Abstract The structure of the Bogoljubov -Valatin transformation and the HFB state is investigated by group-theoretical methods. A direct proof of the Bloch-Messiah factorization theorem shows that the second factor which contains the BCS part of the BV transformation can be chosen always real. An unitary representation of the Bogoljubov-Valatin group in the Fock space is constructed which allows to write down various equivalent expressions for the HFB state vector. The relation between the "unitary form" and the usual form of | HFB〉 is clarified using the "quasispin" concept in connection with a theorem about the factorization of the D(3) Lie group.
We find a new property of determinants very useful for the computation of the generating function of the recouplings coefficients of the angular momentum. Using the generating function method, we simply transform the treatment of Schwinger's couplings in the bosons basis to the analytic space (Fock space). We find the exact generating function of the 9-j coefficients using the property of the determinants and the Gaussian integral. We also give a simple proof of the celebrated Mehler's formula.
We find a new property of determinants very useful for the computation of the generating function of the recouplings coefficients of the angular momentum. Using the generating function method, we simply transform the treatment of Schwinger's couplings in the bosons basis to the analytic space (Fock space). We find the exact generating function of the 9-j coefficients using the property of the determinants and the Gaussian integral. We also give a simple proof of the celebrated Mehler's formula.