  Systems of interacting charges and fields are ubiquitous in physics.
Recently, it has been shown that Hamiltonians derived using different gauges
can yield different physical results when matter degrees of freedom are
truncated to a few low-lying energy eigenstates. This effect is particularly
prominent in the ultra-strong coupling regime. Such ambiguities arise because
transformations reshuffle the partition between light and matter degrees of
freedom and so level truncation is a gauge dependent approximation. To avoid
this gauge ambiguity, we redefine the electromagnetic fields in terms of
potentials for which the resulting canonical momenta and Hamiltonian are
explicitly unchanged by the gauge choice of this theory. Instead the
light/matter partition is assigned by the intuitive choice of separating an
electric field between displacement and polarisation contributions. This
approach is an attractive choice in typical cavity quantum electrodynamics
situations.
