Parameter Invariance in Field Theory and the Hamiltonian Formalism

The deparametrization problem for parameter‐invariant Lagrangian densities defined over J¹(N, F), is solved in terms of a projection onto a suitable jet bundle. The Hamilton‐Cartan formalism for such Lagrangians is then introduced and the pre‐symplectic structure of such variational problems is proved to be projectable through the aforementioned projection. Specific examples with physical meaning are also analyzed. 1998 PACS codes. 02.20.Tw Infinite‐dimensional Lie groups, 02.30.Wd Calculus of variations and optimal control, 02.40.Ky Riemannian geometries, 02.40.Ma Global differential geometry, 02.40.Vh Global analysis and analysis on manifolds, 04.20.Fy Canonical formalism, Lagrangians, and variational principles, 11.10.Ef Lagrangian and Hamiltonian approach, 11.10.Kk Field theories in dimensions other than four, 11.25.Sq Nonperturbative techniques; string field theory. 1991 Mathematics Subject Classification. Primary: 58E30 Variational principles; Secondary: 53B20 Local Riemannian geometry, 58A20 Jets, 58E12 Applications to minimal surfaces (problems in two independent variables), 58G35 Invariance and symmetry properties, 81S10 Geometric quantization, symplectic methods, 83E30 String and superstring theories.     

Lorentz-Covariant Hamiltonian Formalism

The dynamics of a classical system can be expressed by means of Poissonbrackets. In this paper we generalize the relation between the usual noncovariantHamiltonian and the Poisson brackets to a covariant Hamiltonian and new bracketsin the frame of Minkowski space. These brackets can be related to those usedby Feynman in his derivation Maxwell's equations. The case of curved space isalso considered with the introduction of Christoffel symbols, covariant derivatives,and curvature tensors.     

Hamiltonian Formalism in Supermechanics

The Hamilton–Cartan formalism in supermechanics is developed, the graded structure on the manifold of solutions of a variational problem defined by a regular homogeneous Berezinian Lagrangian density is determined and its graded symplectic structure is analyzed. The graded symplectic structure on the manifold of solutions of a classical regular Lagrangian is compared with the Koszul–Schouten brackets.     

Hamiltonian Formalism in Quantum Mechanics

Abstract

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures.

To Vladimir Igorevich Arnol’d with admiration, on occasion of his 60^th birthday.     

Hamiltonian Formalism of de-Sitter Invariant Special Relativity

The Lagrangian of Einstein's special relativity with universal parameter c (SR_c) is invariant under Poincaré transformation, which preserves Lorentz metric η_μν. The SR_c has been extended to be one which is invariant under de Sitter transformation that preserves so-called Beltrami metric B_μν. There are two universal parameters, c and R, in this Special Relativity (denoted as SR_cR). The Lagrangian-Hamiltonian formulism of SR_cR is formulated in this paper. The canonic energy, canonic momenta, and 10 Noether charges corresponding to the space-time's de Sitter symmetry are derived. The canonical quantization of the mechanics for SR_cR-free particle is performed. The physics related to it is discussed.     

Hamiltonian Formalism of de-Sitter Invariant Special Relativity

The Lagrangian of Einstein's special relativity with universal parameter c (SRc) is invariant under Poincaré transformation, which preserves Lorentz metric ημν. The SRc has been extended to be one which is invariant under de Sitter transformation that preserves so-called Beltrami metric Bμν. There are two universal parameters, c and R, in this Special Relativity (denoted as SRcR). The Lagrangian-Hamiltonian formulism of SRcR is formulated in this paper.The canonic energy, canonic momenta, and 10 Noether charges corresponding to the space-time's de Sitter symmetry are derived. The canonical quantization of the mechanics for SRcR-free particle is performed. The physics related to it is discussed.     

Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation

A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.     

Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values

Hamiltonian formalism of the mKdV equation with non-vanishing boundary value is re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 （1988） 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.     

Hamiltonian Formalism of mKdV Equation with Non-vanishing Boundary Values

Hamiltonian formalism of the mKdV equation with non-vanishing boundary valueis re-examined by a revised form of the standard procedure. It is known that the previous papers did not give the final results and involved some questionable points [T.C. Au Yeung and P.C.W. Fung, J. Phys. A 21 (1988) 3575]. In this note, simple results are obtained in terms of an affine parameter and a Galileo transformation is introduced to ensure the results compatible with those derived from the inverse scattering transform.     

Stable Interactions Between the Extended Chern-Simons Theory and a Charged Scalar Field with Higher Derivatives: Hamiltonian Formalism

Russian Physics Journal - The constrained Hamiltonian formalism for the extended higher derivative Chern–Simons theory of an arbitrary finite order is considered. It is shown that the n-th...     

Hamiltonian Formalism for Collective Fermionic Waves in a Quark-Gluon Plasma

Russian Physics Journal - A Hamiltonian theory has been developed for collective quark-antiquark excitations with abnormal relation between chirality and helicity in a high-temperature quark-gluon...     

The Hamiltonian Formalism and Quantizations of (Generalized) BFOFW Model

In this article,we discussed the Hamiltonian formalism of BFOFW model,studied its Dirac canonical quantization and BFV-BRST quantization.We proposed a new kind of gauged WZNW theories,which was called in the text generalized BFOFW model.     

Hamiltonian Formalism for Bose Excitations in a Plasma with a Non-Abelian Interaction

Journal of Experimental and Theoretical Physics - We have developed the Hamiltonian theory for collective longitudinally polarized colorless excitations (plasmons) in a high-temperature gluon...     

Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory

On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is an extended notion of the renormalized time-ordered product as a binary product equivalent to the pointwise product of classical field theory. Originally, in causal perturbation theory, the time-ordered product is understood merely as a sequence of multilinear maps on the space of local functionals. Our extended notion of the renormalized time-ordered product (denoted by ${\cdot_{{}^{\mathcal{T}_{\rm r}}}}$ ) is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to ${\cdot_{{}^{\mathcal{T}_{\rm r}}}}$ . On this space the renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and Dütsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.     

WZNW models and gauged WZNW models based on a family of solvable Lie algebras

A family of solvable self-dual Lie algebras that are not double extensions of Abelian algebras and, therefore, cannot be obtained through a Wigner contraction, is presented. We construct WZNW and gauged WZNW models based on the first two algebras in this family. We also analyze some general phenomena arising in such models.     

Comoving Self-Gravitating Scalar Field in the Newman Penrose Formalism

The Einstein field equation, coupled to the scalar field, is studied in a spherically symmetric comoving system. The problem is translated into the language of the Newman Penrose formalism that is based on the choice of a null tetrad frame. The corresponding (tabulated) Einstein field equation, Bianchi identities and scalar field equation are explicited in terms of the Weyl and Ricci scalars and discussed. Spherical symmetry reduces the difficulties but not so far to enable to integrate the scheme in general. The main result is that static self-gravitation is possible only for massless scalar field. The static solution is determined. It depends on an arbitrary function that can be interpreted as radial coordinate. The part of the space–time solution of the problem does not contain black holes. It is remarked that in the part of the space–time not solution of the problem, light rays cannot propagate radially but admit circular orbits.