Building Lagrangians for Classical Field Theory

Question

I've been studying quantum mechanics and classical field theory for quite a while now. However, I still struggle with the idea of building scalars from vectors and tensors for the Lagrangian density. For instance, I have searched everywhere how to arrive at the conclusion that the Lagrangian for the classical electromagnetic field is given by; $$\mathcal{L} = - \frac{1}{4\mu_0}F^{ab}F_{ab} - j^aA_a$$ But no success, no sources or books show the calculations, it's just a given like the electromagnetic field strength tensor $F^{ab}$ (A little bit different, I could find one and only one book which showed the derivation of this tensor, no articles though, and surprisingly, it arrives quite naturally in the search for a contravariant formulation of electromagnetism. The book is "Tensor Calculus for Physics. A Concise Guide" by Dwight E. Neuenschwander.)

Most books (That I've read) also kind of give the interaction term $-j^aA_a$ as a given. It is said that this term comes from Noether's theorem, but, like before, no calculations are ever shown.

Even more questions arise when we take a look at the lagrangian density for quantum chromodynamics (A little jump to quantum field theory) and how the indices of the gluon field strengh tensor are built into the lagrangian; $$\mathcal{L}_{QCD} = - \frac{1}{4}G^a_{\mu\nu}G^{\mu\nu}_a +\bar{\psi_i}(i(\gamma^{\mu}D_{\mu})_{ij}-m\delta_{ij})\psi_j$$ Where $\psi_i(x)$ is the quark field, $D_{\mu}$ is the gauge covariant derivative, $\gamma^{\mu}$ are Dirac matrices and $G^a_{\mu\nu} = \partial_\mu\mathcal{A}^a_\nu-\partial_\nu\mathcal{A}^a_\mu+gf^{abc}\mathcal{A}^b_\mu\mathcal{A}^c_\nu$, where $\mathcal{A}^a_\mu(x)$ are the gluon fields. Most books say that you kind of guess a scalar using the tensors and vectors, vector potentials, spinors, etc. This seems highly unpractical and prone to error and, if it really is by guessing, where do the factors of $-\frac{1}{4\mu_0}$ and of $-\frac{1}{4}$ come from?

Can someone explain to me the derivations and maybe even show the calculations to help clear this doubt of over one year?

Answer 1

Not everything in physics follows from deductive steps. Often the most profound advances in our understanding comes from inductive steps. So, the Lagrangian for the EM field is more or less obtained by answering the question: "what should the Lagrangian look like so that when the Euler-Lagrange equation is applied to it, one can obtain Maxwell's equation's?" To answer such a question, one needs to think inductively. Given the Lagrangian, one can deductively obtain Maxwell's equations. But the opposite requires an inductive step.

Answer 2

Here, I'll make your question simpler and link in a possible way to answer it. The question is: "what model-building techniques may I use to derive or construct Lagrangians with?". The answer (at least in the case of gauge fields) is the Utiyama Theorem, which is - itself - an extension of the Second Noether Theorem. You might even call it the Third Noether Theorem, as Utiyama explicitly tied his work to Noether's.

This is an example deriving the general gauge-invariant Lagrangian density for a Non-Abelian Gauge Field Plus Extra Field, via Utiyama's Theorem. This is an example for the similar, but less complex, case of a derivation for the (Abelian) Maxwell Field. In both cases, the key points are: (1) the gradients for the gauge field may only occur in anti-symmetric combinations (e.g. $∂_μA_ν - ∂_νA_μ$), (2) the gradients for the other fields may only occur in combination as "gauge-covariant" derivatives, (3) no other explicit dependence on the gauge fields may occur and (4) the continuity equation for the total current constructed from the non-gauge fields holds off-shell and serves as a pre-condition for the well-posedness of the field equations. That means: only external sources that have a continuity-equation-satisfying current may couple to the gauge field.