A generalized particle system interacting with a massless Bose field is investigated. We assume regularity conditions for the commutation relations of the interaction and annihilation operators. It is proven that if the ground state exists, its multiplicity is finite.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.
A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.
In this paper we consider an interacting Bose gas at zero temperature, constrained to a finite box and in the mean field limiting regime. The N gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. The (nonzero) Fourier components of the potential are assumed to be sufficiently large with respect to the corresponding kinetic energies of the modes like in the companion papers [Pi1]-[Pi2]. Using the multi-scale technique in the occupation numbers of particle states introduced in [Pi1]-[Pi2], we provide a convergent expansion of the ground state of the Hamiltonian in terms of the bare operators. In the limit N \to \infty the expansion is up to any desired precision.