Relativistic QM - The Klein Gordon equation (1926)
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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).

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Relativistic QM - The Klein Gordon equation (1926)

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Relativistic qm the klein gordon equation 1926

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


Relativistic qm the klein gordon equation 1926

But energy eigenvalues

Feynman – Stuckelberg interpretation

Two different time orderings giving same observable event :

time

space


Relativistic qm the klein gordon equation 1926

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


Relativistic qm the klein gordon equation 1926

Want to solve :

Solution :

where

and

Feynman propagator

Dirac Delta function

In bra- ket- notation


Relativistic qm the klein gordon equation 1926

}


Relativistic qm the klein gordon equation 1926

But energy eigenvalues

Feynman – Stuckelberg interpretation

Two different time orderings giving same observable event :

time

space


Relativistic qm the klein gordon equation 1926

time

space

(p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )


Relativistic qm the klein gordon equation 1926

}


Relativistic qm the klein gordon equation 1926

time

space

(p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )


Relativistic qm the klein gordon equation 1926

time

space

where

are positive and negative energy solutions to free KG equation


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