Given the formal properties of GRVFs on $\mathbf{R}^{3}$ and the existence of stochastic integrals, Wilson lines can be constructed.

Let $K \in\mathbf{D}\subset\mathbf{R}^{3}$ be an open string/knot or smooth curve in the simply connected region $\mathbf{D}$ with Euclidean coordinates $x^{i}(s)\in K \subset \mathbf{D}$, where $s \in [0,1]$ and with ends $x^{i}(0)=\alpha$ and $x^{i}(1)=\beta$. Now let $X_{i}(x)$ with $\mathcal:\mathbf{R}^{3}\rightarrow\mathbf{R}^{3}$ be a smooth non-random vector field defined for all $x\in\mathbf{D}$. Then the classical Wilson-line holonomy is

$W[K]=\mathcal{P} exp\bigg(\mu{\displaystyle \int_{0}^{s}}d\bar{s}{X}_{i}(x(\bar{s})) \frac{dx^{i}}{d\bar{s}}d\bar{s}\bigg)= \vec{\mathcal{P}} exp\bigg ({\displaystyle \int_{K}}{X}_{i}(x)dx^{i} \bigg)$

and for closed strings/curves $K$

$W[K]=\vec{\mathcal{P}} exp\bigg({\displaystyle \oint_{\gamma}}X_{i}(x)dx^{i}\bigg)$

Proof:
Let $W(s)$ with $W:[0,1]\rightarrow \mathbb{R}^{+}$ be a smooth differentiable function on $K$. Given the line integral for $L[K]={\displaystyle \int_{K}}{X}_{i}(x)dx^{i}$, define a second integral

$\Psi(K,W(s))={\displaystyle \int_{K}}{X}_{i}(x)W[s]dx^{i} ={\displaystyle \int_{0}^{s}}d\bar{s}{X}_{i}(x(\bar{s}))\frac{dx^{i}(\bar{s})}{d\bar{s}} {W}(\bar{s})$

then this is equivalent to the differential equation

${\displaystyle \frac{d\Psi(K,W[s])}{ds}=X_{i}(x(s))\frac{dx^{i}(s)}{ds}W[s]}$

Choosing $\Psi(K, W[s])=W[s]$ then

${\displaystyle \frac{dW(s)]}{ds}=X_{i}(x(s))\frac{dx^{i}(s)}{ds}W(s)}$

The integral solution is

$W(s)=W(0)+{\displaystyle \int_{0}^{s}} ds_{1} X_{i}(x(s_{1}))\frac{dx^{i}(s)}{ds_{1}}W[s_{1}]$

and describes transport of the function $W[s]$ along the curve $K$. Performing one iteration $W[s]=W[0]$

$+\int_{0}^{s}ds_{1}{X}_{i}(x(s_{1}))\frac{dx^{i}(s)}{ds}\bigg( W[0]+{\displaystyle \int_{0}^{s_{1}}}ds_{2}{X}_{i}(x(s_{2}))\frac{dx^{i}(s_{2})}{ds_{2}} W[s_{2}]\bigg)$

Successive iterations can be continued indefinitely and the solution $W(s)$ is a convergent infinite series; that is, an exponential so that the general solution is

$W(s)={P} exp\bigg (\int_{0}^{s}d\bar{s}{X}_{i}(x(\bar{s})){dx^{i}(\bar{s})} {d\bar{s}}\bigg)$

$= \sum_{M=0}^{\infty}\frac{1}{m!}{\displaystyle \int_{0}^{s}}ds_{1}...{\displaystyle \int_{0}^{s_{M-1}}}ds_{M} \mathcal{P}[ X_{i_{1}}(x(s_{1}))\times...\times X_{i_{n}}(s_{n})\frac{dx^{i_{1}}(s_{1})}{ds_{1}}\times...\times \frac{dx^{i_{M}}(s_{M})}{ds_{M}}$

where $P$ is a ‘normal-ordering’ operator. This is then a classical Wilson-line operator in $\mathbf{R}^{3}$.

$W[K]=\vec{\mathcal{P}} exp\bigg(\mu{\displaystyle \int_{0}^{s}}d\bar{s}X_{i}(x(\bar{s}))\frac{dx^{i}(\bar{s})}{d\bar{s}}\bigg)= \vec{\mathcal{P}} exp\bigg({\displaystyle \int_{K}}{X}_{i}(x)dx^{i}\bigg)$

and for closed curves $\mathfrak{K}$

$W[K] =\vec{\mathcal{P}} exp \bigg({\displaystyle \oint_{K}} X_{i}(x)dx^{i}\bigg)$

the stochastic Wilson line follows from coupling a random field to $X_{i}(x)$

Let $x\in\mathbb{D}\subset\mathbf{R}^{3}$. Let $K$ be a curve or knot in $\mathbb{D}$ so that $x\in K\subset\mathbb{D}$. The vector field $X_{i}(x)$ exists for all $x\in\mathbb{D}$. Let $\widehat{\mathcal{F}}_{i}(x;\omega)$ be a random vector field defied with respect to a probability space $(\Omega,\mathfrak{F},\mathbf{P})$ such that $\exists$ map $\mathfrak{M}:(x,\omega)\rightarrow\mathcal{X}_{i}(x,\omega)$ for all $x\in\mathbb{D}$ and $\omega\in\Omega$. Then the total field at any $x\in\mathbb{D}$ is

$\widehat{X}_{i}(x)=X_{i}(x)+\widehat{\mathcal{X}}_{i}(x;\omega)$

The total line integral along $K$ is

$\widehat{L}[K]=L[K]+\widehat{\mathcal{L}}[K]= {\displaystyle \int_{K}}X_{i}(x)dx^{i}+{\displaystyle\int_{K}}\widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}$

The corresponding stochastic Wilson line is then

$\widehat{W}[K]=exp\bigg({\displaystyle \int_{K}}X_{i}(x)dx^{i}+{\displaystyle \int_{K}}\widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}\bigg)$

$=exp\bigg({\displaystyle\int_{K}}X_{i}(x)dx^{i}\bigg) exp{\displaystyle \int_{K}}\widehat{\mathcal{X}}_{i}(x,\omega)dx^{i} \bigg)$

The expectation is

\$\mathfrak{W}[K]=\mathbf{E}(\widehat{W}[K])=exp \bigg({\displaystyle \int_{K}}X_{i}(x)dx^{i}+{\displaystyle \int_{K}} \widehat{\mathcal{X}}_{i}(x, \omega)dx^{i} \bigg)$

$= exp({\displaystyle \int_{K}}X_{i}(x)dx^{i})\mathbf{E} exp{\displaystyle \int_{K}} \widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}$

$= exp{\displaystyle \int_{K}}V_{i}(x)dx^{i}){\displaystyle \int_{\Omega}\int_{\Omega}} exp{\displaystyle \int_{K}} \widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}) d\mathbf{P}(\omega)d\mathbf{P}(\xi)$

$= {\displaystyle\int_{\Omega}\int_{\Omega}} exp{\displaystyle \int_{K}}\widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}) d\mathbf{P}(\omega)d\mathbf{P}(\xi)$

$=exp{\displaystyle \int_{K}}\widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}) d\mathbf{P}(\omega)d\mathbf{P}(\xi))$

If the background field is homogeneous and constant then $X_{i}(x)=X_{i}$ for $x\in\mathbb{D}$ so that $\int_{K}X_{i}dx^{i}=X_{i}l=C$. Hence $latex W[K]=W=const.$

The expectation of the stochastic Wilson line is

$\mathbf{\mathfrak{W}}[K]=\mathbf{E}(\widehat{\mathfrak{W}}[K])={\displaystyle \int_{\Omega}} exp{\displaystyle \int_{K}} \widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}d\mathbf{P}(\omega)$

For a closed curve or knot $K$

$\mathbf{\mathfrak{W}}[K]=\mathbf{E}(\widehat{\mathfrak{W}}[K])={\displaystyle \int_{\Omega}} exp{\displaystyle \oint_{K}} \widehat{\mathcal{X}}_{i}(x,\omega)dx^{i}d\mathbf{P}(\omega)$

We can now present the main cluster integral expansion technique that enables the stochastic expectation of the Wilson loop to be evaluated, provided the random vector field $\mathcal{X}_{i}(x;\omega)$ is Gaussian.Such expansions have also been considered in (-