G. Morandi

Geometry from dynamics, classical and quantum

Foreword: The birth and the long gestation of a project.- Some examples of linear and nonlinear physical systems and their dynamical equations.- Equations of the motion for evolution systems.- Linear systems with infinite degrees of freedom.- Constructing nonlinear systems out of linear ones.- The language of geometry and dynamical systems: the linearity paradigm.- Linear dynamical systems: The algebraic viewpoint.- From linear dynamical systems to vector fields.- Exterior differential calculus on linear spaces.- Exterior differential calculus on submanifolds.- A tensorial characterization of linear structures.- Partial linear structures: Vector bundles.- Covariant calculus.- Riemannian and Pseudo-Riemannian metrics on linear vector spaces.- Invariant geometric structures and the classical formulations of dynamics of Poisson, Jacobi, Hamilton and Lagrange.- Linear vector fields.- Additional invariant structures for linear vector fields.- Poisson structures.- The inverse problem for Poisson structures.- Symplectic structures.- Lagrangian structures.- Invariant Hermitean structures and the geometry of quantum systems.- Invariant Hermitean inner products.- Complex structures and complex exterior calculus.- Algebras associated with Hermitean structures.- The geometry of quantum dynamical evolution.- The Geometry of Quantum Mechanics and the GNS construction.- Alternative Hermitean structures for quantum systems.- Folding and unfolding Classical and Quantum systems.- Introduction: separable dynamics.- The geometrical description of reduction.- The algebraic description.- Reduction in Quantum Mechanics.- Integrable and superintegrable systems.- The geometrization of the notion of integrability.- The normal form of an integrable system.- Lax representation.- The Calogero system: inverse scattering.- Lie-Scheffers systems.- The inhomogeneous linear equation revisited.- Inhomogeneous linear systems.- Non-linear superposition rule.- Related maps.- Lie systems on Lie groups and homogeneous spaces.- Some examples of Lie systems.- Hamiltonian systems of Lie type.

Wigner-weyl correspondence in quantum mechanics for continuous and discrete systems-a dirac-inspired view

Drawing inspiration from Diracs work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.

FROM THE EQUATIONS OF MOTION TO THE CANONICAL COMMUTATION RELATIONS

The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as ”The Inverse Problem in the Calculus of Variations”) was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.

Geometric phase for mixed states: a differential geometric approach

Abstract.A new definition and interpretation of the geometric phase for mixed state cyclic unitary evolution in quantum mechanics are presented. The pure state case is formulated in a framework involving three selected principal fiber bundles, and the well-known Kostant-Kirillov-Souriau symplectic structure on (co-) adjoint orbits associated with Lie groups. It is shown that this framework generalizes in a natural and simple manner to the mixed state case. For simplicity, only the case of rank two mixed state density matrices is considered in detail. The extensions of the ideas of null phase curves and Pancharatnam lifts from pure to mixed states are also presented.

Wigner distributions for finite dimensional quantum systems: An algebraic approach

We discuss questions pertaining to the definition of ‘momentum’, ‘momentum space’, ‘phase space’ and ‘Wigner distributions’; for finite dimensional quantum systems. For such systems, where traditional concepts of ‘momenta’ established for continuum situations offer little help, we propose a physically reasonable and mathematically tangible definition and use it for the purpose of setting up Wigner distributions in a purely algebraic manner. It is found that the point of view adopted here is limited to odd dimensional systems only. The mathematical reasons which force this situation are examined in detail

Effective actions for spin ladders

We derive a path-integral expression for the effective action in the continuum limit of an AFM Heisenberg spin ladder with an arbitrary number of legs. The map is onto an

Path integrals for spinning particles, stationary phase and the Duistermaat-Heckmann theorem

O(3)

ALTERNATIVE HAMILTONIAN DESCRIPTIONS AND STATISTICAL MECHANICS

nonlinear

Spin-1 antiferromagnetic Heisenberg chains in an external staggered field

\sigma

Entanglement and nonclassicality for multimode radiation-field states

-model (NL