Momentum space eigenvectors of the Dirac operator

Momentum-space eigenvectors of the free Dirac operator 

It is not difficult to obtain the eigenvectors of the matrix h(p) with the standard methods of linear algebra. We start with the eigenvectors of the Dirac matrix /3, which are particularly easy to find in the standard representation. For example, take the four-dimensional unit vectors 

The vectors and belong to the eigenvalue +1, the vectors and belong to the eigenvalue —I of 0 (in the standard representation). Notice that each eigenvalue of the matrix 0 has two linearly independent eigenvectors. 

The matrix u(p) that diagonalizes the free Dirac operator in momentum space is unitary. The inverse of this matrix is given by 

The eigenvectors of the matrix h(p) can be obtained with the transformation u(p) from the eigenvectors of the diagonal matrix 0. Consider, for example, a vector /pos, which is some linear combination of the vectors and Then 0fpos — fpos snd 

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