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 (-

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