A How was the symmetry used here? ("QFT and the SM" by Schwartz)

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TL;DR Summary
In this derivation of energy-momentum conservation, on what step the symmetry of space-time translation was used?
Here are steps.
Consider shift of field ##\phi## by a constant 4-vector ##\xi##:
(1) ##\phi(x) \rightarrow \phi(x+\xi)=\phi(x)+\xi^{\nu} \partial_{\nu} \phi(x) + ... ##
The infinitesimal transformation,
(2) ##\frac {\delta \phi} {\delta \xi^{\nu}} = \partial_{\nu} \phi##
and
(3) ##\frac {\delta \mathcal L} {\delta \xi^{\nu}} = \partial_{\nu} \mathcal L##
Using the E-L equations, the variation of Lagrangian is
(4) ##\frac {\delta \mathcal L[\phi, \partial_{\mu} \phi]} {\delta \xi^{\nu}}=\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \frac {\delta \phi} {\delta \xi^{\nu}})##
Using (2) and (3),
(5) ##\partial_{\nu} \mathcal L =\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi)##
or equivalently
(6) ##\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi - g_{\mu \nu} \mathcal L) = 0##
The conclusion is,
"The four symmetries have produced four Noether currents, one for each ##\nu##:
##\mathcal T_{\mu \nu} = \frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi - g_{\mu \nu} \mathcal L##
all of which are conserved: ##\partial_{\mu} \mathcal T_{\mu \nu}=0##."

My question: where in this derivation the assumption was used that the transformation is a symmetry?
 
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The assumption of symmetry is hidden in the use of EL equation. If you did not use EL equation, that would mean that ##\phi(x)## is treated as a non-dynamical fixed background, the explicit ##x##-dependence of which would violate the symmetry. Indeed, the EL equation can be written as
$$\frac{\delta A}{\delta \phi(x)}=0$$
where ##A## is the action, which can be interpreted as a statement that you only consider such ##\phi(x)## for which the action does "not depend" on ##\phi(x)##. Since the dependence on ##x## arises only from dependence on ##\phi(x)##, this means that EL equation expresses also independence on ##x##, which is nothing but the symmetry.
 
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Demystifier said:
The assumption of symmetry is hidden in the use of EL equation. If you did not use EL equation, that would mean that ##\phi(x)## is treated as a non-dynamical fixed background, the explicit ##x##-dependence of which would violate the symmetry. Indeed, the EL equation can be written as
$$\frac{\delta A}{\delta \phi(x)}=0$$
where ##A## is the action, which can be interpreted as a statement that you only consider such ##\phi(x)## for which the action does "not depend" on ##\phi(x)##. Since the dependence on ##x## arises only from dependence on ##\phi(x)##, this means that EL equation expresses also independence on ##x##, which is nothing but the symmetry.
Thank you.
I think also that the symmetry validates the equation (3), because this equation makes the variation of Lagrangian a total derivative, and this makes the variation of action vanish: $$\delta A = \int d^4 x \delta \mathcal L = \delta \xi^{\nu} \int d^4x \partial_{\nu} \mathcal L = 0$$ IOW, without the symmetry, we can't go from (4) to (5).
 
Hill said:
TL;DR Summary: In this derivation of energy-momentum conservation, on what step the symmetry of space-time translation was used?

Here are steps.
Consider shift of field ##\phi## by a constant 4-vector ##\xi##:
(1) ##\phi(x) \rightarrow \phi(x+\xi)=\phi(x)+\xi^{\nu} \partial_{\nu} \phi(x) + ... ##
The infinitesimal transformation,
(2) ##\frac {\delta \phi} {\delta \xi^{\nu}} = \partial_{\nu} \phi##
and
(3) ##\frac {\delta \mathcal L} {\delta \xi^{\nu}} = \partial_{\nu} \mathcal L##
Using the E-L equations, the variation of Lagrangian is
(4) ##\frac {\delta \mathcal L[\phi, \partial_{\mu} \phi]} {\delta \xi^{\nu}}=\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \frac {\delta \phi} {\delta \xi^{\nu}})##
Using (2) and (3),
(5) ##\partial_{\nu} \mathcal L =\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi)##
or equivalently
(6) ##\partial_{\mu} (\frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi - g_{\mu \nu} \mathcal L) = 0##
The conclusion is,
"The four symmetries have produced four Noether currents, one for each ##\nu##:
##\mathcal T_{\mu \nu} = \frac {\partial \mathcal L}{\partial (\partial_{\mu} \phi)} \partial_{\nu} \phi - g_{\mu \nu} \mathcal L##
all of which are conserved: ##\partial_{\mu} \mathcal T_{\mu \nu}=0##."

My question: where in this derivation the assumption was used that the transformation is a symmetry?
I believe that the symmetry is used in step 3. Under the translation symmetry the Lagrangian change is a total derivative. I think this is essentially what @Demystifier is saying.

Incidentally, i wrote a long blog post on this derivation but omitted this question. Thanks for bringing it up.
 
jbergman said:
I believe that the symmetry is used in step 3. Under the translation symmetry the Lagrangian change is a total derivative. I think this is essentially what @Demystifier is saying.

Incidentally, i wrote a long blog post on this derivation but omitted this question. Thanks for bringing it up.
Thank you. That was my understanding also, in the post #3.
 
jbergman said:
I believe that the symmetry is used in step 3. Under the translation symmetry the Lagrangian change is a total derivative. I think this is essentially what @Demystifier is saying.

Incidentally, i wrote a long blog post on this derivation but omitted this question. Thanks for bringing it up.
Step 3 is just saying the Lagrangian is a scalar (density) under translations, right?

I also find this stuff treacherous, and many textbooks completely miss these subtleties and make it look easy. My confusion for a long time was that these symmetries are derived "on shell", but then EL=0 for any variation of the field. And I know from experience that you can even confuse post docs working on string theory with this stuff.😋
 
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