Finite-size scaling properties and Casimir forces in an exactly solvable quantum statistical-mechanical model

A d-dimensional finite quantum model system confined to a general hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T - temperature of the system) is considered in the spherical approximation under periodic boundary conditions. Because of its close relation with the system of quantum rotors it represents an effective model for studying the low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the zero-temperature quantum critical point the ideas of finite-size scaling are used for studying the critical behaviour of the model. For a film geometry in different space dimensions $\half\sigma <d<\threehalf\sigma$, where $0<\sigma\leq2$ controls the long-ranginess of the interactions, an analysis of the free energy and the Casimir forces is given.