Quantum Brownian motion induced by fluctuating boundaries and compactification

In this work, we investigate the quantum Brownian motion of a point charge arising as a consequence of two fluctuating point-like boundaries. The study considers Dirichlet, Neumann, and mixed boundary conditions imposed on a real massless scalar field. Additionally, we analyze the effects of a fluctuating compactification length on the random motion of the point charge, induced by the imposition of a quasi-periodic condition on the scalar field. By associating a wave function with the length scale of each system, we demonstrate that typical divergences, which commonly appear in scenarios with fixed boundaries and compactification size, are effectively smoothed out. This approach generalizes and extends previous results found in the literature, offering new insights into the regularization of divergences appearing in idealized systems.