Path-ordering

 ( Quantum Field Theory ) 

In theoretical physics, path-ordering is the procedure (or a meta-operator $\backslash mathcal\; P$) that orders a product of operators according to the value of a chosen parameter:

$\backslash mathcal\; P\; \backslash left\backslash \{O\_1(\backslash sigma\_1)\; O\_2(\backslash sigma\_2)\; \backslash cdots\; O\_N(\backslash sigma\_N)\backslash right\backslash \}$

   \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:

$p\; :\; \backslash \{1,\; 2,\; \backslash dots,\; N\backslash \}\; \backslash to\; \backslash \{1,\; 2,\; \backslash dots,\; N\backslash \}$

$\backslash sigma\_\{p\_1\}\; \backslash leq\; \backslash sigma\_\{p\_2\}\; \backslash leq\; \backslash cdots\; \backslash leq\; \backslash sigma\_\{p\_N\}.$

For example:

$\backslash mathcal\; P\; \backslash left\backslash \{\; O\_1(4)\; O\_2(2)\; O\_3(3)\; O\_4(1)\; \backslash right\backslash \}\; =\; O\_4(1)\; O\_2(2)\; O\_3(3)\; O\_1(4)\; .$

In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below.

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Examples 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.

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Time ordering In quantum field theory it is useful to take the time-ordered product of operators. This operation is denoted by $\backslash mathcal\; T$. (Although $\backslash mathcal\; T$ is often called the "time-ordering operator", strictly speaking it is neither an linear 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 $\backslash tau\_x$ and $\backslash tau\_y$ denote the invariant scalar time-coordinates of the points x and y.Steven Weinberg, The Quantum Theory of Fields, Vol. 3, Cambridge University Press, 1995, , p. 143.

Explicitly we have

$\backslash mathcal\; T\; \backslash left\backslash \{A(x)\; B(y)\backslash right\backslash \}\; :=\; \backslash theta\; (\backslash tau\_x\; -\; \backslash tau\_y)\; A(x)\; B(y)\; \backslash pm\; \backslash theta\; (\backslash tau\_y\; -\; \backslash tau\_x)\; B(y)\; A(x),$

where $\backslash theta$ denotes the Heaviside step function and the $\backslash pm$ depends on if the operators are or 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 Commutativity. This is why it is necessary to use $\backslash tau$ 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 p's 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

$\backslash exp\; h\; =\; \backslash lim\_\{N\backslash to\backslash infty\}\; \backslash left(1\; +\; \backslash frac\{h\}\{N\}\backslash right)^N.$

Now consider the discretized evolution operator

$S\; =\; \backslash cdots\; (1+h\_\{+3\})(1+h\_\{+2\})(1+h\_\{+1\})(1+h\_0)(1+h\_\{-1\})(1+h\_\{-2\})\backslash cdots$

where $1+h\_\{j\}$ is the evolution operator over an infinitesimal time interval $j\backslash varepsilon,(j+1)\backslash varepsilon$. The higher order terms can be neglected in the limit $\backslash varepsilon\backslash to\; 0$. The operator $h\_j$ is defined by

$h\_j\; =\backslash frac\{1\}\{i\backslash hbar\}\; \backslash int\_\{j\backslash varepsilon\}^\{(j+1)\backslash varepsilon\}\; \backslash ,\; dt\; \backslash int\; d^3\; x\; \backslash ,\; H(\backslash vec\; x,t).$

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

$S\; =\; \{\backslash mathcal\; T\}\; \backslash exp\; \backslash left(\backslash sum\_\{j=-\backslash infty\}^\backslash infty\; h\_j\backslash right)\; =\; \backslash mathcal\; T\; \backslash exp\; \backslash left(\backslash int\; dt\backslash ,\; d^3\; x\; \backslash ,\; \backslash frac\{H(\backslash vec\; x,t)\}\{i\backslash hbar\}\backslash right).$

The only subtlety we had to include was the time-ordering operator $\backslash mathcal\; T$ because the factors in the product defining S above were time-ordered, too (and operators do not commute in general) and the operator $\backslash mathcal\; T$ ensures that this ordering will be preserved.

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See also

• Ordered exponential (essentially the same concept)
• Dyson series
• Gauge theory
• S-matrix

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