Casimir forces in transmission-line circuits: QED and fluctuation-dissipation formalisms

It was recently shown that transmission-line waveguides can mediate long-range fluctuation forces between neutral objects, potentially leading to novel Casimir forces in electric circuits. Here we present two approaches for the general description of these forces between electric components embedded in transmission-line circuits. The first, following ordinary quantum electrodynamics (QED), consists of the quantization and scattering theory of voltage and current waves inside transmission lines. The second approach relies on a simple circuit analysis with additional noisy current sources due to resistors in the circuit, as per the fluctuation-dissipation theorem (FDT). We apply the latter approach to derive a general formula for the Casimir force induced by circuit fluctuations between any two impedances. The application of this formula, considering the sign of the resulting force, is discussed. While both QED and FDT approaches are equivalent, we conclude that the latter is simpler to generalize and solve.