Tentative one-particle interpretation

We note that the choice of a Hilbert space of square-integrable functions as the state space of the evolution equation is perfectly natural for the Schrodinger equation. The solutions of the Schrodinger equation are in the Hilbert space L (R ) (they have only one component), and the expression tj x,t) is interpreted as a density for the position probability at time t. Hence the norm of a Schrodinger wave packet, 

For the Dirac equation, this reasoning is more problematic, because a generally accepted interpretation for 

at some time t, the relativistic particle is described by a C -valued square-integrable wave function ij), then 

Note For vector-valued wave functions the Fourier transformation is applied to each component separately, 

In that way the Fourier transformation becomes a one-to-one mapping between square-integrable 4-spinors if) and ip- The inverse Fourier transformation expresses a square-integrable function ipj as a continuous superposition of plane waves exp(ip - /h). 

---