B How do you find the Lagrangians for different fields?

Wrichik Basu

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2018 Award
Is there any specific procedure to find the Lagrangian of different fields?
I am currently studying QFT from this book.

I have progressed to the chapter of QED. In the course, the authors have been writing the Lagrangian for different fields as and when necessary. For example, the Lagrangian for the complex scalar field is $$\mathcal{L} \ = \ (\partial ^\mu \phi^\dagger)(\partial _\mu \phi) \ - \ m^2 \phi^\dagger \phi,$$ while the gauge-fixing Lagrangian for the EM field is $$\mathcal{L}_{GF} \ = \ - \frac{1}{2\xi}(\partial ^\mu A_\mu)^2,$$ where symbols have their usual meanings and ##\xi## is an arbitrary parameter.

Everything is fine, but how does one get these Lagrangians? The authors have simply written down the answers, but say I am trying to find a new theory in QFT. In that case, how do I find the correct Lagrangian? Is this kind of guess work - if you chance upon the correct one, you are good to go, otherwise not? When I was studying the Lagrangian for QED, the authors kind of derived it from the Dirac and photon fields using simple arguments. But what about the other cases that I have studied before? Any specific method to find the Lagrangians?

Honestly, I haven't studied classical dynamics to a great depth (only that much which will help me in studying QM), but when I was listening to an online lecture series, the professor once commented that often an amount of guessing goes into finding the Lagrangian. Is that true?


Gold Member
There are several routes that you can take. First, the number of different fields is finite, and characterized by their spin. Spin zero is scalar field. Spin 1 is a vector field (Electromagnetic field), spin 2 is a tensor field (Gravity). The Dirac Lagrangian is the simplest Lagrangian whose Euler-Lagrange equations give the Dirac equation. The number of field types is finite and their equations are already known. For the complex scalar field, the derivative with respect to ##\phi## minus the derivative wrt the derivative wrt \phi is equal to $$(\partial_\mu\partial^\mu - m^2)\phi = 0$$ which is the wave equation for a massive field. The Gauge breaking term is a Landau gauge term. The philosophy of quantization is that you first write down a gauge invariant Lagrangian, then you break gauge invariance, and then you restore it at the end! The book you're reading may have missed some steps :)

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