Generalized symmetries generating Noether currents and canonical conserved quantities

We determine the condition for a Noether-Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.


Introduction
As it is well known, invariance properties of field dynamics are an effective tool to understand a physical system without solving the equations themselves: the existence of conservation laws associated with symmetries of equations strongly simplifies their study and corresponding conserved currents along solutions (on-shell) appear to be the most significant for the description of a system. The Noether Theorems [15] stated that, for the description of systems the equations of which arise from a variational problem, off-shell conserved currents are even more relevant since, for a large class of field theories, potentials (also called superpotentials in natural and gauge-natural theories) of such currents provide conserved quantities such as energy, momenta and charges. Therefore, it turns out to be fundamental to understand whether conserved currents associated with invariance of equations could be identified with Noether conserved currents for a certain Lagrangian; in fact, a symmetry of a Lagrangian is also a symmetry of its Euler-Lagrange form, but the converse in general is not true. We are interested to this converse problem which belongs to aspects of inverse problems the calculus of variations. The expression for the socalled canonical Noether current associated with a Lagrangian is the outcome of an integration by parts, related with the precise form taken by the Lie derivative of a Lagrangian (i.e. the Lie derivative of an horizontal n-form, n being the dimension of the base manifold). Of course, the particular type of inverse problem we shall consider will strictly interplay with the formula for a variational Lie derivative; analogously to what happens in the study of variationality of equations, the solution to this problem involves a kind of variational integrating factor.
Consider then conserved currents associated with invariance properties of (locally) variational global field equations, i.e. with so-called generalized or Bessel-Hagen symmetries [1]. Noether currents for different local Lagrangian presentations and corresponding conserved currents associated with each local presentation have been characterized in [3,4,5,17]. There exist cohomological obstructions for such local currents be globalized and such obstructions are also related with the existence of global solutions for a given global field equation [6].
We refer to the geometric formulation of the calculus of variations as a subsequence of the de Rham sequence of differential forms on finite order prologations of fibered manifolds. We assume the r-th order prolongation of a fibered manifold π : Y → X, with dim X = n and dim Y = n + m, to be the configuration space; this means that fields are assumed to be (local) sections of π r : J r Y → X. Due to the affine bundle structure of π r+1 , which induces natural spittings in horizontal and vertical parts of vector fields, forms and of the exterior differential on J r Y. Let ρ be a q-form on J r Y; in particular we obtain a natural decomposition of the pullback by the affine projections of a q-form ρ, as (π r+1 r ) * ρ = q i=0 p i ρ, where p i ρ is the i-contact component of ρ (by definition a contact form is zero along any holonomic section of J r Y). For 1 ≤ q ≤ n, the representation mapping is just given by the horizontalization p 0 ρ = hρ. For q > n, say, q = n + k, it is clear that, in this case, p k ρ denotes the component of ρ with the lowest degree of contactness. Starting from this splitting one can define sheaves of contact forms Θ * r , suitably characterized by the kernel of p i [9]; the sheaves Θ * r form an exact subsequence of the de Rham sequence on J r Y and one can define the quotient sequence the r-th order variational sequence on Y → X which is an acyclic resolution of the constant sheaf IR Y ; see [9]. In the following, if ρ ∈ Λ q r , [ρ] ∈ V r q denotes the equivalence class of ρ modulo ker p i , i = 0, 1, . . . , m, q = n + i. The quotient sheaves V k r ≡ Λ k r /Θ k r in the variational sequence can be represented as sheaves of k-forms on jet spaces of higher order by the interior Euler operator which is uniquely intrinsically defined by the decomposition p k ρ = I(ρ) + p k dp k R(ρ), together with the properties where N = dim J r Y and P is the maximal degree of non trivial contact forms on J r Y (see e.g. [7,8,9,12,18], whereby also local coordinate expressions can be found). The representation sequence {0 → R * (V * r ) , E * }, is also exact and we have ). Currents are sheaf sections of V n−1 r and E n−1 = d H is the total divergence; Lagrangians are sections λ of V n r , while E n is called the Euler-Lagrange morphism; Sections η of V n+1 r are called source forms or also dynamical forms, E n+1 is called the Helmholtz morphism.

Currents associated with (locally) variational dynamical forms
In the following for the sake of generality, we will consider locally variational dynamical forms. We will denote by a subscript i the fact that in general a sheaf section is defined locally. A variational Lie derivative operator L jrΞ is well defined acting on the sections of sheaves in the variational sequence: it can be characterized as a quotient Cartan formula (which in fact provides a variation formula); the basic idea is to factorize modulo contact structures [10,11,12,13]. This enable us to define symmetries of classes of forms of any degree in the variational sequence and corresponding conservation theorems; see [2] and [14] for details. We shall study the interplay of such an operator with the representation by forms. Let j r Ξ be a projectable vector field on J r Y, ρ a q-form defined (locally) on J r Y. We have a commutative diagram defined byR q (L jrΞ [ρ]) .
This enables us to deal with ordinary Lie derivatives of forms on Λ q s , then apply the Cartan formula for differential forms, therefore return back to the classes of forms to obtain the following variational Cartan formulae [14].
Note that, for q = n − 1, the formula above defines a 'momentum' associated with a current, we shall denote such a momentum byp; it is clear that, for q < n, E q = d H .
Let for simplicity η λ i denote a global Euler-Lagrange class of forms for a (local) variational problem represented by (local) sheaf sections λ i . Proposition 1 (Noether Theorem I) reads Definition 1 A generalized symmetry of a (locally variational) dynamical form η λ i is a projectable vector field j s Ξ on J s Y such that L jsΞ η λ i = 0.
Since we assume η λ i to be closed, Proposition 2 reduces (case q = n+1) to L jsΞ η λ = E n (Ξ V η λ i ), and if j s Ξ is such that L jsΞ η λ i = 0, then E n (Ξ V η λ i ) = 0; therefore, locally we have Ξ V η λ = d H ν i . Notice that, although Ξ V η λ is global, in general it defines a non trivial cohomology class [3]; it is clear that ν i is a (local) current which is conserved on-shell (i.e. along critical sections). On the other hand, and independently (see [15]), we get locally L Ξ λ i = d H β i thus we can write Ξ V η λ + d H ( i − β i ) = 0, where i is the usual canonical Noether current.

Definition 2 We call the (local) current i − β i a Noether-Bessel-Hagen current.
A Noether-Bessel-Hagen current i − β i is a current associated with a generalized symmetry (conserved along critical sections); in [5,6] we proved that a Noether-Bessel-Hagen current is variationally equivalent to a global (conserved) current if and only if 0 = [Ξ V E n (λ i )] ∈ H n dR (Y).

Generalized symmetries generating Noether currents
In the following we investigate under which conditions a Noether-Bessel-Hagen current is variationally equivalent to a Noether conserved current for a suitable Lagrangian. We shall see that this is involved with the existence of a variationally trivial local Lagrangian d H µ i , and with a condition on the current associated with it.

Proposition 3 A Noether-Bessel-Hagen current λ i −β i associated with a generalized symmetry of η λ i is a Noether conserved current if and only if is of the form
Proof. From L jsΞ η λ i = 0, we get L jsΞ λ i = d H β i . It is easy to see that the current ) (see [7]).
Proof. As it is well known, along any section pulling back to zero Ξ V η λ i we get the on-shell conservation law d H ( λ i −β i ) = 0. If there exists a current µ i such that is closed on-shell. By an uniqueness argument, we see that the latter expression must be equal Remark 1 It turns out that, on-shell, a canonical potential of the Noether current λ i −d H µ i , then a corresponding canonical conserved quantity, is defined.
Remark 2 An off-shell exact Noether current associated with the invariance of λ i − d H µ i would be generated by a generalized symmetry j s Ξ such that Ξ V η λ i = 0; the corresponding cohomology class would be, therefore, trivial (see the discussion in [5,6]).