The Maslov indices of Hamiltonian periodic orbits

Abstract

We use the properties of the Leray index to give precise formulas in arbitrary dimensions for the Maslov index of the monodromy matrix arising in periodic Hamiltonian systems. We compare our index with other indices appearing in the literature.

Symplectic Geometry and Quantum Mechanics

“…Here is a direct argument; in more complicated cases the formulas we proved in [68] are useful. When t goes from 0 to r the lineS t P describes a loop in Lag (1) going from P toS r P .…”

On the Symplectic Dirac Equation

“…He was motived by the problem of finding a way to improve the quantum statistical mechanics, based on the desity matrix, to treat the transport equations for superfluids [2][3][4]. Since then, the formalism proposed by Wigner has been applied in different contexts, such as quantum optics [5,6], condensed matter [7][8][9], quantum computing [10][11][12], quantum tomography [13], plasma physics [14][15][16][17][18][19]. Wigner introduced his formalism by using a kind of Fourier transform of the density matrix, ρ(q, q ′ ), giving rise to what in nowadays called the Wigner function, f W (q, p), where (q, p) are coordinates of a phase space manifold (Γ).…”

Solutions for the Landau Problem Using Symplectic Representations of the Galilei Group

“…[47][48][49] The scalar representation of Lorentz group for spin 0 and spin 1/2 leads to, for instance, the Klein-Gordon and Dirac equations in phase space, such that the wave functions are closely associated with the Wigner function. 47,48 This provides a fundamental ingredient for the physical interpretation of the formalism, showing its advantage in relation to other attempts to explore, for instance, the Schrödinger equation in phase space; [13][14][15] such an association is not evident, a fact that represents a hindrance to reach a physical interpretation. In terms of nonrelativistic quantum mechanics, the proposed formalism has been used to treat a nonlinear oscillator perturbatively, to study the notion of coherent states and to introduce a nonlinear Schrödinger equation from the point of view of phase space.…”