Abstract
We define a family of observables for abelian Yang-Mills fields associated to compact regions U ⊆ M with smooth boundary in Riemannian manifolds. Each observable is parametrized by a first variation of solutions and arises as the integration of gauge invariant conserved current along admissible hypersurfaces contained in the region. The Poisson bracket uses the integration of a canonical multisymplectic current.
Highlights
IntroductionIn Classical Covariant Field Theory two desirable conditions are required for a family of observables: In one side we require this function to separate solutions of the Euler-Lagrange equations
In Classical Covariant Field Theory two desirable conditions are required for a family of observables: In one side we require this function to separate solutions of the Euler-Lagrange equations.On the other hand, we need the Jacobi identity in order to have a Lie (Poisson) bracket
Under locality assumptions, Jacobi identity is well established but generically there are few observables associated with conservation laws given by Noether’s First Theorem, see for instance [3]
Summary
In Classical Covariant Field Theory two desirable conditions are required for a family of observables: In one side we require this function to separate solutions of the Euler-Lagrange equations. For a divergence (non-autonomous) free vector field, ξ = ξ (t) ∈ X∂Σ (Σ) in a three-dimensional Riemannian manifold Σ tangent to the boundary ∂Σ, helicity is defined as g(v, ξ )νΣ (1). If we adopt v ∈ X∂Σ (Σ) divergence-free or d?Σ (α) = 0, respectively, the property of isovorticity holds for v(t) for the magnetic potential, as well as for any solution of the Euler equation of hydrodynamics This means that ξ (t2 ) can be constructed as the image of ξ (t1 ) under a diffeomorphism and if we consider a space-time domain Σ × [t1 , t2 ], helicity does not depend on the parameter t of the non-autonomous flow.
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