Physlib.Relativity.CliffordAlgebra

The $\gamma^0$ gamma matrix in the Dirac representation is a $4 \times 4$ complex matrix defined as: $$\gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$$

The $\gamma^1$ gamma matrix in the Dirac representation is defined as the $4 \times 4$ complex matrix: $$\gamma^1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$$

The $\gamma^2$ gamma matrix in the Dirac representation is defined as the $4 \times 4$ complex matrix: $$\gamma^2 = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{pmatrix}$$

The $\gamma^3$ gamma matrix in the Dirac representation is defined as the $4 \times 4$ complex matrix: $$\gamma^3 = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$$

For any type $\alpha$ equipped with $0$ and $1$, the $4 \times 4$ identity matrix over $\alpha$ is equal to $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$.

The square of the gamma matrix $\gamma^0$ in the Dirac representation is equal to the $4 \times 4$ identity matrix: $$\gamma^0 \gamma^0 = 1$$ where $1$ denotes the $4 \times 4$ identity matrix.

In the Dirac representation of gamma matrices, the product of the $\gamma^1$ matrix with itself is equal to the negative of the $4 \times 4$ identity matrix $I$: $$\gamma^1 \gamma^1 = -I$$ where $\gamma^1$ is the complex $4 \times 4$ matrix defined as: $$\gamma^1 = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$$

The square of the gamma matrix $\gamma^2$ in the Dirac representation is equal to the negative of the identity matrix, i.e., $\gamma^2 \gamma^2 = -I$.

In the Dirac representation of the Clifford algebra for Minkowski spacetime with signature $(+, -, -, -)$, the square of the gamma matrix $\gamma^3$ is equal to the negative of the $4 \times 4$ identity matrix, i.e., $\gamma^3 \gamma^3 = -I_4$.

The Dirac gamma matrices $\gamma^1$ and $\gamma^0$ satisfy the anticommutation relation $\gamma^1 \gamma^0 = -(\gamma^0 \gamma^1)$.

The Dirac gamma matrices $\gamma^2$ and $\gamma^0$ satisfy the anticommutation relation: $$\gamma^2 \gamma^0 = -\gamma^0 \gamma^2$$

For the gamma matrices $\gamma^3$ and $\gamma^0$ in the Dirac representation, the following anticommutation relation holds: $$\gamma^3 \gamma^0 = -(\gamma^0 \gamma^3)$$

For the gamma matrices $\gamma^1$ and $\gamma^2$ in the Dirac representation, the following anticommutation relation holds: $$\gamma^2 \gamma^1 = -\gamma^1 \gamma^2$$

For the gamma matrices $\gamma^3$ and $\gamma^1$ in the Dirac representation, the anticommutation relation $\gamma^3 \gamma^1 = -\gamma^1 \gamma^3$ holds.

The gamma matrices $\gamma^3$ and $\gamma^2$ in the Dirac representation satisfy the anticommutation relation: $$\gamma^3 \gamma^2 = -\gamma^2 \gamma^3$$

The $\gamma^5$ gamma matrix in the Dirac representation is the $4 \times 4$ complex matrix defined as the product of the imaginary unit $i$ and the four gamma matrices $\gamma^0, \gamma^1, \gamma^2,$ and $\gamma^3$: $$\gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3$$

The function $\gamma$ maps an index $\mu \in \{0, 1, 2, 3\}$ to its corresponding $4 \times 4$ complex gamma matrix in the Dirac representation, specifically $\gamma(0) = \gamma^0$, $\gamma(1) = \gamma^1$, $\gamma(2) = \gamma^2$, and $\gamma(3) = \gamma^3$.

The set $\text{γSet}$ is the collection of the four $4 \times 4$ complex matrices $\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$ in the Dirac representation, which are elements of $\text{Mat}_4(\mathbb{C})$.

For any index $\mu \in \{0, 1, 2, 3\}$, the Dirac gamma matrix $\gamma^\mu$ is an element of the set $\text{γSet} = \{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$.

The Dirac algebra is the $\mathbb{R}$-subalgebra of the space of $4 \times 4$ complex matrices $M_4(\mathbb{C})$ generated by the set of gamma matrices $\gamma\text{Set} = \{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$.

The set of gamma matrices $\gamma\text{Set} = \{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$ is a subset of the Dirac algebra, where the Dirac algebra is defined as the $\mathbb{R}$-subalgebra of $4 \times 4$ complex matrices $M_4(\mathbb{C})$ generated by $\gamma\text{Set}$.

For any index $\mu \in \{0, 1, 2, 3\}$, the Dirac gamma matrix $\gamma^\mu$ is an element of the Dirac algebra, where the Dirac algebra is defined as the $\mathbb{R}$-subalgebra of $4 \times 4$ complex matrices $M_4(\mathbb{C})$ generated by the set of gamma matrices $\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}$.

The quadratic form of the Clifford algebra corresponding to the $\gamma$ matrices. It is defined on $\mathbb{R}^4$ (specifically, the space of functions from $\{0, 1, 2, 3\}$ to $\mathbb{R}$) with Minkowski signature $(+, -, -, -)$, mapping a vector $x = (x_0, x_1, x_2, x_3)$ to $x_0^2 - x_1^2 - x_2^2 - x_3^2$.

This definition constructs the $\mathbb{R}$-algebra homomorphism from the Clifford algebra associated with the Dirac quadratic form (the Minkowski signature $(+,-,-,-)$ on $\mathbb{R}^4$) to the Dirac algebra. Specifically, the map is defined using the universal property of Clifford algebras: it identifies a vector $v = (v_0, v_1, v_2, v_3) \in \mathbb{R}^4$ with the matrix sum $\sum_{\mu=0}^3 v_\mu \gamma^\mu$, where $\gamma^\mu$ are the $4 \times 4$ complex Dirac matrices. The codomain, the Dirac algebra, is the $\mathbb{R}$-subalgebra of $M_4(\mathbb{C})$ generated by these gamma matrices.

Let $\Phi$ be the $\mathbb{R}$-algebra homomorphism from the Clifford algebra associated with the Dirac quadratic form (on $\mathbb{R}^4$) to the Dirac algebra (the subalgebra of $M_4(\mathbb{C})$ generated by the gamma matrices). For any index $i \in \{0, 1, 2, 3\}$ and any real scalar $r \in \mathbb{R}$, let $e_i$ be the $i$-th standard basis vector in $\mathbb{R}^4$ (where the $i$-th component is $1$ and others are $0$). The homomorphism $\Phi$ maps the element $\iota(r e_i)$ in the Clifford algebra to the scaled gamma matrix $r \gamma^i$, where $\iota$ is the canonical linear map from the vector space to the Clifford algebra.

For each index $i \in \{0, 1, 2, 3\}$, the Dirac gamma matrix $\gamma^i$, considered as an element of the Dirac algebra, belongs to the range of the $\mathbb{R}$-algebra homomorphism $\Phi: \text{CliffordAlgebra}(\text{diracForm}) \to \text{diracAlgebra}$. Here, $\text{diracForm}$ is the quadratic form on $\mathbb{R}^4$ with Minkowski signature $(+, -, -, -)$, and the Dirac algebra is the $\mathbb{R}$-subalgebra of $M_4(\mathbb{C})$ generated by the gamma matrices.

Let $\text{diracForm}$ be the quadratic form on $\mathbb{R}^4$ with Minkowski signature $(+,-,-,-)$. Let $\text{diracAlgebra}$ be the $\mathbb{R}$-subalgebra of $M_4(\mathbb{C})$ generated by the Dirac gamma matrices $\gamma^0, \gamma^1, \gamma^2, \gamma^3$. Consider the $\mathbb{R}$-algebra homomorphism $\phi: \text{CliffordAlgebra}(\text{diracForm}) \to \text{diracAlgebra}$ which identifies vectors in $\mathbb{R}^4$ with their representation in terms of gamma matrices. The range of $\phi$ is equal to the entire Dirac algebra.

Let $Q$ be the Dirac quadratic form on $\mathbb{R}^4$ with Minkowski signature $(+,-,-,-)$, defined by $Q(x) = x_0^2 - x_1^2 - x_2^2 - x_3^2$. Let $\text{Cl}(Q)$ be the Clifford algebra associated with this quadratic form. Let the Dirac algebra be the $\mathbb{R}$-subalgebra of $4 \times 4$ complex matrices $M_4(\mathbb{C})$ generated by the gamma matrices $\gamma^0, \gamma^1, \gamma^2, \gamma^3$. The $\mathbb{R}$-algebra homomorphism $\phi: \text{Cl}(Q) \to \text{diracAlgebra}$, which maps a vector $v = (v_0, v_1, v_2, v_3) \in \mathbb{R}^4$ to the matrix sum $\sum_{\mu=0}^3 v_\mu \gamma^\mu$, is surjective.