Homogeneous spaces in Hartree-Fock-Bogoliubov theory (Q6611188)

The article under review studies the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree-Fock-Bogoliubov theory. For this the authors first exhibit the structure of the group \(U_{\text{Bog}}\) of Bogoliubov transformations satisfying the Shale-Stinespring condition as a Banach Lie group. This infinite-dimensional Lie group structure arises by interpreting the group \(U_{\text{Bog}}\) as a subgroup of a Lie group in a certain class of restricted Lie groups à la [\textit{K.-H. Neeb}, Banach Cent. Publ. 55, 87--151 (2002; Zbl 1010.22024)].\N\NNow the authors investigate the Lie group action (by conjugation) of \(U_{\text{Bog}}\) on the set of all admissible generalized one-particle density matrices \(\mathcal{D}\). The orbits of this action have previously been investigated from a physics perspective (assuming that they could be modelled as finite-dimensional manifolds, see [\textit{G. Rosensteel}, ``Hartree-Fock-Bogoliubov theory without quasiparticle vacua'', Phys. Rev. A 23, No. 6, 2794--2801 (1981)]). The article under review shows that the orbits of this action are reductive homogeneous spaces. Among the results of the article, several equivalences that characterize when the orbits are embedded submanifolds of natural ambient spaces are established. In addition, Lie theoretic arguments are shown to provide the orbits with an invariant symplectic form. It turns out that if the operators in the orbits have finite spectrum, then the orbits are actually Kähler homogeneous spaces.