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: In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below.
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.
denote the invariant scalar time-coordinates of the points x and y.
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.
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), …, An(tn) the time-ordered product of operators are defined as follows: where the sum runs all over p's and over the symmetric group of n degree permutations and The S-matrix in quantum field theory is an example of a time-ordered product.
We obtain a time-ordered expression because of the following reason: We start with this simple formula for the exponential Now consider the discretized evolution operator where
The higher order terms can be neglected in the limit
is defined by Note that the evolution operators over the "past" time intervals appears on the right side of the product.
We see that the formula is analogous to the identity above satisfied by the exponential, and we may write The only subtlety we had to include was the time-ordering operator