Variational formulation of Chern–Simons theory for arbitrary Lie groups

We show that the Chern–Simons theory for a principal G-bundle P over a three-dimensional manifold, with G an arbitrary Lie group, can be formulated as a variational problem defined by local data on the bundle of connections C(P) of P. By means of the theory of variational problems defined by local data we prove that the Euler–Lagrange operator and the differential of the Poincaré–Cartan form can be intrinsically expressed in terms of the symplectic form and the curvature morphism of C(P). These facts and the theory of the global inverse problem of the Calculus of Variations allow us to prove that there is indeed a global Lagrangian density for these theories. We also prove that every infinitesimal automorphism of P produces in a natural way an infinitesimal symmetry of the variational problem defined by the Chern–Simons theory. We therefore conclude that the algebra of infinitesimal symmetries of these theories is infinite dimensional.     

A model Lagrangian for strong interactions at large distances and BRS symmetries: gluon sectors

Within the covariant operator formalism respecting extended BRS symmetries, we propose an effective Lagrangian (a dipole field theory) for strong interactions at large distances which accommodates the color confinement of quarks and gluons. This is also derived from the stochastic treatment of the classical massive Yang-Mills system.     

A modular invariant partition function for the fivebrane

We compute an invariant partition function for the chiral two-form of the M theory fivebrane compactified on the six-torus T⁶. From a manifestly invariant formalism, we prove that the partition function has an additional symmetry. The combination of these two symmetries ensures invariance. Thus, whether or not a fully covariant Lagrangian is available, the fivebrane on the six-torus has a consistent quantum theory.     

The field copy problem: to what extent do curvature (gauge field) and its covariant derivatives determine connection (gauge potential)?

We show that a connection of a principal bundle is determined up to (global) gauge equivalence by the curvature and its covariant derivatives provided that the infinitesimal holonomy group is of constant dimension and the base space is simply connected. If the dimension of the infinitesimal holonomy group varies, there may be obstructions of a topological nature to the existence of a global or even local gauge equivalence between two connections whose curvatures and covariant derivatives of curvature agree everywhere. These obstructions are analyzed and illustrated by examples.     

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R^p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.     

The Hamilton-Cartan formalism for higher order conserved currents: 1. Regular first order Lagrangians

The Hamilton–Cartan formalism for regular first order Lagrangian field theories is extended to deal with conserved currents which depend on higher order derivatives of the field variables. These conserved currents are characterized. Exterior differential systems I^{(k + 1)} and I equivalent to the k-th and infinite prolongations of the Euler-Lagrange equations are defined. It is shown that to each conserved current is associated an equivalence class of infinitesimal symmetries of I. Conserved charges are defined and a Poisson bracket is constructed by analogy with the usual definition. The sine-Gordon equation is treated briefly as an application of the formalism.     

Gauge interaction as periodicity modulation

The paper is devoted to a geometrical interpretation of gauge invariance in terms of the formalism of field theory in compact space–time dimensions (Dolce, 2011) [8]. In this formalism, the kinematic information of an interacting elementary particle is encoded on the relativistic geometrodynamics of the boundary of the theory through local transformations of the underlying space–time coordinates. Therefore gauge interactions are described as invariance of the theory under local deformations of the boundary. The resulting local variations of the field solution are interpreted as internal transformations. The internal symmetries of the gauge theory turn out to be related to corresponding space–time local symmetries. In the approximation of local infinitesimal isometric transformations, Maxwell’s kinematics and gauge invariance are inferred directly from the variational principle. Furthermore we explicitly impose periodic conditions at the boundary of the theory as semi-classical quantization condition in order to investigate the quantum behavior of gauge interaction. In the abelian case the result is a remarkable formal correspondence with scalar QED.     

Frölicher-smooth geometries,quantum jet bundles and BRST symmetry

We attempt a clarification of geometric aspects of quantum field theory by using the notion of smoothness introduced by Frölicher and exploited by several authors in the study of functional bundles. A discussion of momentum and position representations in curved spacetime, in terms of generalized semi-densities, leads to a definition of quantum configuration bundle which is suitable for a treatment of that kind. A consistent approach to Lagrangian field theories, vertical infinitesimal symmetries and related currents is then developed, and applied to a formulation of BRST symmetry in a gauge theory of the Yang–Mills type.     

Rigid symmetries and BRST-invariance in gauge theories

The structure of the symmetry algebra of theories with simultaneous local and rigid symmetries is analyzed. BRST-invariant Faddeev-Popov gauge-fixing in such theories is discussed and it is proven that the BRST-transformations can always be made to commute with the rigid symmetries by assigning specific transformation rules to the ghosts. The problem of keeping the rigid symmetries manifest in the quantum theory is shown to reduce to the problem of finding covariant gauge conditions. Such covariant gauges exist only if the algebra of local and rigid symmetries has a semi-direct product structure.     

Second order Hamiltonian formalism and constraints algebra for non-polynomial supergravities

The second order Hamiltonian formalism for a non-polynomial N = 1D = 10 supergravity coupled to super Yang-Mills theory is developed. This is done by starting from the first order canoncial covariant formalism on group manifold. The Hamiltonian, generator of time evolution, is found as a functional of the first class constraints of this coupled system. These contraints close the constraint algebra and they are the generators of all the Hamiltonian gauge symmetries.     

Variational wave equations of two fermions interacting via scalar,pseudoscalar, vector,pseudovector and tensor fields

We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.     

Covariant derivative on non-linear fiber bundles

A gauge field is usually described as a connection on a principal bundle. It induces a covariant derivative on associated vector bundles, sections of which represent matter fields. In general, however, it is not possible to define a covariant derivative on non-linear fiber bundles, i.e. on those which are not vector bundles. We definelogarithmic covariant derivatives acting on two special non-linear fiber bundles — on the principal bundle and on the local gauge group bundle. The logarithmic derivatives map from sections of these bundles to the sections of the local gauge algebra bundle. Some properties of the logarithmic derivatives are formulated.     

Variational Equations and Symmetries in the Lagrangian Formalism; Arbitrary Vector Fields

We continue the study of symmetries in the Lagrangian formalism of arbitrary order with the help of the generalized Helmholtz equations (sometimes called the Anderson-Duchamp-Krupka equations). For the case of second-order equations and arbitrary vector fields we are able to establish a polynomial structure in the second-order derivatives. This structure is based on the some linear combinations of Olver hyper-Jacobians. We use as the main tools Fock space techniques and induction. This structure can be used to analyze Lagrangian systems with groups of Noetherian symmetries. As an illustration we analyze the case of Lagrangian equations with Abelian gauge invariance.     

Faddeev-Jackiw Quantization and Its Application to the Linear σ Model for Disoriented Chiral Condensates

The equivalence between the Faddeev-Jackiw formalism and Dirac-Bergmann algorithm is proved. A two-dimensional constrained system and a charged vector field are quantized in the Faddeev-Jackiw formalism. This symplectic method is technically developed, without recourse to Hamiltonian or Lagrangian, to quantize systems whose equations of motion are known. Examples are given to show this role. For constructing quantum approaches to the disoriented chiral condensates, the linear σ model is quantized in the instant form, light-cone form and covariant form.     

A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's exact set of fields for the vacuum equations.Dedicated to the memory of H. Rund     

Symmetries and constants of the motion for singular Lagrangian systems

A classification of infinitesimal symmetries of singular autonomous and nonautonomous Lagrangian systems is obtained. The relationship between infinitesimal symmetries and constants of the motion is given.     

一般完整系统Mei对称性的逆问题

动力学逆问题是星际航行学、火箭动力学、规划运动学理论的基本问题. Mei对称性是力学系统的动力学函数在群的无限小变换下仍然满足系统原来的运动微分方程的一种新的不变性. 本文研究广义坐标下一般完整系统的Mei对称性以及与Mei对称性相关的动力学逆问题. 首先, 给出系统动力学正问题的提法和解法. 引入时间和广义坐标的无限小单参数变换群, 得到无限小生成元向量及其一次扩展. 讨论由n个广义坐标确定的一般完整力学系统的运动微分方程, 将其Lagrange函数和非势广义力作无限小变换, 给出系统运动微分方程的Mei对称性定义, 在忽略无限小变换的高阶小量的情况下得到Mei对称性的确定方程, 借助规范函数满足的结构方程导出系统Mei对称性导致的Noether守恒量. 其次, 研究系统Mei对称性的逆问题. Mei对称性的逆问题的提法是: 由已知守恒量来求相应的Mei对称性. 采取的方法是将已知积分当作由Mei对称性导致的Noether守恒量, 由Noether逆定理得到无限小变换的生成元, 再由确定方程来判断所得生成元是否为Mei对称性的. 然后, 讨论生成元变化对各种对称性的影响. 结果表明, 生成元变化对Noether和Lie对称性没有影响, 对Mei 对称性有影响, 但在调整规范函数时, 若满足一定条件, 生成元变化对Mei对称性也可以没有影响. 最后, 举例说明结果的应用.     

Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem

The hierarchy of integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of linearization of these equations and their conservation law in the terms of solutions of corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and conservation laws explicitly expressed through the variables of the nonlinear equations are derived.