How can I find conserved current for a Lagrangian involving vector fields?

[phywithAK]

Homework Statement

I am stuck on a problem and wold love to see any insights that i can get about this. To also begin with i am a beginner in the course on quantum field theory and don't have much experience working with vector fields and have only done examples regarding scalar fields. It concerns with finding the symmetries of a massive vector field lagrangian.

Relevant Equations

$$L = -\partial_{\mu}A^{\nu} \partial^{\mu}A_{\nu}-M^2* A^{\nu}A_{\nu}$$

Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be :

$$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$

I have seen examples based on the electromagnetic fields and how to verify gauge invariance, but since i am very new to this i have not much idea how to begin looking for symmetries of such kind of lagrangian involving vector fields. To be frank i haven't proceeded much in this and i would really appreciate any hints on how to begin examining such problems. Thank you

 

[PeroK]

If you are a beginner in physics and have no experience of vector fields, then QFT is not the place to start!

PS you need two dollar signs either side of Latex.

 

[PeroK]

If you want to learn about symmetries in physics I would recommend the book by Neuenschwander:

https://link.springer.com/article/10.1007/s10701-011-9562-3

Note that this is still for advanced undergraduate students.

 

[phywithAK]

If you are a beginner in physics and have no experience of vector fields, then QFT is not the place to start!

PS you need two dollar signs either side of Latex.

About the latex: I realized that a few moments ago so i updated it.

I also updated about the equation of motion which i found as a first step to this problem, so kindly give a look into that too.

Also by beginner i meant i have just started a course in QFT and since than we have been introduced to only scalar fields and some basic examples on how to find symmetries in them not much regarding how to deal with vector fields has even been done, so clearly i don't have ideas now on how to try and find symmetries for such a lagrangian. Only recently i also viewed example on the electromagnetic field lagrangian and tried to understand it's calculations for symmetries but i haven't been able to apply it into this lagrangian. That's the reason i wanted some insights on how to approach such a system.

 

[PeroK]

Homework Statement:: I am stuck on a problem and wold love to see any insights that i can get about this. To also begin with i am a beginner in quantum field theory and don't have much experience working with vector fields. It concerns with finding the symmetries of a massive vector field lagrangian.
Relevant Equations:: $$L = -\partial_{\mu}A^{\nu} \partial^{\mu}A_{\nu}-M^2* A^{\nu}A_{\nu}$$

Untill now i have only been able to derive the equations of motion for this lagrangian when the field $\phi$ in the Euler-Lagrange equation covariant field $A_{\nu}$ which came out to be :

$$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$

I have seen examples based on the electromagnetic fields and how to verify gauge invariance, but since i am very new to this i have not much idea how to begin looking for symmetries of such kind of lagrangian involving vector fields. To be frank i haven't proceeded much in this and i would really appreciate any hints on how to begin examining such problems. Thank you

I've fixed your Latex anyway.

 

[PeroK]

That Lagrangian effectively involves four scalar fields: $A^{\mu}$ and the four partial derivatives of each. You should end up with four E-L equations: one for each of $A^{\mu}$.

You could write everything out with specific indices and then look at $A^0$ and $\partial_{\nu}A^0$, for example. See what you get for that. Then do the same for $A^1$ and you should see how the E-L equations come out in general.

 

[phywithAK]

Yes @PeroK ,I tried writing my equations for each of the four components and thus obtaining 4 E-L equations. But what about the symmetries of this lagrangian, i tried showing it's invariance for space-time symmetries but i am still stuck on how to properly proceed for that. Any direction towards the calculation would be appreciated.

 

[PeroK]

Yes @PeroK ,I tried writing my equations for each of the four components and thus obtaining 4 E-L equations. But what about the symmetries of this lagrangian, i tried showing it's invariance for space-time symmetries but i am still stuck on how to properly proceed for that. Any direction towards the calculation would be appreciated.

Are you able to post some working? It's difficult to help when we can't see what you're actually trying to do.

 

[phywithAK]

@PeroK I was busy with other subjects so replied late . I did try proving yesterday the invariance under lorentz transformation in this way, now that it's invariant i want to know the current due to lorentz symmetry but i don't know how to proceed for that. Is calculation till here sufficient to proceed for finding the conserved current ? Can anyone tell how should i proceed next ?