The Exercise

The product of two Pauli matrices is again the identity or a Pauli matrix, up to a factor. The square of each is the identity. Two different ones give $i$ times the third, with a sign that flips when their order is reversed: for the three Pauli matrices $\sigma_1$, $\sigma_2$, $\sigma_3$, one has $\sigma_1 \sigma_2 = i \sigma_3$, while $\sigma_2 \sigma_1 = -i \sigma_3$. The exercise asks for the whole table as one formula in the Kronecker delta and the Levi-Civita symbol.

The Pauli Matrices

The Pauli matrices $\sigma_1$, $\sigma_2$, $\sigma_3$ are the complex $2 \times 2$ matrices $X$, $Y$ and $Z$, in that order, with $i$ the imaginary unit: $$\sigma_1 = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_2 = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_3 = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)

The Levi-civita Symbol

The Levi-Civita symbol $\epsilon_{jkl}$ is the integer attached to three indices $j$, $k$, $l$, each one of $1, 2, 3$, by $$\epsilon_{jkl} = \begin{cases} \phantom{-}1 & (j, k, l) = (1, 2, 3),\ (2, 3, 1),\ (3, 1, 2), \\ -1 & (j, k, l) = (1, 3, 2),\ (2, 1, 3),\ (3, 2, 1), \\ \phantom{-}0 & \text{otherwise.} \end{cases}$$(2)^✦

The Claim

Write $I$ for the $2 \times 2$ identity matrix and $\delta_{jk}$ for the Kronecker delta, equal to $1$ when $j = k$ and to $0$ otherwise.

What Is Checked

The claim is stated over the Pauli matrices of (1), given as explicit complex $2 \times 2$ matrices, the Levi-Civita symbol of (2), and the identity matrix. About these the library proves (3) for each of the nine ordered pairs $(j, k)$.