Relativistic QM - The Klein Gordon equation (1926)

1. Relativistic QM - The Klein Gordon equation (1926) Scalar particle (field) (J=0) : Energy eigenvalues 1927 Dirac tried to eliminate negative solutions by writing a relativistic equation linear in E (a theory of fermions) 1934 Pauli and Weisskopf revived KG equation with E<0 solutions as 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). As we shall see the antiparticle states make the field theory causal

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

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

4. Want to solve : Solution : where and Feynman propagator Dirac Delta function In bra- ket- notation

5. }

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

7. time space (p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )

8. }

9. time space (p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )

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