Spacetime translations and general Lagrangian density for Field Theory

[Paulpaulpa]

Homework Statement

Find the variation of a general Lagrangian density, independent of spacetime coordinates, under a translation in space time.

Relevant Equations

The space time translation 4-vector is $e^{\mu}$ and we have $\frac{\partial \mathcal{L}}{\partial x}=0$ with $\mathcal{L}$ the Lagrangian density

In Sydney Coleman Lectures on Quantum field Theory (p48), he finds : $$D\mathcal{L} = e^{\mu} \partial _{\mu} \mathcal{L}$$

My calulation, with $\phi$ my field and the variation of the field under space time tranlation $D\phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}$ :

$$D\mathcal{L} = \frac{\partial \mathcal{L}}{\partial \phi}D\phi + \frac{\partial \mathcal{L}}{\partial \partial _{\nu} \phi} \partial _{\nu} D\phi$$

Then using the Euler Lagrange equations $\frac{\partial\mathcal{L}}{\partial\phi}=\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)$

I find $$D\mathcal{L} =
\partial_\mu
\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)
D\phi + \frac{\partial \mathcal{L}}{\partial \partial _{\nu} \phi} \partial _{\nu} D\phi$$

Then using the fact that $\partial _{\mu} (ab) = \partial _{\mu} (a)b + a\partial _{\mu} (b)$

$$D\mathcal{L} = \partial _{\nu} (\frac{\partial \mathcal{L}}{\partial \partial _{\nu} \phi} D\phi)$$

I don't understand where I am wrong. Moreover how can $\partial _{\mu} \mathcal{L}$ be any different than zero since the lagrangian doesn't depend explicitely on spacetime ?

 

[LCSphysicist]

$$\delta L = \frac{\partial L}{\partial \phi} \delta \phi + \frac{\partial L}{\partial (\partial_{\mu} \phi)} \delta (\partial_{\mu} \phi)$$
As you pointed, first term is zero
$$\delta L = \frac{\partial L}{\partial (\partial_{\mu} \phi)} (\partial_{\mu} \delta \phi)$$
$$\delta \phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}$$

$$\delta L = \frac{\partial L}{\partial (\partial_{\mu} \phi)} \partial_{\mu} (e^{\mu}{\partial_{\mu} \phi}) = e^{\mu} \frac{\partial L}{\partial (\partial_{\mu} \phi)} \partial_{\mu}{\partial_{\mu} \phi} = e^{\mu} \partial_{\mu} L$$

I used chain rule.

 

[Paulpaulpa]

$$\delta L = \frac{\partial L}{\partial \phi} \delta \phi + \frac{\partial L}{\partial (\partial_{\mu} \phi)} \delta (\partial_{\mu} \phi)$$
As you pointed, first term is zero
$$\delta L = \frac{\partial L}{\partial (\partial_{\mu} \phi)} (\partial_{\mu} \delta \phi)$$
$$\delta \phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}$$

$$\delta L = \frac{\partial L}{\partial (\partial_{\mu} \phi)} \partial_{\mu} (e^{\mu}{\partial_{\mu} \phi}) = e^{\mu} \frac{\partial L}{\partial (\partial_{\mu} \phi)} \partial_{\mu}{\partial_{\mu} \phi} = e^{\mu} \partial_{\mu} L$$

I used chain rule.

Thanks for you answer. The first term is zero because of the equation of motion and the fact that
$\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right) = 0$ since $\mathcal{L}$ do not depend on x. Is that right ? But why is $\partial_\mu \mathcal{L}$ different than zero ? $\mathcal{L}$ is supposed to be independant of x. Or should I interpret $\partial _{\mu} \mathcal{L}$ as a total derivative and not a partial one ?

 

[samalkhaiat]

I don't understand where I am wrong. Moreover how can $\partial _{\mu} \mathcal{L}$ be any different than zero since the lagrangian doesn't depend explicitely on spacetime ?

1) $$\partial_{\nu}\mathcal{L} = \frac{\partial \mathcal{L}}{\partial x^{\nu}}|_{\phi , \partial \phi} + \frac{\partial \mathcal{L}}{\partial \phi}\partial_{\nu}\phi + \frac{\partial \mathcal{L}}{\partial \partial_{\mu}\phi}\partial_{\nu}\partial_{\mu}\phi .$$ The first term on the RHS vanishes when $\mathcal{L}$ has no explicit dependence on $x^{\mu}$.

2) Don’t use the equation of motion when you want to find the change in the Lagrangian under a specific (given) transformation of the coordinates. Under infinitesimal translation, $x \to x - e$, the field changes according to $$\delta \phi = e^{\nu}\partial_{\nu}\phi .$$ This induces the following (infinitesimal) change in the Lagrangian: $$\delta \mathcal{L} = e^{\nu} \left(\frac{\partial \mathcal{L}}{\partial \phi}\partial_{\nu}\phi + \frac{\partial \mathcal{L}}{\partial \partial_{\mu}\phi}\partial_{\nu}\partial_{\mu}\phi \right) .$$ Since $\mathcal{L}$ does not depend (explicitly) on $x$, the above is written as $$\delta \mathcal{L} = e^{\mu}\partial_{\mu}\mathcal{L} . \ \ \ \ \ \ (2)$$ This is called “off-shell” change in the Lagrangian, meaning that Eq(2) has been obtained without using the equation of motion.

3) An arbitrary (infinitesimal) change in the field $\delta \phi$ induces the following infinitesimal change in the Lagrangian: $$\delta \mathcal{L} = \left[ \frac{\partial \mathcal{L}}{\partial \phi} - \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)} \right) \right] \delta \phi + \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi)} \delta \phi \right) .$$ When you use the equation of motion, you get the so-called “on-shell” change in the Lagrangian: $$\delta \mathcal{L} = \partial_{\mu}\left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi)} \delta \phi \right) . \ \ \ \ \ \ \ \ \ (3)$$

4) In this method, the (Noether) symmetry current is obtained by equating the off-shell variation Eq(2) with the on-shell variation Eq(3) with $\delta \phi = e^{\nu}\partial_{\nu}\phi$: $$\partial_{\mu} \left( \frac{\partial \mathcal{L}}{\partial (\partial_{\mu}\phi )} \partial_{\nu}\phi - \delta^{\mu}_{\nu} \mathcal{L} \right) = 0 .$$

 

[Paulpaulpa]

1) $$\partial_{\nu}\mathcal{L} = \frac{\partial \mathcal{L}}{\partial x^{\nu}}|_{\phi , \partial \phi} + \frac{\partial \mathcal{L}}{\partial \phi}\partial_{\nu}\phi + \frac{\partial \mathcal{L}}{\partial \partial_{\mu}\phi}\partial_{\nu}\partial_{\mu}\phi .$$ The first term on the RHS vanishes when $\mathcal{L}$ has no explicit dependence on $x^{\mu}$.

Thank you very much this is much more clear. I always noted total derivative $d \mathcal{L}$ so the $\partial _{\nu}\mathcal{L}$ confused me a lot. Your explanation of Noether's Current is more understandable than the one in my lecture.

 

[realanswers]

$$\delta L = \frac{\partial L}{\partial \phi} \delta \phi + \frac{\partial L}{\partial (\partial_{\mu} \phi)} \delta (\partial_{\mu} \phi)$$
As you pointed, first term is zero
$$\delta L = \frac{\partial L}{\partial (\partial_{\mu} \phi)} (\partial_{\mu} \delta \phi)$$
$$\delta \phi = e^{\mu} \frac{\partial \phi}{\partial x^{\mu}}$$

I used chain rule.

Where was the first term shown to be 0 / why would it be 0, this seems incorrect...?

I get your second reply specifying the off-shell condition but I do not see where you showed that $\frac{\partial L}{\partial \phi} \delta \phi= 0$