The free propagator of strongly anisotropic systems with free surfaces

A brief overview of fluctuation-induced forces in statistical systems with film geometry at the critical point and the calculation of Casimir amplitudes, which characterize these forces quantitatively, is presented. Particular attention is paid to the special features of strongly anisotropic $m$-axis systems at the Lifshitz point, specifically, in the case of a "$perpendicular$" orientation of surfaces with free boundary conditions. Beyond the simplest one-loop approximation, calculations of Casimir amplitudes are impossible without knowledge of the Gaussian propagator, which corresponds to the lines of Feynman diagrams in the perturbation theory. We present an explicit expression for such a propagator in the case of an anisotropic system confined by parallel surfaces $perpendicular$ to one of the anisotropy axes. Using this propagator, we reproduce the one-loop result derived earlier in an essentially different way. The knowledge of the propagator provides the possibility of higher-order calculations in perturbation theory.