| The microscopic theory of superconductivity by Bardeen, Cooper and Schrieffer (BCS) is one of the most successful theories in condensed matter physics. Bogoliubov gave a very elegant mathematical description of BCS theory from the point of view of the quasi-particle picture. In this picture, quasi-particles are not eigenstates of charge and particle number. This, however, creates theoretical difficulties. For example, the original derivation of the Meissner effect is not gauge-invariant. This became more obvious in Bogoliubov's scheme. In 1960, Nambu applied ideas and techniques known in quantum electrodynamics to BCS theory. He considered gauge-invariant diagrammatic corrections to the electrodynamic interacting vertex function. This vertex function must be consistent with the self-energy of quasi-particles. Thus the Ward identity (with a generalized form in the 2x2 Nambu space) is obtained between vertex functions and the self-energy. In this way, the Meissner effect calculation is rendered strictly gauge-invariant, and essentially keeps the BCS result unaltered for transverse external fields.;This thesis addresses how Nambu's brilliant ideas can be generalized to the rho -- J response (charge) theory of BCS-BEC crossover at any temperature, by including pairing fluctuations and pseudogap effects. BCS-BEC crossover is a very important topic in ultracold Fermi gases. Also considered is the spin counterpart, via the nS - JS response. In this thesis, we choose a proper set of correcting diagrams to the electrodynamic and spin-magnetic field interacting vertex functions respectively, and include effects of collective modes due to symmetry breaking in the charge channel. Finally we obtain gauge invariant charge and spin response functions which can be compared with experiment. Importantly, the sum-rules derived from conservation laws for charge and spin provide stringent consistency checks on our approximation schemes.;In particular, here we derive the density and spin structure factors, which can be measured by Bragg two photon scattering experiments, using our "conserving'' approximations for rho -- J and nS - JS response theories; this study of the density response functions required some numerical techniques. An important finding was our observation that there is a way to use these Bragg experiments to establish the presence of coherence in neutral, atomic Fermi superfluids. Once we understood the experimental implications of the density channel we turned to the experimental implications for the current component. Here the experimental implications involve studies of the viscosities (for a neutral system) and conductivity (for a charged system). Studies of the viscosity in ultracold Fermi gases are highly topical and recently discussed in Physics Today. For viscosities, our microscopic approach was based on the Kubo formalism. Our numerical results appear consistent with experiments.;For conductivity, there is a great amount of interest from the high temperature superconductivity community. Our theory shows the close relationship between viscosity and o → 0 conductivity. We argue there are commonalities between ultracold Fermi gases and high temperature superconductors. Interestingly, the ultracold gases are known as "perfect fluids'', while the high Tc superconductors as known as "bad metals''. In this thesis we address the relation between the two. |
