Given a Lagrangian Density

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 \(e^{iS}=e^{-S_E}\).

  1. Find equation of motion (Euler-Lagrange Equation)
  2. Identify symmetries
  3. Find conserved currents for symmetry transformations (field and coordinates)
  4. Interpret currents by acting charges on the fields
  5. Find generating functional
  6. Calculate n-point correlation functions