Noether’s second theorem – Part I

In her most famous (among physicists) paper [1], the Artemisia Gentileschi of mathematics, Emmy Noether, did not only prove the theorem which had probably the greatest influence over the twentieth century physics, but also proved one which had probably one of the smallest. The first is obviously the one best known as Noether’s theorem, but some people refers to it as Noether’s first theorem, to distinguish it from the theorem we are going to talk about today.

The theorem

Consider $n$ fields labeled with the subscript $k$ , whose dynamics is described by the action funcional:

$S = \int d^4 x \mathcal{L} [\phi_k(x)] ~,$

which is invariant under a local transformation of the fields, written in infinitesimal form as:

$\phi'_k (x) \simeq \phi_k (x) + \sum_\alpha \left( U_{k,\alpha}[\phi] ~ \Theta_\alpha (x) + V^\mu_{k,\alpha} [\phi] ~ \partial_\mu \Theta_\alpha (x) \right) ~ ,$

where we assumed that the transformation depends only on the $m$ functions $\Theta_\alpha (x)$ and their first derivatives $\partial_\mu \Theta_\alpha (x)$ . The variation of the lagrangian under the above transformation is (sum over repeated indexes is understood, also for the $\alpha$ and $k$ index):

$\delta \mathcal{L} = \frac{\delta \mathcal{L}}{\delta \phi_k} \delta \phi_k + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu \delta \phi_k =$ $\frac{\delta \mathcal{L}}{\delta \phi_k} U_{k,\alpha}[\phi] ~ \Theta_\alpha(x) + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu ( U_{k,\alpha}[\phi] ~ \Theta_\alpha (x)) +$ $\frac{\delta \mathcal{L}}{\delta \phi_k} V^\nu_{k,\alpha}[\phi] ~ \partial_\nu\Theta_\alpha(x) + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu ( V^\nu_{k,\alpha}[\phi] ~ \partial_\nu \Theta_\alpha (x)) ~,$

factorizing the functions $\Theta_\alpha (x)$ and its first and second derivatives, with some integrations by parts, one obtains:

$\delta \mathcal{L} = \left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} U_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu U_{k,\alpha}[\phi] \right\} \Theta_\alpha (x) +$
$\left\{ \frac{\delta \mathcal{L}}{\delta \partial_\nu \phi_k} U_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \phi_k} V^\nu_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu V^\nu_{k,\alpha}[\phi] \right\} \partial_\nu \Theta_\alpha (x) +$
$\frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} V^\nu_{k,\alpha}[\phi] ~ \partial_\mu \partial_\nu \Theta_\alpha (x) ~ ,$

the invariance of the lagrangian $\delta \mathcal{L} = 0$ under a generic local transformation (defined by an arbitrary choice of the functions $\Theta_\alpha (x)$ ) imply the following constraints (the last of the three equations follows from the symmetry of $\partial_\mu \partial_\nu \Theta_\alpha (x)$ with respect to the indices $\mu$ , $\nu$ ):

$\frac{\delta \mathcal{L}}{\delta \phi_k} U_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu U_{k,\alpha}[\phi] = 0 ~,$
$\frac{\delta \mathcal{L}}{\delta \partial_\nu \phi_k} U_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \phi_k} V^\nu_{k,\alpha}[\phi] + \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \partial_\mu V^\nu_{k,\alpha}[\phi] = 0 ~,$
$\frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} V^\nu_{k,\alpha}[\phi] = - \frac{\delta \mathcal{L}}{\delta \partial_\nu \phi_k} V^\mu_{k,\alpha}[\phi] ~ .$

Let’s rationally rearrange the above equations. If we call:

$K^{\mu\nu}_\alpha = \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} V^\nu_{k,\alpha}[\phi] ~ .$

we have that $K^{\mu\nu}_\alpha = - K^{\nu\mu}_\alpha$ , and the second equation becomes:

$J^\nu_\alpha = \frac{\delta \mathcal{L}}{\delta \partial_\nu \phi_k} U_{k,\alpha}[\phi] + \left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \right) \right\} V^\nu_{k,\alpha}[\phi] = - \partial_\mu K^{\mu\nu}_\alpha ~,$

so that it express the conservation of the current $J^\nu_\alpha$ , since:

$J^\mu_\alpha + \partial_\nu K^{\nu\mu}_\alpha = 0~,$

and, from the antisymmetry of $K^{\nu\mu}_\alpha$ we have $\partial_\mu \partial_\nu K^{\nu\mu}_\alpha = 0$ , so that:

$\partial_\mu J^\mu_\alpha = 0 ~.$

It is noteworthy that this conservation law we found does not apply only on solutions of the Euler-Lagrange equations (some people would say that it holds off-shell).
The first equation of the trio can be rephrased in a more convenient way:

$\left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \right) \right\} U_{k,\alpha}[\phi] = - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} U_{k,\alpha}[\phi] \right) ~,$

and from the equation for $J^\mu_\alpha$ :

$- \partial_\nu \left( \frac{\delta \mathcal{L}}{\delta \partial_\nu \phi_k} U_{k,\alpha}[\phi] \right) = \partial_\nu \left( \left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \right) \right\} V^\nu_{k,\alpha}[\phi] + \partial_\mu K^{\mu\nu}_\alpha \right) =$ $\partial_\nu \left( \left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \right) \right\} V^\nu_{k,\alpha}[\phi] \right) ~,$

so that we obtain an equation in terms only of the Euler-Lagrange expressions $\Pi_k[\mathcal{L}] = \left\{ \frac{\delta \mathcal{L}}{\delta \phi_k} - \partial_\mu \left( \frac{\delta \mathcal{L}}{\delta \partial_\mu \phi_k} \right) \right\}$ :

$\Pi_k[\mathcal{L}] ~ U_{k,\alpha}[\phi] = \partial_\nu \left(\Pi_k[\mathcal{L}] ~ V^\nu_{k,\alpha}[\phi] \right) ~.$

The last equation express the main content of the theorem: it enforces the role of redundancies played by the symmetries. In fact the physical content of the theory, more than in the lagrangian, is contained in the equations of motion:

$\Pi_k[\mathcal{L}] = 0 ~ ,$

they simply state that for every field $\phi_k$ its corresponding Euler-Lagrange expression vanishes on the solutions (or on-shell). There are $n$ of these equations. But the equation before establishes $m$ relations (recall that $\alpha = 1 , 2, \dots , m$ ) one for each independent local symmetry between the $n$ Euler-Lagrange expressions. So there are only $n - m$ independent equations of motion.

References

[1] E. Noether’s 1919 paper: arXiv:physics/0503066v1 (in 1919 the arXiv didn’t have hep-th… thanks to M. A. Tavel for the translation)

F. Mercati

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6 Responses to “Noether’s second theorem – Part I”

vittorio Says:
December 17, 2008 at 1:21 pm
Astonishing!
ASAP you will be one of my links.
Vale, bone amice, vale.
V
Il barbarico re Says:
December 17, 2008 at 2:10 pm
Che dire…da paura (stessa)!
Il barbarico re Says:
December 17, 2008 at 2:18 pm
Altra cosa per quando vuoi fare le referenze in fondo alla pagina:
devi mettere un tag a name=qualchecosa tra parentesi acute e poi chiudi con /a tra parentesi acute nell’elemento di bibliografia in fondo e un tag a href=qualchecosa chiuso come sopra dove vuoi il link.
Ciao ciao
metumipsum Says:
December 17, 2008 at 3:15 pm
Grazie Carlo!

però mi hai fatto impazzire… lo potevi dire che nell’ href ci andava il maledetto asterisco!!

Comunque da metum…
Il barbarico re Says:
December 17, 2008 at 5:17 pm
asterisco?
metumipsum Says:
December 21, 2008 at 12:09 am
Si si fa così:
<a href=”#qualcosa” rel=”nofollow”> Blabla<\a>
…
<a name=”qualcosa” rel=”nofollow”> qualcosa<\a>

Metum Ipsum