  Non-canonical degrees of freedom provide one of the most promising routes
towards characterising a range of important phenomena in condensed matter
physics. Potential candidates include the pseudogap regime of the cuprates,
heavy-fermion behaviour, and also indeed magnetically ordered systems.
Nevertheless it remains an open question whether non-canonical algebras can in
fact provide legitimate quantum degrees of freedom. In this manuscript we
survey progress made on this topic, complementing distinct approaches so as to
obtain a unified description. In particular we obtain a novel closed-form
expression for a self-energy-like object for non-canonical degrees of freedom.
We further make a resummation of density correlations to obtain analogues of
the RPA and GW approximations commonly employed for canonical degrees of
freedom. We discuss difficulties related to generating higher-order
approximations which are consistent with conservation laws, which represents an
outstanding issue. We also discuss how the interplay between canonical and
non-canonical degrees of freedom offers a useful paradigm for organising the
phase diagram of correlated electronic behaviour.
