What is the mathematical meaning and condition for the Noether's charge to be conserved locally?

Question

Noether's theorem for continuous symmetries gives rise to a conserved charge. Mathematiccally, what does it mean for a charge to be (i) locally and (ii) globally conserved? I think, a locally conserved charge needs not arise only from a local gauge symmetry. For example, conservation of energy is a local conservation law which arises from translational invariance in time.

Answer 1

A quantity $Q$ with a density $\rho(t,x)$ (such that $Q(V) = \int_V \rho(t,x)\mathrm{d}x$ is the amount of $Q$ inside $V$) is globally conserved if the global amount of $Q$ is constant, i.e. $\int\rho(x)$ does not depend on time if we integrate over the entire space $M$: $$\frac{\mathrm{d}}{\mathrm{d}t}\int_M \rho(t,x)\mathrm{d}x=0\tag{1}$$ Such a quantity is locally conserved if for every volume $V$ we have that $$\frac{\mathrm{d}}{\mathrm{d}t}\rho = \nabla j,\tag{2}$$ where $j$ is the current density of $Q$, i.e. the function says how much of $Q$ crosses a given unit area in a given unit time.

Noether's theorem says that to every quasi-symmetry of the action there exists a locally conserved quantity. The notion of "local conservation" is unrelated to the notion of "local symmetries" (i.e. gauge symmetries).

There is a slightly different notion of local vs. global conservation in the context of e.g. general relativity (The following is paraphrased from this article by Baez, for more discussion on global conservation of energy in GR, see this question and its linked questions): Here, you'll often find people stating that something (like energy or momentum) is locally but not globally conserved, and what it means is that while the differential statement like $\dot{\rho} = \nabla j$ holds, its classically equivalent integral formulation $$\int_V \rho(t_1,x)\mathrm{d}x - \int_V \rho(t_2,x)\mathrm{d}x = \int_{\partial V}\int_{t_0}^{t_1} \rho(t,x)\mathrm{d}t\mathrm{d}S\tag{3}$$ doesn't hold if $\rho,Q$ are not scalars, but instead vectors/tensors as in the case of the energy-momentum tensor. In this case, the differential equation is a "local" statement, and the integral formulation is a "global" statement.