Path-ordering
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In theoretical physics, path-ordering is the procedure (or a meta-operator <math>{\mathcal P}<math>) of ordering a product of many operators according to the value of one chosen parameter:
- <math>{\mathcal P} \left[O_1(\sigma_1)O_2(\sigma_2)\dots O_N(\sigma_N)\right]:= O_{p_1}(\sigma_{p_1}) O_{p_2}(\sigma_{p_2})\dots O_{p_N}(\sigma_{p_N}) <math>
Here <math>p:\{1,2,\dots ,N\} \to \{1,2,\dots, N\}<math> is a permutation that orders the parameters:
- <math>\sigma_{p_1}\leq \sigma_{p_2}\leq \dots \leq \sigma_{p_N}<math>
Examples
If an operator is not simply expressed as a product, but as a function of another operator, we must first perform Taylor expansion of this function. This is the case of the Wilson loop that is defined as a path-ordered exponential; this guarantees that the Wilson loop encodes the holonomy of the gauge connection. The parameter <math>\sigma<math> 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 become gauge-invariant.
Time ordering
The S-matrix in quantum field theory is another example of a path-ordered product and the parameter governing the ordering is time; therefore this type of ordering is called time ordering. The S-matrix, transforming the state at <math>t=-\infty<math> to a state at <math>t=+\infty<math>, can also be thought of as a kind of "holonomy", analogous to the Wilson loop. We obtain a time-ordered expression because of the following reason:
We start with this simple formula for the exponential:
- <math>\exp(h) = \lim_{N\to\infty} \left(1+\frac hN\right)^N<math>
Now consider the discretized evolution operator
- <math>S = \dots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_0)(1+h_{-1})(1+h_{-2})\dots<math>
where <math>1+h_{j}<math> is the evolution operator over an infinitesimal time interval <math>[j\epsilon,(j+1)\epsilon]<math>. The higher order terms can be neglected in the limit <math>\epsilon\to 0<math>. The operator <math>h_j<math> is defined by
- <math>h_j =\frac{1}{i\hbar} \int_{j\epsilon}^{(j+1)\epsilon} dt \int d^3 x \, H(\vec x,t).<math>
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
- <math> S = {\mathcal T} \exp \left(\sum_{j=-\infty}^\infty h_j\right) = {\mathcal T} \exp \left(\int dt\, d^3 x \, \frac{H(\vec x,t)}{i\hbar}\right)<math>
The only subtlety we had to include was the time-ordering operator <math>{\mathcal T}<math> because the factors in the product defining <math>S<math> above were time-ordered, too (and operators do not commute in general) and the operator <math>{\mathcal T}<math> guarantees that this ordering will be preserved.