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In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we calculate any change in Lagrangian density we observe that under the conditions of equation of motion being satisfied, it only changes by a four-divergence term.

If ${\cal L}(x) $ changes to $ {\cal L}(x) + \alpha \partial_\mu J^{\mu} (x) $ then action is invariant. But isn't this only in the case of extremization of action to obtain Euler-Lagrange equations.

Comparing this to $ \delta {\cal L}$

$$ \alpha \delta {\cal L} = \frac{\partial {\cal L}}{\partial \phi} (\alpha \delta \phi) + \frac{\partial {\cal L}}{\partial \partial_{\mu}\phi} \partial_{\mu}(\alpha \delta \phi) $$

$$= \alpha \partial_\mu(\frac{\partial {\cal L}}{\partial \partial_{\mu}\phi} \delta \phi) + \alpha [ \frac{\partial {\cal L}}{\partial \phi} - \partial_\mu (\frac{\partial {\cal L}}{\partial \partial_{\mu}\phi})] \delta \phi. $$

Getting the second term to zero assuming application of equations of motion. Doesn't this imply that the noether's current itself is zero, rather than its derivative? That is:

$$J^{\mu} (x) = \frac{\partial {\cal L}}{\partial \partial_{\mu}\phi} \delta \phi .$$

I add that my doubt is why changing ${\cal L}$ by a four divergence term lead to invariance of action globally when that idea itself was derived while extremizing the action which I assume is a local extremization and not a global one.

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3 Answers

Here's what I perceive to be a mathematically and logically precise presentation of the theorem, let me know if this helps.

Mathematical Preliminaries

First let me introduce some precise notation so that we don't encounter any issues with "infinitesimals" etc. Given a field $\phi$, let $\hat\phi(\alpha, x)$ denote a smooth one-parameter family of fields for which $\hat \phi(0, x) = \phi(x)$. We call this family a flow of $\phi$. Then we can define the variation of $\phi$ under this flow as the first order approximation to the change in $\phi$ as follows:

Definition 1. (Variation of field) $$ \delta\phi(x) = \frac{\partial\hat\phi}{\partial\alpha}(0,x) $$

This definition then implies the following expansion $$ \hat\phi(\alpha, x) = \phi(x) + \alpha\delta\phi(x) + \mathcal O(\alpha^2) $$ which makes contact with the notation in many physics books like Peskin and Schroeder.

Note: In my notation, $\delta\phi$ is NOT an "infinitesimal", it's the coefficient of the parameter $\alpha$ in the first order change in the field under the flow. I prefer to write things this way because I find that it leads to a lot less confusion.

Next, we define the variation of the Lagrangian under the flow as the coefficient of the change in $\mathcal L$ to first order in $\alpha$;

Definition 2. (Variation of Lagrangian density) $$ \delta\mathcal L(\phi(x), \partial_\mu\phi(x)) = \frac{\partial}{\partial\alpha}\mathcal L(\hat\phi(\alpha, x), \partial_\mu\hat\phi(\alpha, x))\Big|_{\alpha=0} $$

Given these definitions, I'll leave it to you to show

Lemma 1. For any variation of the fields $\phi$, the variation of the Lagrangian density satisfies \begin{align} \delta\mathcal L &= \left(\frac{\partial \mathcal L}{\partial\phi} - \partial_\mu\frac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\right)\delta\phi + \partial_\mu K^\mu,\qquad K^\mu = \frac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\delta\phi \end{align} You'll need to use (1) The chain rule for partial differentiation, (2) the fact $\delta(\partial_\mu\phi) = \partial_\mu\delta\phi$ which can be proven from the above definition of $\delta\phi$ and (3) the product rule for partial differentiation.

Noether's theorem in steps

  1. Let a particular flow $\hat\phi(\alpha, x)$ be given.

  2. Assume that for this particular flow, there exists some vector field $J^\mu\neq K^\mu$ such that $$ \delta\mathcal L = \partial_\mu J^\mu $$

  3. Notice, that for any field $\phi$ that satisfies the equation of motion, Lemma 1 tells us that $$ \delta \mathcal L = \partial_\mu K^\mu $$

  4. Define a vector field $j^\mu$ by $$ j^\mu = K^\mu - J^\mu $$

  5. Notice that for any field $\phi$ satisfying the equations of motion steps 2+ 3 + 4 imply $$ \partial_\mu j^\mu = 0 $$

Q.E.D.

Important Notes!!! If you follow the logic carefully, you'll see that $\delta L = \partial_\mu K^\mu$ only along the equations of motion. Also, part of the hypothesis of the theorem was that we found a $J^\mu$ that is not equal to $K^\mu$ for which $\delta\mathcal L = \partial_\mu J^\mu$. This ensures that $j^\mu$ defined in the end is not identically zero! In order to find such a $J^\mu$, you should not be using the equations of motion. You should be applying the given flow to the field and seeing what happens to it to first order in the "flow parameter" $\alpha$.

Cheers!

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What is the difference between a flow and a Homotopy? Apart from the fact that in the Homotopy the parameter lies on a compact interval and the "final" field is given (we usually say Homotopy between functions f and g) –  Barefeg Mar 15 '13 at 4:16
    
Well there really are a lot of things that are, in general, different; have you looked at en.wikipedia.org/wiki/Flow_(mathematics) and en.wikipedia.org/wiki/Homotopy and compared the definitions? –  joshphysics Mar 15 '13 at 5:04
    
Thanks joshphysics for such a clear detailed answer. You say we assume that there is some $J^\mu$ such that $\delta L = \partial_\mu J^\mu $ and then we get a noether's current. So incase we can't see such four divergence change in lagrangian we don't have any nice noether current expression anymore right? –  excitedaboutphysics Mar 15 '13 at 19:48
    
Yeah that's exactly right. The assumption of the existence of such a $J$ for which $\delta \mathcal L = \partial_\mu J^\mu$ is precisely what one means when on says that the variation being considered is a symmetry. Note also that $J^\mu = 0$ is a totally fine option. This would happen, for example, if the Lagrangian is invariant under the transformation under consideration. –  joshphysics Mar 15 '13 at 20:58
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This is the best derivation of Nother's theorem for physicists I've seen on the web. Most are hopelessly vague about what's being held constant, what a deformation is, and the difference between the quantities called $J^\mu$ and $K^\mu$ here. The others overwhelm physicists with needless technical complications, and lack generality. Well done. –  Jess Riedel Jan 30 at 20:06
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First, the differential operation is called "four-divergence" (the four-dimensional divergence), not "fourth divergence".

Second, the action obviously does change under a generic change of the fields i.e. if the change of the Lagrangian is not a four-divergence. It is a completely general functional of the fields so it does change.

Third, the action is stationary when the equations of motion are satisfied. These two conditions are ultimately equivalent. But when deriving the equations of motion, you can't assume that the equations of motion are satisfied. That would be a circular reasoning and you couldn't derive anything.

Fourth, yes, the equations of motion are being used when one derives $\partial_\mu J^\mu=0$ (i.e. action is stationary) but no, the derivation of the Noether's current does not imply that $J^\mu=0$. Your mistake is to confuse what is extremized. The equations of motion only mean $\delta S =0$, not $\delta L=0$ or $\delta{\mathcal L}=0$.

Fifth, your last equation is completely meaningless because the left hand side is finite but the right hand side is infinitesimal. Much like with problems of dimensional analysis (incompatible units), a manipulation with these expressions that obey the basic rules can never end up with a similar mismatch. Your previous "calculation" is also off because you're writing some bizarre expressions that are of second-order. In the variations, $\alpha$ itself is supposed to be infinitesimal, and in valid derivations, there is never a product of $\alpha$ with another infinitesimal quantity such as $\delta \phi$. In effect, your terms are of second-order (doubly infinitesimal) but your analysis doesn't have this higher-order precision so it's wrong.

I think it's a better idea to follow the actual correct derivation instead of your personal attempts to revise the functional calculus that you haven't mastered yet.

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Well I knew that it shouldn't be an infinitesimal but if we assume a significant change in the field, wouldn't that raise an issue with using the equations of motion since they are valid in the case of a local extremization. –  excitedaboutphysics Mar 14 '13 at 14:17
    
OK, I don't fully understand what you're trying to ask but you're failing. But both the equations of motion and the derivation of Noether's current only depend on infinitesimal variations of the fields. That doesn't mean that fields can't vary by a finite amount, but a finite (greater than infinitesimal variations) are irrelevant for the derivation of equations of motion; and they're irrelevant for the derivation of Noether's current (the current is linked to the Lie algebra of symmetries which correspond to group elements infinitesimally close to the identity). –  Luboš Motl Mar 15 '13 at 7:02
    
thanks Luboš Motl, this is exactly the kind of answer that i was aiming for: "the current is linked to the Lie algebra of symmetries which correspond to group elements infinitesimally close to the identity" . But that again pinches back on my insufficient understanding of math while trying to do physics. So though i appreciate the answer in some way but I don't completely understand what is meant by symmetries corresponding to group elements inf close to identity –  excitedaboutphysics Mar 15 '13 at 15:31
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@excitedaboutphysics, you don't need that much math to understand what he's saying. Modern treatments of QFT include an introductory chapter/appendices about group theory, Lie groups/algebras, representation theory, Clifford algebras, etc. Without that you are not going to get very far on QFT unless you only want to learn the rules to calculate amplitudes, etc. –  Barefeg Mar 15 '13 at 19:25
    
hi Barefeg, any texts you might refer to which give a decent introduction to such math required for qft. –  excitedaboutphysics Mar 16 '13 at 14:26
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Lagrangian invariant upto a overall 4-divergence and Euler Lagrange equation they together give you $\partial_{\mu}\left(\frac{\partial L}{\partial\partial_{\mu}\phi}\delta\phi\right)=\partial_{\mu}\left(J^{\mu}(x)\right)$

Now if I understood you correctly you are saying essentially if $\dfrac{df}{dx}=\dfrac{dg}{dx}$ then $f=g$ which in general is not true all one can say is $\dfrac{d(f-g)}{dx}=0$ i.e. $f-g=constant$.

Similarly here $\partial_{\mu}\left(J^{\mu}(x)-\frac{\partial L}{\partial\partial_{\mu}\phi}\delta\phi\right)=0$ would imply

$j^{\mu}(x)=J^{\mu}(x)-\frac{\partial L}{\partial\partial_{\mu}\phi}\delta\phi$ such that $\partial_{\mu}(j^{\mu}(x))=0$

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Well that is fine, I understand that but I don't understand why one should write $J_\mu$ at all in the change of lagrangian and then equate it and then define a noether's current –  excitedaboutphysics Mar 14 '13 at 14:13
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