QFT - Derivative in Equation of Motion

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SUMMARY

The discussion revolves around deriving the equations of motion (EOM) for a generalized Lagrangian in Quantum Field Theory (QFT). The participant seeks clarification on the notation involving upper indices in the Euler-Lagrange (E-L) equation. They conclude that the upper indices are dummy indices, and the switching of indices (mu and nu) is to prevent cancellation between terms. The participant references Griffiths' Problem 10.2 as a resource that clarifies these concepts effectively.

PREREQUISITES
  • Understanding of Quantum Field Theory (QFT)
  • Familiarity with the Euler-Lagrange equation
  • Knowledge of tensor notation and index manipulation
  • Basic principles of Lagrangian mechanics
NEXT STEPS
  • Study Griffiths' Problem 10.2 for a practical example of EOM derivation
  • Review advanced tensor calculus for better comprehension of index notation
  • Explore additional resources on Quantum Field Theory to solidify foundational concepts
  • Practice deriving equations of motion from various Lagrangians
USEFUL FOR

Students and researchers in physics, particularly those focusing on Quantum Field Theory and Lagrangian mechanics, will benefit from this discussion.

Adoniram
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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!
 
Physics news on Phys.org
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...
 

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