  We make a conceptual overview of a particular approach to the initial-value
problem in canonical gauge theories. We stress how the first-class phase-space
constraints may be relaxed if we interpret them as fixing the values of new
degrees of freedom. This idea goes back to Fock and Stueckelberg, leading to
restrictions of the gauge symmetry of a theory, and it corresponds, in certain
cases, to promoting constants of Nature to physical fields. Recently, different
versions of this formulation have gained considerable attention in the
literature, with several independent iterations, particularly in classical and
quantum descriptions of gravity, cosmology, and electromagnetism. In
particular, in the case of canonical quantum gravity, the Fock--Stueckelberg
approach is relevant to the so-called problem of time. Our overview recalls the
work of Fock and Stueckelberg and its physical interpretation with the aim of
conceptually unifying the different iterations of the idea that appear in the
literature and of motivating further research.
