By Javier Sanchez Mondragon, Kurt B. Wolf
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44) Theory of the Ground State The principle of the compensation of dangerous diagrams leads to the vanishing of terms contributing to the creation of pairs with momenta q Φ 0. Hence v(p,a)u(q-p) = 0. 39) give the trivial solution for the functions {u, v}. Therefore, a general canonical transformation has been proposed which allows for compensated terms leading to the creation of pairs with q Φ 0 in a non-trivial way. lib) takes the form Ω(ν,μ) = Ω(ρ)δ(ν-μ) where ß(p)-/C2W+C2W. 37). 46). If E(p) x E(p), then ζ(ρ) is the energy of the particles on a scale where the Fermi energy is zero.
58). In this case, for the H—F solution eqns. U ~σ\Ρ2> -σ)+ίλο(Ρ3)+λο(Ρ4)]δΕ(ρ3, σ\ρ4, σ) P\>P2 = EF)ÔF(p3,a\p4,a). 90) On the other hand, for the generalized solution eqns. 91c) = Ε$δΦ(ρ3,σ\ρ49-σ)9 [λα(ρ*)+λα(Ρ4)]άΦ(Ρ3, * Ι Α , <*) = Ε$(σ, σ)δΦ(ρ39 σ|ρ4, σ). 91a-d) we can omit the spin dependence, since there is no preferred direction. 9 lc) it is best to retain the spin indices since the functions Φ are antisymmetric in f=(p, a). 59) which follows from the equation of compensation of dangerous diagrams for pairs of quasi-particles with opposite momenta.
The mathematical formulation of this fact is, in the variant proposed by Bogoliubov, a special canonical transformation. By means of this transformation we switch from originally considered Fermi particles (electrons) to Fermi quasi-particles. A special Bogoliubov transformation connects fermions with opposite spins and momenta. Correspondingly, instead of the particle vacuum we obtain the quasi-particle vacuum. From the "point of view" of particles, the quasi-particle vacuum is a ground state for correlated pairs considered in the BCS theory.
