Klein-Gordon equation

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The Klein-Gordon equation is given by

where is a fixed mass and f is a scalar or vector-valued field on Minkowski space. When the mass is zero this is just the free wave equation. When the mass is positive, one usually rescales spacetime to normalize the mass to equal one, unless one is studying the vanishing mass limit.

In practice, this mass term makes absolutely no difference to the local well-posedness theory of an equation (since the mass term is negligible for high frequencies), but often plays a key role in the global theory (because of the improved decay and dispersion properties, and because the Hamiltonian controls the low frequencies more effectively).

The Maxwell-Klein-Gordon system is the massless Klein-Gordon equation (i.e. the free wave equation) coupled with an electromagnetic field.

In the non-relativistic limit the Klein-Gordon equation decouples into two copies of the free Schrodinger equation.