  The thermal field theory is applied to fermionic superfluids by doubling the
degrees of freedom of the BCS theory. We construct the two-mode states and the
corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The
expectation values with respect to the BCS thermal vacuum produce the
statistical average of the thermodynamic quantities. The BCS thermal vacuum
allows a quantum-mechanical perturbation theory with the BCS theory serving as
the unperturbed state. We evaluate the leading-order corrections to the order
parameter and other physical quantities from the perturbation theory. A direct
evaluation of the pairing correlation as a function of temperature shows the
pseudogap phenomenon results from the perturbation theory. The BCS thermal
vacuum is shown to be a generalized coherent and squeezed state. The
correspondence between the thermal vacuum and purification of the density
matrix allows a unitary transformation, and we found the geometric phase in the
parameter space associated with the transformation.
