Physlib.Relativity.PauliMatrices.AsTensor

The tensor $\sigma$ is an element of the tensor product space $V_{\text{Lorentz}} \otimes V_L \otimes V_{\dot{R}}$, where $V_{\text{Lorentz}}$ is the representation space of complex contravariant Lorentz vectors, $V_L$ is the fundamental representation space for left-handed Weyl fermions (with indices $a$), and $V_{\dot{R}}$ is the representation space for right-handed Weyl fermions (with dotted indices $\dot{a}$). It is defined as the sum $$\sigma = \sum_{\mu=0}^3 e_\mu \otimes \sigma^\mu$$ where $\{e_\mu\}_{\mu=0}^3$ is the standard basis for $V_{\text{Lorentz}}$, and $\sigma^\mu$ are the Pauli matrices—specifically $\sigma^0 = I$ (the identity matrix) and $\sigma^{1,2,3}$ as the standard Pauli matrices—which are identified with elements of $V_L \otimes V_{\dot{R}}$ via a linear equivalence with the space of $2 \times 2$ complex matrices. In index notation, this corresponds to the tensor $\sigma^{\mu a \dot{a}}$.

The Pauli matrix tensor $\sigma \in V_{\text{Lorentz}} \otimes V_L \otimes V_{R}$ can be expanded in terms of the standard basis $\{e_\mu\}_{\mu=0}^3$ of the complex contravariant Lorentz representation space as: $$\sigma = e_0 \otimes \Phi^{-1}(\sigma^0) + e_1 \otimes \Phi^{-1}(\sigma^1) + e_2 \otimes \Phi^{-1}(\sigma^2) + e_3 \otimes \Phi^{-1}(\sigma^3)$$ where: - $\{e_0, e_1, e_2, e_3\}$ is the standard basis for complex contravariant Lorentz vectors (indexed by $0$ in `Sum.inl` and $0, 1, 2$ in `Sum.inr`). - $\{\sigma^0, \sigma^1, \sigma^2, \sigma^3\}$ are the standard Pauli matrices (where $\sigma^0 = I_{2\times 2}$). - $\Phi^{-1} : \text{Mat}_{2 \times 2}(\mathbb{C}) \cong V_L \otimes V_R$ is the inverse linear equivalence between $2 \times 2$ complex matrices and the tensor product of left-handed and right-handed Weyl fermion representation spaces.

Let $V_L$ and $V_R$ be the complex vector spaces associated with left-handed and right-handed Weyl fermions, and let $\{e^L_0, e^L_1\}$ and $\{e^R_0, e^R_1\}$ be their respective standard bases. Let $\sigma^0$ denote the zeroth Pauli matrix (the $2 \times 2$ identity matrix $I$). The expansion of $\sigma^0$ in the basis of the tensor product space $V_L \otimes V_R$, via the inverse linear equivalence $\Phi^{-1} : \text{Mat}_{2 \times 2}(\mathbb{C}) \cong V_L \otimes V_R$, is given by: $$\Phi^{-1}(\sigma^0) = e^L_0 \otimes e^R_0 + e^L_1 \otimes e^R_1$$ where $\otimes$ denotes the tensor product over $\mathbb{C}$.

Let $V_L$ and $V_R$ be the complex vector spaces associated with left-handed and right-handed Weyl fermions, and let $\{e^L_0, e^L_1\}$ and $\{e^R_0, e^R_1\}$ be their respective standard bases. Let $\sigma_1$ denote the first Pauli matrix. The expansion of $\sigma_1$ in the basis of the tensor product space $V_L \otimes V_R$, via the inverse linear equivalence $\Phi^{-1} : \text{Mat}_{2 \times 2}(\mathbb{C}) \cong V_L \otimes V_R$, is given by: $$\Phi^{-1}(\sigma_1) = e^L_0 \otimes e^R_1 + e^L_1 \otimes e^R_0$$ where $\otimes$ denotes the tensor product over $\mathbb{C}$.

Let $V_L$ and $V_R$ be the complex vector spaces associated with left-handed and right-handed Weyl fermions, and let $\{e^L_0, e^L_1\}$ and $\{e^R_0, e^R_1\}$ be their respective standard bases. Let $\Phi^{-1} : \text{Mat}_{2 \times 2}(\mathbb{C}) \cong V_L \otimes V_R$ be the inverse of the linear equivalence between $2 \times 2$ matrices and the tensor product of these spinor spaces. The expansion of the second Pauli matrix $\sigma_2$ in terms of the tensor product basis is given by: $$\Phi^{-1}(\sigma_2) = -i(e^L_0 \otimes e^R_1) + i(e^L_1 \otimes e^R_0)$$ where $i$ is the imaginary unit and $\otimes$ denotes the tensor product over $\mathbb{C}$.

Let $V_L$ and $V_R$ be the complex vector spaces for left-handed and right-handed Weyl fermions with standard bases $\{e^L_0, e^L_1\}$ and $\{e^R_0, e^R_1\}$, respectively. Let $\Phi^{-1} : \text{Mat}_{2 \times 2}(\mathbb{C}) \to V_L \otimes V_R$ be the inverse of the linear equivalence between $2 \times 2$ matrices and the tensor product of spinor spaces. The expansion of the third Pauli matrix $\sigma_3$ in terms of the tensor product basis is given by: $$\Phi^{-1}(\sigma_3) = e^L_0 \otimes e^R_0 - e^L_1 \otimes e^R_1$$

Let $V_{\text{Lorentz}}$ be the complex vector space of contravariant Lorentz vectors with standard basis $\{e_0, e_1, e_2, e_3\}$. Let $V_L$ and $V_R$ be the complex vector spaces for left-handed and right-handed Weyl fermions with standard bases $\{e^L_0, e^L_1\}$ and $\{e^R_0, e^R_1\}$, respectively. The Pauli matrix tensor $\sigma \in V_{\text{Lorentz}} \otimes V_L \otimes V_R$ is expanded in terms of these bases as: $$\begin{aligned} \sigma &= e_0 \otimes (e^L_0 \otimes e^R_0 + e^L_1 \otimes e^R_1) \\ &+ e_1 \otimes (e^L_0 \otimes e^R_1 + e^L_1 \otimes e^R_0) \\ &+ e_2 \otimes (-i e^L_0 \otimes e^R_1 + i e^L_1 \otimes e^R_0) \\ &+ e_3 \otimes (e^L_0 \otimes e^R_0 - e^L_1 \otimes e^R_1) \end{aligned}$$ where $i$ is the imaginary unit and $\otimes$ denotes the tensor product over $\mathbb{C}$.

Let $\text{Rep}(\mathbb{C}, SL(2, \mathbb{C}))$ be the category of finite-dimensional complex representations of the special linear group $SL(2, \mathbb{C})$. Let $\mathbb{1}$ denote the trivial representation (the monoidal unit) on $\mathbb{C}$. This definition constructs a morphism of representations (an equivariant linear map) $$\sigma : \mathbb{1} \longrightarrow V_{\text{Lorentz}} \otimes V_L \otimes V_{\dot{R}}$$ where $V_{\text{Lorentz}}$ is the contravariant Lorentz vector representation, $V_L$ is the fundamental representation for left-handed Weyl spinors, and $V_{\dot{R}}$ is the conjugate representation for right-handed Weyl spinors. The map is defined by sending $a \in \mathbb{C}$ to $a \cdot \sigma^{\mu a \dot{a}}$, where $\sigma^{\mu a \dot{a}}$ is the Pauli matrix tensor. This morphism manifests the invariance of the Pauli matrices under the joint action of the Lorentz group and $SL(2, \mathbb{C})$.

Let $\text{asConsTensor} : \mathbb{1} \longrightarrow V_{\text{Lorentz}} \otimes V_L \otimes V_{\dot{R}}$ be the $SL(2, \mathbb{C})$-equivariant morphism (morphism of representations) defined by the Pauli matrices, where $\mathbb{1}$ is the trivial representation on $\mathbb{C}$. This theorem states that the underlying linear map of $\text{asConsTensor}$ evaluated at the unit $1 \in \mathbb{C}$ is equal to the Pauli matrix tensor $\sigma^{\mu a \dot{a}}$, denoted by `asTensor`.