The Lagrangian Density and Equations of Motion

[bleist88]

Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be assumed. Then from there, other physics may be derived. Is it possible, instead, to start with Lagrangian ahead of time and use it to derive new equations of motion and new physics?

In a similar manor, it seems that in many books (Zee, Peskin & Schroeder and others) that there is an argument for where mass comes from. I see statements along the lines of "m is simply a parameter in our Lagrangian, and later we will show that this term must be the mass". Sure enough, they will work through Lagrangian and Hamiltonian mechanics to arrive at some physical argument on why m must be the mass. However, this seems silly to me. The Lagrangian comes directly from knowing what it must be in order to reproduce the equations of motion that you're confident (again the Klein-Gordon and Dirac), so it seems to me that it must have been the mass already.

I'd be very interested in the insight you might share. (Also, I must apologize. I have been sitting on this question for a year and have been planning to have notes, quotations and specific examples from books but I never got around to producing them. I would have liked to have prepared a more educated question but life has not permitted me the time.)

Thanks!

 

[king vitamin]

It sounds like you've only studied the very beginning of a quantum field theory course, where free fields are studied. Once you've included interactions, the EoMs are no longer the simple Klein-Gordon or Dirac, and the parameter "m" in the free part of the Lagrangian is no longer the physical mass.

The usual principle in constructing interacting Lagrangians is symmetry. If you specify that you need a certain field/particle content (e.g. either massive or massless fields, and specify the field's spin), and then ask what interactions you can write down, you generate some infinitely many interactions. But then, from studying the renormalization group of the theory, you actually find that all but a few finite terms will be irrelevant at low energies (here I'm assuming this is an effective quantum field theory). Then you'll have some well-defined theory where you can do calculations.

In the resulting theory, you must input the "measured" physics to get further predictions. For example, you need to input the physical masses and charges, before you can compute how other physical quantities behave. So the Lagrangian very much follows from the physics you're trying to model.

 

[bleist88]

Thanks King Vitamin. Yes I've only had a semester of QFT and even that is a rare offer at the University I went to. We worked up to and through being able to do a few QED processes and deriving Feynman Diagrams from operator methods but did not go further than that. When I've gone back to the subject in my own free time I usually don't get much further than wrestling with spinor fields. I look forward to getting deeper into the texts!

 

[vanhees71]

From a more fundamental theoretical point of view the great merit with the Hamilton principle of least action is that it allows to directly use the ideas of symmetries in terms of Lie groups and Lie algebras. E.g., from a detailed analysis of the irreducible unitary representations of the proper orthochronous Poincare group, you find that the "standard equations of motion" are an inevitable consequence of the very structure of the underlying spacetime model, i.e., Minkowski space with its Lorentz fundamental form. Making the additional assumptions of locality, microcausality, you get the two classes of particles leading to well-definied S-matrices, namely those with a positive $m^2$ (mass is a Casimir operator of the Poincare algebra) and those with $m^2=0$. For massless vector particles it also follows that it must be described as a gauge theory, if you don't want to have continuous polarization-like intrinsic degrees of freedom, which never has been observed in Nature. So you are quite straight forwardly pointed to the very foundations of relativistic QFT, underlying the Standard Model. The only additional idea is the generalization of the U(1) gauge symmetry implied by the structure of the Poincare group for massless vector bosons to non-Abelian (compact) gauge groups like SU(2) and SU(3), as well as the Higgs Mechanism which enable to describe consistently also massive vector bosons within a local gauge theory. From this structure again also the other intrinsic quantum numbers follow (i.e., the various charges). Of course, we still have to plug in the collected empirical knowledge to get the specific gauge group $\text{SU}(3)_{\text{color}} \times \text{SU}(2)_{\text{wiso L}} \times \text{U}(1)_{\text{w-hyper}}$ (with the latter chiral group Higgsed to $\text{U}(1)_{\text{electromagnetic}}$ and couplings of the Standard Model. Despite the very strong and persuasive symmetry principles, physics after all still is an empirical science!