Functional integrals for Hartree-Fock-Bogoliubov many-body matrix elements

We discuss exact functional integral expressions for many-body matrix elements of the type, left angle bracketψA vertical bar e-βH-circumflex vertical bar ψ, A right angle bracket, between particle number projected Hartree-Fock-Bogoliubov (HFB) wavefunctions with time-reversal symmetry. A proof of positivity is given for a class of Hamiltonians and when the HFB wavefunctions with time-reversal symmetry are particle number projected to an even number of particles. We show explicitly how to reduce the propagator in the functional integral to a Hermitian positive definite propagator for particle pairs. This result generalizes that previously obtained using Bardeen-Cooper-Schrieffer wavefunctions. (author)