Lorentz invariant lagrangian density

[naos]

Hi,

Would someone know where I can find a derivation of the lorentz-invariant lagrangian density?
This lagrangian often pops-up in books and papers and they take it for granted, but I was actually wondering if there's a "simple" derivation somewhere... Or does it take a whole theory and tens of pages to get there?

As a reminder, it can be found on slide 5 of this paper:
http://www.physics.indiana.edu/~dermisek/QFT_08/qft-I-2-1p.pdf

Sorry for this question, I'm pretty new in Lagrangian and field theory (come from maths)...

Thanks

 

[Physics Monkey]

It's not so much derived as it is assumed or obtained from experiment, but here is one simple route.

In relativity we want a dispersion relation like $$E^2 = p^2 + m^2$$ with energy E and momentum p. Combining this with quantum mechanics we replace $$E \rightarrow i \partial_t$$ and $$p \rightarrow -i \partial_x$$ and demand our fields satisfy an equation of motion like $$(-\partial_t^2 + \partial_x^2 - m^2 ) \phi = 0$$. This equation is Lorentz invariant provided $$\phi$$ is a scalar. Now just look for a Lagrangian that gives this equation of motion and then you have what you were looking for. There are plenty of more sophisticated points of view, but I think this one is nice and direct.

Hope this helps.

 

[ParticleGrl]

Maybe take this approach- we want to build a lagrangian that is a Lorentz scalar. Now, the lagrangian we build depends on the fields we are working with. The slide you posted has a scalar field $$\phi$$, since this is already a lorentz scalar, any function $$f\left(\phi\right)$$ is also lorentz invariant.

The other object we have to work with is $$\partial_\mu \phi$$, that transforms as a vector. We can build an invariant object out of two of them. $$\partial_\mu \phi \partial^\mu \phi$$. So our lagrangian could be $$\mathcal{L} = f'\left(\partial_\mu \phi \partial^\mu \phi, \phi\right)$$. Now, we can write an infinite number of terms in the theory, but when you learn about renormalization you'll discover that at long wavelengths, we only have to consider the lowest order terms.

$$\mathcal{L} = \frac{1}{2}\partial^\mu \phi \partial_\mu \phi - \frac{1}{2}m^2\phi^2-\frac{1}{4!}\phi^4$$

Think of m and lambda as constants to be decided by experiment.

 

[naos]

Thank you for your answers!

Renormalization is a bit far from now, but I have a feeling that'll be an elegant approach. Thank for that. I definitely like the simple approach you're having, Physics Monkey, although it looks like a baking recipe ;)

In addition to those proofs, would you know if there's one which goes from a study of the lorentz group, then look for all Lagrangians which are invariant under any transformation belonging to that group? For instance, it could be an analysis of a particular and simpler case, leading to the correct form, then using physics' arguments for a generalization.

Also, I'm currently reading the book of W. N. Cottingham on Standard Model (ISBN 0521852498) and I think I'm going to need another book on classical and quantum field theory. Which one would you recommend me? Apparently, the one from Tai Kai Ng (Introduction to classical and quantum field theory) seems to be good and cover what my needs (basics field theory and their quantization). What do you think?

Thanks