Miscellenea

tdg.PauliMatrix

tdg.pauli.PauliMatrix = tensor([[[ 1.+0.j,  0.+0.j],          [ 0.+0.j,  1.+0.j]],          [[ 0.+0.j,  1.+0.j],          [ 1.+0.j,  0.+0.j]],          [[ 0.+0.j, -0.-1.j],          [ 0.+1.j,  0.+0.j]],          [[ 1.+0.j,  0.+0.j],          [ 0.+0.j, -1.+0.j]]], dtype=torch.complex128)

A \(4\times2\times2\) tensor of Pauli matrices \(\sigma\).

\[\begin{split}\begin{align} \texttt{PauliMatrix[0]} &= \begin{pmatrix} +1 & 0 \\ 0 & +1 \end{pmatrix} & \texttt{PauliMatrix[1]} &= \begin{pmatrix} 0 & +1 \\ +1 & 0 \end{pmatrix} \\ \texttt{PauliMatrix[2]} &= \begin{pmatrix} 0 & -i \\ +i & 0 \end{pmatrix} & \texttt{PauliMatrix[3]} &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{align}\end{split}\]

They form a complete basis for \(2\times2\) Hermitian matrices and satisfy

\[\begin{align} \sigma_j \sigma_k = \delta_{jk} \sigma_0 + i \epsilon_{jkl} \sigma_l \end{align}\]

for \(j,k,l \in {1,2,3}\).

tdg.epsilon

tdg.epsilon.epsilon = tensor([[[ 0,  0,  0],          [ 0,  0,  1],          [ 0, -1,  0]],          [[ 0,  0, -1],          [ 0,  0,  0],          [ 1,  0,  0]],          [[ 0,  1,  0],          [-1,  0,  0],          [ 0,  0,  0]]])

The totally-antisymmetric Levi-Civita symbol \(\epsilon\) with three indices.

\(\epsilon[0,1,2]=+1\). Even permutations of the indices are also +1; odd permutations give -1. All other entries vanish.