Noether's identities

Question

I have some questions about the Noether's second theorem (generally not covered by field theory books):

1. What is the most general Noether identity for (classical) field theories?

2. Why are Noether identities important for quantum field theories?

Answer 1

Let us use Einstein's summation convention, DeWitt's condensed notation, and follow Ref. 1. Let there be given an action $S_0[\varphi]$ for a classical field theory. Let

$$\tag{17.1a} \delta\varphi^i~=~ R^i{}_{\alpha} \varepsilon^{\alpha} $$

be infinitesimal gauge transformations. Here $\varepsilon^{\alpha}$ are infinitesimal gauge parameters, and $R^i{}_{\alpha}$ are gauge generators. [Expanding DeWitt's condensed notations, eq. (17.1a) becomes

$$\tag{17.1a'} \delta\varphi^i(x)~=~\int\! d^dy~ R^i{}_{\alpha}(x,y) \varepsilon^{\alpha}(y) $$

In local field theories, $R^i{}_{\alpha}(x,y)$ are typically $\varphi$-dependent integral kernels consisting of differential operators of finite order acting on delta functions $\delta^d(x-y)$.] The Noether identities$^1$

$$\tag{17.8} \frac{\delta^{R} S_0}{\delta\varphi^i}R^i{}_{\alpha}~=~0 $$

encode the gauge invariance of the action

$$ \delta S_0~=~0. $$

The gauge generator $R^i{}_{\alpha}$ form a gauge algebra, which may be a reducible gauge algebra ("gauge transformation of gauge transformations leading to ghosts-for-ghosts"), or an open gauge algebra ("gauge algebra closes only on-shell"). Quantization of the most general gauge algebras (and the question of e.g. renormalization) is in general best studied via the Batalin-Vilkovisky formalism. Here the analogue of Zinn-Justin equation is encoded via the classical master equation

$$\tag{17.27c} (S,S)~=~0 ,$$

where $S$ is the so-called minimal proper action, whose first terms reads

$$\tag{17.28d} S~=~S_0 + \varphi^{*}_i R^i{}_{\alpha} c^{\alpha}+\ldots.$$

Here $(\cdot,\cdot)$ is the antibracket; $\varphi^{*}_i$ are antifields; and $c^{\alpha}$ are Faddeev-Popov ghosts. The classical master equation (17.27c) can be expanded in so-called antifield number into a tower of equations. The first equation in this tower is precisely the Noether identities (17.8).

Finally, let us mention that Noether identities are closely related to Ward identities.

References:

1. Henneaux and Teitelboim, Quantization of Gauge Systems 1994, Chap. 17.

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$^1$ We do not completely adapt the sign conventions of Ref. 1. The derivative in eq. (17.8) acts in our convention from the right, hence the superscript "$R$" in eq. (17.8). (This sign becomes important if the original fields $\varphi^i$ are Grassmann-odd.)