Physlib.Relativity.PauliMatrices.ToTensor

This definition provides an equivalence between the component indices of a tensor with the index configuration $[.up, .upL, .upR]$—consisting of one Lorentz vector index, one left-handed spinor index, and one right-handed spinor index—and the product type $(\text{Fin } 1 \oplus \text{Fin } 3) \times \text{Fin } 2 \times \text{Fin } 2$. Specifically, the 4-dimensional Lorentz index $i \in \{0, 1, 2, 3\}$ is decomposed into a temporal component ($\text{Fin } 1$) and three spatial components ($\text{Fin } 3$), while the spinor indices remain in $\text{Fin } 2 \times \text{Fin } 2$. This correspondence aligns the abstract tensor indices with the physical representation of Pauli matrices $\sigma^\mu_{a\dot{b}}$.

This definition establishes a tensorial structure for the Pauli matrices by providing a linear isomorphism between the abstract tensor space in the `complexLorentzTensor` species with indices of colors $[.up, .upL, .upR]$ (representing a contravariant Lorentz vector index, a left-handed spinor index, and a right-handed spinor index) and the concrete space of matrix-valued functions $(\text{Fin } 1 \oplus \text{Fin } 3) \to \text{Mat}_{2 \times 2}(\mathbb{C})$. The isomorphism maps an abstract tensor $T$ to its components $T^\mu_{a\dot{b}}$, identifying the 4-dimensional Lorentz index $\mu$ (decomposed into temporal and spatial parts) with the function's domain and the spinor indices $a, \dot{b}$ with the matrix indices.

Let $p$ be a tensor in the complex Lorentz tensor space $\mathbb{C}T^{[\text{up, upL, upR}]}$, which has one contravariant Lorentz vector index and two spinor indices. The inverse of the tensorial isomorphism, $(\text{toTensor})^{-1}(p)$, maps this abstract tensor to its concrete representation as a matrix-valued function $f: (\text{Fin } 1 \oplus \text{Fin } 3) \to \text{Mat}_{2 \times 2}(\mathbb{C})$. This theorem states that this function $f$ is obtained by: 1. Taking the coefficients of $p$ with respect to the canonical tensor basis. 2. Reindexing these components using the equivalence `indexEquiv`, which maps the abstract multi-indices to the physical product space $(\text{Fin } 1 \oplus \text{Fin } 3) \times \text{Fin } 2 \times \text{Fin } 2$. 3. Currying the resulting indices so that the Lorentz index $(\text{Fin } 1 \oplus \text{Fin } 3)$ selects a $2 \times 2$ matrix indexed by the spinor components.

Let $b = (b_0, b_1, b_2)$ be a multi-index belonging to the component index set for the color sequence $[\text{up}, \text{upL}, \text{upR}]$, and let $e_b$ be the corresponding basis tensor in the complex Lorentz tensor space $\mathbb{C}T^{[\text{up, upL, upR}]}$. This theorem states that the component representation of $e_b$ as a matrix-valued function $(\text{toTensor})^{-1}(e_b)$, evaluated at Lorentz index $\mu$ and spinor indices $\alpha, \beta$, is 1 if the indices match the components of $b$ and 0 otherwise. Specifically: $$((\text{toTensor})^{-1}(e_b))(\mu)_{\alpha, \beta} = \begin{cases} 1 & \text{if } b_0 \text{ matches } \mu, b_1 = \alpha, \text{ and } b_2 = \beta \\ 0 & \text{otherwise} \end{cases}$$ where the match for $b_0$ and $\mu$ is defined by the equivalence between the abstract four-dimensional index and the decomposed temporal/spatial index $\text{Fin } 1 \oplus \text{Fin } 3$.

The notation $\sigma^{\mu \alpha \beta}$ (denoted as `σ^^^`) represents the Pauli matrices, including the identity, as a tensor in the complex tensor space $\mathbb{C}T^{[\text{up}, \text{upL}, \text{upR}]}$. This tensor is defined by applying the `toTensor` mapping to the standard Pauli matrices `pauliMatrix`, where $\mu$ corresponds to the Lorentz vector index (type `up`) and $\alpha, \beta$ correspond to the spinor indices (types `upL` and `upR`).

Let $\sigma \in \text{Tensor}_{\mathbb{C}}([\text{up}, \text{upL}, \text{upR}])$ be the Pauli tensor. Let $e_{\mu, a, b}$ denote the canonical basis elements of the tensor space indexed by the multi-index $(\mu, a, b)$, where $\mu \in \{0, 1, 2, 3\}$ is the contravariant Lorentz index and $a, b \in \{0, 1\}$ are the left-handed and right-handed spinor indices, respectively. The Pauli tensor is expressed as the following linear combination of basis elements: $$\sigma = e_{0,0,0} + e_{0,1,1} + e_{1,0,1} + e_{1,1,0} - i e_{2,0,1} + i e_{2,1,0} + e_{3,0,0} - e_{3,1,1}$$ where $i$ is the imaginary unit.

Let $\sigma$ be the Pauli tensor within the `complexLorentzTensor` species, associated with the sequence of index colors $[\text{up}, \text{upL}, \text{upR}]$ (representing a contravariant Lorentz vector index, a left-handed spinor index, and a right-handed spinor index, respectively). Let $\text{asConsTensor} : \mathbb{1} \to V_{\text{up}} \otimes V_{\text{upL}} \otimes V_{\text{upR}}$ be the $SL(2, \mathbb{C})$-invariant morphism of representations that defines the Pauli matrices. This theorem states that the Pauli tensor $\sigma$ is equal to the rank-3 tensor constructed from this invariant morphism via the `fromConstTriple` mapping.

The Pauli tensor $\sigma^{\mu}_{a\dot{b}}$ is equal to the tensor constructed via the `ofRat` mapping from the following component function $f$ of the multi-index $b = (\mu, a, \dot{b})$: - $f(b) = 1$ if $\mu = 0$ and $a = \dot{b}$; - $f(b) = 1$ if $\mu = 1$ and $a \neq \dot{b}$; - $f(b) = -i$ if $\mu = 2, a = 0,$ and $\dot{b} = 1$; - $f(b) = i$ if $\mu = 2, a = 1,$ and $\dot{b} = 0$; - $f(b) = 1$ if $\mu = 3, a = 0,$ and $\dot{b} = 0$; - $f(b) = -1$ if $\mu = 3, a = 3,$ and $\dot{b} = 3$; - $f(b) = 0$ otherwise. Here, $i$ denotes the imaginary unit, and the components are represented as rational complex numbers $\langle \text{Re}, \text{Im} \rangle$.

For any element $\Lambda \in SL(2, \mathbb{C})$, the action of $\Lambda$ on the Pauli matrices $\sigma$ leaves them invariant: $$\Lambda \cdot \sigma = \sigma$$ Here, $\sigma$ is the collection of Pauli matrices $\sigma^\mu$ (for $\mu \in \{0, 1, 2, 3\}$), considered as a tensor with one contravariant Lorentz vector index and two spinor indices (one left-handed and one right-handed conjugate). The group action is defined via the tensorial isomorphism between the space of matrix-valued functions and the abstract Lorentz tensor space.

For any transformation $\Lambda \in SL(2, \mathbb{C})$, the Pauli tensor $\sigma^{\text{^^^}}$ (representing the Pauli matrices $\sigma^\mu_{a\dot{b}}$ with one contravariant Lorentz vector index and two spinor indices) is invariant under the group action, such that $\Lambda \cdot \sigma^{\text{^^^}} = \sigma^{\text{^^^}}$.

The definition `pauliCo` represents the covariant version of the Pauli tensor, denoted by $\sigma_{\mu}{}^{\alpha \dot{\beta}}$. It is an element of the complex Lorentz tensor space $\mathbb{C}T$ with three indices: a covariant Lorentz vector index (color `.down`), a left-handed spinor index (color `.upL`), and a right-handed spinor index (color `.upR`). Mathematically, this tensor is defined by contracting the covariant Minkowski metric $\eta_{\mu\nu}$ with the contravariant Pauli tensor $\sigma^{\nu \alpha \dot{\beta}}$: $$\sigma_{\mu}{}^{\alpha \dot{\beta}} = \eta_{\mu\nu} \sigma^{\nu \alpha \dot{\beta}}$$ where the index $\nu$ is the internal index over which the contraction occurs, $\mu$ is the resulting covariant vector index, and $\alpha, \dot{\beta}$ are the spinor indices.

The notation `σ_^^` represents the Pauli matrix tensor $\sigma$, which is formally defined as `PauliMatrix.pauliCo`. This tensor maps spacetime indices to their corresponding $2 \times 2$ complex Pauli matrices within a tensorial framework.

The complex Lorentz tensor $\sigma_{\mu}{}^{\alpha \dot{\beta}}$, obtained by dualizing the contravariant Lorentz index of the Pauli tensor $\sigma^{\mu \alpha \dot{\beta}}$, is equal to the tensor constructed from the rational-complex components defined by the function `pauliContrDownComponent`. Specifically, the tensor with index colors $[.down, .upL, .upR]$ is given by the image of `pauliContrDownComponent` under the `ofRat` map.

The complex Lorentz tensor obtained by dualizing the Lorentz index $\mu$ and the left-handed Weyl index $\alpha$ of the contravariant Pauli tensor $\sigma^{\mu \alpha \dot{\beta}}$ has components $\sigma_{\mu \alpha}{}^{\dot{\beta}}$ given by the formula: $$\sigma_{\mu \alpha}{}^{\dot{\beta}} = \sum_{x \in \{0, 1\}} \sigma_{\mu}{}^{x \dot{\beta}} \epsilon_{x \alpha}$$ where $\sigma_{\mu}{}^{x \dot{\beta}}$ represent the components of the Pauli tensor with a lowered Lorentz index, and $\epsilon$ is the two-dimensional Levi-Civita tensor (spinor metric) defined such that $\epsilon_{01} = 1$ and $\epsilon_{10} = -1$.

The definition `pauliCoDown` represents the Pauli matrices as a complex Lorentz tensor with three covariant (lower) indices, denoted as $\sigma_{\mu \dot{\beta} \alpha}$. It is an element of the complex Lorentz tensor space $\mathbb{C}T$ with indices corresponding to: 1. A covariant Lorentz vector index (color `.down`), represented by $\mu$. 2. A covariant right-handed (dotted) spinor index (color `.downR`), represented by $\dot{\beta}$. 3. A covariant left-handed (undotted) spinor index (color `.downL`), represented by $\alpha$. Mathematically, this tensor is constructed by lowering the indices of the contravariant Pauli tensor $\sigma^{\nu \alpha' \dot{\beta'}}$ using the covariant Minkowski metric $\eta_{\mu \nu}$ and the covariant spinor metrics $\epsilon_{\dot{\beta} \dot{\beta'}}$ and $\epsilon_{\alpha \alpha'}$: $$\sigma_{\mu \dot{\beta} \alpha} = \eta_{\mu \nu} \epsilon_{\dot{\beta} \dot{\beta'}} \epsilon_{\alpha \alpha'} \sigma^{\nu \alpha' \dot{\beta'}}$$ where $\eta$ is the metric for Lorentz vectors and $\epsilon$ denotes the invariant metrics for the respective spinor representations.

The notation `σ___` represents the Pauli matrices as a tensor with three covariant (lower) indices, typically denoted in physics as $\sigma_{\mu \alpha \dot{\beta}}$. This notation is defined to refer to the formal term `PauliMatrix.pauliCoDown`.

The Pauli tensor $\sigma^\mu_{\dot{\beta}\alpha}$ is a complex Lorentz tensor of type $(1, 2)$ defined over the species `complexLorentzTensor`. It possesses: - A contravariant Lorentz vector index $\mu$ (corresponding to the color `.up`). - A covariant right-handed (dotted) Weyl spinor index $\dot{\beta}$ (corresponding to the color `.downR`). - A covariant left-handed (undotted) Weyl spinor index $\alpha$ (corresponding to the color `.downL`). The tensor is constructed by taking the product of the contravariant Pauli matrices $\sigma^{\mu \alpha \dot{\beta}}$ with the right-handed and left-handed spinor metrics ($\epsilon_{\dot{\beta}\dot{\beta}'}$ and $\epsilon_{\alpha\alpha'}$) and performing the appropriate contractions to lower the spinor indices.

The notation $\sigma^\mu_{ab}$ represents the Pauli tensor `PauliMatrix.pauliContrDown`, which is a complex tensor with one contravariant index (typically corresponding to Minkowski space) and two covariant indices (representing the spinor row and column indices). This notation provides a shorthand for the tensorial representation of the Pauli matrices $(\sigma^0, \sigma^1, \sigma^2, \sigma^3)$.

The covariant Pauli tensor $\sigma_{\mu}{}^{\alpha \dot{\beta}}$ is equal to the tensor constructed via the `ofRat` mapping from the component function $f$ of the multi-index $b = (\mu, \alpha, \dot{\beta})$ defined as: - $f(b) = 1$ if $\mu = 0$ and $\alpha = \dot{\beta}$; - $f(b) = -1$ if $\mu = 1$ and $\alpha \neq \dot{\beta}$; - $f(b) = i$ if $\mu = 2, \alpha = 0,$ and $\dot{\beta} = 1$; - $f(b) = -i$ if $\mu = 2, \alpha = 1,$ and $\dot{\beta} = 0$; - $f(b) = -1$ if $\mu = 3, \alpha = 0,$ and $\dot{\beta} = 0$; - $f(b) = 1$ if $\mu = 3, \alpha = 1,$ and $\dot{\beta} = 1$; - $f(b) = 0$ otherwise. Here, $i$ denotes the imaginary unit, and the components are represented as rational complex numbers $\langle \text{Re}, \text{Im} \rangle$ where $\mu$ corresponds to the covariant Lorentz index (color `.down`), $\alpha$ to the left-handed spinor index (color `.upL`), and $\dot{\beta}$ to the right-handed spinor index (color `.upR`).

The covariant Pauli tensor $\sigma_{\mu \dot{\beta} \alpha}$ is equal to the tensor constructed via the `ofRat` mapping from the component function $f$ of the multi-index $b = (\mu, \dot{\beta}, \alpha)$ defined as: - $f(b) = 1$ if $\mu = 0$ and $\dot{\beta} = \alpha$; - $f(b) = 1$ if $\mu = 1$ and $\dot{\beta} \neq \alpha$; - $f(b) = -i$ if $\mu = 2, \dot{\beta} = 0,$ and $\alpha = 1$; - $f(b) = i$ if $\mu = 2, \dot{\beta} = 1,$ and $\alpha = 0$; - $f(b) = -1$ if $\mu = 3, \dot{\beta} = 1,$ and $\alpha = 1$; - $f(b) = 1$ if $\mu = 3, \dot{\beta} = 0,$ and $\alpha = 0$; - $f(b) = 0$ otherwise. Here, $i$ denotes the imaginary unit, and the components are represented as rational complex numbers $\langle \text{Re}, \text{Im} \rangle$ where $\mu$ corresponds to the covariant Lorentz index (color `.down`), $\dot{\beta}$ to the covariant right-handed spinor index (color `.downR`), and $\alpha$ to the covariant left-handed spinor index (color `.downL`).

The Pauli tensor $\sigma^\mu_{\dot{\beta}\alpha}$ is equal to the tensor constructed via the `ofRat` mapping from the following component function $f$ of the multi-index $b = (\mu, \dot{\beta}, \alpha)$, where $\mu \in \{0, 1, 2, 3\}$ is the Lorentz vector index, $\dot{\beta} \in \{0, 1\}$ is the right-handed (dotted) spinor index, and $\alpha \in \{0, 1\}$ is the left-handed (undotted) spinor index: - $f(b) = 1$ if $\mu = 0$ and $\dot{\beta} = \alpha$; - $f(b) = -1$ if $\mu = 1$ and $\dot{\beta} \neq \alpha$; - $f(b) = i$ if $\mu = 2, \dot{\beta} = 0,$ and $\alpha = 1$; - $f(b) = -i$ if $\mu = 2, \dot{\beta} = 1,$ and $\alpha = 0$; - $f(b) = 1$ if $\mu = 3, \dot{\beta} = 1,$ and $\alpha = 1$; - $f(b) = -1$ if $\mu = 3, \dot{\beta} = 0,$ and $\alpha = 0$; - $f(b) = 0$ otherwise. Here, $i$ denotes the imaginary unit, and the components are represented as rational complex numbers $\langle \text{Re}, \text{Im} \rangle$ via the `ofRat` map.

For any transformation $g \in SL(2, \mathbb{C})$, the covariant Pauli tensor $\sigma_{\mu}{}^{\alpha \dot{\beta}}$ (represented by `pauliCo`) is invariant under the group action, such that $g \cdot \sigma_{\mu}{}^{\alpha \dot{\beta}} = \sigma_{\mu}{}^{\alpha \dot{\beta}}$.

For any transformation $g \in SL(2, \mathbb{C})$, the covariant Pauli tensor $\sigma_{\mu \dot{\beta} \alpha}$ (represented by `pauliCoDown`) is invariant under the group action, such that $g \cdot \sigma_{\mu \dot{\beta} \alpha} = \sigma_{\mu \dot{\beta} \alpha}$. Here, the tensor $\sigma_{\mu \dot{\beta} \alpha}$ consists of a covariant Lorentz vector index $\mu$, a covariant right-handed spinor index $\dot{\beta}$, and a covariant left-handed spinor index $\alpha$.

For any transformation $g$ in the group $SL(2, \mathbb{C})$, the Pauli tensor $\sigma^\mu_{\dot{\beta}\alpha}$ (representing the Pauli matrices with one contravariant Lorentz vector index $\mu$, one covariant dotted spinor index $\dot{\beta}$, and one covariant undotted spinor index $\alpha$) is invariant under the group action, such that $g \cdot \sigma^\mu_{\dot{\beta}\alpha} = \sigma^\mu_{\dot{\beta}\alpha}$.

Dualizing (lowering) both Weyl (spinor) indices of the contravariant Pauli tensor $\sigma^{\mu \alpha \dot{\beta}}$ results in the Pauli tensor $\sigma^\mu_{\dot{\beta} \alpha}$, which possesses one contravariant Lorentz vector index and two covariant spinor indices.

Dualizing all three indices of the contravariant Pauli tensor $\sigma^{\mu \alpha \dot{\beta}}$—the Lorentz vector index $\mu$, the left-handed spinor index $\alpha$, and the right-handed spinor index $\dot{\beta}$—results in the covariant Pauli tensor $\sigma_{\mu \dot{\beta} \alpha}$. This dualization corresponds to lowering the indices using the Minkowski metric $\eta_{\mu \nu}$ and the invariant spinor metrics $\epsilon_{\alpha \alpha'}$ and $\epsilon_{\dot{\beta} \dot{\beta'}}$, such that: $$\eta_{\mu \nu} \epsilon_{\alpha \alpha'} \epsilon_{\dot{\beta} \dot{\beta'}} \sigma^{\nu \alpha' \dot{\beta'}} = \sigma_{\mu \dot{\beta} \alpha}$$ where the indices on the right-hand side follow the covariant ordering (Lorentz vector, right-handed spinor, left-handed spinor).

Let $\sigma^{\mu \alpha \dot{\beta}}$ be the contravariant Pauli tensor, where $\mu$ is a contravariant Lorentz vector index, $\alpha$ is a left-handed spinor index, and $\dot{\beta}$ is a right-handed spinor index. Let $\sigma_{\mu}{}^{\alpha \dot{\beta}}$ be the covariant Pauli tensor. The theorem states that applying the duality map $\tau$ to the Lorentz index $\mu$ of $\sigma^{\mu \alpha \dot{\beta}}$ (which corresponds to lowering the index via the Minkowski metric $\eta_{\mu\nu}$) results in the covariant Pauli tensor $\sigma_{\mu}{}^{\alpha \dot{\beta}}$.

Lowering the contravariant Lorentz index $\mu$ of the Pauli tensor $\sigma^\mu_{\dot{\beta} \alpha}$ (where $\dot{\beta}$ and $\alpha$ are covariant spinor indices) results in the covariant Pauli tensor $\sigma_{\mu \dot{\beta} \alpha}$. This operation uses the metric transformation $\tau$ (associated with the Minkowski metric $\eta_{\mu\nu}$) to map the index from the contravariant representation to the covariant representation, such that $\sigma_{\mu \dot{\beta} \alpha} = \eta_{\mu \nu} \sigma^\nu_{\dot{\beta} \alpha}$.