Path-ordering

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

${\displaystyle {\mathcal {P}}\left\{O_{1}(\sigma _{1})O_{2}(\sigma _{2})\cdots O_{N}(\sigma _{N})\right\}\equiv O_{p_{1}}(\sigma _{p_{1}})O_{p_{2}}(\sigma _{p_{2}})\cdots O_{p_{N}}(\sigma _{p_{N}}).}$

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

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

${\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}$

For example:

${\displaystyle {\mathcal {P}}\left\{O_{1}(4)O_{2}(2)O_{3}(3)O_{4}(1)\right\}=O_{4}(1)O_{2}(2)O_{3}(3)O_{1}(4).}$

In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below.