Construction of a relativistic field theory

1. Construction of a relativistic field theory (Nonrelativistic mechanics) Lagrangian Action Feynman lectures Classical path … minimises action Quantum mechanics … sum over all paths with amplitude Lagrangian invariant under all the symmetries of nature -makes it easy to construct viable theories

2. Lagrangian formulation of the Klein Gordon equation Klein Gordon field Manifestly Lorentz invariant } } T V Classical path : Dimensions? Euler Lagrange equation

3. Euler Lagrange equs Principle of least action : 0 (surface integral) Euler Lagrange equation

4. Lagrangian formulation of the Klein Gordon equation Klein Gordon field Manifestly Lorentz invariant } } T V Euler Lagrange equation Klein Gordon equation

5. The Lagrangian and Feynman rules Associate with the various terms in the Lagrangian a set of propagators and vertex factors The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. The remaining terms in the Lagrangian are associated with interaction vertices. The Feynman vertex factor is just given by the coefficient of the corresponding term in }

6. New symmetries …an Abelian (U(1)) gauge symmetry Is invariant under A symmetry implies a conserved current and charge. e.g. Translation Momentum conservation Angular momentum conservation Rotation What conservation law does the U(1) invariance imply?

7. Noether current …an Abelian (U(1)) gauge symmetry Is invariant under 0 (Euler lagrange eqs.) Noether current

8. The Klein Gordon current …an Abelian (U(1)) gauge symmetry Is invariant under This is of the form of the electromagnetic current we used for the KG field

9. The Klein Gordon current …an Abelian (U(1)) gauge symmetry Is invariant under This is of the form of the electromagnetic current we used for the KG field is the associated conserved charge

10. Suppose we have two fields with different U(1) charges : ..no cross terms possible (corresponding to charge conservation)

11. U(1) local gauge invariance and QED not invariant due to derivatives To obtain invariant Lagrangian look for a modified derivative transforming covariantly Need to introduce a new vector field

12. is invariant under local U(1) is equivalent to Note : universal coupling of electromagnetism follows from local gauge invariance

13. The electromagnetic Lagrangian Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations ! `

14. The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve In momentum space (‘t Hooft Feynman gauge ξ=1)

16. The inclusion of fermions Dirac spinor Weyl spinors 2-component spinors of SU(2) Rotations and Boosts Dirac Gamma matrices Weyl basis

17. The Dirac equation Fermions described by 4-cpt Dirac spinors Lorentz invariant New 4-vector The Lagrangian Dimension? From Euler Lagrange equation obtain the Dirac equation Feynman rules U(1) symmetry