Time ordering

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:

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

For example:

If an operator is not simply expressed as a product, but as a function of another operator, we must first perform a Taylor expansion of this function. This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection. The parameter σ that determines the ordering is a parameter describing the contour, and because the contour is closed, the Wilson loop must be defined as a trace in order to be gauge-invariant.

In quantum field theory it is useful to take the time-ordered product of operators. This operation is denoted by ${\displaystyle {\mathcal {T}}}$. For two operators A(x) and B(y) that depend on spacetime locations x and y we define:

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