Physlib.Relativity.Tensors.ComplexTensor.Units.Symm

In the complex Lorentz tensor species, swapping the indices of the covariant-contravariant unit tensor $\delta_\mu{}^\nu$ (denoted `coContrUnit`) yields the contravariant-covariant unit tensor $\delta^\nu{}_\mu$ (denoted `contrCoUnit`). That is, $$(\delta_\mu{}^\nu)^T = \delta^\nu{}_\mu$$ where $\delta_\mu{}^\nu$ is the unit tensor with index sequence $(\text{down}, \text{up})$, $\delta^\nu{}_\mu$ is the unit tensor with index sequence $(\text{up}, \text{down})$, and the superscript $T$ denotes the permutation (transposition) of the tensor indices.

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor $\delta^\mu{}_\nu$ (represented by `contrCoUnit`), which has a contravariant index $\mu$ and a covariant index $\nu$, is equal to the unit tensor $\delta_\nu{}^\mu$ (represented by `coContrUnit`) under a permutation that swaps the first and second indices.

In the context of the complex Lorentz tensor species, the unit tensor associated with the alt-left spinor representation, denoted $\delta_{L'}$, is equal to the unit tensor associated with the left-handed spinor representation, denoted $\delta_L$, with its indices swapped. Specifically: $$(\delta_{L'})_{\alpha \alpha'} = (\delta_L)_{\alpha' \alpha}$$ where $\alpha$ and $\alpha'$ are indices representing the dual and primary left-handed Weyl spinor representations respectively.

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor $\delta_L$ (representing `leftAltLeftUnit`) and the dual unit tensor $\delta_{L'}$ (representing `altLeftLeftUnit`) satisfy a symmetry relation such that swapping the indices of one yields the other. Mathematically, this is expressed as: $$(\delta_L)_{\alpha \alpha'} = (\delta_{L'})_{\alpha' \alpha}$$ where $\alpha$ and $\alpha'$ are indices corresponding to the left-handed spinor representation and its dual (alt) representation, respectively.

In the tensor species of complex Lorentz representations for $SL(2, \mathbb{C})$, the unit tensor $\delta_{R'}$ associated with the "alt" right-handed representation and the unit tensor $\delta_R$ associated with the standard right-handed representation are related by a permutation of indices. Specifically, swapping the indices of $\delta_{R'}$ yields $\delta_R$: $$(\delta_{R'})_{\beta \beta'} = (\delta_R)_{\beta' \beta}$$ where $\beta$ and $\beta'$ are indices corresponding to right-handed spinor representations.

In the context of the complex Lorentz tensor species for $SL(2, \mathbb{C})$, the unit tensor for right-handed representations `rightAltRightUnit` (denoted $\delta_R$) and the `altRightRightUnit` tensor (denoted $\delta_{R'}$) are related by a permutation of indices. Specifically, swapping the indices of $\delta_R$ yields $\delta_{R'}$: $$\delta_R = \text{permT}_{[1, 0]} \delta_{R'}$$ where $\text{permT}_{[1, 0]}$ denotes the transposition of the two tensor indices. In index notation, this is expressed as $(\delta_R)_{\beta \beta'} = (\delta_{R'})_{\beta' \beta}$.