Physlib.Relativity.PauliMatrices.SelfAdjoint

For any Pauli matrix $\sigma_\mu$ (where $\mu \in \{0, 1, 2, 3\}$) and any $2 \times 2$ self-adjoint complex matrix $A$, the trace of the product $\sigma_\mu A$ is a real number. That is, $\text{Re}(\text{Tr}(\sigma_\mu A)) = \text{Tr}(\sigma_\mu A)$.

Let $A$ and $B$ be $2 \times 2$ self-adjoint matrices over the complex numbers $\mathbb{C}$. Let $\sigma_0, \sigma_1, \sigma_2, \sigma_3$ denote the Pauli matrices. If the traces of the products of each Pauli matrix with $A$ and $B$ are equal, specifically: $\text{Tr}(\sigma_0 A) = \text{Tr}(\sigma_0 B)$, $\text{Tr}(\sigma_1 A) = \text{Tr}(\sigma_1 B)$, $\text{Tr}(\sigma_2 A) = \text{Tr}(\sigma_2 B)$, and $\text{Tr}(\sigma_3 A) = \text{Tr}(\sigma_3 B)$, then $A = B$.

Let $A$ and $B$ be $2 \times 2$ self-adjoint matrices over the complex numbers $\mathbb{C}$. Let $\sigma_0, \sigma_1, \sigma_2, \sigma_3$ denote the Pauli matrices. If the real parts of the traces of the products of each Pauli matrix with $A$ and $B$ are equal, specifically: $\text{Re}(\text{Tr}(\sigma_0 A)) = \text{Re}(\text{Tr}(\sigma_0 B))$, $\text{Re}(\text{Tr}(\sigma_1 A)) = \text{Re}(\text{Tr}(\sigma_1 B))$, $\text{Re}(\text{Tr}(\sigma_2 A)) = \text{Re}(\text{Tr}(\sigma_2 B))$, and $\text{Re}(\text{Tr}(\sigma_3 A)) = \text{Re}(\text{Tr}(\sigma_3 B))$, then $A = B$.

The function maps an index $i \in \{0, 1, 2, 3\}$ (represented by the type $\text{Fin } 1 \oplus \text{Fin } 3$) to the corresponding Pauli matrix $\sigma_i$ viewed as an element of the set of self-adjoint $2 \times 2$ complex matrices.

The collection of Pauli matrices $\{\sigma_i\}_{i \in \{0, 1, 2, 3\}}$, viewed as elements of the real vector space of self-adjoint $2 \times 2$ complex matrices, is linearly independent over $\mathbb{R}$.

The real linear span of the set of Pauli matrices $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$ is the entire space of $2 \times 2$ self-adjoint complex matrices. Here, $\sigma_0$ denotes the identity matrix $I_2$ and $\sigma_1, \sigma_2, \sigma_3$ denote the standard spatial Pauli matrices.

The basis for the real vector space of $2 \times 2$ self-adjoint complex matrices formed by the set of Pauli matrices $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$. Here, $\sigma_0$ represents the $2 \times 2$ identity matrix $I_2$, and $\sigma_1, \sigma_2, \sigma_3$ represent the standard spatial Pauli matrices. The indexing set $\text{Fin } 1 \oplus \text{Fin } 3$ corresponds to the indices $\{0, 1, 2, 3\}$.

The function maps an index $i \in \{0\} \oplus \{0, 1, 2\}$ to a self-adjoint $2 \times 2$ complex matrix. The mapping is defined as follows: - For the index $i = \text{inl } 0$, it returns the identity matrix $\sigma_0$. - For the indices $i = \text{inr } j$ where $j \in \{0, 1, 2\}$, it returns the negative of the corresponding Pauli matrix, $-\sigma_{j+1}$. In this context, $\sigma_1, \sigma_2, \sigma_3$ represent the standard Pauli matrices and $\sigma_0$ represents the identity matrix $I_2$.

The family of $2 \times 2$ self-adjoint complex matrices $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ is linearly independent over the real numbers $\mathbb{R}$. Here, $\sigma_0$ denotes the $2 \times 2$ identity matrix, and $\sigma_1, \sigma_2, \sigma_3$ denote the standard Pauli matrices.

Let $\sigma_0$ be the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ be the standard Pauli matrices. The set of self-adjoint matrices $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ spans the space of all $2 \times 2$ complex self-adjoint matrices over the real numbers $\mathbb{R}$.

This definition provides an $\mathbb{R}$-basis for the space of $2 \times 2$ complex self-adjoint matrices, indexed by $\{0, 1, 2, 3\}$. The basis consists of the matrices $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$, where $\sigma_0$ is the $2 \times 2$ identity matrix $I_2$, and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices. These are referred to as the covariant Pauli matrices.

Let $M$ be a $2 \times 2$ complex self-adjoint matrix. Let $\sigma_0$ be the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ be the standard Pauli matrices. Let $b_0, b_1, b_2, b_3$ be the basis elements of the covariant Pauli basis, defined as $b_0 = \sigma_0, b_1 = -\sigma_1, b_2 = -\sigma_2,$ and $b_3 = -\sigma_3$. Then $M$ can be decomposed as: $$M = \frac{1}{2} \text{Re}(\text{Tr}(\sigma_0 M)) b_0 - \frac{1}{2} \text{Re}(\text{Tr}(\sigma_1 M)) b_1 - \frac{1}{2} \text{Re}(\text{Tr}(\sigma_2 M)) b_2 - \frac{1}{2} \text{Re}(\text{Tr}(\sigma_3 M)) b_3$$

Let $M$ be a $2 \times 2$ self-adjoint complex matrix. The coordinate (or component) of $M$ corresponding to the identity matrix $\sigma_0$ in the covariant Pauli basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ is given by $\frac{1}{2} \text{Tr}(\sigma_0 M)$. Here, $\sigma_0$ denotes the $2 \times 2$ identity matrix and $\text{Tr}$ denotes the matrix trace.

For any $2 \times 2$ complex self-adjoint matrix $M$, the component of $M$ with respect to the basis element $-\sigma_1$ in the covariant Pauli basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ is equal to $-\frac{1}{2} \text{Tr}(\sigma_1 M)$. Here, $\sigma_0$ is the identity matrix, $\sigma_1$ is the first Pauli matrix, and $\text{Tr}$ denotes the matrix trace.

Let $M$ be a $2 \times 2$ self-adjoint complex matrix. Consider the covariant Pauli basis $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ for the space of such matrices, where $\sigma_0$ is the identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices. The coordinate of $M$ corresponding to the basis vector $-\sigma_2$ (indexed by the second spatial component) is equal to $-\frac{1}{2} \text{Tr}(\sigma_2 M)$.

Let $M$ be a $2 \times 2$ complex self-adjoint matrix. Let $\{\sigma_0, -\sigma_1, -\sigma_2, -\sigma_3\}$ be the covariant Pauli basis for the space of $2 \times 2$ self-adjoint matrices, where $\sigma_0$ is the identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices. The coordinate of $M$ corresponding to the basis element $-\sigma_3$ is given by: $$[M]_{-\sigma_3} = -\frac{1}{2} \operatorname{tr}(\sigma_3 M)$$ where $\operatorname{tr}$ denotes the matrix trace.

Let $\{\sigma^i\}_{i=0}^3$ be the basis of contravariant Pauli matrices (`pauliBasis`) and $\{\sigma_i\}_{i=0}^3$ be the basis of covariant Pauli matrices (`pauliBasis'`) for the real vector space of $2 \times 2$ self-adjoint complex matrices. Let $\eta$ be the $4 \times 4$ Minkowski matrix defined as $\mathrm{diag}(1, -1, -1, -1)$. For any index $i \in \{0, 1, 2, 3\}$, the relationship between the basis elements is given by $\sigma^i = \eta_{ii} \cdot \sigma_i$, where $\eta_{ii}$ denotes the $i$-th diagonal component of the Minkowski matrix.

Let $A$ be a $2 \times 2$ complex self-adjoint matrix. Then $A$ can be uniquely decomposed into the sum of a scalar part and a vector part: $$A = a_0 I + A_{\text{vec}}$$ where $a_0$ is the scalar coefficient (`scalarCoeff A`), $I$ is the $2 \times 2$ identity matrix, and $A_{\text{vec}}$ is the vector part (`vectorPart A`) representing the linear combination of the spatial Pauli matrices $\sigma_1, \sigma_2, \sigma_3$.

Let $A$ be a $2 \times 2$ self-adjoint complex matrix. The trace of $A$ is equal to twice its scalar Pauli coefficient $a_0$. That is, $$\operatorname{Tr}(A) = 2 a_0.$$

Let $A$ be a $2 \times 2$ self-adjoint complex matrix. The Pauli vector part of $A$, denoted as $\text{vectorPart}(A)$, has zero trace. That is, $$\operatorname{Tr}(\text{vectorPart}(A)) = 0.$$

Let $A$ be a $2 \times 2$ self-adjoint complex matrix. The square of the vector part of $A$ is equal to the square of its Pauli radius times the $2 \times 2$ identity matrix $I$. That is, $$(\text{vectorPart } A)^2 = (\text{pauliRadius } A)^2 I.$$

Let $\sigma'_a$ and $\sigma'_b$ be elements of the covariant Pauli basis for $2 \times 2$ self-adjoint complex matrices, where the indices $a, b \in \{0, 1, 2, 3\}$. This basis is defined such that $\sigma'_0 = I_2$ (the identity matrix) and $\sigma'_j = -\sigma_j$ for $j \in \{1, 2, 3\}$, where $\sigma_j$ are the standard Pauli matrices. The trace of the product of any two such matrices satisfies: $$\text{Tr}(\sigma'_a \sigma'_b) = 2 \delta_{ab}$$ where $\delta_{ab}$ is the Kronecker delta, equal to $1$ if $a = b$ and $0$ otherwise.

Let $\sigma_0$ be the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ be the standard Pauli matrices: $$\sigma_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ Define the covariant Pauli matrices $\sigma'_l$ for $l \in \{0, 1, 2, 3\}$ such that $\sigma'_0 = \sigma_0$ and $\sigma'_j = -\sigma_j$ for $j \in \{1, 2, 3\}$. For any $2 \times 2$ complex matrix $N$, the trace of the product $\sigma'_l N$ is expressed in terms of the matrix entries $N_{ij}$ as: - For $l = 0$: $\text{Tr}(\sigma'_0 N) = N_{00} + N_{11}$ - For $l = 1$: $\text{Tr}(\sigma'_1 N) = -(N_{01} + N_{10})$ - For $l = 2$: $\text{Tr}(\sigma'_2 N) = -i(N_{01} - N_{10})$ - For $l = 3$: $\text{Tr}(\sigma'_3 N) = -(N_{00} - N_{11})$