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 rote usually played by the a and a† operators; therefore, breaking the symmetry of particles conservation is not necessary.