Invariant Lagrangian or action

[Physiana]

"invariant" Lagrangian or action

Hello everyone,

I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question...

Are invariance of the Lagrangian under a transformation and invariance of the action equivalent to each other? or even the same?

Physiana

P.S.: I know it sounds stupid. But I was wondering if there is a difference in physics by demanding the Lagrangian being invariant or just transforming in a certain way. As for example for time dilatation $t \rightarrow t' = \lambda t$ the Lagrangian transforms as$L \rightarrow L'= \frac{L}{\lambda}$ which is not "exactly" invariant, although "a" is constant and probably does not really matter. So I wondered if there are more transformations that "change" the Lagrangian but leave the action invariant.

 

[Tomsk]

They're not quite the same, the Lagrangian can vary by a total derivative which would leave the action invariant, since it can be integrated over the boundary by stokes' theorem, and it is normally assumed fields all go to zero at infinity.

The variation of the lagrangian $\delta L$ contributes one term to the Noether current of that symmetry.

 

[Physiana]

So there is a difference between a Lagrangian being invariant and it "transforming as".

Does it say anything about the underlying symmetry, if a Lagrangian is invariant or "only" transforming as? (e.g. space-time, external, internal)

In field theories one generally speaks of invariance, right?

It is just, I have read too much these days and I ram starting to get really confused and loose what I believed to know/ understand.

 

[Physiana]

To specify; For space time transformations I get a $\delta L$, while for all the symmetries of the Standard model (QCD, GWS and chiral) the Lagrangian remains invariant. Space time symmetries are external symmetries, while QCD, GWS and chiral symmetries are internal symmetries. So can I generalize the above "observed" transformation properties of the Lagrangian to all external resp. internal symmetry transformations?

I just ask because I am writing my thesis and I am not exactly sure which words to use and it is important to me to be as clear as possible in my use of words.

 

[samalkhaiat]

To specify; For space time transformations I get a $\delta L$, while for all the symmetries of the Standard model (QCD, GWS and chiral) the Lagrangian remains invariant. Space time symmetries are external symmetries, while QCD, GWS and chiral symmetries are internal symmetries. So can I generalize the above "observed" transformation properties of the Lagrangian to all external resp. internal symmetry transformations?

I just ask because I am writing my thesis and I am not exactly sure which words to use and it is important to me to be as clear as possible in my use of words.

Under a spacetime symmetry the action is invariant if and only if

$$\delta \mathcal{L} + \partial_{\mu}(\delta x^{\mu} \mathcal{L}) = 0$$

For internal symmetries; $\delta x^{\mu} = 0$, therefore, the action is invariant if and only if the Lagrangian is invariant. See post #12 in

www.physicsforums.com/showthread.php?t=172461

regards

sam

 

[Physiana]

:) Thank you.