Solutions of the Dirac equation in field-free space

Solutions of the Dirac equation in field-free space 

The wave function P=P(R, t) may be written as a four-component spinor 

The condition for non-trivial solutions is that the determinant of the coefficients is zero, yielding 

Note that this relationship is in agreement with our previous equation (3.6) connecting the energy and momentum. Now equation (3.22) reveals the existence of positive and negative energy solutions, i.e. 

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