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On page 28 of Pierre Ramond Field theory - A modern primer the following is written:

"we remark that a conserved current does not have a unique definition since we can always add to it the four-divergence of an antisymmetric tensor [...] Also since $j$ [the Noether current] is conserved only after use of the equations of motion we have the freedom to add to it any quantity which vanishes by virtue of the equations of motion".

I do not understand what he means by saying, any quantity which vanishes by virtue of the equations of motion.

Qmechanic
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2 Answers2

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In Noether's first theorem, the continuity equation$^1$ $$ d_{\mu} J^{\mu}~\approx~0 \tag{*}$$ is an on-shell equation, i.e. it holds if the EOMs [= Euler-Lagrange (EL) equations] are satisfied. It does not necessarily hold off-shell.

Hence we can modify the Noether current $J^{\mu}$ with

  1. terms that vanish on-shell, and/or

  2. terms of the form $d_{\nu}A^{\nu\mu}$, where $A^{\nu\mu}=-A^{\mu\nu}$ is an antisymmetric tensor,

without spoiling the continuity eq. (*).

--

$^1$ The $\approx$ symbol means equality modulo EOMs.

Qmechanic
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  • @Qmechanic I noticed you put a full derivative on the $J^{\mu}$ and $\partial_{\mu}A^{\mu\nu}$ term so it's $d_{\mu}J^{\mu}$ and $d_{\nu}A^{\nu\mu}$ Is that intentional? – joseph h Jul 04 '21 at 19:36
  • @joseph h: Yes, $d_{\mu}$ is a total rather than an explicit spacetime derivative, cf. e.g. this Phys.SE post. – Qmechanic Jul 04 '21 at 19:48
  • Very good. Sloppy notation is everywhere then! Thanks. – joseph h Jul 04 '21 at 20:11
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I can give you an example:

$$ S=\int dt \left(\dot x^+\dot x^--x^+ x^-\right) $$

has a conserved current associated with $x^{\pm}\rightarrow e^{\pm\rho} x^{\pm}$ given by

$$ j=x^+\dot x^--x^-\dot x^+ $$

This means that the current above will be conserved if the equations of motion are satisfied. Now if we add to this current a term of the form $\ddot x^++x^+$ the statement will stil be true, i.e. the current will stil be conserved. This is due to the fact that this term that I added is precisely the equations of motion.

Nogueira
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  • If I am not wrong, noether current is a function of spacetime coordinates only. In the example you gave, this means that the current is only a function of time, so the contribution from the equation of motions will be zero. –  Jul 04 '21 at 10:28
  • You can add more dimensions if you want. The current in d=0+1 is given by a function of time only since there is only time to propagate. I choose this case because it is simpler and captures the question. The idea is very simple, if you add things that vanish when the equations of motion are satisfied to the current you are actually adding zeros to the conservation law since the conservation is only true when the equations of motion are satified. – Nogueira Jul 04 '21 at 11:54
  • Incidentally, the same symmetry is found, albeit for subtly different reasons, in the FP-ghost sector. – J.G. Jul 04 '21 at 18:06
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    @J.G. I think this symmetry apear everywhere. It is a U(1) symmetry. – Nogueira Jul 04 '21 at 20:55