On Covariant Phase Space and the Variational Bicomplex

The notion of a phase space in classical mechanics is well known. The extension of this concept to field theory however, is a challenging endeavor, and over the years numerous proposals for such a generalization have appeared in the literature. In this paper We review a Hamiltonian formulation of Lagrangian field theory based on an extension to infinite dimensions of J.-M. Souriau's symplectic approach to mechanics. Following G. Zuckerman, we state our results in terms of the modern geometric theory of differential equations and the variational bicomplex. As an elementary example, we construct a phase space for the Monge–Ampere equation.     

A Generally Covariant Wave Equation for Grand Unified Field Theory

A generally covariant wave equation is derived geometrically for grand unified field theory. The equation states most generally that the covariant d'Alembertian acting on the vielbein vanishes for the four fields which are thought to exist in nature: gravitation, electromagnetism, weak field and strong field. The various known field equations are derived from the wave equation when the vielbein is the eigenfunction. When the wave equation is applied to gravitation the wave equation is the eigenequation of wave mechanics corresponding to Einstein's field equation in classical mechanics, the vielbein eigenfunction playing the role of the quantized gravitational field. The three Newton laws, Newton's law of universal gravitation, and the Poisson equation are recovered in the classical and nonrelativistic, weak-field limits of the quantized gravitational field. The single particle wave-equation and Klein-Gordon equations are recovered in the relativistic, weak-field limit of the wave equation when scalar components are considered of the vielbein eigenfunction of the quantized gravitational field. The Schrödinger equation is recovered in the non-relativistec, weak-field limit of the Klein-Gordon equation). The Dirac equation is recovered in this weak-field limit of the quantized gravitational field (the nonrelativistic limit of the relativistic, quantezed gravitational field when the vielbein plays the role of the spinor. The wave and field equations of O(3) electrodynamics are recovered when the vielbein becomes the relativistic dreibein (triad) eigenfunction whose three orthonormal space indices become identified with the three complex circular indices (1), (2), (3), and whose four spacetime indices are the indices of non-Euclidean spacetime (the base manifold). This dreibein is the potential dreibein of the O(3) electromagnetic field (an electromagnetic potential four-vector for each index (1), (2), (3)). The wave equation of the parity violating weak field is recovered when the orthonormal space indices of the relativistic dreibein eigenfunction are identified with the indices of the three massive weak field bosons. The wave equation of the strong field is recovered when the orthonormal space indices of the relativistic vielbein eigenfunction become the eight indices defined by the group generators of the SU (3) group.     

The double exponential map and covariant derivation

We study the double exponential map which is a composition of a special form of two exponential maps on a manifold with connection. We relate this map and the composition of covariant derivations as well as the composition of pseudodifferential operators on these manifolds.     

The Waring Locus of Binary Forms

Abstract

For integers m, d, n, we study the locus of m-dimensional subspaces of degree d binary forms whose elements admit simultaneous decompositions as sums of powers of n linear forms. We show that this locus has rational singularities and it is arithmetically Cohen–Macaulay in its natural Plücker imbedding. We describe invariant theoretic formulations for the equations of these loci.     

Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Search for exotic state 1-+ π1(1400) at BESⅢ

The J/ψ hadronic decays provide good laboratory to search for the hybrid states with exotic quantum numbers. A full Partial Wave Analysis (PWA) is performed to the generated Monte Carlo J/ψ→ pηπ data, based on the design of BESⅢ detector, to study the sensitivity of searching for a possible exotic state at BESⅢ.     

Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.As an application of the investigated general scheme, (pseudo) Riemannian spaces with contravariant and covariant affine connections (whose components differ not only by sign) are considered as a special case of -spaces with Riemannian metric, symmetric covariant connection and a weaker definition of dual vector basis with conformal noncanonical contraction operator . The geodesic and autoparallel equations in -spaces are found as different equations in contrast to the case of V ₄-spaces. The Euler–Lagrange equations as Einstein's field equations in -spaces and the corresponding energy-momentum tensors (EMTs) are obtained and compared with the Einstein equations and the EMTs in V ₄-spaces. The geodesic and the auto-parallel equations are discussed.     

Wavelets in Banach Spaces

We describe a construction of wavelets (coherent states) in Banach spaces generated by admissible group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example, we consider operator-valued Segal–Bargmann-type spaces and the Weyl functional calculus.