Traditionally it is assumed that the Casimir vacuum pressure does not depend
on the ultraviolet cut-off. There are, however, some arguments that the effect
actually depends on the regularization procedure and thus on the
trans-Planckian physics. We provide the condensed matter example where the
Casimir forces do explicitly depend on the microscopic (correspondingly
trans-Planckian) physics due to the mesoscopic finite-N effects, where N is the
number of bare particles in condensed matter (or correspondingly the number of
the elements comprising the quantum vacuum). The finite-N effects lead to
mesoscopic fluctuations of the vacuum pressure. The amplitude of the mesoscopic
flustuations of the Casimir force in a system with linear dimension L is larger
by the factor N^{1/3}\sim L/a than the traditional value of the Casimir force
given by effective theory, where a is the interatomic distance which plays the
role of the Planck length.Comment: LaTeX file, 13 pages, no figures, submitted to JETP Letter
