A system of coordinates for quantum many body problems

Jerome K. Percus, George J. Yevick

Valdir C. Aguilera-Navarro

Érik Amorim

In the context of classical or quantum many-body problems involving identical bodies, a linear change of coordinates can be constructed with the properties that it includes the center-of-mass as one of the new coordinates and preserves the inherent permutation symmetry of both the Hamiltonian and the admissible states. This has advantages over the usual system of Jacobi coordinates in the study of many-body problems for which permutation symmetry of the bodies plays an important role. This paper contains the details of the construction of this system and the proof that these properties uniquely determine it up to trivial modifications. Examples of applications to both classical and quantum problems are explored, including a generalization to problems involving groups of different species of bodies.

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.

Israel Michael Sigal