Canonical transformation

About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

Abstract: A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

Plasma frequency of the electron gas in layered structures

Abstract: The plasma frequency of a complete degenerate electron gas in the layered model of Visscher and Falicov is calculated by means of both the Bohm-Pines canonical transformation method and the equation-of-motion method in the RPA. The dispersion equation for plasmons is obtained in a finite system ofn planes with both the cyclic condition and free ends. It is shown that the thermodynamic limit (n→∞) of the plasma frequency is independent of the boundary conditions. The previous results obtained by various authors in different ways are shown to be certain limits of our result.

Local canonical transformations of fermions

Abstract: By permitting canonical transformations that are nonlinear in fermion creation and annihilation operators, we show that the space of canonical transformations of ordinary spin-1/2 operators local to a point in space is SU(2)\ensuremath{\bigotimes}SU(2)\ensuremath{\bigotimes}U(1)\ensuremath{\bigotimes}${\mathit{Z}}_{2}$. We identify those subgroups that form local and global gauge symmetries of the Hubbard-Heisenberg model on and off half filling. Our systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling. The paper presents this group together with all other known canonical transformations in a unified framework.

A multisymplectic approach to defects in integrable classical field theory

Abstract: We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1+1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schrodinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen Backlund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.