Antilinear Operators in Hartree-Bogolyubov Theory. I

Abstract

Hartree-Bogolyubov (HB) theory is formulated in a basis-independent way, i.e., in terms of linear and antilinear operators acting in the one-particle space. For that purpose, some basic antilinear algebra is presented. The pairing tensor and the pairing potential are shown to represent two antilinear skew-Hermitian operators. The polar factorization of the first of them (the correlation operator ${\stackrel{^}{t}}_{a}$), i.e., ${\stackrel{^}{t}}_{a}={(\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}{\stackrel{^}{\ensuremath{\rho}}}^{2})}^{\frac{1}{2}}{\stackrel{^}{P}}_{a}$, shows that HB theory has only two variational (trial) operators: the density operator $\stackrel{^}{\ensuremath{\rho}}$ and the antilinear pairing operator ${\stackrel{^}{P}}_{a}$ which is defined by the properties ${{\stackrel{^}{P}}_{a}}^{+}={{\stackrel{^}{P}}_{a}}^{\ifmmode\dagger\else\textdagger\fi{}1}=\ensuremath{-}{\stackrel{^}{P}}_{a}$. These two operators commute. The former is the unique and very well-known variational operator of Hartree-Fock (HF) theory, and the latter represents a new variational freedom typical of HB theory. Most calculations, as for instance the Bardeen-Cooper-Schrieffer (BCS) approximation, restrict this freedom by choosing ${\stackrel{^}{P}}_{a}$ to be the time-reversal operator. The basic dynamical (Euler-Lagrange) equations of HB theory are obtained directly by varying linear and antilinear operators. They are expressed in a compact form, using only commutators and anticommutators of the kinematical and the dynamical operators: ${\stackrel{^}{A}}_{a}\ensuremath{\equiv}{[\stackrel{^}{h},{\stackrel{^}{t}}_{a}]}_{+}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},\stackrel{^}{\ensuremath{\rho}}\ensuremath{-}\frac{1}{2}]}_{+}=0, \stackrel{^}{B}\ensuremath{\equiv}{[\stackrel{^}{h},\stackrel{^}{\ensuremath{\rho}}]}_{\ensuremath{-}}\ensuremath{-}{[{\stackrel{^}{\ensuremath{\Delta}}}_{a},{\stackrel{^}{t}}_{a}]}_{\ensuremath{-}}=0,$ where ${\ensuremath{\Delta}}_{a}$ is the pairing potential and $\stackrel{^}{h}$ is the one-particle Hamiltonian.