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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?

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This sounds like it's related to Ward identities, doesn't it? – A friendly helper Apr 18 '13 at 12:27
More on Noether's second theorem: – Qmechanic Jun 19 '13 at 23:55

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.


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


$^1$ 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.)

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