Centers of mass and rotational kinematics for the relativistic N-body problem in the rest-frame instant form
Abstract
A relativistic kinematics for the N-body problem which solves all the problems raised until now on this topic is constructed by exploting the Wigner-covariant rest-frame instant form of dynamics in the context of parametrized Minkowski theories. The Wigner hyperplanes, orthogonal to the total timelike four-momentum of any N-body configuration, define the intrinsic rest frame and realize the separation of the center-of-mass motion. The point chosen as origin in each Wigner hyperplane can be made to coincide with the covariant noncanonical Fokker-Pryce center of inertia. As is well known, the latter is distinct from the canonical pseudo- four-vector describing the decoupled motion of the center of mass (which possess the same Euclidean covariance as the quantum Newton-Wigner three-position operator) and from the noncanonical pseudo-four-vector known as Møller's center of energy. Our approach leads to the splitting of the notion of center of mass into an external one, defined in terms of an external Poincaré group realization, and an internal one defined in terms of an internal unfaithful realization of the group inside the Wigner hyperplane. Because of the first class constraints defining the rest frame (vanishing of the internal three-momentum), the latter three internal concepts of center of mass weakly coincide. The resulting unique internal center of mass is thereby a gauge variable which, by a suitable gauge fixing, can be localized at the origin of the Wigner hyperplane. An adapted canonical basis of relative variables is constructed by means of the classical counterpart of the Gartenhaus-Schwartz transformation. The invariant mass of the N-body configuration is the Hamiltonian for the relative motions. Within this general framework, the rotational kinematics can be developed in terms of the same dynamical body frames, orientation-shape variables, spin frame, and canonical spin bases already introduced in the case of the nonrelativistic N-body problem.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- April 2002
- DOI:
- 10.1063/1.1435424
- arXiv:
- arXiv:hep-th/0102087
- Bibcode:
- 2002JMP....43.1677A
- Keywords:
-
- 03.30.+p;
- 05.30.-d;
- Special relativity;
- Quantum statistical mechanics;
- High Energy Physics - Theory;
- Astrophysics;
- General Relativity and Quantum Cosmology;
- Nuclear Theory
- E-Print:
- 78 pages, revtex file