Diffeomorphism Noether current identity and local charge

Let $M$ be an oriented smooth four-manifold, let $\pi :E\to M$ be the chosen smooth field bundle, and let $\mathcal{L}(\varphi )$ be a local Lagrangian four-form for its smooth sections, with a chosen first-variation decomposition $\delta \mathcal{L}=E_{\varphi }[\delta \varphi ]+d_M\theta (\varphi ;\delta \varphi )$. Let $\xi$ be a smooth spacetime vector field with local flow ${\mathrm{\Phi }}_t$. Suppose there is a smooth local lift ${\tilde{\mathrm{\Phi }}}_t$ to fibre-preserving local diffeomorphisms of $E$ over ${\mathrm{\Phi }}_t$, with ${\tilde{\mathrm{\Phi }}}_0=\mathrm{i}\mathrm{d}$ and the local flow composition law. On the common domain define its induced pullback action on sections by $(A_t\varphi )(x):={\tilde{\mathrm{\Phi }}}_{-t}(\varphi ({\mathrm{\Phi }}_t(x)))\mathrm{.}$ Write ${\delta }_{\xi }\varphi :={\frac{d}{dt}\mid }_{t=0}{A}_t\varphi ,$ the vertical field variation determined by this smooth family; at each $x$ it lies in $\ker (d\pi {)}_{\varphi (x)}$, the tangent space to the fibre $E_x$. In the tensor-field case where $A_t\varphi ={\mathrm{\Phi }}_t^{\ast }\varphi$, it is the Lie derivative ${\mathcal{L}}_{\xi }\varphi$ defined by the local-flow derivative. Assume the action preserves the chosen field domain, and assume covariance in the precise sense $\mathcal{L}(A_t\varphi )={\mathrm{\Phi }}_t^{\ast }\mathcal{L}(\varphi )$ wherever both sides are defined; thus its derivative satisfies ${\delta }_{\xi }\mathcal{L}={\mathcal{L}}_{\xi }\mathcal{L}$. Define the Noether current three-form

$J_{\xi }:=\theta (\varphi ;{\delta }_{\xi }\varphi )-{\iota }_{\xi }\mathcal{L}\mathrm{.}$

Then

$d_M{J}_{\xi }=-E_{\varphi }[{\delta }_{\xi }\varphi ]\mathrm{.}$

Here a solution means a field configuration $\varphi$ at which the Euler functional vanishes, that is, $E_{\varphi }[\eta ]=0$ for every smooth variation $\eta$ in its domain. Thus $J_{\xi }$ is closed on solutions. For every chart $\chi :U\to V\subseteq {\mathbb{R}}^4$ whose open image $V$ is star-shaped, a solution has a local two-form $Q_{\xi }\in {\mathrm{\Omega }}^2(U)$ with $J_{\xi }{\mathrm{\mid }}_U=d_M{Q}_{\xi }$.

Some Properties of Noether Charge, §4

Advanced Lectures on General Relativity, §1.3.3

Covariant phase space with boundaries, §§2.3–2.4

Covariant field families and variations

Covariant presymplectic current

Closed differential forms are exact on star-shaped coordinate domains

Cartan's magic formula

The Lie derivative of a differential form

The Lie derivative of a tensor field

The pullback of a differential form

The exterior derivative commutes with pullback

The primitive is local. This proposition gives no global Noether charge without a separate cohomological argument. It asserts a classical identity only; it constructs no measure, Hilbert space, or anomaly-free quantization.

The current sign in the cited sources is not uniform. Iyer--Wald's equation (49) and Harlow--Wu's equation (2.44) use the current defined here. Compère--Fiorucci's equation (1.59) uses $J_{\xi }^{\mathrm{C}\mathrm{F}}:={\iota }_{\xi }\mathcal{L}-\theta (\varphi ;{\mathcal{L}}_{\xi }\varphi )$, which is $-J_{\xi }$ in the tensor-field case; its on-shell current identity and local primitive therefore require the same sign conversion. The proof above uses only the explicit local hypotheses stated here.