Klein-Gordon equation

free massive scalar field which describes the spin-0 particles
in condensed matter physics it describes long wavelength optical phonons
formulated as a relativistic generalization of Schrodinger equation
there are real KG equation and complex KG equation
- real case describes electrically neutral particles
- complex case describes charged particles
\((\Box + m^2) \psi = 0\) i.e. \((\Box + m^2) \psi =\psi_{tt}-\psi_{xx}-\psi_{yy}-\psi_{zz}+m^2\psi=0\)
people found 2 problems of KG equations
- negative energy states
- negative probability density
correct interpretations of \(\phi\) requires the idea of quantum field rather than the particle wavefunction
- negative probability density -> charge density
Dirac suggested Dirac sea by invoking the exclusion principle and then KG equation only applicable to spinless particles
- for example, \(\pi\)-meson
Thus the Dirac equation comes in to deal with spin-\(1/2\) particles.

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