1. Quantum Mechanics Classical – non relativistic Quantum Mechanical : Schrodinger eq

2. Quantum Mechanics Classical – non relativistic Quantum Mechanical : Schrodinger eq Classical – relativistic Quantum Mechanical - relativistic : Klein-Gordon (Schrodinger) equation (natural units)

3. Relativistic QM - The Klein Gordon equation (1926) Scalar particle (field) (J=0) : Energy eigenvalues 1934 Pauli and Weisskopf revived KG equation with E 0 solutions for particles of opposite charge (antiparticles). Unlike Dirac’s hole theory this interpretation is applicable to bosons (integer spin) as well as to fermions (half integer spin). 1927 Dirac tried to eliminate negative solutions by writing a relativistic equation linear in E (a theory of fermions) As we shall see the antiparticle states make the field theory causal (natural units)

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

5. Physical interpretation of Quantum Mechanics Schrödinger equation (S.E.) “probability current”“probability density” Normalised free particle solutions Klein Gordon equation Negative probability?

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

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

8. Relativistic QM - The Klein Gordon equation (1926) Scalar particle (field) (J=0) : (natural units) Relativistic notation :

9. 4 vector notation contravariant covariant 4 vectors

10. 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

11. Exchange Force Pion Propagator

12. Solution : where Feynman propagatorDirac Delta function Want to solve : and The pion propagator

13. Solution : where Want to solve : and The pion propagator Simplest to solve for propagator in momentum space by taking Fourier transform

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

15. Feynman – Stuckelberg interpretation time space Two different time orderings giving same observable event : But energy eigenvalues

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

17. }

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