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 t̂a), i.e., t̂a=(ρ̂-ρ̂2)12P̂a, shows that HB theory has only two variational (trial) operators: the density operator ρ̂ and the antilinear pairing operator P̂a which is defined by the properties P̂a+=P̂a+1=-P̂a. 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 P̂a 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: A[ĥ,t̂a]+-[Δ̂a,ρ̂-12]+=0,B[ĥ,ρ̂] - [Δ̂a,t̂a]-=0, where Δa is the pairing potential and ĥ is the one-particle Hamiltonian. Two identities are found between Âa and B̂ which turn out to be very useful for obtaining solutions. Symmetries of the trial operators and of the solutions are discussed in great detail, special attention being paid to real HB solutions and their connection with some antiunitary symmetries. Several simple solutions are analyzed: (1) the case where ρ̂ is restricted to be a projector, i.e., t̂a=0 (HF case); (2) the case where ρ̂ and P̂a are restricted to commute with a complete set of observables, which determines the eigensubspaces of ρ̂, and which, in particular cases of rotational and translational symmetries, fixes P̂a to be equal to the time-reversal operator (BCS case); and (3) the case where P̂a is any given symmetry operator of the Hamiltonian. © 1968 The American Physical Society.
