Collective Excitations in the Theory of Superconductivity

The complex dielectric constant of a superconductor and the Meissner effect are derived in a manner which is gauge invariant, from the theory of superconductivity due to Bardeen, Cooper, and Schrieffer. The collective excitations are important in maintaining gauge invariance; the longitudinal collective excitations ensure that a static vector potential produces no longitudinal current and the transverse collective excitations contribute to the Meissner current an amount which depends on the angular properties of the two-body interaction. This contribution is estimated to be small. An earlier calculation of ultrasonic absorption in superconductors is justified. The whole investigation is based upon the generalized random-phase approximation introduced by Anderson and applies whether or not the Coulomb interaction between the electrons is taken into account. The equations of motion are linearized in such a way that the exchange terms are automatically screened if the Coulomb interaction is, in fact, taken into account. The region of applicability of most of the results is limited by the approximations to temperatures at or near absolute zero.