Physlib.Relativity.PauliMatrices.CliffordAlgebra

The definition `PauliMatrix.form` is the quadratic form $Q: \mathbb{R}^3 \to \mathbb{R}$ that maps a vector $v = (v_1, v_2, v_3)$ to the sum of the squares of its components, given by $Q(v) = \sum_{i=1}^3 v_i^2$. This represents the square of the standard Euclidean norm on $\mathbb{R}^3$.

This definition is the $\mathbb{R}$-algebra homomorphism from the Clifford algebra associated with the Euclidean quadratic form $Q$ on $\mathbb{R}^3$ to the subalgebra of $2 \times 2$ complex matrices generated by the Pauli matrices $\{\sigma_1, \sigma_2, \sigma_3\}$. Given the standard Euclidean form $Q(v) = \sum_{i=1}^3 v_i^2$, the map is uniquely determined by sending each basis vector $e_i$ of $\mathbb{R}^3$ to the corresponding Pauli matrix $\sigma_i$.

Let $\Phi: C\ell(\mathbb{R}^3, Q) \to \text{Mat}_{2 \times 2}(\mathbb{C})$ be the $\mathbb{R}$-algebra homomorphism from the Clifford algebra associated with the standard Euclidean quadratic form $Q$ on $\mathbb{R}^3$ to the Pauli matrix algebra. For any index $i \in \{0, 1, 2\}$ and any scalar $r \in \mathbb{R}$, let $v \in \mathbb{R}^3$ be the vector with $r$ at the $i$-th component and $0$ elsewhere (i.e., $v = r e_i$). Then the image of the generator $\iota(v)$ under $\Phi$ is given by: $$\Phi(\iota(r e_i)) = r \sigma_{i+1}$$ where $\iota: \mathbb{R}^3 \to C\ell(\mathbb{R}^3, Q)$ is the canonical embedding and $\sigma_{i+1}$ is the corresponding Pauli matrix ($\sigma_1, \sigma_2$, or $\sigma_3$).