1. The Klein Gordon equation (1926) Scalar field (J=0) :

2. 4 vector notation contravariant covariant 4 vectors

3. Field theory of Classical electrodynamics, motion of charge –e in EM potential is obtained by the substitution : Quantum mechanics : The Klein Gordon equation becomes: The smallness of the EM coupling,, means that it is sensible to Make a “perturbation” expansion of V in powers of Scalar particle – satisfies KG equation

4. Physical interpretation of Quantum Mechanical equations Schrödinger equation (S.E.) “probability current”“probability density”

5. Physical interpretation of Quantum Mechanics Schrödinger equation (S.E.) “probability current”“probability density” Klein Gordon equation Pauli and Weisskopf Negative energy? Negative probability??

6. Want to solve : Solution : where Feynman propagatorDirac Delta function and

7. Want to solve : Solution : where Feynman propagatorDirac Delta function Simplest to solve for propagator in momentum space by taking Fourier transform and

8. The Born series Since V(x) is small can solve this equation iteratively : Interpretation :

9. but what about negative energy eigenvalues

10. Feynman – Stuckelberg interpretation time space Two different time orderings giving same observable event :

11. time space (p 0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )

12. }

13. time space where are positive and negative energy solutions to free KG equation

15. Scattering in Quantum Mechanics Prepare state at Time evolution (possibly scattering) Observe resulting system in state QM : probability amplitude : Theory confronts experiment - Cross sections and decay rates

16. S matrix for Klein Gordon scattering Relativistic probability density

17. Feynman rules Feynman rule associated with Feynman diagram