Given the formal properties of GRVFs on and the existence of stochastic integrals, Wilson lines can be constructed.

Let be an open string/knot or smooth curve in the simply connected region with Euclidean coordinates , where and with ends and . Now let with be a smooth non-random vector field defined for all . Then the classical Wilson-line holonomy is

and for closed strings/curves

Proof:

Let with be a smooth differentiable function on . Given the line integral for , define a second integral

then this is equivalent to the differential equation

Choosing then

The integral solution is

and describes transport of the function along the curve . Performing one iteration

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

where is a ‘normal-ordering’ operator. This is then a classical Wilson-line operator in .

and for closed curves

the stochastic Wilson line follows from coupling a random field to

Let . Let be a curve or knot in so that . The vector field exists for all . Let be a random vector field defied with respect to a probability space such that map for all and . Then the total field at any is

The total line integral along is

The corresponding stochastic Wilson line is then

The expectation is

\

If the background field is homogeneous and constant then for so that . Hence $latex W[K]=W=const.$

The expectation of the stochastic Wilson line is

For a closed curve or knot

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 is Gaussian.Such expansions have also been considered in (-

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