https://hal.archives-ouvertes.fr/jpa-00239583/document

Il nuovo cimento

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 Moller’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 Poincare 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.

Centers of mass and rotational kinematics for the relativistic N-body problem in the rest-frame instant form

A modern re-visitation of the consequences of the lack of an intrinsic notion of instantaneous 3-space in relativistic theories leads to a reformulation of their kinematical basis emphasizing the role of non-inertial frames centered on an arbitrary accelerated observer. In special relativity the exigence of predictability implies the adoption of the 3 + 1 point of view, which leads to a well posed initial value problem for field equations in a framework where the change of the convention of synchronization of distant clocks is realized by means of a gauge transformation. This point of view is also at the heart of the canonical approach to metric and tetrad gravity in globally hyperbolic asymptotically flat space-times, where the use of Shanmugadhasan canonical transformations allows the separation of the physical degrees of freedom of the gravitational field (the tidal effects) from the arbitrary gauge variables. Since a global vision of the equivalence principle implies that only global non-inertial frames can exist in general relativity, the gauge variables are naturally interpreted as generalized relativistic inertial effects, which have to be fixed to get a deterministic evolution in a given non-inertial frame. As a consequence, in each Einstein's space-time in this class the whole chrono-geometrical structure, including also the clock synchronization convention, is dynamically determined and a new approach to the Hole Argument leads to the conclusion that "gravitational field" and "space-time" are two faces of the same entity. This view allows to get a classical scenario for the unification of the four interactions in a scheme suited to the description of the solar system or our galaxy with a deparametrization to special relativity and the subsequent possibility to take the non-relativistic limit.

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The Chrono-geometrical structure of special and general relativity: A Re-visitation of canonical geometrodynamics

In non-relativistic mechanics the center of mass of an isolated system is easily separated out from the relative variables. For a N-body system these latter are usually described by a set of Jacobi normal coordinates, based on the clustering of the centers of mass of sub-clusters. The Jacobi variables are then the starting point for separating {\it orientational} variables, connected with the angular momentum constants of motion, from {\it shape} (or {\it vibrational}) variables. Jacobi variables, however, cannot be extended to special relativity. We show by group-theoretical methods that two new sets of relative variables can be defined in terms of a {\it clustering of the angular momenta of sub-clusters} and directly related to the so-called {\it dynamical body frames} and {\it canonical spin bases}. The underlying group-theoretical structure allows a direct extension of such notions from a non-relativistic to a special- relativistic context if one exploits the {\it rest-frame instant form of dynamics}. The various known definitions of relativistic center of mass are recovered. The separation of suitable relative variables from the so-called {\it canonical internal} center of mass leads to the correct kinematical framework for the relativistic theory of the orbits for a N-body system with action -at-a-distance interactions. The rest-frame instant form is also shown to be the correct kinematical framework for introducing the Dixon multi-poles for closed and open N-body systems, as well as for continuous systems, exemplified here by the configurations of the Klein-Gordon field that are compatible with the previous notions of center of mass.

New Directions in Non-Relativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems

After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the nonrelativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of body frame for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian left action. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a spin frame for the N-body system. Then, the existence of appropriate nonsymmetry Hamiltonian right actions for nonrigid systems leads to the construction of an N-dependent discrete number of dynamical body frames for the N-body system, hence to the associated notions of dynamical and measurable orientation and shape variables, angular velocity, rotational and vibrational configurations. For N=3 the dynamical body frame turns out to be unique and our approach reproduces the xxzz gauge of the gauge theory assoc...

Dynamical body frames, orientation-shape variables and canonical spin bases for the nonrelativistic N-body problem

The construction of the Dirac observables in the stratum for a system of N relativistic free particles is carried out on the basis of a quasi-Shanmugadhasan canonical transformation related to the existence of a Poincare group action. The explicit form of the Dirac observables is derived by exploiting an internal Euclidean group having the Poincare canonical spin as generator of rotations. This procedure provides the symplectic version of the conventional angular momentum composition.

A Lucenti

Papers

Dirac observables and spin bases for N relativistic particles