  Multiple bases are presented for the conclusion that potentials are
fundamental in electrodynamics, with electric and magnetic fields as quantities
auxiliary to the scalar and vector potentials -- opposite to the conventional
ordering. One foundation for the concept of basic potentials and auxiliary
fields consists of examples where two sets of gauge-related fields are such
that one is physical and the other is erroneous, with the information for the
proper choice supplied by the potentials. A major consequence is that a change
of gauge is not a unitary transformation in quantum mechanics; a principle
heretofore unchallenged. The primacy of potentials over fields leads to the
concept of a hierarchy of physical quantities, where potentials and energies
are primary, while fields and forces are secondary. Secondary quantities
provide less information than do primary quantities. Some criteria by which
strong laser fields are judged are based on secondary quantities, making it
possible to arrive at inappropriate conclusions. This is exemplified by several
field-related misconceptions as diverse as the behavior of charged particles in
very low frequency propagating fields, and the fundamental problem of pair
production at very high intensities. In each case, an approach based on
potentials gives appropriate results, free of ambiguities. The examples
encompass classical and quantum phenomena, in relativistic and nonrelativistic
conditions. This is a major extension of the quantum-only Aharonov-Bohm effect,
both in supporting the primacy of potentials over fields, and also in showing
how field-based conceptions can lead to errors in basic applications.
