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 $\phi(g \cdot \psi) = g \cdot \phi(\psi)$.