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Concept
Path-ordering
Summary
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:
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 . (Although is often called the "time-ordering operator", strictly speaking it is neither an operator on states nor a superoperator on operators.)
For two operators A(x) and B(y) that depend on spacetime locations x and y we define:
Here and denote the invariant scalar time-coordinates of the points x and y.
Explicitly we have
where denotes the Heaviside step function and the depends on if the operators are bosonic or fermionic in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering. Note that the statistical factors do not enter here.
Since the operators depend on their location in spacetime (i.e. not just time) this time-ordering operation is only coordinate independent if operators at spacelike separated points commute. This is why it is necessary to use rather than , since usually indicates the coordinate dependent time-like index of the spacetime point. Note that the time-ordering is usually written with the time argument increasing from right to left.
In general, for the product of n field operators A1(t1), ...
Official source
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