Polynomial In Momenta Research Articles

Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations

We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn’s). The moments of non-equilibrium classical distributions, implied by the Hn’s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly.

Bi-Poisson Structures and Integrability of Geodesic Flow on Homogeneous Spaces

Let G=K be a semisimple orbit of the adjoint representation of a real connected reductive Lie group G. Let K1 be any closed subgroup of K containing the commutant of the identity component of K. We prove that the geodesic flow on the symplectic manifold T ⁄ (G=K1), corresponding to a G-invariant pseudo-Riemannian metric on G=K1 which is in- duced by a bi-invariant pseudo-Riemannian metric on G, is completely integrable in the class of real analytic functions, polynomial in momenta. To this end we study the Poisson geometry of the space of G-invariant functions on T ⁄ (G=K) using a one-parameter family of moment maps.

Complete sets of invariants for dynamical systems that admit a separation of variables

Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated Hamilton–Jacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1=H, P2,…,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj}={Pi,Pj}=0, {Qi,Pj}=δij. The 2n−1 functions Q2,…,Qn,P1,…,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n=2 we go further and consider all cases where the Hamilton–Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.

Nonlinear supersymmetry, quantum anomaly and quasi-exactly solvable systems

The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ``cured'' by the specific superpotential-dependent term of order $\hbar^2$. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.

Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space

The problem considered here is that of finding conditions ensuring that a reversible Hamiltonian system has integrals polynomial in momenta. The kinetic energy is a zero-curvature Riemannian metric and the potential a smooth function on a two-dimensional torus. It is known that the existence of integrals of degrees 1 and 2 is related to the existence of cyclic coordinates and the separation of variables. The following conjecture is also well known: if there exists an integral of degree n independent of the energy integral, then there exists an additional integral of degree 1 or 2. In the present paper this result is established for n = 3 (which generalizes a theorem of Byalyi), and for n = 4, 5, and 6 this is proved under some additional assumptions about the spectrum of the potential.

Unbounded growth of energy in nonautonomousHamiltonian systems

The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincaré-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian. AMS classification scheme number: 58F05

Quantization of Forms on the Cotangent Bundle

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on T ⋆ M is made into a space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the (“super”)commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenböck like identities appear here.) Formulae for the traces corresponding to natural gradings of Ω (T ⋆ M ) are established. Using these formulae, we give a simple direct proof of the Gauss–Bonnet–Chern Theorem. We discuss these results in connection with a general question of the quantization of forms on a Poisson manifold.

Pure quantum integrability

The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not 1-1, although some close relationships exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta, one can construct a corresponding commuting set of differential operators by adding suitable O( h ̵ ) 2 correction terms. Here we discuss some two- or three-dimensional purely quantum integrable systems, that is, quantum integrable potentials whose amplitude is not free but rather proportional to h ̵ 2 , and which therefore vanish in the classical limit. Furthermore it turns out that some of these systems have actually N + 1 commuting differential operators, connected by a nontrivial algebraic relation. The same phenomenon has been discussed recently by Veselov et al. from the point of view of Baker-Akheizer functions.

First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus

In a large collection of examples of completely integrable Hamiltonian systems, the first integrals are polynomial in momenta, lhtegrals that are linear in momenta correspond to one-parameter groups of symmetries of a dynamical system [i]. In the case of separation of variables, in the Hamilton-Jacobi equation integrals naturally arise that are quadratic in momenta. The problem of the existence of linear and quadratic integrals for a natural mechanical system with two degrees of freedom on the plane was considered in [2]. In [3], linear and quadratic integrals on the two-dimensional sphere are investigated. In addition, it has been proved that for natural mechanical systems with a two-dimensional compact configuration space of genus greater than one, integrals that are analytic in momenta are lacking [4].

Current-Algebra Calculation of Hard-Pion Processes:A1→ρ+πandρ→π+π

New techniques are developed for treating the $n$-point functions of currents, which are interrelated by means of Ward identities obtained from the equal-time current commutation relations. The $n$-point functions are sorted out so as to define proper vertices which describe the reactions of particles of definite spin. A meson-dominance assumption is made by approximating the proper vertices by simple polynomials in momenta, with the coefficients determined by the Ward identities. The method is discussed in detail for $n=3$ and the currents of chiral $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$, and then applied to the decay processes ${A}_{1}\ensuremath{\rightarrow}\ensuremath{\rho}+\ensuremath{\pi}$ and $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}$.