QFT - Derivative in Equation of Motion

In summary, the conversation is about deriving the equations of motion for a generalized Lagrangian. The person is trying to refresh their understanding of derivatives and is using an online example to help. They are confused by the notation, specifically the upper indices in the first term of the E-L equation. They question whether the indices are just dummy indices and why they are changed on the right side. They mention that they are not a fan of QFT. Finally, they mention finding the answer in a problem from Griffiths.
  • #1
Adoniram
94
6

Homework Statement


As part of a problem, I need to derive the EOM for a generalized Lagrangian. Before I get there, I'm trying to refresh myself on exactly how these derivatives work because the notation is so bizarre. I am trying to follow a simple example I found online:

Start with:
47eb7f2fdafef195081aceeb8f1c62a4.png


Homework Equations


The E-L eq:
c5a5a879417dd0515c350505abb55b97.png


The Attempt at a Solution


The actual solution is:
83077f5697553ab11092b8f121f6df28.png


I totally understand the second term, but the first term is bothering me (namely the part INSIDE the parentheses where there are all upper indices). How, pray tell, do we end up with all upper indices? Is it because the mu and nu are just dummy indices, and all that matters is that they are both lower and different? If so, then I can see how you end up with all upper indices. But, why are the upper indices on the right changed (mu <---> nu) if that's the case? Just to distinguish is from the left term so that they don't cancel? QFT is not my favorite subject...

Once I get this, I think I can tackle the actual problem I am solving, which is to find the EOM of the following Lagrangian:
la.png


Thank you!
 
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  • #2
Looks like I found my own answer! Griffiths has a nice example (Problem 10.2) that shows exactly this. That guy has the answer to everything...
 

What is QFT - Derivative in Equation of Motion?

QFT stands for Quantum Field Theory, which is a theoretical framework that combines quantum mechanics and special relativity to describe the behavior of particles at the subatomic level. The derivative in the equation of motion refers to the mathematical operation used to calculate the rate of change of a particle's position or momentum over time.

What is the importance of the derivative in the equation of motion in QFT?

The derivative in the equation of motion is crucial in QFT as it allows us to predict the behavior of particles over time. It is used to calculate the trajectory of a particle and understand how it will interact with other particles in a given system. Without the derivative, we would not be able to accurately describe the dynamics of particles in quantum field theory.

How is the derivative in the equation of motion calculated in QFT?

In QFT, the derivative in the equation of motion is calculated using mathematical tools and equations such as the Dirac equation, Klein-Gordon equation, and Feynman diagrams. These equations take into account the principles of quantum mechanics and special relativity to accurately describe the behavior of particles at the subatomic level.

What are some practical applications of QFT - Derivative in Equation of Motion?

QFT and the derivative in the equation of motion have many practical applications in fields such as particle physics, nuclear physics, and condensed matter physics. They are used to study and understand the behavior of particles in high-energy accelerators, nuclear reactors, and superconductors, among others. The predictions made by QFT have also been experimentally verified, making it an essential tool in modern physics.

Are there any limitations to using QFT - Derivative in Equation of Motion?

While QFT and the derivative in the equation of motion have been successful in describing the behavior of particles at the subatomic level, they have limitations. For example, they do not take into account gravity, and there is currently no unified theory that combines quantum mechanics and general relativity. Additionally, QFT relies on mathematical calculations and approximations, which may not always accurately reflect real-world observations.

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