Physlib.Relativity.Tensors.LeviCivita.Complex

The identity map is a valid reindexing that relates the colors of the complexified real Levi-Civita tensor to the sequence of four complex contravariant Lorentz indices. Specifically, if $c_{\mathbb{R}} = (\text{up}, \text{up}, \text{up}, \text{up})$ is the sequence of four contravariant colors for a real Lorentz tensor and $\phi$ is the mapping `colorToComplex`, then the identity map $\text{id}$ satisfies the reindexing property between the complexified colors $\phi \circ c_{\mathbb{R}}$ and the sequence of four complex contravariant colors $c_{\mathbb{C}} = (\text{up}, \text{up}, \text{up}, \text{up})$.

The Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is a rank-4 contravariant complex Lorentz tensor, an element of the tensor space $\mathbb{C}T^{(\text{up, up, up, up})}$. It is defined as the complexification of the real Lorentz Levi-Civita tensor, mapping its real components into the complex field. Its components are determined by the sign of the permutation of the indices $(0, 1, 2, 3)$, specifically satisfying $\epsilon^{0123} = 1$, and it is totally antisymmetric under the exchange of any two indices.

The notation $\epsilon_{4\mathbb{C}}$ denotes the complex Lorentz Levi-Civita tensor, which is a tensor with four contravariant indices, an element of the tensor space $\mathbb{C}T^{\text{up, up, up, up}}$.

The rank-4 complex contravariant Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is equal to the tensor constructed from rational components where the real part is the generalized Kronecker delta $\delta^{\mu\nu\rho\sigma}_{0123}$ and the imaginary part is zero. Here, the indices $\mu, \nu, \rho, \sigma$ range over $\{0, 1, 2, 3\}$, and the generalized Kronecker delta $\delta^{\mu\nu\rho\sigma}_{0123}$ represents the Levi-Civita symbol.

The rank-4 complex contravariant Levi-Civita tensor $\epsilon^{\mu\nu\rho\sigma}$ is equal to the tensor constructed from rational components where, for a multi-index $b = (b_0, b_1, b_2, b_3)$ with $b_i \in \{0, 1, 2, 3\}$, the real part is given by the product over pairs $i < j$ of the sign of the difference of the indices: $$\prod_{0 \le i < j \le 3} \text{sgn}(b_j - b_i)$$ where $\text{sgn}(x)$ is $1$ if $x > 0$, $0$ if $x = 0$, and $-1$ if $x < 0$, and the imaginary part of each component is zero.