Noether's theorem and gauge transformations. Application to the bosonic string and CP(2,n-1) model

New results on the theory of constrained systems are applied to characterize the generators of Noethers symmetry transformations. As a byproduct, an algorithm to construct gauge transformations in Hamiltonian formalism is derived. This is illustrated with two relevant examples

Dynamical Hartree-Fock-Bogoliubov Theory of Vortices in Bose-Einstein Condensates at Finite Temperature

We present a method utilizing the continuity equation for the condensate density to make predictions of the precessional frequency of single off-axis vortices and of vortex arrays in Bose-Einstein condensates at finite temperature. We also present an orthogonalized Hartree-Fock-Bogoliubov (HFB) formalism. We solve the continuity equation for the condensate density self-consistently with the orthogonalized HFB equations, and find stationary solutions in the frame rotating at this frequency. As an example of the utility of this formalism we obtain time-independent solutions for quasi-two-dimensional rotating systems in the co-rotating frame. We compare these results with time-dependent predictions where we simulate stirring of the condensate.Comment: 13 pages, 11 figures, 1 tabl

Symmetries and conservation laws in non-Hermitian field theories

Anti-Hermitian mass terms are considered, in addition to Hermitian ones, for PT-symmetric complex-scalar and fermionic field theories. In both cases, the Lagrangian can be written in a manifestly symmetric form in terms of the PT-conjugate variables, allowing for an unambiguous definition of the equations of motion. After discussing the resulting constraints on the consistency of the variational procedure, we show that the invariance of a non-Hermitian Lagrangian under a continuous symmetry transformation does not imply the existence of a corresponding conserved current. Conserved currents exist, but these are associated with transformations under which the Lagrangian is not invariant and which reflect the well-known interpretation of PT-symmetric theories in terms of systems with gain and loss. A formal understanding of this unusual feature of non-Hermitian theories requires a careful treatment of Noether’s theorem, and we give specific examples for illustration

An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg chains

Quantum spin chains are prototype quantum many-body systems. They are employed in the description of various complex physical phenomena. The goal of this paper is to provide an introduction to the subject by focusing on the time evolution of a Heisenberg spin-1/2 chain and interpreting the results based on the analysis of the eigenvalues, eigenstates, and symmetries of the system. We make available online all computer codes used to obtain our data.Comment: 8 pages, 3 figure

Symmetries of the Energy-Momentum Tensor: Some Basic Facts

It has been pointed by Hall et al. [1] that matter collinations can be defined by using three different methods. But there arises the question of whether one studies matter collineations by using the ${\cal L}_\xi T_{ab}=0$, or ${\cal L}_\xi T^{ab}=0$ or ${\cal L}_\xi T_a^b=0$. These alternative conditions are, of course, not generally equivalent. This problem has been explored by applying these three definitions to general static spherically symmetric spacetimes. We compare the results with each definition.Comment: 17 pages, accepted for publication in "Communications in Theoretical Physics

Nonlinear physics of the ionosphere and LOIS/LOFAR

The ionosphere is the only large-scale plasma laboratory without walls that we have direct access to. From results obtained in systematic, repeatable experiments in this natural laboratory, where we can vary the stimulus and observe its response in a controlled, repeatable manner, we can draw conclusions on similar physical processes occurring naturally in the Earth's plasma environment as well as in parts of the plasma universe that are not easily accessible to direct probing. Of particular interest is electromagnetic turbulence excited in the ionosphere by beams of particles (photons, electrons) and its manifestation in terms of secondary radiation (electrostatic and electromagnetic waves), structure formation (solitons, cavitons, alfveons, striations), and the associated exchange of energy, linear momentum, and angular momentum. We present a new diagnostic technique, based on vector radio allowing the utilization of EM angular momentum (vorticity), to study plasma turbulence remotely.Comment: Six pages, two figures. To appear in Plasma Physics and Controlled Fusio

Noether symmetries for two-dimensional charged particle motion

We find the Noether point symmetries for non-relativistic two-dimensional charged particle motion. These symmetries are composed of a quasi-invariance transformation, a time-dependent rotation and a time-dependent spatial translation. The associated electromagnetic field satisfy a system of first-order linear partial differential equations. This system is solved exactly, yielding three classes of electromagnetic fields compatible with Noether point symmetries. The corresponding Noether invariants are derived and interpreted

Bogomol'nyi Decomposition for Vesicles of Arbitrary Genus

We apply the Bogomol'nyi technique, which is usually invoked in the study of solitons or models with topological invariants, to the case of elastic energy of vesicles. We show that spontaneous bending contribution caused by any deformation from metastable bending shapes falls in two distinct topological sets: shapes of spherical topology and shapes of non-spherical topology experience respectively a deviatoric bending contribution a la Fischer and a mean curvature bending contribution a la Helfrich. In other words, topology may be considered to describe bending phenomena. Besides, we calculate the bending energy per genus and the bending closure energy regardless of the shape of the vesicle. As an illustration we briefly consider geometrical frustration phenomena experienced by magnetically coated vesicles.Comment: 8 pages, 1 figure; LaTeX2e + IOPar

Covariant gauge-natural conservation laws

When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equations for the gauge-natural invariant Lagrangian. {\em Vice versa} all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms which are in the kernel of generalized Jacobi morphisms are generators of canonical covariant currents and superpotentials. In particular, only a few gauge-natural lifts can be considered as {\em canonical} generators of covariant gauge-natural physical charges.Comment: 16 pages; presented at XXXVI Symposium on Math. Phys., Torun 09/06-12/06/04; the last paragraph of Section 3 has been reformulated, in particular a mistake in the equation governing the vertical part of gauge-natural lifts with respect to prolongations of principal connections (appearing e.g. in the vertical superpotential) has been correcte

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian $\Lambda_A$ including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate \ba. Without $\Lambda_A$, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian $\Lambda_A$ yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the $\Lambda_A$ characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian \ba space and the Eulerian \bx space. In contrast, with translation symmetry alone, there is an arbitrariness in the ransformation between these spaces.Comment: 34 pages, Fluid Dynamics Research (2008), accepted on 1st Dec. 200