Pauli Algebra and Anticommutators

The Pauli matrices obey two complementary product rules: a commutator relation that encodes how rotations about different axes fail to commute, and an anticommutator relation that encodes their orthogonality. Together they generate the entire algebra of spin-½.

The commutator relation

The commutator of two operators is $[A, B] = AB - BA$. For the Pauli matrices,

These are cyclic: advancing $x \to y \to z \to x$ reproduces the next line. Compactly, with the Levi-Civita symbol $\varepsilon_{jkl}$ and the Einstein summation convention,

Let us verify the first relation directly. Using $\sigma_x\sigma_y = i\sigma_z$ and $\sigma_y\sigma_x = -i\sigma_z$,

The fact that $\sigma_x$ and $\sigma_y$ do not commute is the algebraic origin of the uncertainty relation between spin components: you cannot have a definite value of $S_x$ and $S_y$ simultaneously.

The anticommutator relation

The anticommutator is $\{A, B\} = AB + BA$. The Paulis satisfy

So distinct Paulis anticommute ($\{\sigma_x, \sigma_y\} = 0$), while each squares to the identity ($\{\sigma_x, \sigma_x\} = 2I$, i.e. $\sigma_x^2 = I$). Anticommutation is the matrix analogue of two unit vectors being orthogonal.

The master product identity

Adding half the commutator and half the anticommutator gives a single formula for any product of two Paulis:

For example $\sigma_x\sigma_y = i\sigma_z$ (here $j\neq k$, so only the $\varepsilon$ term survives), and $\sigma_z\sigma_z = I$ (here $j = k$, so only the $\delta$ term survives). This one identity lets you multiply Paulis without ever writing out a matrix.

Try it

Compute the commutator $[\sigma_x, \sigma_y] = \sigma_x\sigma_y - \sigma_y\sigma_x$ as a complex $2\times 2$ matrix and return the imaginary part of its $(0,0)$ entry. Since $[\sigma_x,\sigma_y] = 2i\sigma_z$ and $(\sigma_z)_{00} = 1$, the answer should be $2$.

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