We know that the Nevanlinna (Herglotz) functions arise in physics in a fundamental way as an alternative representation of causality. We have discovered recently that they also relevant to the amplification phenomenon occurring in traveling wave tubes (TWTs). The Nevanlinna (Herglotz) functions represent there the so-called slow wave structure - a significant a component of the TWT. Though an ideal TWT system is perfectly conservative its energy functional turns out to be not positive definite. This explains the existence of exponentially growing in space eigenfunctions manifesting the amplification. The corresponding theory is intimately related to the Krein spectral theory.