About "conserved quantities" in a diffeomorphism-invariant theory by Wald and Zoupas

Question

In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism.

Let $\xi^a$ be a vector field on the $n$-dimensional spacetime manifold $M$. Physical fields are collectively represented by $\phi$, which could be the metric $g_{ab}$ and the 4-potential $A_a$, etc.. $\xi^a$ will generate an infinitesimal diffeomorphism, which is a local symmetry of the theory, so the variation of the Lagrangian $n$-form $\boldsymbol L$ is

$$\mathscr L_\xi\boldsymbol L=\boldsymbol E\hat\delta\phi+d\boldsymbol\Theta(\phi,\mathscr L_\xi\phi)=d(i_\xi\boldsymbol L),$$

where $\hat\delta\phi=\mathscr L_\xi\phi$, $\boldsymbol E$ represent the equations of motion, $d$ means the exterior derivative, $\boldsymbol\Theta$ is a $(n-1)$-form called the presymplectic potential current, $i_\xi$ refers to the interior product.

The Noether current $(n-1)$-form $\boldsymbol j$ is given by

$$\boldsymbol j[\xi]=\boldsymbol\Theta(\phi,\mathscr L_\xi\phi)-i_\xi\boldsymbol L.\tag{9}$$

On shell, one can show that $d\boldsymbol j=0$ and locally, one can find a $(n-2)$-form $\boldsymbol Q[\xi]$ such that $\boldsymbol j[\xi]=d\boldsymbol Q[\xi]$. $\boldsymbol Q$ is the Noether charge $(n-2)$-form. In this reference, the general form of $\boldsymbol Q$ was given in Proposition 4.1, which is really complicated.

Now, in Wald & Zoupas, a different vector field $\eta^a$ is considered. Its effect on $\boldsymbol Q$ is given by Eq. (33),

$$\delta\boldsymbol Q[\xi]=\mathscr L_\eta\boldsymbol Q[\xi]-\boldsymbol Q[\mathscr L_\eta\xi].\tag{33}$$

However, I do not understand how this relation was obtained. It looks to me that the left-hand side is nothing but

$$\delta\boldsymbol Q[\xi]=\mathscr L_\eta(\boldsymbol Q[\xi]).$$

Clearly, this is not the option.

Please help me with this problem.

Answer 1

It seems Wald & Zoupas mean

$$(\delta\boldsymbol{Q})[\xi]~=~(\mathscr{L}_\eta\boldsymbol{Q})[\xi]~=~\mathscr{L}_\eta (\boldsymbol{Q}[\xi]) ~-~\boldsymbol{Q}[\mathscr{L}_\eta\xi].\tag{33}$$

Answer 2

I contacted R. Wald. He told me that the variation operation $\delta$ does not act on $\xi^a$. So

$$\delta\boldsymbol Q[\xi]\ne\mathscr L_\eta (\boldsymbol Q[\xi]).$$

But they wanted to use Lie derivative to carry out the actual computation, so they allowed $\delta$ acts on $\xi^a$, and then remove the action of $\delta$ on $\xi^a$ by subtracting $\boldsymbol Q[\mathscr L_\eta\xi]$ from $\mathscr L_\eta\boldsymbol Q[\xi]=\mathscr L_\eta(\boldsymbol Q[\xi])$. This is why Eq. (33) holds in Wald & Zoupas.