  The Radio Interferometer Measurement Equation (RIME) is a matrix-based
mathematical model that describes the response of a radio interferometer. The
Jones calculus it employs is not suitable for describing the analogue
components of a telescope. This is because it does not consider the effect of
impedance mismatches between components. This paper aims to highlight the
limitations of Jones calculus, and suggests some alternative methods that are
more applicable. We reformulate the RIME with a different basis that includes
magnetic and mixed coherency statistics. We present a microwave network
inspired 2N-port version of the RIME, and a tensor formalism based upon the
electromagnetic tensor from special relativity. We elucidate the limitations of
the Jones-matrix-based RIME for describing analogue components. We show how
measured scattering parameters of analogue components can be used in a 2N-port
version of the RIME. In addition, we show how motion at relativistic speed
affects the observed flux. We present reformulations of the RIME that correctly
account for magnetic field coherency. These reformulations extend the standard
formulation, highlight its limitations, and may have applications in
space-based interferometry and precise absolute calibration experiments.
