Bogoliubov Transformation for Distinguishable Particles

The Bogoliubov transformation is generally derived in the context of identical bosons with the use of " second quantized " a and a † operators (or, equivalently, in field theory). Here, we show that the transformation , together with its characteristic energy spectrum, can also be derived within the Hilbert space of distinguishable particles, obeying Boltzmann statistics; in this derivation, ordinary dyadic operators play the role usually played by the a and a † operators; therefore, breaking the symmetry of particle conservation is not necessary. The Bogoliubov transformation [1][2] is an essential tool in the theory of Bose-Einstein condensation of identical bosons 1. It modifies the quadratic energy spectrum of free particles into a quasiparticle spectrum which includes a linear variation for small momenta; this feature is generally associated with the existence of phonons and, since it introduces a non zero minimum value for the ratio between the energy and momentum, it allows a natural derivation of the notion of critical velocity (a maximum velocity for the system to remain superfluid). Usually, the mathematics of the Bogoliubov transformation is performed within the formalism of creation and annihilation operators (often called " second quantization " for historical reasons); assuming that the system is entirely made of identical particles, one then uses a formalism which automatically ensures a full symmetriza-tion of the state vector. Nevertheless, the notion of phonons has a much broader scope in physics than just identical quantum particles; it is even often discussed in the context of classical systems, solids or even fluids. One can therefore wonder whether it is possible to re-derive the Bogoliubov spectrum in a context where the particles are considered as distinguishable and where, as a consequence, the effect of exchange operators remains completely 1 Historically, the first introduction of the mathematical transformation seems to be the work of Holstein and Primakoff in 1940 [3], in the context of magnetic systems.