Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 33

One easily proves the following results for both representations:

(a)Orthogonality relations

The positive (negative) energy spinors are seen to have positive (negative) norm and to be orthogonal, respectively. One furthermore has

We thus conclude that the “positive” and “negative” energy solutions⁸ for halfintegral spin are also mutually orthogonal with respect to the “Dirac scalar product”.

(b)Projectors on positive and negative energy states

According to (a) the matrices

have the properties of projectors on the positive and negative energy solutions, respectively. In particular, the property

follows from the completeness relation

for the spinors. We have for both representations