The Lagrangian density is the basis of any field theory (covered in this class). If someone hands us a Lagrangian density, there are quite a few mechanical steps we can do to find out interesting properties.
To fix notation, I will denote Lagrangian density by \(\mathcal{L}\). It can be integrated over space to obtain the Lagrangian $$ L = \int d^3x \mathcal{L},$$ which can in turn be integrated over time to obtain the real-time action $$ S = \int dt L = \int d^4x \mathcal{L}. $$ The imaginary-time or Euclidean action can be found by doing a Wick rotation \(t\rightarrow i\tau\) and require the weight of path integral to stay unchanged \(e^{iS}=e^{-S_E}\).
- Find equation of motion (Euler-Lagrange Equation)
- Identify symmetries
- Find conserved currents for symmetry transformations (field and coordinates)
- Interpret currents by acting charges on the fields
- Find generating functional
- Calculate n-point correlation functions