Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.

$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$

However there is a more general class of transformations which leave the equations of motion invariant, and that is a divergence term along with an overall scaling:

$\mathcal{L} \rightarrow \alpha\mathcal{L} + \partial_{\mu}f^{\mu}$

but Noether's theorem does not seem to deal with these types of symmetries, which are exhibited by, for example, the Klein-Gordon Lagrangian:

$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m^2\phi^2$

under the transformation $\phi \rightarrow \alpha\phi$ (giving $\mathcal{L} \rightarrow \alpha^2\mathcal{L}$).

The action similarly transforms as $S \rightarrow \alpha^2 S$. I feel this is important to emphasize as there are cases in which the Lagrangian scales by a factor but the action remains strictly invariant. This is not the case here, as the action scales by $\alpha^2$ but the equations of motion and extrema of the action are invariant, which is ultimately the only physical thing that matters.

Is there a generalization to Noether's theorem for these "scaling-type" symmetries? I.e. is there a conserved quantity/current arising from scale invariance?

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.

$$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$

However there is a more general class of transformations which leave the equations of motion invariant, and that is a divergence term along with an overall scaling:

$$\mathcal{L} \rightarrow \alpha\mathcal{L} + \partial_{\mu}f^{\mu}$$

but Noether's theorem does not seem to deal with these types of symmetries, which are exhibited by, for example, the Klein-Gordon Lagrangian:

$$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m^2\phi^2$$

under the transformation $\phi \rightarrow \alpha\phi$ (giving $\mathcal{L} \rightarrow \alpha^2\mathcal{L}$).

The action similarly transforms as $S \rightarrow \alpha^2 S$. I feel this is important to emphasize as there are cases in which the Lagrangian scales by a factor but the action remains strictly invariant. This is not the case here, as the action scales by $\alpha^2$ but the equations of motion and extrema of the action are invariant, which is ultimately the only physical thing that matters.

Is there a generalization to Noether's theorem for these "scaling-type" symmetries? I.e. is there a conserved quantity/current arising from scale invariance?

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.

$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$

However there is a more general class of transformations which leave the equations of motion invariant, and that is a divergence term along with an overall scaling:

$\mathcal{L} \rightarrow \alpha\mathcal{L} + \partial_{\mu}f^{\mu}$

but Noether's theorem does not seem to deal with these types of symmetries, which are exhibited by, for example, the Klein-Gordon Lagrangian:

$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m^2\phi^2$

under the transformation $\phi \rightarrow \alpha\phi$ (giving $\mathcal{L} \rightarrow \alpha^2\mathcal{L}$).

Is there a generalization to Noether's theorem for these "scaling-type" symmetries? I.e. is there a conserved quantity/current arising from scale invariance?

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.

$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$

However there is a more general class of transformations which leave the equations of motion invariant, and that is a divergence term along with an overall scaling:

$\mathcal{L} \rightarrow \alpha\mathcal{L} + \partial_{\mu}f^{\mu}$

but Noether's theorem does not seem to deal with these types of symmetries, which are exhibited by, for example, the Klein-Gordon Lagrangian:

$\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m^2\phi^2$

under the transformation $\phi \rightarrow \alpha\phi$ (giving $\mathcal{L} \rightarrow \alpha^2\mathcal{L}$).

The action similarly transforms as $S \rightarrow \alpha^2 S$. I feel this is important to emphasize as there are cases in which the Lagrangian scales by a factor but the action remains strictly invariant. This is not the case here, as the action scales by $\alpha^2$ but the equations of motion and extrema of the action are invariant, which is ultimately the only physical thing that matters.

Is there a generalization to Noether's theorem for these "scaling-type" symmetries? I.e. is there a conserved quantity/current arising from scale invariance?