Hamiltonian Field Research Articles

First-order Hamiltonian field theory and mechanics

This article deals with the geometric analysis of the evolutionary and the polysymplectic approach in first-order Hamiltonian field theory. Based on a variational formulation in the Lagrangian picture, two possible counterparts in a Hamiltonian formulation are discussed. The main difference between these two approaches, which are important for the application, is besides a different bundle construction, the different Legendre transform as well as the analysis of the conserved quantities. Furthermore, the role of the boundary conditions in the Lagrangian and in the Hamiltonian pictures will be addressed. These theoretical investigations will be completed by the analysis of several examples, including the wave equation, a beam equation and a special subclass of continuum mechanics in the presented framework.

Existence of solutions for Hamiltonian field theories by the Hamilton–Jacobi technique

The paper is devoted to prove the existence of a local solution of the Hamilton–Jacobi equation in field theory, whence the general solution of the field equations can be obtained. The solution is adapted to the choice of the submanifold where the initial data of the field equations are assigned. Finally, a technique to obtain the general solution of the field equations, starting from the given initial manifold, is deduced.

Synthesis, Characterization, and Quantum-Chemical Studies of Ni(CN)2MX (M = Rb, Cs; X = Cl, Br)

A new series of coordination-network compounds containing Ni(CN)(2) and MX (M = Rb, Cs; X = Cl, Br) quasi two-dimensional sheets has been synthesized and structurally characterized. The tetragonal crystal structure (I4/mmm, no. 139) can be derived from the distorted perovskite type. Chemically, the Lewis-acidic Ni(CN)(2) moieties accept halide ligands, resulting both in slightly elongated [Ni(NC)(4)X(2)](4-) octahedra and strongly elongated [Ni(CN)(4)X(2)](4-) octahedra that are charge-balanced by M(+) cations. Quantum-chemical calculations of the effective Hamiltonian crystal field (EHCF) type indicate the simultaneous presence of high- and low-spin Ni(2+) in a 1:1 ratio, in agreement with GGA+U and UV studies reported here. Our magnetic susceptibility data also corroborate the theoretical findings. An analysis of the magnetic superexchange paths in the new series of compounds is performed as well as that of the tentative magnetic state of the series.

Reduced classical field theories: k-cosymplectic formalism on Lie algebroids

In this paper we introduce a geometric description of Lagrangian and Hamiltonian classical field theories on Lie algebroids in the framework of k-cosymplectic geometry. We discuss the relation between Lagrangian and Hamiltonian descriptions through a convenient notion of Legendre transformation. The theory is a natural generalization of the standard one; in addition, other interesting examples are studied, mainly on reduction of classical field theories.

Hamiltonian light-front field theory in a basis function approach

Hamiltonian light-front quantum field theory constitutes a framework for the nonperturbative solution of invariant masses and correlated parton amplitudes of self-bound systems. By choosing the light-front gauge and adopting a basis function representation, a large, sparse, Hamiltonian matrix for mass eigenstates of gauge theories is obtained that is solvable by adapting the ab initio no-core methods of nuclear many-body theory. Full covariance is recovered in the continuum limit, the infinite matrix limit. There is considerable freedom in the choice of the orthonormal and complete set of basis functions with convenience and convergence rates providing key considerations. Here we use a two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall anti-de Sitter/quantum chromodynamics (AdS/QCD) model obtained from light-front holography. We outline our approach and present illustrative features of some noninteracting systems in a cavity. We illustrate the first steps toward solving quantum electrodynamics (QED) by obtaining the mass eigenstates of an electron in a cavity in small basis spaces and discuss the computational challenges.

Hamiltonian light-front field theory within an AdS/QCD basis

Non-perturbative Hamiltonian light-front quantum field theory presents opportunities and challenges that bridge particle physics and nuclear physics. Fundamental theories, such as Quantum Chromodynmamics (QCD) and Quantum Electrodynamics (QED) offer the promise of great predictive power spanning phenomena on all scales from the microscopic to cosmic scales, but new tools that do not rely exclusively on perturbation theory are required to make connection from one scale to the next. We outline recent theoretical and computational progress to build these bridges and provide illustrative results for nuclear structure and quantum field theory. As our framework we choose light-front gauge and a basis function representation with two-dimensional harmonic oscillator basis for transverse modes that corresponds with eigensolutions of the soft-wall AdS/QCD model obtained from light-front holography.

Experimental and Quantum‐Chemical Investigations of the UV/Vis Absorption Spectrum of Manganese Carbodiimide, MnNCN

AbstractThe UV/Vis absorption spectrum of a single crystal of manganese carbodiimide, MnNCN, has been measured and theoretically analyzed using both the angular‐overlap model (AOM) and the effective Hamiltonian crystal field (EHCF) method. Independent from the method used, we find a somewhat higher ligand‐field splitting (10 Dq ranging from 7300 up to 8842 cm–1 depending on the estimation procedure used) compared to Mn–O chromophores and a smaller nephelauxetic ratio (β = 0.62 ÷ 0.67). In addition, the Mn–N bond within the MnIIN6 octahedra is lacking significant π interaction.

Straight line orbits in Hamiltonian flows

We investigate periodic straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar H\'enon-Heiles problem. It is shown that SLO's can exist for arbitrary finite superpositions of $N$-forms. The results are applied to a family of generalized H\'enon-Heiles potentials having discrete rotational symmetry. SLO's are also found for superpositions of these potentials.

Classical R-matrix theory for bi-Hamiltonian field systems

This is a survey of the application of the classical R-matrix formalism to the construction of infinite-dimensional integrable Hamiltonian field systems. The main point is the study of bi-Hamiltonian structures. Appropriate constructions on Poisson, noncommutative and loop algebras as well as the central extension procedure are presented. The theory is developed for (1 + 1)- and (2 + 1)-dimensional field and lattice soliton systems as well as hydrodynamic systems. The formalism presented contains sufficiently many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite-dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.

K-symplectic formalism on Lie algebroids

In this paper we introduce a geometric description of Lagrangian and Hamiltonian classical field theories on Lie algebroids in the framework of k-symplectic geometry. We discuss the relation between the Lagrangian and Hamiltonian descriptions through a convenient notion of Legendre transformation. The theory is a natural generalization of the standard one; in addition, other interesting examples are studied, in particular, systems with symmetry and Poisson-sigma models.

Classical isometrodynamics

A generalization of non-Abelian gauge theories of compact Lie groups is developed by gauging the noncompact group of volume-preserving diffeomorphisms of a $D$-dimensional space ${\mathbf{R}}^{D}$. This group is represented on the space of fields defined on ${\mathbf{M}}^{4}\ifmmode\times\else\texttimes\fi{}{\mathbf{R}}^{D}$. As usual the gauging requires the introduction of a covariant derivative, a gauge field, and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws of the new gauge theory are developed. Finally, the theory's Hamiltonian in the axial gauge and its Hamiltonian field dynamics are derived.

Dynamical thermalization and vortex formation in stirred two-dimensional Bose-Einstein condensates

We present a quantum-mechanical treatment of the mechanical stirring of Bose-Einstein condensates using classical field techniques. In our approach the condensate and excited modes are described using a Hamiltonian classical field method in which the atom number and (rotating frame) energy are strictly conserved. We simulate a $T=0$ quasi-two-dimensional condensate perturbed by a rotating anisotropic trapping potential. Vacuum fluctuations in the initial state provide an irreducible mechanism for breaking the initial symmetries of the condensate and seeding the subsequent dynamical instability. Highly turbulent motion develops and we quantify the emergence of a rotating thermal component that provides the dissipation necessary for the nucleation and motional damping of vortices in the condensate. Vortex lattice formation is not observed, rather the vortices assemble into a spatially disordered vortex liquid state. We discuss methods we have developed to identify the condensate in the presence of an irregular distribution of vortices, determine the thermodynamic parameters of the thermal component, and extract damping rates from the classical field trajectories.

COVARIANT HAMILTONIAN FIELD THEORY

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.

HAMILTONIAN FIELDS AND ENERGY IN CONTACT MANIFOLDS

To the contact distribution of a contact manifold we associate Hamiltonian type vector fields, called contact Hamiltonian fields. Their properties are investigated and the existence of such vector fields nowhere tangent to a given submanifold is proved. Time-depending contact Hamiltonian vector fields allow us to define the contact energy whose properties are studied. A class of submanifolds in relation to the study of contact Hamiltonian fields is also analyzed.

SYMMETRIES AND CONSERVATION LAWS IN THE GÜNTHER k-SYMPLECTIC FORMALISM OF FIELD THEORY

This paper is devoted to studying symmetries of k-symplectic Hamiltonian and Lagrangian first-order classical field theories. In particular, we define symmetries and Cartan symmetries and study the problem of associating conservation laws to these symmetries, stating and proving Noether's theorem in different situations for the Hamiltonian and Lagrangian cases. We also characterize equivalent Lagrangians, which lead to an introduction of Lagrangian gauge symmetries, as well as analyzing their relation with Cartan symmetries.

Extended Hamiltonian systems in multisymplectic field theories

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of nonautonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.

Hermiticity of the Dirac Hamiltonian in gravitational field

The hermiticity properties of the Dirac Hamiltonian are discussed, both on the quantum mechanical and on the quantum field level. In a first step, it is shown that the Hamiltonian generating the time evolution of the Dirac wave function in relativistic quantum mechanics is not hermitian with respect to the covariantly defined inner product. A hermitian Hamiltonian is then defined and is shown to be directly related to the canonical field energy. In a second step, we use a manifestly covariant form of canonical Hamiltonian field theory in curved spacetime and show that, for the Dirac field, the canonical field momentum does not coincide with the generators of spacetime translations. Moreover, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.

Contact degeneracies of closed 2-forms

Consider a closed 2-form that is degenerate at the points of a hypersurface and is non-degenerate outside it. In the neighbourhood of a singularity (which is called contact under certain natural conditions) the limit behaviour of Hamiltonian fields is investigated and a canonical form of the 2-form is found (Darboux's theorem). Connections with regular Lie structures are established. Properties of integrable structures on Liouville tori containing contact degeneracies are studied. Bibliography: 16 titles.

Generally covariant quantization and the Dirac field

Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that the modification of the Dirac Lagrangian by a surface term leads to a momentum transfer between the Dirac field and the gravitational background field, resulting in a theory that is free of constraints, but not manifestly hermitian.

Exact one- and two-particle excitation spectra of acute-angle helimagnets above their saturation magnetic field

The two-magnon problem for the frustrated XXZ spin-$1∕2$ Heisenberg Hamiltonian and external magnetic fields exceeding the saturation field ${B}_{s}$ is considered. We show that the problem can be exactly mapped onto an effective tight-binding impurity problem. It allows to obtain explicit exact expressions for the two-magnon Green's functions for arbitrary dimension and number of interactions. We apply this theory to a quasi-one-dimensional helimagnet with ferromagnetic nearest-neighbor ${J}_{1}<0$ and antiferromagnetic next-nearest neighbor ${J}_{2}>0$ interactions. An outstanding feature of the excitation spectrum is the existence of two-magnon bound states. This leads to deviations of the saturation field ${B}_{s}$ from its classical value ${B}_{s}^{\mathrm{cl}}$ which coincides with the one-magnon instability. For the refined frustration ratio $\ensuremath{\mid}{J}_{2}∕{J}_{1}\ensuremath{\mid}>0.374\phantom{\rule{0.2em}{0ex}}661$ the minimum of the two-magnon spectrum occurs at the boundary of the Brillouin zone. Based on the two-magnon approach, we propose general analytic expressions for the saturation field ${B}_{s}$, confirming known previous results for one-dimensional isotropic systems, but explore also the role of interchain and long-ranged intrachain interactions as well as of the exchange anisotropy.