Utilizing group property of Bogoliubov transformation

We revisit the Bogoliubov transformation as a representation of the group of unitary operators of Balian and Brezin. We show that the group property is best utilized when we treat successive transformations of quasiparticles and their vacuum at the same time. In particular, we establish a one-to-one correspondence between sets of quasiparticle operators and their vacua using the group property. The correspondence determines the quasiparticle vacuum uniquely including the phase, which is inevitable in treating probability amplitudes in a consistent fashion.