The time-dependent Hartree-Fock-Bogoliubov equations for Bosons
  In this article, we use quasifree reduction to derive the time-dependent Hartree-Fock-Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate in $\mathbb R^d$. We prove global well-posedness for the HFB equations for sufficiently regular pair interaction potentials, and establish key conservation laws. Moreover, we show that the solutions to the HFB equations exhibit a symplectic structure, and have a form reminiscent of a Hamiltonian system. In particular, this is used to relate the HFB equations to the HFB eigenvalue equations encountered in the physics literature. Furthermore, we construct the Gibbs states at positive temperature associated with the HFB equations, and establish criteria for the emergence of Bose-Einstein condensation. 
Volker Bach, S\'ebastien Breteaux, Thomas Chen, J\"urg Fr\"ohlich,   Israel Michael Sigal
math-ph math.AP math.MP