# Noether charge and equivalence class of Noether currents

Let some field theory be described by the Lagrangian density $${\cal L}$$ on spacetime. Noether's first theorem asserts that given a quasisymmetry $$\hat{\delta}\phi$$ there is a class of currents $$j^\mu$$ such that $$\partial_\mu j^\mu =E\hat{\delta}\phi\tag{1}$$ where $$E$$ are the equations of motion.

Two currents in the same class differ by a trivial current which can be either (1) a current that identically vanishes on-shell, (2) a current which is conserved even off-shell and (3) any combination of these.

Noether's second theorem states that when the quasisymmetry is local, i.e., parameterized by a function $$f$$, one such current associated to it, verifying (1), is some $$S^\mu$$ which vanishes on-shell $$S^\mu\approx 0$$. Therefore any other current in the class $$j^\mu$$ verifies

$$\partial_\mu(j^\mu - S^\mu)=0\Longrightarrow j^\mu=S^\mu+\partial_\nu k^{[\mu\nu]}\tag{2}.$$

In this paper by G. Barnich & F. Brandt the authors say that this gives rise to a "Noether charge puzzle":

Note that the superpotential is completely arbitrary because it drops out of (1.1) [Eq. (1) of this post] owing to $$\partial_\mu\partial_\nu k^{[\nu\mu]}=0$$. This implies that the Noether charge corresponding $$\delta_f$$ is undefined because it is given by the surface integral of an arbitrary $$(n-2)$$ form.

1. How the same problem does not happen for a global symmetry for which Noether's second theorem does not apply? I mean, the current class of such symmetry is not the trivial one anymore. Still, if $$j^\mu$$ is a current in the class we can always add some $$\partial_\nu k^{[\mu\nu]}$$. How is this any different than the local case?

2. More importantly if we define the Noether charge by integrating $$j^\mu$$ over a Cauchy surface $$\Sigma$$ is the charge, in the global case, well-defined? Because I see the same issue taking place in the global case. Let $$j^\mu$$ be a current in the class. We get another one by adding $$\partial_\nu k^{[\mu\nu]}$$, then the charge changes by a boundary term at $$\partial \Sigma$$.