A Hilbert space for the Dirac equation

In order to apply the methods and techniques of (nonrelativistic) quantum theory to an evolution equation of the form (8), one has to define a Hilbert space as an appropriate state space. At every instance of time, the solution has to be an element of this Hilbert space. 

A suitable state space for the Dirac equation must consists of vector-valued functions with four components, in order to match the dimension of the Dirac matrices. If one assumes that each of the four components is a square-integrable (in the sense of Lebesgue) function of the position x, then we obtain the Hilbert space denoted by Its elements 0 are called 4-spinors. They satisfy 

The integrand in the integral above is just the ordinary C -scalar product at the point X, 

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