Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the NonRelativistic N-Body Problem (original) (raw)

Spinning particles, coadjoint orbits and Hamiltonian formalism

Nuclear Physics B, 2022

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

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

2002

In the Wigner-covariant rest-frame instant form of dynamics it is possible to develop a relativistic kinematics for the N-body problem which solves all the problems raised till now on this topic. The Wigner hyperplanes, orthogonal to the total timelike 4-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 of each Wigner hyperplane can be made to coincide with the covariant non-canonical Fokker-Pryce center of inertia. This is distinct from the canonical pseudo-vector describing the decoupled motion of the center of mass (having the same Euclidean covariance as the quantum Newton-Wigner 3-position operator) and the non-canonical pseudo-vector for the Møller center of energy. These are the only external notions of relativistic center of mass, definable only in terms of the external Poincaré group realization. Inside the Wigner hyperplane, an internal unfaithful realization of the Poincaré group is defined while the analogous three concepts of center of mass weakly coincide due to the first class constraints defining the rest frame (vanishing of the internal 3-momentum). This unique internal center of mass is consequently a gauge variable which, through a gauge fixing, can be localized atthe origin of the Wigner hyperplane. An adapted canonical basis of relative variables is found 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. In this framework we can introduce the same dynamical body frames, orientation-shape variables, spin frame and canonical spin bases for the rotational kinematics developed for the non-relativistic N-body problem.

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

2005

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.

Rotating frames and gauge invariance in three-dimensional many-body quantum systems

Journal of Physics A: Mathematical and General, 2004

We study the quantization of many-body systems in three dimensions in rotating coordinate frames using a gauge invariant formulation of the dynamics. We consider reference frames defined by linear gauge conditions, and discuss their Gribov ambiguities and commutator algebra. We construct the momentum operators, inner-product and Hamiltonian in those gauges, for systems with and without translation invariance. The analogy with the quantization of non-Abelian Yang-Mills theories in noncovariant gauges is emphasized. Our results are applied to quasi-rigid systems in the Eckart frame.

Dirac observables and spin bases for N relativistic particles

Journal of Physics A: Mathematical and General, 1998

The construction of the Dirac observables in the P 2 > 0 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 Poincar e group action. The explicit form of the Dirac observables is derived by exploiting an internal Euclidean group having the Poincar e canonical spin as generator of rotations. This procedure provides the symplectic version of the conventional angular momentum composition. This work has been supported partly by I.N.F.N., Italy Iniziativa speci ca FI-41, and partly by the Network Constrained Dynamical Systems of the E.U. Programme`Human Capital and Mobility'.

Relativistic dynamics for N-body systems

Physics Letters B, 1988

N spin-0 particle systems are described by means of N manifestly covariant relativistic wave equations. These have the following properties: The relative time evolution laws are determined in a kinematic way. The dynamics is governed by a single 3 N-~dimensional wave equation. An approximate form of separability is satisfied.

On the equations of motion for particles with arbitrary spin in nonrelativistic mechanics

1975

It is well known that the electron motion in the external electromagnetic field is described by the relativistic Dirac equation. In this case, in the Foldy-Wouthuysen representation, the Hamiltonian includes the terms corresponding to the interaction of the particle magnetic moment with a magnetic field (∼ (1/m)(σH)) and the terms which are interpreted as a spin-orbit coupling (∼ (σ/m 2 ){(p − eA) × E)). Apart from these constituents the Hamiltonian includes the Darwin term (∼ (1/m 2 ) div E) .

Relativistic N-body problem in a separable two-body basis

Physical Review C, 2001

We use Dirac's constraint dynamics to obtain a Hamiltonian formulation of the relativistic N -body problem in a separable two-body basis in which the particles interact pair-wise through scalar and vector interactions. The resultant N -body Hamiltonian is relativistically covariant. It can be easily separated in terms of the center-of-mass and the relative motion of any twobody subsystem. It can also be separated into an unperturbed Hamiltonian with a residual interaction. In a system of two-body composite particles, the solutions of the unperturbed Hamiltonian are relativistic two-body internal states, each of which can be obtained by solving a relativistic Schrödingerlike equation. The resultant two-body wave functions can be used as basis states to evaluate reaction matrix elements in the general N -body problem.

Equations of motion for particles of arbitrary spin invariant under the Galileo group

Theoretical and Mathematical Physics, 1980

Systems of differential equations of first and second order are derived that are invariant under the Galileo group and describe the motion of a particle with arbitrary spin. These equations admit a Lagrangian formulation and describe the dipole, spin-orbit, and Darwin couplings of the particle to an external electromagnetic field; traditionally, these have been regarded as purely relativistic effects. Examples are given of infinite-component equations that are invariant under the Galileo group. The problem of the motion of a nonrelativistic particle with spin s = ~ in a homogeneous magnetic field is solved exactly.