A Collective Coordinate Method in the Many-Body Problem

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 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.

Physical ReviewFeb 1, 195616 citations

An Approximate Method for Solving the Quantum Many-Body Problem

CRC Press eBooksMay 21, 200419 citations

The Treatment of Collective Coordinates in Many-Body Systems

Daniel R. Bes, Jorge Kurchan B

These lecture notes deal with the problem of collective coordinates in many-body systems, which are treated as gauge systems in (0+1) dimensions. The resulting classical Dirac brackets are discussed, as well as the structure of the quantal space of wavefunctions. Emphasis is made on the application of the BRST formalism. Several systems displaying an approximate breakdown of symmetries are treated. Many-body physicists may find attractive both a rigorous formulation of the problem of collective coordinates and being introduced to the BRST formalism, which has become a fundamental tool in gauge theories. Field-theorists may find illuminating the applications to simpler mechanical examples. The notes are self-contained. In particular, they do not require a previous knowledge of either the BRST formalism or of collective transformations.These lecture notes deal with the problem of collective coordinates in many-body systems, which are treated as gauge systems in (0+1) dimensions. The resulting classical Dirac brackets are discussed, as well as the structure of the quantal space of wavefunctions. Emphasis is made on the application of the BRST formalism. Several systems displaying an approximate breakdown of symmetries are treated. Many-body physicists may find attractive both a rigorous formulation of the problem of collective coordinates and being introduced to the BRST formalism, which has become a fundamental tool in gauge theories. Field-theorists may find illuminating the applications to simpler mechanical examples. The notes are self-contained. In particular, they do not require a previous knowledge of either the BRST formalism or of collective transformations.

World Scientific lecture notes in physicsJul 1, 199060 citations

Large-amplitude collective motion in the Lipkin model

Eugene R. Marshalek

It is shown how a simple reinterpretation of the "transition-operator boson method" eliminates the difficulty noted by Okubo. The revised method is applied to the problem of anharmonic motion in the Lipkin model. The occurrence of a non-positive-definite mass parameter, as obtained by Holzwarth and by Giraud and Grammaticos, is then exposed as the easily avoidable consequence of an inauspicious choice of collective coordinate.NUCLEAR STRUCTURE Approximate many-body method applied to a solvable model.It is shown how a simple reinterpretation of the "transition-operator boson method" eliminates the difficulty noted by Okubo. The revised method is applied to the problem of anharmonic motion in the Lipkin model. The occurrence of a non-positive-definite mass parameter, as obtained by Holzwarth and by Giraud and Grammaticos, is then exposed as the easily avoidable consequence of an inauspicious choice of collective coordinate.NUCLEAR STRUCTURE Approximate many-body method applied to a solvable model.

Physical Review CJul 1, 19762 citations

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.