Procedure of ordering a product operators
This article is about rearranging a product of operators in physics. For the well-orderings on mathematical terms, see
Path ordering (term rewriting).
In theoretical physics, path-ordering is the procedure (or a meta-operator
) 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}}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79a800ec76e580884c480b9c59088ac8bb4b8c9b)
Here p is a permutation that orders the parameters by value:
![{\displaystyle p:\{1,2,\dots ,N\}\to \{1,2,\dots ,N\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a0184a268a4ba862cda1bf0c6a8207a330cf0e)
![{\displaystyle \sigma _{p_{1}}\leq \sigma _{p_{2}}\leq \cdots \leq \sigma _{p_{N}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c565f1f8df03bba63e4165a3da3597d85c428f0)
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc9c8c5153052425c19312ae46258ed5b10fc99)