  We aim to give a pedagogical introduction to those elementary aspects of
superconductivity which are not treated in the classic textbooks. In
particular, we emphasize that global U(1) phase rotation symmetry, and not
gauge symmetry, is spontaneously violated, and show that the BCS wave function
is, contrary to claims in the literature, fully gauge invariant. We discuss the
nature of the order parameter, the physical origin of the many degenerate
states, and the relation between formulations of superconductivity with fixed
particle numbers vs. well defined phases. We motivate and to some extend derive
the effective field theory at low temperatures, explore symmetries and
conservation laws, and justify the classical nature of the theory. Most
importantly, we show that the entire phenomenology of superconductivity
essentially follows from the single assumption of a charged order parameter
field. This phenomenology includes Anderson's characteristic equations of
superfluidity, electric and magnetic screening, the Bernoulli Hall effect, the
balance of the Lorentz force, as well as the quantum effects, in which Planck's
constant manifests itself through the compactness of the U(1) phase field. The
latter effects include flux quantization, phase slippage, and the Josephson
effect.
