# How do you find the Lagrangians for different fields?

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• Wrichik Basu
In summary, the conversation discusses the process of finding Lagrangians for different fields in QFT. The authors provide examples of Lagrangians for complex scalar and electromagnetic fields, but the question is raised about how to find the correct Lagrangian for new theories. It is mentioned that there are a finite number of field types with known equations, and the process of quantization involves breaking and restoring gauge invariance. The book being studied may have missed some steps in this process.
Wrichik Basu
Gold Member
TL;DR Summary
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?

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 :)

Wrichik Basu

## 1. How do you determine the Lagrangian for a specific field?

The Lagrangian for a field can be determined by considering the symmetries and interactions of the field. This involves analyzing the equations of motion for the field and using mathematical techniques such as Noether's theorem to identify conserved quantities. The Lagrangian is then constructed using these conserved quantities and the symmetries of the field.

## 2. Can the Lagrangian be derived from first principles?

Yes, the Lagrangian can be derived from first principles by considering the fundamental laws of physics, such as conservation of energy and momentum, and using mathematical techniques such as variational calculus. This approach is known as the Lagrangian formalism and is commonly used in theoretical physics.

## 3. Are there different methods for finding the Lagrangian for different fields?

Yes, there are different methods for finding the Lagrangian for different fields. Some fields, such as scalar fields, can be described using a simple Lagrangian, while others, such as gauge fields, require more complex formulations. The specific method used depends on the properties and interactions of the field being studied.

## 4. How does the Lagrangian relate to the Hamiltonian?

The Lagrangian and the Hamiltonian are two different mathematical formulations used to describe the dynamics of a physical system. The Hamiltonian is derived from the Lagrangian and is used to calculate the equations of motion for a system. The two formulations are closely related, but the Hamiltonian approach is often more convenient for certain types of problems.

## 5. Can the Lagrangian be used to predict the behavior of a system?

Yes, the Lagrangian can be used to predict the behavior of a system by solving the equations of motion derived from it. This approach is commonly used in theoretical physics to study the dynamics of complex systems, such as particles interacting with each other through various forces. However, the accuracy of these predictions depends on the accuracy of the Lagrangian and the assumptions made in its construction.

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