Covariant field theory on frame bundles of fibered manifolds

Abstract

We show that covariant field theory for sections of π : E → M lifts in a natural way to the bundle of vertically adapted linear frames L π E. Our analysis is based on the fact that L π E is a principal fiber bundle over the bundle of 1-jets J 1 π. On L π E the canonical soldering 1-forms play the role of the contact structure of J 1 π. A lifted Lagrangian L:L π E → R is used to construct modified soldering 1-forms, which we refer to as the Cartan-Hamilton-Poincaré 1-forms. These 1-forms on L π E pass to the quotient to define the standard Cartan-Hamilton-Poincaré m-form on J 1 π. We derive generalized Hamilton-Jacobi and Hamilton equations on L π E, and show that the Hamilton-Jacobi and canonical equations of Carathéodory-Rund and de DonderWeyl are obtained as special cases. The manifold L π E emerges as a natural arena for a unified theory that contains, in addition to the sector for sections of π, dynamical sectors for a geometry for M and a geometry for the fibers of E.

n -symplectic algebra of observables in covariant Lagrangian field theory

“…In section III we present the relevant details of the canonical n-symplectic geometry on LE, and the reduced subbundle of adapted linear frames L π E. In section IV we recall the bundle structure ρ : L π E → J 1 π and the definition [9] of the modified soldering 1-forms on L π E, which we refer to as the Cartan-HamiltonPoincaré (CHP) R n -value 1-form. The CHP 1-form is defined herein as the pull-back, under a lifted Legendre transformation, of the canonical R n -valued soldering 1-form on LE to L π E. We then prove the theorem that (L π E, dθ L ) is an n-symplectic manifold provided L is non-zero.…”

“…, m + k.) A point in L π E is a triple (e, e i , e A ) where e ∈ E and (e i , e A ) is a linear frame for the tangent space to E at e in which the last k vectors (e A ) are vertical on π : E → M . The lifting of L : J 1 π → R to L π E is natural since L π E is known [9] to be an H = GL(m) × GL(k) principal fiber bundle over J 1 π. If ρ : L π E → J 1 π then we put L := ρ * (L).…”

“…Using a lifted Legendre transformation we construct the Cartan-Hamilton-Poincaré (CHP) 1-formsθ L first introduced in reference [9], and prove the theorem that (L π E, dθ L ) is an n-symplectic manifold provided the Lagrangian is non-zero. The observables of the theory are then the ⊗ p R n -valued functionsf on L π E that satisfy the n-symplectic structure equation…”

“…In this paper we construct an algebra of observables for covariant Lagrangian field theories using the n-symplectic theory [3,4,5,6,7,8,9] as the engine for the construction.…”

“…To use the n-symplectic theory to construct an algebra of observables we lift the Lagrangian on J 1 π to the bundle of adapted linear frames L π E, the subbundle of LE that arises [15,8,9] due to the fiber structure of E → M . (Throughout the paper we use the index range convention i, j = 1, 2, .…”