Path-ordering

In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P ) that orders a product of operators according to the value of a chosen parameter: : \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}). Here p is a permutation that orders the parameters by value: : p : \{1, 2, \dots, N\} \to \{1, 2, \dots, N\} : \sigma_{p_1} \leq \sigma_{p_2} \leq \cdots \leq \sigma_{p_N}. For example: : \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) . In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below.

In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P ) that orders a product of operators according to the value of a chosen parameter: \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}). Here p is a permutation that orders the parameters by value: p : \{1, 2, \dots, N\} \to \{1, 2, \dots, N\} \sigma_{p_1} \leq \sigma_{p_2} \leq \cdots \leq \sigma_{p_N}. For example: \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) . In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below. \mathcal T \left\{A(x) B(y)\right\} := \begin{cases} A(x) B(y) & \text{if } \tau_x > \tau_y, \\ \pm B(y)A(x) & \text{if } \tau_x Here \tau_x and \tau_y denote the invariant scalar time-coordinates of the points x and y. Explicitly we have \mathcal T \left\{A(x) B(y)\right\} := \theta (\tau_x - \tau_y) A(x) B(y) \pm \theta (\tau_y - \tau_x) B(y) A(x), where \theta denotes the Heaviside step function and the \pm 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 \tau rather than t_0 , since t_0 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 the time-ordered product of operators are defined as follows: \begin{align} \mathcal T \{ A_1(t_1) A_2(t_2) \cdots A_n(t_n) \} &= \sum_p \theta(t_{p_1} > t_{p_2} > \cdots > t_{p_n}) \varepsilon(p) A_{p_1}(t_{p_1}) A_{p_2}(t_{p_2}) \cdots A_{p_n}(t_{p_n}) \\ &= \sum_p \left( \prod_{j=1}^{n-1} \theta(t_{p_j} - t_{p_{j+1}}) \right) \varepsilon(p) A_{p_1}(t_{p_1}) A_{p_2}(t_{p_2}) \cdots A_{p_n}(t_{p_n}) \end{align} where the sum runs all over ps and over the symmetric group of n degree permutations and \varepsilon(p) \equiv \begin{cases} 1 & \text{for bosonic operators,} \\ \text{sign of the permutation} & \text{for fermionic operators.} \end{cases} The S-matrix in quantum field theory is an example of a time-ordered product. The S-matrix, transforming the state at to a state at , 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 \exp h = \lim_{N\to\infty} \left(1 + \frac{h}{N}\right)^N. Now consider the discretized evolution operator S = \cdots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_0)(1+h_{-1})(1+h_{-2})\cdots where 1+h_{j} is the evolution operator over an infinitesimal time interval [j\varepsilon,(j+1)\varepsilon] . The higher order terms can be neglected in the limit \varepsilon\to 0 . The operator h_j is defined by h_j =\frac{1}{i\hbar} \int_{j\varepsilon}^{(j+1)\varepsilon} \, dt \int d^3 x \, H(\vec x,t). 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 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). The only subtlety we had to include was the time-ordering operator \mathcal T because the factors in the product defining S above were time-ordered, too (and operators do not commute in general) and the operator \mathcal T ensures that this ordering will be preserved. ==See also== Ordered exponential (essentially the same concept) Dyson series Gauge theory S-matrix ==References==

Text derived from https://en.wikipedia.org/w/index.php?oldid=1371383531 under CC BY-SA 4.0; modifications have been made.