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Progress of Theoretical Physics

Collective Representation of the Many-Particle System Interacting through Arbitrary Two-Body Potential

Naomi Shohno

Published Nov 1, 1960• 1 citation

Abstract

We develope a collective description of the many-particle system interacting through an arbitrary two-body potential. The description is given both in terms of the second quantized form and in terms of an auxiliary field analogous to the longitudinal electromagnetic field. As Bohm-Pines theory of the collective description has some unsatisfactory features concerning the subsidiary condition imposed on the auxiliary field, we propose here a strict method of treatment on the subsidiary condition. To obtain the useful result of the collective representation, we have to take the three steps of the canonical transformations. Though the first two transformations are the same as those of B-P, the third transformation, which is not applied by B-P and is characteristic of our theory, is necessary for obtaining a meaningful result. By using both the strict subsidiary condition and the three canonical transformations, the collective field coordinate and the particle coordinate are separated not only in the total Hamiltonian but also in the subsidiary condition. Moreover, we are able to clarify the correct meaning of the subsidiary condition and clarify also the connections between our theory and the theory of Sawada-Brueckner-Fukuda-Brout.

DOI: 10.1143/ptp.24.991

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We develope a collective description of the many-particle system interacting through an arbitrary two-body potential. The description is given both in terms of the second quantized form and in terms of an auxiliary field analogous to the longitudinal electromagnetic field. As Bohm-Pines theory of the collective description has some unsatisfactory features concerning the subsidiary condition imposed on the auxiliary field, we propose here a strict method of treatment on the subsidiary condition. To obtain the useful result of the collective representation, we have to take the three steps of the canonical transformations. Though the first two transformations are the same as those of B-P, the third transformation, which is not applied by B-P and is characteristic of our theory, is necessary for obtaining a meaningful result. By using both the strict subsidiary condition and the three canonical transformations, the collective field coordinate and the particle coordinate are separated not only in the total Hamiltonian but also in the subsidiary condition. Moreover, we are able to clarify the correct meaning of the subsidiary condition and clarify also the connections between our theory and the theory of Sawada-Brueckner-Fukuda-Brout.

An elementary theory is developed of a classical treatment with the ‘Collective Motion’ of many-particle systems. The fundamental concept of collective motion can be reduced into a certain transformation property of Hamiltonian. Regarded as the dynamical variables, the continuous parameters of such a transformation describe the collective motion of the system, while the individual motions are partly suppressed by the same numbers of subsidiary conditions, or constraints, logically derived in, as the numbers of transformation parameters. Then the general procedure of formulation is illustrated; as an example, the incompressible motion with three modes can be systematically formulated. As another example, the unified treatment is performed also with the vortex motion. The digression in the approximating procedure is also presumed.

A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the

S

matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.

The dynamical behavior of

N

interacting particles may be simplified by the introduction of

3N

collective coordinates. A geometric interpretation of collective coordinates is described which is analogous to a type of random walk problem. Considerations on the validity of the transformation from ordinary spatial coordinates to symmetrical collective coordinates are discussed. The difficult problem of the boundary of collective coordinate space is touched upon.In the second half of the paper, the Lagrangian formulation for the dynamical problem is carried out by means of collective coordinates. Conditions for the equivalence of the physical Lagrangian and the collective coordinate Lagrangian are established. This leads to the problem of the Fourier representation of a potential by a finite number of terms. Finally, preliminary remarks on a modified Dirac $\ensuremath{\delta}$ function are presented.

The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.