Relativistic QM The Klein Gordon equation 1926

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Relativistic QM - The Klein Gordon equation (1926)
Scalar particle (field) (J0)
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
Elt0 solutions as Egt0 solutions for particles of
opposite charge (antiparticles). Unlike Diracs
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
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But energy eigenvalues
Feynman Stuckelberg interpretation
Two different time orderings giving same
observable event
time
space
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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
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Want to solve
Solution
where
and
Feynman propagator
Dirac Delta function
In bra- ket- notation
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6
But energy eigenvalues
Feynman Stuckelberg interpretation
Two different time orderings giving same
observable event
time
space
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(p0 integral most conveniently evaluated using
contour integration via Cauchys theorem )
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9
(p0 integral most conveniently evaluated using
contour integration via Cauchys theorem )
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time
space
where
are positive and negative energy solutions to
free KG equation