On the Phonon-Phonon Interaction in a System of Bose Particles

Abstract

127The definite expression of the phonon-phonon interaction in a Bose $ystem is derived up to the order of N-1 in terms of the density operator and its exact canonical conjugate which was introduced in the previous papers. We find some mathematical ambiguity in the course of calculation, and clarify the reason why the different results were obtained so far. The validity of our formalism is discussed from the physical point of view.

Velocity operator approach to a Fermion system

“…which already appeared in the preceding papers [4,5] and is of the same form as the Bogoliubov transformation for the usual Bosons [6] except the introduction of the Grassmann numbers θ and θ. The above kind of the diagonalization (3.27) has also been given by Sunakawa in the Boson system [11].…”

“…k ′ ] = 0. He transformed the quantum-fluid Hamiltonian (3.24) to one in configuration space and obtained the classical-fluid Hamiltonian for the case of irrotational flow [7,11]. From this view point, very recently, we have developed a quantum-fluid approach to vortex motion in nuclei [14].…”

Exact canonically conjugate momenta approach to a one-dimensional neutron–proton system, I

“…given in [34]. In (6.8), we must emphasize that the term k 2 k 2 2m is interpreted as a kinetic energy in the state of perfect degeneracy if the term dx m 2 ∂ x Π(x)• ρ(x)•∂ x Π(x) is regarded as the "excitation kinetic energy" in the sense of Tomonaga [20].…”

“…In this context, a new field of exploration of elementary excitations of a one-dimensional Fermi system, which is intended to be presented elsewhere, may be opened. Finally, we emphasize the possibility of extending the present exact canonical momenta approach to the three-dimensional case which would lead to the isospin T = 0 quantum hydrodynamics, by introducing the velocity operator v k instead of canonical conjugate momentum Π k as has been done by Sunakawa [33,34]. …”

“…which leads to the following coordinate representation of the collective field Hamiltonian: given in [34]. In (6.8), we must emphasize that the term k…”