Physlib.Relativity.Fermions.Dirac.Slash

For a Lorentz vector $k$, the Dirac slash $k\!\!\!/$ is the $4 \times 4$ complex matrix defined by the contraction $k\!\!\!/ = \sum_{\mu=0}^3 k_\mu \gamma^\mu$, where $k_\mu$ are the covariant (lowered) components of $k$ and $\gamma^\mu$ are the Dirac gamma matrices. This matrix is the representation of the Clifford multiplication operator $\gamma(k)$ in the Dirac basis.

For a Lorentz vector $k$, the notation $k\mkern-8.5mu/$ represents the Dirac slash operator, defined as the contraction $k_\mu \gamma^\mu$ of the vector's components with the Dirac gamma matrices $\gamma^\mu$. This operator is represented as a $4 \times 4$ complex matrix.

The Dirac slash $\slashed{k}$ is an $\mathbb{R}$-linear map from the space of Lorentz vectors to the space of $4 \times 4$ complex matrices $\text{Mat}_{4 \times 4}(\mathbb{C})$. It is defined by the contraction $\slashed{k} = \sum_{\mu=0}^3 k_\mu \gamma^\mu$, where $k_\mu$ are the covariant (lowered) components of the Lorentz vector $k$ and $\gamma^\mu$ are the Dirac gamma matrices. This map is the matrix representation of Clifford multiplication by a Lorentz vector.

For any Lorentz vector $k$, the value of the $\mathbb{R}$-linear map $\text{slashLinear}$ applied to $k$ is equal to the Dirac slash matrix $\slashed{k}$, which is defined by the contraction $\slashed{k} = \sum_{\mu=0}^3 k_\mu \gamma^\mu$ of the covariant components of $k$ with the Dirac gamma matrices $\gamma^\mu$.

For a Lorentz vector $k$, the Dirac slash $\slashed{k}$ is equal to the sum over spacetime indices $\mu$ of the covariant components $k_\mu$ multiplied by the matrix representation of the Dirac gamma matrices $\gamma^\mu$: \[\slashed{k} = \sum_{\mu} k_\mu \gamma^\mu\] where $k_\mu = \eta_{\mu\mu} k^\mu$ are the lowered coordinates of $k$ obtained via the Minkowski metric $\eta = \text{diag}(1, -1, -1, -1)$, and $\gamma^\mu$ are the Dirac gamma matrices.

The Dirac slash of the zero Lorentz vector is the zero matrix. That is, $0\!\!\!/ = 0$, where the $0$ on the left is the zero Lorentz vector and the $0$ on the right is the zero $4 \times 4$ complex matrix.

For any two Lorentz vectors $k$ and $l$, the Dirac slash of their sum is equal to the sum of their individual Dirac slashes: $$\not{(k + l)} = \not{k} + \not{l}$$ where $\not{k} = \sum_{\mu} k_\mu \gamma^\mu$ is the $4 \times 4$ complex matrix formed by contracting the covariant components of the Lorentz vector $k$ with the Dirac gamma matrices $\gamma^\mu$.

For any real number $c \in \mathbb{R}$ and any Lorentz vector $k$, the Dirac slash of the scalar product $c \cdot k$ is equal to the scalar multiplication of the matrix $\not{k}$ by $c$. That is: $$\not{(ck)} = c \not{k}$$ where $\not{k} = \sum_{\mu=0}^3 k_\mu \gamma^\mu$ is the $4 \times 4$ complex matrix formed by the contraction of the covariant components of $k$ with the Dirac gamma matrices $\gamma^\mu$.

For a Lorentz vector $k$, this is the $\mathbb{C}$-linear endomorphism of the space of $4 \times 4$ complex matrices $M_4(\mathbb{C})$ that maps a matrix $A$ to the product $k\!\!\!/ A$, where $k\!\!\!/ = \sum_{\mu=0}^3 k_\mu \gamma^\mu$ is the Dirac slash matrix.

Given a list of Lorentz vectors $[k_1, k_2, \dots, k_n]$, this function computes the product of their corresponding Dirac slash matrices in the order they appear. The result is the $4 \times 4$ complex matrix: $$\not{k}_1 \not{k}_2 \dots \not{k}_n$$ where each $\not{k}_i = \sum_{\mu=0}^3 k_{i,\mu} \gamma^\mu$ is the Dirac slash of the vector $k_i$ and $\gamma^\mu$ are the Dirac gamma matrices. If the list is empty, the result is the identity matrix $I_4$.

The product of the Dirac slashes of an empty list of Lorentz vectors is the $4 \times 4$ identity matrix $I_4$.

For any Lorentz vector $k$ and any list of Lorentz vectors $ks = [k_1, k_2, \dots, k_n]$, the product of the Dirac slashes of the list formed by prepending $k$ to $ks$ (denoted $k :: ks$) is equal to the matrix product of the Dirac slash $k\!\!\!/ = \sum_{\mu=0}^3 k_\mu \gamma^\mu$ and the product of the Dirac slashes of the list $ks$.