyl fermion with a lower index, typically denoted as $\psi_a$ in physics notation.

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the vector space of left-handed Weyl fermions $\psi_a$ (which carries the representation of $SL(2, \mathbb{C})$ where $M$ acts as $(M^{-1})^T$). For each $i \in \{0, 1\}$, the representation of the $i$-th basis element as a vector in $\mathbb{C}^2$ is the standard unit vector $\mathbf{e}_i$ (the vector with 1 at index $i$ and 0 elsewhere).

Let $\rho$ be the representation of the group $SL(2, \mathbb{C})$ on the complex vector space of left-handed Weyl fermions (often denoted by $\psi_a$ in physics notation). In the standard basis $\{e_0, e_1\}$, the matrix representation of an element $M \in SL(2, \mathbb{C})$ is given by the transpose of its inverse. Specifically, for any $M \in SL(2, \mathbb{C})$ and indices $i, j \in \{0, 1\}$, the $(i, j)$-th entry of the representation matrix $\rho(M)$ is equal to the $(i, j)$-th entry of $(M^{-1})^T$.

This definition constructs the representation of the group $SL(2, \mathbb{C})$ on the complex vector space $\mathbb{C}^2$ that describes right-handed Weyl fermions. For an element $M \in SL(2, \mathbb{C})$, the action on a spinor $\psi \in \mathbb{C}^2$ is given by $\psi \mapsto \bar{M} \psi$, where $\bar{M}$ denotes the matrix $M$ with complex conjugated entries. This corresponds to the conjugate representation of $SL(2, \mathbb{C})$, which in physics notation is associated with spinors carrying dotted indices, denoted as $\psi^{\dot{a}}$.

The standard basis $\{e_0, e_1\}$ for the complex vector space of right-handed Weyl fermions. This space is isomorphic to $\mathbb{C}^2$, and the basis is defined such that each $e_i$ corresponds to the standard unit vector in $\mathbb{C}^2$ (the vector with $1$ at index $i$ and $0$ elsewhere). In the context of the right-handed representation of $SL(2, \mathbb{C})$, this basis allows spinors to be expressed in terms of their dotted-index components $\psi^{\dot{a}}$.

Let $\{e_0, e_1\}$ denote the standard basis (`rightBasis`) for the complex vector space of right-handed Weyl fermions. Let $\Phi$ be the canonical isomorphism (`toFin2ℂ`) that maps these fermions to the coordinate space $\mathbb{C}^2$. For any index $i \in \{0, 1\}$, the image of the basis vector $e_i$ under $\Phi$ is the standard unit vector in $\mathbb{C}^2$ with $1$ at position $i$ and $0$ elsewhere.

Let $M \in SL(2, \mathbb{C})$ and let $\rho$ be the right-handed Weyl representation of $SL(2, \mathbb{C})$ on the complex vector space $\mathbb{C}^2$. Let $\{e_0, e_1\}$ be the standard basis for this representation. Then the matrix of the linear transformation $\rho(M)$ with respect to this basis is equal to the entry-wise complex conjugate of $M$. That is, for any indices $i, j \in \{0, 1\}$, the $(i, j)$-th entry of the matrix is given by $(\rho(M))_{ij} = \overline{M_{ij}}$.

The representation of the special linear group $SL(2, \mathbb{C})$ on the complex vector space $\mathbb{C}^2$ where each matrix $M \in SL(2, \mathbb{C})$ acts on a spinor $\psi$ via the mapping $\rho(M)\psi = (M^{-1})^\dagger \psi$, where $(M^{-1})^\dagger$ denotes the conjugate transpose of the inverse of $M$. In physics, this corresponds to the representation for a right-handed Weyl fermion carrying a dotted index $\psi_{\dot{a}}$.

The standard basis $\{e_i\}_{i \in \{0, 1\}}$ for the complex vector space associated with the alt-right-handed Weyl fermion representation. In this representation, the special linear group $SL(2, \mathbb{C})$ acts on the space via $\rho(M) = (M^{-1})^\dagger$, where $(M^{-1})^\dagger$ is the conjugate transpose of the inverse of $M$. This basis corresponds to the standard unit vectors in $\mathbb{C}^2$ and is used to represent spinors with dotted indices $\psi_{\dot{a}}$.

For any index $i \in \{0, 1\}$, the $i$-th vector of the standard basis for the alt-right-handed Weyl fermion representation, when identified with an element of $\mathbb{C}^2$, is the standard unit vector $e_i$ (the vector with 1 at position $i$ and 0 elsewhere).

For any matrix $M \in SL(2, \mathbb{C})$, the matrix representing the alt-right-handed Weyl representation $\rho(M)$ with respect to the standard basis $\{e_i\}_{i \in \{0, 1\}}$ is the conjugate transpose of the inverse of $M$. That is, for any indices $i, j \in \{0, 1\}$, the $(i, j)$-entry of the matrix is given by $$[(\text{altRightHanded.ρ } M)]_{ij} = ((M^{-1})^\dagger)_{ij}$$ where $M^{-1}$ is the inverse of $M$ and $\dagger$ denotes the conjugate transpose.

This definition defines a morphism of $SL(2, \mathbb{C})$-representations from the fundamental representation `leftHanded` (representing Weyl fermions with an upper index $\psi^a$) to the alternative representation `altLeftHanded` (representing fermions with a lower index $\psi_a$). The morphism is a linear map that transforms a spinor $\psi \in \mathbb{C}^2$ by multiplying it with the $2 \times 2$ antisymmetric matrix $$\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ In the context of spinor calculus, this map corresponds to the lowering of a spinor index using the Levi-Civita symbol, $\psi_a = \epsilon_{ab} \psi^b$. The definition includes a proof that this map is an intertwining operator, satisfying $\epsilon(M \psi) = (M^{-1})^T (\epsilon \psi)$ for any $M \in SL(2, \mathbb{C})$.

For a left-handed Weyl fermion $\psi$ in the fundamental representation `leftHanded` (corresponding to a spinor with an upper index $\psi^a$), the morphism `leftHandedToAlt` acts by multiplying the spinor's component vector by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. That is, $$\text{leftHandedToAlt}(\psi) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} \psi^1 \\ \psi^2 \end{pmatrix}$$ where the result is interpreted as a spinor in the `altLeftHanded` representation (corresponding to a lower index $\psi_a$).

This morphism defines a linear map from the "alternative" left-handed Weyl fermion representation `altLeftHanded` to the fundamental left-handed Weyl fermion representation `leftHanded`. Both representations act on the complex vector space $\mathbb{C}^2$ under the group $\text{SL}(2, \mathbb{C})$. The map transforms a spinor $\psi$ (representing a fermion with a lower index $\psi_a$) by multiplying it with the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. In the language of physics, this map corresponds to the operation of raising the index of a left-handed Weyl fermion, $\psi^a = \epsilon^{ab} \psi_b$.

For a left-handed Weyl fermion $\psi$ in the representation `altLeftHanded` (corresponding to a spinor with a lower index $\psi_a$), the morphism `leftHandedAltTo` (which represents the index-raising operation) acts by multiplying the spinor's component vector by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. That is, $$\text{leftHandedAltTo}(\psi) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix}$$ in terms of its components in $\mathbb{C}^2$.

This definition establishes an isomorphism between the representation `leftHanded` and the representation `altLeftHanded` of the group $SL(2, \mathbb{C})$ on the vector space $\mathbb{C}^2$. The isomorphism is defined by the antisymmetric matrix $$\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ which acts as the intertwiner (the "hom" part of the equivalence) by mapping a spinor $\psi^a$ in the fundamental representation to a spinor $\psi_a$ in the alternative representation via index lowering, $\psi_a = \epsilon_{ab} \psi^b$. The inverse map (the "inv" part) is defined by the matrix $$\epsilon^{-1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ which corresponds to index raising, $\psi^a = \epsilon^{ab} \psi_b$.

For a left-handed Weyl spinor $\psi$ in the fundamental representation of $SL(2, \mathbb{C})$, the isomorphism `leftHandedAltEquiv` (which corresponds to the index-lowering operation) acts on $\psi$ by multiplying its component vector in $\mathbb{C}^2$ by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Mathematically, this is expressed as: $$\text{leftHandedAltEquiv}(\psi) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} \psi^1 \\ \psi^2 \end{pmatrix}$$ where the result represents the spinor with a lower index, often denoted as $\psi_a = \epsilon_{ab} \psi^b$.

For a left-handed Weyl spinor $\psi$ in the alternative representation `altLeftHanded` (representing a spinor with a lower index $\psi_a$), the inverse of the isomorphism `leftHandedAltEquiv` acts by multiplying the component vector of $\psi$ by the matrix $$\epsilon^{-1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.$$ This operation corresponds to raising the index of the spinor, $\psi^a = \epsilon^{ab} \psi_b$.

The linear equivalence between the space of right-handed Weyl fermions (denoted `rightHandedWeyl`) and the alternative representation of right-handed Weyl fermions (denoted `altRightHandedWeyl`) is defined by the mapping $\psi \mapsto \epsilon \psi$, where $\epsilon$ is the $2 \times 2$ antisymmetric matrix $$\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$

The linear equivalence `rightHandedWeylAltEquiv` is equivariant with respect to the action of $SL(2, \mathbb{C})$ on the space of right-handed Weyl spinors (`rightHandedWeyl`) and its alternative representation (`altRightHandedWeyl`). That is, for any $g \in SL(2, \mathbb{C})$ and any spinor $\psi \in \text{rightHandedWeyl}$, the equivalence $\phi$ satisfies $\phi(g \cdot \psi) = g \cdot \phi(\psi)$.

For any element $M \in SL(2, \mathbb{C})$ and any right-handed Weyl spinor $\psi$, the action of $M$ on $\psi$ (denoted as $\text{rep} \, M \, \psi$) is given by the formula: $$\text{rep} \, M \, \psi = \sum_{i} \left( \sum_{j} \bar{M}_{ij} \psi_j \right) e_i$$ where $\bar{M}_{ij}$ denotes the complex conjugate of the $(i, j)$-th entry of the matrix $M$, $\psi_j$ is the $j$-th component of the spinor $\psi$, and $\{e_i\}$ is the standard basis for the space of right-handed Weyl spinors.

Let $M$ be an element of the special linear group $SL(2, \mathbb{C})$ and let $\{e_0, e_1\}$ be the basis for the space of right-handed Weyl spinors. The representation of $M$ acting on the basis vector $e_i$ is given by $$\text{rep}(M)(e_i) = \sum_{j \in \{0, 1\}} \bar{M}_{ji} e_j$$ where $\bar{M}_{ji}$ is the complex conjugate of the $(j, i)$-th entry of the matrix $M$.

For any matrix $M$ in the special linear group $\mathrm{SL}(2, \mathbb{C})$ (the group of $2 \times 2$ complex matrices with determinant 1), the matrix of the representation $\rho(M)$ acting on the space of right-handed Weyl fermions, relative to the standard basis, is equal to the entry-wise complex conjugate of the matrix $M$, denoted by $\overline{M}$.

Let $SL(2, \mathbb{C})$ be the special linear group of $2 \times 2$ complex matrices and let $M \in SL(2, \mathbb{C})$. Let $\{e_0, e_1\}$ be the standard basis for the space of right-handed Weyl fermions. For any indices $i, j \in \{0, 1\}$, the $j$-th coordinate of the image of the basis vector $e_i$ under the representation of $M$ is equal to the complex conjugate of the matrix entry $M_{ji}$, denoted as $(M_{ji})^*$.

The definition provides the standard basis for the space of dual right-handed Weyl fermions, which is a two-dimensional vector space over the complex numbers $\mathbb{C}$. The basis is indexed by $i \in \{0, 1\}$ (represented by $\text{Fin } 2$). According to its properties, the $i$-th basis vector corresponds to the element whose $j$-th component is $1$ if $i = j$ and $0$ otherwise (the Kronecker delta $\delta_{ij}$).

Let $\{e_i\}_{i \in \{0, 1\}}$ be the standard basis for the space of dual right-handed Weyl fermions. For any indices $i, j \in \{0, 1\}$, the $j$-th component of the $i$-th basis vector $e_i$ is equal to $1$ if $j = i$ and $0$ otherwise.

For any index $i \in \{0, 1\}$, the $i$-th vector of the standard basis for the space of dual right-handed Weyl fermions is equal to the standard basis vector $e_i \in \mathbb{C}^2$, which has a $1$ at index $i$ and $0$ elsewhere.

This definition establishes the group representation of the special linear group $SL(2, \mathbb{C})$ on the space of dual right-handed Weyl fermions (denoted as $\text{DualRightHandedWeyl}$). The space is identified with $\mathbb{C}^2$, and the representation maps an element $M \in SL(2, \mathbb{C})$ to the linear operator defined by the conjugate transpose of its inverse, $(M^{-1})^\dagger$. In physical index notation, this corresponds to the transformation law for a Weyl fermion with a dotted lower index $\psi_{\dot{a}}$.

For any $M \in SL(2, \mathbb{C})$ and any dual right-handed Weyl fermion $\psi \in \mathbb{C}^2$, the representation of $SL(2, \mathbb{C})$ acts on $\psi$ according to the formula $\text{rep}(M) \psi = (M^{-1})^\dagger \psi$, where $(M^{-1})^\dagger$ denotes the conjugate transpose of the inverse of the matrix $M$. In physical index notation, this corresponds to the transformation of a spinor with a dotted lower index $\psi_{\dot{a}}$.

Let $M \in SL(2, \mathbb{C})$ and let $\psi$ be a dual right-handed Weyl fermion (spinor). The action of the group representation $\rho(M)$ on the spinor $\psi$ can be expressed in terms of the standard basis $\{e_i\}_{i \in \{0, 1\}}$ and the spinor components $\psi_j$ as: $$\rho(M)\psi = \sum_{i=0}^1 \left( \sum_{j=0}^1 (M^{-1})^\dagger_{ij} \psi_j \right) e_i$$ where $(M^{-1})^\dagger_{ij}$ denotes the $ij$-th entry of the conjugate transpose of the inverse of the matrix $M$.

Let $SL(2, \mathbb{C})$ be the special linear group of $2 \times 2$ complex matrices with determinant 1. Let $\text{DualRightHandedWeyl}$ be the two-dimensional complex vector space representing dual right-handed Weyl fermions, equipped with a standard basis $\{e_0, e_1\}$. For any $M \in SL(2, \mathbb{C})$, the representation $\rho(M)$ (which corresponds to the transformation $(M^{-1})^\dagger$) acts on the $i$-th basis vector as: $$\rho(M) e_i = \sum_{j=0}^1 \left((M^{-1})^\dagger\right)_{ji} e_j$$ where $\left((M^{-1})^\dagger\right)_{ji}$ is the entry in the $j$-th row and $i$-th column of the conjugate transpose of the inverse of $M$.

Let $M \in SL(2, \mathbb{C})$ and let $\rho(M)$ be the representation of $SL(2, \mathbb{C})$ acting on the space of dual right-handed Weyl fermions. The matrix of the linear operator $\rho(M)$ relative to the standard basis is equal to $(M^{-1})^\dagger$, where $\dagger$ denotes the conjugate transpose.

Let $\text{DualRightHandedWeyl}$ be the two-dimensional complex vector space representing dual right-handed Weyl fermions (corresponding to dotted lower indices $\psi_{\dot{a}}$). Let $\{e_0, e_1\}$ be the standard basis for this space. For any matrix $M \in SL(2, \mathbb{C})$, the representation $\rho(M)$ acts on the basis vectors such that the $j$-th component of the transformed basis vector $e_i$ is given by the complex conjugate of the $(i, j)$-th entry of the inverse matrix $M^{-1}$: $$[\rho(M) e_i]_j = \overline{(M^{-1})_{ij}}$$ where $i, j \in \{0, 1\}$.

The definition provides an $SL(2, \mathbb{C})$-intertwining map from the representation of left-handed Weyl spinors to its dual representation. For a left-handed spinor $\psi$, the map is defined by the matrix multiplication $\epsilon \psi$, where $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

Let $\psi$ be a left-handed Weyl spinor. Let $[\psi] \in \mathbb{C}^2$ denote its representation as a column vector. The action of the map $\text{dual}$ (which maps left-handed Weyl spinors to their dual representation) on $\psi$ is given by: $$\text{dual}(\psi) = \Phi^{-1} \left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} [\psi] \right)$$ where $\Phi^{-1}$ is the inverse of the standard isomorphism from the dual left-handed Weyl spinor space to $\mathbb{C}^2$, and the matrix multiplication utilizes the Levi-Civita tensor $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

The morphism `dual` is an intertwining map (a representation morphism) from the representation of dual left-handed Weyl spinors to the representation of left-handed Weyl spinors. For an element $\psi$ of the dual left-handed Weyl spinor space, the map is defined by multiplying its component vector by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

For a dual left-handed Weyl spinor $\psi$, the application of the dual morphism $\text{dual} : \text{DualLeftHandedWeyl} \to \text{LeftHandedWeyl}$ is defined by taking the coordinate representation of $\psi$ in $\mathbb{C}^2$, multiplying it by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, and mapping the result back to the space of left-handed Weyl spinors. Specifically, $$\text{dual}(\psi) = \Phi^{-1} \left( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \Phi(\psi) \right)$$ where $\Phi$ denotes the equivalence between the spinor vector space and $\mathbb{C}^2$.

This definition establishes an equivalence of $SL(2, \mathbb{C})$ representations between the space of left-handed Weyl spinors and the space of dual left-handed Weyl spinors. The isomorphism is defined by multiplying a left-handed spinor $\psi$ by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Its inverse is given by multiplying a dual left-handed spinor by the matrix $\epsilon^{-1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

For a left-handed Weyl spinor $\psi$, the application of the equivalence map $\text{dualEquiv} : \text{LeftHandedWeyl} \to \text{DualLeftHandedWeyl}$ is defined by taking the coordinate representation of $\psi$ in $\mathbb{C}^2$, multiplying it by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, and mapping the result back to the space of dual left-handed Weyl spinors. Specifically, $$\text{dualEquiv}(\psi) = \Phi_D^{-1} \left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \cdot \Phi_L(\psi) \right)$$ where $\Phi_L$ and $\Phi_D$ denote the isomorphisms between the respective spinor vector spaces and $\mathbb{C}^2$.

For any dual left-handed Weyl spinor $\psi \in \text{DualLeftHandedWeyl}$, the inverse of the isomorphism $\text{dualEquiv}$, denoted as $\text{dualEquiv}^{-1}(\psi)$, is calculated by representing $\psi$ as a vector in $\mathbb{C}^2$ and multiplying it by the matrix $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ Specifically, if $\mathbf{v}_\psi$ is the vector representation of $\psi$, then the result is the spinor in $\text{LeftHandedWeyl}$ corresponding to the vector $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \mathbf{v}_\psi$.

The linear equivalence between the space of right-handed Weyl spinors and its dual space. For a right-handed Weyl spinor $\psi$, the map is defined by multiplying its vector representation by the antisymmetric matrix $\epsilon = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$.

The linear equivalence `rightHandedWeylDualEquiv` between the space of right-handed Weyl spinors and its dual space is equivariant with respect to the action of the group $SL(2, \mathbb{C})$. This means that for any transformation $M \in SL(2, \mathbb{C})$ and any right-handed Weyl spinor $\psi$, the equivalence commutes with the group action: $\text{dualEquiv}(M \cdot \psi) = M \cdot \text{dualEquiv}(\psi)$, where the action on the dual space is the appropriate induced action.