Fundamental Poisson Bracket Relations Research Articles

Spin and statistics in classical mechanics

The spin–statistics connection is obtained for classical particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and that exhibits classical analogs of both spin and statistics. Classical realizations of Lie groups are constructed in a canonical formalism generalized to include Grassmann variables. The theory of irreducible canonical realizations of the Poincaré group is developed in this framework, with particular emphasis on the rotation subgroup. The behavior of irreducible realizations under time inversion and charge conjugation is obtained. The requirement that the Lagrangian retain its form under the combined operation of charge conjugation and time reversal leads directly to the spin–statistics connection by an adaptation of Schwinger’s 1951 proof to irreducible canonical realizations of the Poincaré group of spin j: Generalized spin coordinates and momenta satisfy fundamental Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.

A Class of Integrable Spin Calogero-Moser Systems

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of $A_{n}$-type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids.

Spin Calogero–Moser systems associated with simple Lie algebras

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. Our analysis is based on a group-theoretic framework similar in spirit to the standard classical $r$-matrix theory for constant $r$-matrices.

THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into the $r$-$s$-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matrices $r$ and $s$ are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrix~$c$. It is proposed that all these Poisson brackets taken together are representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.

INTEGRABILITY AND SYMMETRIC SPACES I: THE GROUP MANIFOLD

It is shown that any nonsingular Lagrangian describing the motion of a particle on a semisimple Lie group possesses a Fundamental Poisson bracket Relation (FPR) and consequently charges in involution. This property is independent of the dynamics of the model and can be derived in a quite simple and general way from the geometric and algebraic structures of the group manifold. The conditions a Hamiltonian has to satisfy in order those charges are to be conserved are discussed. These conditions lead to an algebra which plays an important role in the construction of conserved charges. In the second paper of the series, this work is extended to the coset spaces which are symmetric spaces.

INTEGRABILITY AND SYMMETRIC SPACES II: THE COSET SPACES

It is shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a Fundamental Poisson bracket Relation, and consequently charges in involution, is that it must be a symmetric space. The conditions, a Hamiltonian, or any functions of the canonical variables, has to satisfy in order to commute with these charges, are studied. It is shown that, for the case of the noncompact symmetric spaces, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities.

Non-compact symmetric spaces and the Toda molecule equations

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, GN, by its maximal compact subgroup. This is explained in more detail and it is shown that the “fundamental Poisson bracket relation” involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forGN endows the symmetric space with a hidden group theoretic structure.

Canonical Realizations of the Galilei Group

The general theory of the realizations of finite Lie groups by means of canonical transformations in classical mechanics, which has been developed in a preceding paper and already applied to the rotation group, is now applied to the Galilei group. Some complements to the general theory are introduced; in particular, a new kind of possible canonical realizations connected with the singularity surfaces of the functions Q(y),P(y),J(y) are discussed (singular realizations). In agreement with the situation encountered in quantum mechanics, the constants dρσ appearing in the fundamental Poisson bracket relations among the infinitesimal generators ({yρ,yσ}=cρστyτ+dρσ) cannot all be reduced to zero. There remains a single independent constant m, which, in the physically significant cases (m > 0), represents the mass of the system. No physical interpretation seems to be attachable to the realizations corresponding to m = 0. For m ≠ 0, two different kinds of irreducible realizations exist: one of a singular type which describes the free mass-point, and another of a regular type which describes a classical particle spin. A number of physical significant examples corresponding to nonirreducible realizations are thereafter discussed and the related typical forms are constructed: specifically, the cases dealt with are the rigid rod (linear rotator), the rigid body, and a system of two interacting mass points. It is shown that the problem of the construction of the variables of the typical form is equivalent to the determination of an appropriate solution of the time-independent Hamilton-Jacobi equation.