Path integrals have been used to solve problems in polymer physics that have yet to be solved through purely Newtonian treatments. However, despite successful empirical verification of predictions made using path integrals, there remains much skepticism over the theoretical validity of their use to model polymer physics due to concerns over mathematical rigor and a disconnect between the dynamical picture of unitary evolution of a particle with the static picture of polymer configuration specifications. All prevailing attempts to mathematically formalize path integrals use Wiener measure. However, many situations arise, in both quantum field theory and in polymer physics, which involve paths of Wiener measure zero and/or forbid contributions from Brownian paths. Additionally, a Wiener measure-theoretic formalism of path integrals does not provide a physical explanation as to why one should expect to be able to model polymer physics using path integrals. We propose a general Feynman path integral model for topologically linear polymers by identifying the spatial configuration of the polymer with a path that a single particle may take while undergoing unitary evolution, the amount of time elapsed when a particle undergoes unitary evolution with some physical observable of the polymer that is invariant with respect to the number of segments the polymer may be discretized into, such that for any fixed value of this observable, the length of each segment is a positive-definite injective function with respect to the number of segments. We show that for the freely-jointed chain, the square-modulus of the propagator, under our path integral model, is able to reproduce the well-known probability density function of the end-to-end displacement vector. We then discuss the implications our path integral model has for more complicated polymer systems.
Timothy Leong (Sun,) studied this question.