查看“︁路径排序”︁的源代码

{{Unreferenced|date=2009年12月}}

'''路径排序'''在[[理论物理]]中表示将多个算符的乘积按照某个给定的[[参数]]重新排序的过程（可以视作一个{{link-en|元算符|meta-operator}}）：

<center><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></center>

式中 <math>p:\{1,2,\dots ,N\} \to \{1,2,\dots, N\}</math> 是一个对参数排序的[[置换]]，使得：

<center><math>\sigma_{p_1}\leq \sigma_{p_2}\leq \dots \leq \sigma_{p_N}.</math></center>

例如：

<center><math>{\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) .</math></center>

如果算符并非上面这种简单乘积的形式，就需要先作[[泰勒展开]]，然后对展开式中的每一项进行路径排序。
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==範例==
If an [[Operator (physics)|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 function|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 integration|contour]], and because the contour is closed, the Wilson loop must be defined as a [[trace (linear algebra)|trace]] in order to become [[gauge-invariant]].
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==时间排序==
在[[量子场论]]中经常需要对算符进行时间排序，这一操作用原算符 <math>{\mathcal T}</math> 表示。对于分别依赖于两个时空点 x 和 y 的算符 <math>A(x)</math> 和 <math>B(y)</math> 而言，<math>{\mathcal T}</math> 的定义如下：

<center><math>{\mathcal T} \left[A(x) B(y)\right] := \left\{ \begin{matrix} A(x) B(y) & \textrm{ if } & x_0 > y_0 \\ \pm B(y)A(x) & \textrm{ if } & x_0 < y_0. \end{matrix} \right.</math></center>
这里 <math>x_0</math> 和 <math>y_0</math> 分别表示点 <math>x</math> 和 <math>y</math> 的时间坐标。

也可以写成：
<center><math>{\mathcal T} \left[A(x) B(y)\right] := \theta (x_0 - y_0) A(x) B(y) \pm \theta (y_0 - x_0) B(y) A(x), </math></center>
这里 <math>\theta</math> 表示[[单位阶跃函数]]，而 <math>\pm</math> 取决于算符是[[玻色子]]体系的还是[[费米子]]体系的。对玻色子体系总是取正号。对于费米子体系取决于前述置换的奇偶性，对偶置换取正号，对奇置换取负号。<!-- Note that the statistical factors do not enter here. -->

因为算符依赖于具体的时空点（不仅仅依赖于时间），因此仅当这些算符在任意两个[[类空间隔]]的点上的取值对易时，最终的表达式才会与具体的时空点无关。一般来说，在时间排序中，自右往左，时间依次增大。

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The [[S-matrix]] in [[quantum field theory]] is an example of a time-ordered product. 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.

==See also==
* [[Ordered exponential]] describes essentially the same concept.
* [[Gauge theory]]
* [[S-matrix]]
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[[Category:量子场论]]
[[Category:规范理论]]