What is "path-ordering"

In theoretical physics, path-ordering is the procedure (or a meta-operator $\mathcal P$) that orders a product of operators according to the value of a chosen parameter:

$$\mathcal P \left\{O_1(\sigma_1) O_2(\sigma_2) \cdots O_N(\sigma_N)\right\} \equiv O_{p_1}(\sigma_{p_1}) O_{p_2}(\sigma_{p_2}) \cdots O_{p_N}(\sigma_{p_N}).$$

Here p is a permutation that orders the parameters by value:

$$p \mathrel{:} \{1, 2, \dots, N\} \to \{1, 2, \dots, N\}$$

σ ≤ σ ≤ ⋯ ≤ σ.

For example:

$$\mathcal P \left\{ O_1(4) O_2(2) O_3(3) O_4(1) \right\} = O_4(1) O_2(2) O_3(3) O_1(4) .$$