Quantum Mechanics of a Many-Boson System and the Representation of Canonical Variables

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Abstract
We present a method of treating the assembly of interacting bosons under Bose‐Einstein condensation. Without applying the Bogoliubov approximation in which the creation and the annihilation operators of zero‐momentum particles are replaced by a c number, we keep the quantum nature of these operators—thus the title, ``Quantum Mechanics.'' The method is a quantum mechanical adaptation of the theory of small oscillation. The oscillation means the fluctuation of the number of condensed particles. The interaction between particles determines the stability of this oscillation. When it is stable and its amplitude is not macroscopic, the Bogoliubov approximation is valid. In this way, our method provides a validity criterion for the Bogoliubov approximation as well as an estimation of the errors thereby committed. We have to note that the excitations associated with the fluctuation of condensed particles can never be obtained within that approximation. Our method is applied to the Huang model, the assembly of bosons interacting through a hard core plus weak attractive potential. Having found that, within the physically accessible range of the particle density, the above‐mentioned oscillation is stable, we can conclude that Huang's treatment is well founded. We have discussed the mathematical background of our approximation by invoking the representation theory of canonical variables of an infinitely large system.