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.