# The Dirac and K-G Solutions

The component solutions of the Dirac equation are also solutions of the Klein-Gordon equation.
But these solutions are not scalars since the coefficients contain quantities like energy and momentum[the phase part is of course an invariant]
These are neither zero spin nor half spin particles[we are treating them as solutions of the K-G equation]. Is it possible for such particles to exist independently, in the normal or in extreme conditions?

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dextercioby
Homework Helper
The component solutions of the Dirac equation are also solutions of the Klein-Gordon equation.

True.

But these solutions are not scalars since the coefficients contain quantities like energy and momentum[the phase part is of course an invariant]

No, the solutions of either Dirac's equation or KG's equation are invariant if described from an inertial reference system which is not rotated, nor boosted, but only space-time displaced. In other words, the solutions of these equations are invariant wrt to the subgroup of space-time translations, the only difference separating the equations and their solutions comes from the behavior under (restricted) Lorentz transformations.

The solutions to the Dirac equation are invariant in form but not in value.One may consider the solution for a Dirac particle at rest in some inertial frame and the corresponding solution in some frame wrt which it is in motion. In certain types of standard treatment we start from the Dirac solution for a particle at rest and then move on to the more general type by some suitable boost to other inertial frames[where the particle is not at rest].

[As an illustration/argument I might say that the quantity $${Et}{-}{p}{.}{x}$$ does not change in value when we move from one inertial frame to another. The scalar nature of the dot product conforms to such invariance of value]

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