  A value of the cosmological constant in a toy model of the five-dimensional
universe is calculated in such a manner that it remains in agreement with both
astronomical observations and the quantum field theory concerning the
zero-point fluctuations of the vacuum. The (negative) cosmological constant is
equal to the inverse of the Planck length squared, which means that in the toy
model the vanishing of the observed value of the cosmological constant is a
consequence of the existence of an energy cutoff exactly at the level of the
Planck scale. In turn, a model for both a virtual and a real
particle-antiparticle pair is proposed which describes properly some energetic
properties of both the vacuum fluctuations and created particles, as well as it
allows one to calculate the discrete "bare" values of an elementary-particle
mass, electric charge and intrinsic angular momentum (spin) at the energy
cutoff. The relationships between the discussed model and some phenomena such
as the Zitterbewegung and the Unruh-Davies effect are briefly analyzed, too.
The proposed model also allows one to derive the Lorentz transformation and the
Maxwell equations while considering the properties of the vacuum filled with
the sea of virtual particles and their antiparticles. Finally, the existence of
a finite value of the vacuum-energy density resulting from the toy model leads
us to the formulation of dimensionless Einstein field equations which can be
derived from the Lagrangian with a dimensionless (naively renormalized)
coupling constant.
