Scarf for Lifshitz
University of Alberta, T6G 2E1, Edmonton, Canada
Accepted: 30 June 2021
Published online: 18 July 2021
Polarization of dispersive and dissipative dielectric media with smoothed-out inhomogeneities is studied with the goal to clarify the question of renormalizability of electromagnetic stress–energy tensor. The stress tensor is computed with the Lifshitz approach to van der Waals forces in the non-retarded limit, which accounts for dominant effects at the distances from the interface shorter than the absorption wavelength. After the substraction of the leading free space ultraviolet divergencies, there still remain two types of divergencies. First, contributions diverging in the sharp interface case become finite once it is smoothed out. Second, new subleading ultraviolet cut-off-dependent contributions appear due to the interface internal structure. The Hadamard expansion, based on the heat kernel method, is applied to systematically single out both finite and subleading contributions and to demonstrate incomplete renormalizability of the Lifshitz theory. The above approach also allows us to reveal the purely quantum mechanical nature of surface tension, which consists of finite cut-off-independent as well as cut-off-dependent contributions. The deduced theory of surface tension and its calculations for real dielectric media are favourably compared to the available experimental data. The problem of surface tension proves to be of a distinguished limit type because the sharp interface formulation loses the critical information about the internal structure of an interface. The general theory offered here is illustrated with an exactly solvable model representing a smooth transition between two different dielectric media, which relies upon a solution of the Schrödinger equation with the Scarf potential.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021