  The Casimir-Lifhitz force acts between neutral material bodies and is due to
the fluctuations (around zero) of the electrical polarizations of the bodies.
This force is a macroscopic manifestation of the van der Waals forces between
atoms and molecules. In addition to being of fundamental interest, the
Casimir-Lifshitz force plays an important role in surface physics,
nanotechnology and biophysics. There are two different approaches in the theory
of this force. One is centered on the fluctuations inside the bodies, as the
source of the fluctuational electromagnetic fields and forces. The second
approach is based on finding the eigenmodes of the field, while the material
bodies are assumed to be passive and non-fluctuating. In spite of the fact that
both approaches have a long history, there are still some misconceptions in the
literature. In particular, there are claims that (hypothetical) materials with
a strictly real dielectric function $\varepsilon(\omega)$ can give rise to
fluctuational Casimir-Lifshitz forces. We review and compare the two
approaches, using the simple example of the force in the absence of
retardation. We point out that also in the second (the "field-oriented")
approach one cannot avoid introducing an infinitesimal imaginary part into the
dielectric function, i.e. introducing some dissipation. Furthermore, we
emphasize that the requirement of analyticity of $ \varepsilon(\omega)$ in the
upper half of the complex $\omega$ plane is not the only one for a viable
dielectric function. There are other requirements as well. In particular,
models that use a strictly real $\varepsilon(\omega)$ (for all real positive
$\omega)$ are inadmissible and lead to various contradictions and
inconsistencies. Specifically, we present a critical discussion of the
"dissipation-less plasma model".
