Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 28

Since in a manifestly Lorentz-covariant wave equation, space and time variables should appear on equal footing, Dirac demanded that the hamiltonian in the equation

should depend linearly on the momentum canonically conjugate to . This led him to the Ansatz¹

The triple and β in this equation cannot be just numbers, since this would already be inconsistent with rotational covariance. Hence they are expected to be given by matrices. These matrices must be hermitian, in order to warrant the hermiticity of the Hamilton operator. Furthermore, Eq. (4.2) should lead to the correct relation between energy and momentum for free particles. In order to see what this implies, we differentiate Eq. (4.2) with respect to time, thus obtaining

Here the bracket {A, B} denotes the anticommutator of two objects:

In order to get the desired energy momentum relation, this equation has to reduce to the Klein–Gordon equation, which is the case if

From here we deduce the following properties of the matrices: