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Journal of Mathematical Physics

Three-Particle Nonrelativistic Kinematics and Phase Space

Jean-Marc Lévy-Leblond, Monique Lévy-Nahas

Published Oct 1, 1965• 39 citations
Publisher

Abstract

The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.

DOI
10.1063/1.1704696

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quant-pharXiv:2512.18262•Dec 23, 2025
The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (democracy'' arguments), we are led to choose the SU3 group as a particularly convenient great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.
This paper is a group-theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase-space manifold and preserves the phase-space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three-dimensional rotation group. Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.
The ``global method'' for describing N-particle systems (which relies on the existence of a large invariance group of the total Hamiltonian for N noninteracting particles, the ``great group'', whose Lie algebra is generated by the ``grand angular momentum tensor''), is adapted to describe systems of identical particles by means of basis states with simple symmetry properties (with respect to permutations of the particles). We are led to define and study the concept of ``democracy'' among the particles, from which we obtain the ``democratic'' subgroups of the great group. The eigenvectors of a complete set of commuting observables, consisting essentially of Casimir operators of democratic subgroups, may furnish the desired basis. Unfortunately the scheme is seen to be sufficient only in the 3- and 4-particle cases, which, however, are most important. The Appendix contains a discussion of the possible relativistic generalizations of the global method.
The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.
Recent developments in the unification of quantum mechanics and relativity have emphasized the necessity of generalizing classical phase space into a relativistic quantum phase space which is a framework that inherently incorporates the uncertainty principle and relativistic covariance. In this context, the present work considers the derivation of linear and quadratic Casimir operators corresponding to representations of the Linear Canonical Transformations (LCT) group associated with a five-dimensional spacetime of signature (1,4). This LCT group, which emerges naturally as the symmetry group of the relativistic quantum phase space, is isomorphic to the symplectic group Sp(2,8). The latter notably contains the de Sitter group SO(1,4) as a subgroup. This geometric setting provides a unified framework for extending the Standard Model of particle physics while incorporating cosmological features. Previous studies have shown that the LCT group admits both fermionic-like and bosonic-like representations. Within this framework, a novel classification of quarks and leptons, including sterile neutrinos, has also been proposed. In this work, we present a systematic derivation of the linear and quadratic Casimir operators associated with these representations, motivated by their fundamental role in the characterization of symmetry groups in physics. The construction is based on the relations between the LCT group and the pseudo-unitary group U(1,4). Three linears and three quadratics Casimir operators are identified: two corresponding to the fermionic-like representation, two to the bosonic-like representation, and two hybrid operators linking the two representations. The complete eigenvalue spectra and corresponding eigenstates for each operator are subsequently computed and identified