Systems like superconductors and superfluids are often treated by specifying some phenomenological mean field theory where the free energy is given as a functional of some order parameter field. Given this mean field theory we can look for topological defect solutions (in particular, strings) that minimize the free energy. And so we can calculate the tension of a straight string in terms of the temperature dependent coefficients in the free energy. We can then consider solutions that are small perturbations from the straight string solution, and by integrating out the spatial directions perpendicular to the string axis we get a one-dimensional field theory of perturbations on the string itself.

Now the string is sitting in a thermal bath and so the excited modes on the string contribute to the equilibrium state. And these modes are typically gapless since they arise from breaking of global symmetries. So in particular there are massless scalar fields on the string associated to translational moduli. If we construct a statistical field theory of a massless scalar in 1d we get a typical configuration being a random walk.

So my question is whether this is physical - are the vortices in something like liquid Helium random walks on a large enough scale? I was under the impression that the vortices in a non-turbulent rotating superfluid were like straight strings, and the mean field theory approach described in my first paragraph seems standard in the literature.

My related question is how to calculate anything with this statistical field theory. If I want to find something like the heat capacity I can calculate the vacuum energy. But then as the cutoff is taken to infinity the contribution from the modes of the massless field completely dominates the contribution from the temperature dependence of the `bare' tension. Again since the statistical random walk string doesn't look like a straight string even for low temperatures I am wondering whether what I am doing is correct, and I suspect there needs to be a particular choice of regularization.