Physlib.Relativity.PauliMatrices.Basic

The function $\sigma: (\text{Fin } 1 \oplus \text{Fin } 3) \to \text{Mat}_{2 \times 2}(\mathbb{C})$ defines the Pauli matrices. The mapping is given by: - $\sigma(\text{inl } 0) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ - $\sigma(\text{inr } 0) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ - $\sigma(\text{inr } 1) = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ - $\sigma(\text{inr } 2) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ where $i$ is the imaginary unit.

The symbol $\sigma$ is a notation representing the function `pauliMatrix`, which maps indices in $\text{Fin } 1 \oplus \text{Fin } 3$ (representing the set $\{0, 1, 2, 3\}$) to $2 \times 2$ complex matrices $\text{M}_{2 \times 2}(\mathbb{C})$.

The notation $\sigma_0$ denotes the Pauli matrix $\sigma$ evaluated at the index corresponding to the first summand (index 0), which represents the $2 \times 2$ identity matrix $I$ (or $1$).

The notation $\sigma_1$ denotes the first Pauli matrix, which is represented by the $2 \times 2$ complex matrix $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

The notation $\sigma_2$ represents the second Pauli matrix, defined as the $2 \times 2$ complex matrix: $$\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$ where $i$ is the imaginary unit.

The notation $\sigma_3$ represents the third Pauli matrix, which is defined as the $2 \times 2$ complex matrix: $$\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ This corresponds to the index $2$ in the spatial component (the right side of the sum type `Fin 1 ⊕ Fin 3`) of the Pauli matrix map $\sigma$.

The Pauli matrix $\sigma$ evaluated at the index $\text{inl } 0$ is equal to the $2 \times 2$ identity matrix $I$: $$\sigma(\text{inl } 0) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

For any index $\mu \in \{0, 1, 2, 3\}$ (represented by the type $\text{Fin } 1 \oplus \text{Fin } 3$), the Pauli matrix $\sigma_\mu$ is self-adjoint (Hermitian). That is, $\sigma_\mu^\dagger = \sigma_\mu$, where $\dagger$ denotes the conjugate transpose (Hermitian conjugate).

For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the product of the Pauli matrix $\sigma_\mu$ with itself is the $2 \times 2$ identity matrix $I$: $$\sigma_\mu \sigma_\mu = I$$

For every index $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the corresponding Pauli matrix $\sigma_\mu$ is an invertible matrix.

For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } 3$, the inverse of the Pauli matrix $\sigma_\mu$ is the matrix itself: $$\sigma_\mu^{-1} = \sigma_\mu$$

The Pauli matrices $\sigma_2$ and $\sigma_1$ satisfy the anticommutation relation $\sigma_2 \sigma_1 = -(\sigma_1 \sigma_2)$.

For the Pauli matrices $\sigma_1$ and $\sigma_3$, the following anti-commutation relation holds: $$\sigma_3 \sigma_1 = -(\sigma_1 \sigma_3)$$ where $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Let $\sigma_2$ and $\sigma_3$ be the Pauli matrices. Then their product satisfies the anticommutation relation $\sigma_3 \sigma_2 = -\sigma_2 \sigma_3$.

The trace of the first Pauli matrix $\sigma_1$ is $0$, where $\sigma_1$ is the $2 \times 2$ complex matrix defined as $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

The trace of the Pauli matrix $\sigma_2$ is zero, where $\sigma_2$ is defined as the $2 \times 2$ complex matrix $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$.

The trace of the third Pauli matrix $\sigma_3$ is $0$, where $\sigma_3$ is the $2 \times 2$ complex matrix defined as $$\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

The trace of the product of the Pauli matrix $\sigma_0$ with itself is $2$: $$\text{Tr}(\sigma_0 \sigma_0) = 2$$ where $\sigma_0$ is the $2 \times 2$ identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_0$ and $\sigma_1$ is equal to $0$: $$\text{Tr}(\sigma_0 \sigma_1) = 0$$ where $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is the first spatial Pauli matrix.

The trace of the product of the Pauli matrices $\sigma_0$ and $\sigma_2$ is equal to $0$, which is expressed as $\operatorname{Tr}(\sigma_0 \sigma_2) = 0$. Here, $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_2$ is the second Pauli matrix $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_0$ and $\sigma_3$ is equal to $0$. Here, $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_3$ is the third Pauli matrix, defined as $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_1$ and $\sigma_0$ is $0$: $$\operatorname{Tr}(\sigma_1 \sigma_0) = 0$$ where $\sigma_1$ is the first Pauli matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_0$ is the $2 \times 2$ identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

The trace of the product of the Pauli matrix $\sigma_1$ with itself is equal to $2$: $$\operatorname{Tr}(\sigma_1 \sigma_1) = 2$$ where $\sigma_1$ is the first Pauli matrix, defined as $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_1$ and $\sigma_2$ is $0$: $$\text{Tr}(\sigma_1 \sigma_2) = 0$$ where $\sigma_1$ and $\sigma_2$ are the standard $2 \times 2$ complex Pauli matrices.

The trace of the product of the Pauli matrices $\sigma_1$ and $\sigma_3$ is equal to $0$, denoted as $\text{Tr}(\sigma_1 \sigma_3) = 0$.

The trace of the product of the Pauli matrices $\sigma_2$ and $\sigma_0$ is $0$, which can be expressed as $\text{Tr}(\sigma_2 \sigma_0) = 0$. Here, $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_2$ is the second Pauli matrix $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_2$ and $\sigma_1$ is zero: $$\text{Tr}(\sigma_2 \sigma_1) = 0$$ where $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ are the standard $2 \times 2$ complex Pauli matrices.

The trace of the product of the Pauli matrix $\sigma_2$ with itself is equal to $2$: $$\operatorname{Tr}(\sigma_2 \sigma_2) = 2$$ where $\sigma_2$ is the Pauli matrix defined as $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$.

The trace of the product of the Pauli matrices $\sigma_2$ and $\sigma_3$ is zero: $$\operatorname{Tr}(\sigma_2 \sigma_3) = 0$$ where $\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ and $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Let $\sigma_3$ and $\sigma_0$ be the Pauli matrices in $\text{Mat}_{2 \times 2}(\mathbb{C})$ defined by $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ and $\sigma_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. The trace of the product of $\sigma_3$ and $\sigma_0$ is equal to $0$, that is, $\operatorname{Tr}(\sigma_3 \sigma_0) = 0$.

The trace of the product of the Pauli matrices $\sigma_3$ and $\sigma_1$ is equal to $0$: $$\operatorname{Tr}(\sigma_3 \sigma_1) = 0$$ where $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

Let $\sigma_2$ and $\sigma_3$ be the Pauli matrices in $\text{Mat}_{2 \times 2}(\mathbb{C})$ defined by $\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$ and $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. The trace of the product of $\sigma_3$ and $\sigma_2$ is equal to $0$, that is, $\operatorname{Tr}(\sigma_3 \sigma_2) = 0$.

Let $\sigma_3$ be the Pauli matrix in $\text{Mat}_{2 \times 2}(\mathbb{C})$ defined by $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. The trace of the product of $\sigma_3$ with itself is equal to $2$, that is, $\operatorname{Tr}(\sigma_3 \sigma_3) = 2$.

The Pauli matrices $\sigma_1, \sigma_2,$ and $\sigma_3$ satisfy the commutation relation $$\sigma_1 \sigma_2 - \sigma_2 \sigma_1 = 2i \sigma_3$$ where $i$ is the imaginary unit.

The Pauli matrices $\sigma_1, \sigma_2,$ and $\sigma_3$ satisfy the commutation relation $\sigma_1 \sigma_3 - \sigma_3 \sigma_1 = -2i \sigma_2$, where $i$ is the imaginary unit.

Let $\sigma_1, \sigma_2, \sigma_3$ be the $2 \times 2$ Pauli matrices and $i$ be the imaginary unit. Then the commutation relation between $\sigma_2$ and $\sigma_1$ is given by $$\sigma_2 \sigma_1 - \sigma_1 \sigma_2 = -2i \sigma_3.$$

Let $\sigma_1, \sigma_2, \sigma_3$ be the Pauli matrices and $i$ be the imaginary unit. Then the commutation relation between $\sigma_2$ and $\sigma_3$ is given by $$\sigma_2 \sigma_3 - \sigma_3 \sigma_2 = 2i\sigma_1.$$

Let $\sigma_1, \sigma_2,$ and $\sigma_3$ be the Pauli matrices and $i$ be the imaginary unit. The commutation relation between $\sigma_3$ and $\sigma_1$ is given by: $$\sigma_3 \sigma_1 - \sigma_1 \sigma_3 = 2i \sigma_2$$

Let $\sigma_1, \sigma_2, \sigma_3$ be the Pauli matrices, where $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, and $\sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. The commutation relation between $\sigma_3$ and $\sigma_2$ is given by $\sigma_3 \sigma_2 - \sigma_2 \sigma_3 = -2i \sigma_1$, where $i$ is the imaginary unit.

For any indices $i, j \in \{0, 1, 2\}$ (representing the spatial components of the Pauli matrices), let $\sigma_i$ and $\sigma_j$ be the corresponding Pauli matrices. Their anticommutator satisfies the relation: $$\sigma_i \sigma_j + \sigma_j \sigma_i = 2 \delta_{ij} I$$ where $\delta_{ij}$ is the Kronecker delta and $I$ is the $2 \times 2$ identity matrix.

Given a real three-dimensional vector $a = (a_1, a_2, a_3) \in \mathbb{R}^3$, the function returns the $2 \times 2$ complex matrix defined by the linear combination $a \cdot \sigma = \sum_{i=1}^3 a_i \sigma_i$. Here, $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices: $$\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ where $i$ is the imaginary unit.

Let $\mathbf{a}, \mathbf{b} \in \mathbb{R}^3$ be two real three-dimensional vectors. Let $\mathbf{a} \cdot \sigma$ and $\mathbf{b} \cdot \sigma$ denote the $2 \times 2$ complex matrices (Pauli vectors) defined by the linear combinations $\sum_{i=1}^3 a_i \sigma_i$ and $\sum_{i=1}^3 b_i \sigma_i$ respectively, where $\sigma_i$ are the Pauli matrices. The anticommutator of these matrices satisfies: $$(\mathbf{a} \cdot \sigma)(\mathbf{b} \cdot \sigma) + (\mathbf{b} \cdot \sigma)(\mathbf{a} \cdot \sigma) = 2 (\mathbf{a} \cdot \mathbf{b}) I$$ where $\mathbf{a} \cdot \mathbf{b}$ is the Euclidean dot product and $I$ is the $2 \times 2$ identity matrix.

Let $\mathbf{a}, \mathbf{b} \in \mathbb{R}^3$ be real three-dimensional vectors. Let $\mathbf{a} \cdot \sigma$ denote the matrix $\sum_{i=1}^3 a_i \sigma_i$, where $\sigma_i$ are the Pauli matrices. The commutator of the matrices associated with $\mathbf{a}$ and $\mathbf{b}$ is given by twice $i$ times the matrix associated with their cross product: $$(\mathbf{a} \cdot \sigma)(\mathbf{b} \cdot \sigma) - (\mathbf{b} \cdot \sigma)(\mathbf{a} \cdot \sigma) = 2i ((\mathbf{a} \times \mathbf{b}) \cdot \sigma)$$ where $i$ is the imaginary unit and $\mathbf{a} \times \mathbf{b}$ is the standard vector cross product.

For any two real 3-vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^3$, the product of their corresponding Pauli vector matrices is given by the formula: $$(\mathbf{a} \cdot \boldsymbol{\sigma})(\mathbf{b} \cdot \boldsymbol{\sigma}) = (\mathbf{a} \cdot \mathbf{b}) I + i (\mathbf{a} \times \mathbf{b}) \cdot \boldsymbol{\sigma}$$ where $\mathbf{a} \cdot \mathbf{b}$ denotes the standard dot product, $\mathbf{a} \times \mathbf{b}$ denotes the vector cross product, $I$ is the $2 \times 2$ identity matrix, $i$ is the imaginary unit, and $\mathbf{v} \cdot \boldsymbol{\sigma} = \sum_{j=1}^3 v_j \sigma_j$ represents the $2 \times 2$ complex matrix associated with a vector $\mathbf{v}$.

For any real 3-vector $a \in \mathbb{R}^3$, let $a \cdot \sigma = \sum_{i=1}^3 a_i \sigma_i$ be the $2 \times 2$ complex matrix formed by the linear combination of the Pauli matrices. Then the square of this matrix is equal to the squared Euclidean norm of $a$ times the identity matrix $I$, i.e., $$(a \cdot \sigma)^2 = |a|^2 I$$ where $|a|^2 = \sum_{i=1}^3 a_i^2$.