# About “conserved quantities” in a diffeomorphism-invariant theory by Wald and Zoupas

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.

## 2 Answers

It seems Wald & Zoupas mean

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

• Thanks! But $\delta\boldsymbol Q[\xi]$ should mean $\delta(\boldsymbol Q[\xi])=\mathscr L_\eta(\boldsymbol Q[\xi])$. This can be seen from Wald's earlier paper arxiv.org/abs/gr-qc/9307038. Eq. (14) is related to what I am asking for here. There, $\delta$ acts on $\boldsymbol j[\xi]$, not just on $\boldsymbol j$, otherwise, the right-hand side of Eq. (14) would have an extra term, something like $\boldsymbol j[\delta\xi]$. – Drake Marquis Jan 22 at 2:53
• In most of Wald et al's writings the 2 operations $\delta$ and $\mathscr{L}_{\eta}$ are indeed unrelated. However, they make an exception around eq. (33). – Qmechanic Jan 23 at 13:53

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.