  We consider a massive scalar field with a coordinate-dependent mass in
higher-dimensional spacetime. The field satisfies Dirichlet boundary conditions
on a brane representing the four-dimensional world. Despite being massive, the
theory is scale-invariant. We quantize the theory calculating the zero-point
energy. We find the lower bound for the uncertainty product in the uncertainty
principle. We show that the zero-point energy density could be small if large
extra dimensions exist. Identifying the zero-point energy as a source of dark
energy, we extract the four-dimensional cosmological constant from
higher-dimensional theory, considering quantum fluctuations close to the brane
surface. We examine numerically ten- and eleven-dimensional spaces. The
resulting zero-point energy is parameterized by the number of extra dimensions
and the additional dimensionless {\it saturation parameter}, expressing the
deviation from perfect saturation of the uncertainty principle. Letting the
parameter to be small and of order of the fine-structure constant, we reproduce
the experimental value of the cosmological constant in four dimensions.
