Physical Interpretation of the Solutions to the Dirac Equation

Abstract

In interpreting the Dirac equation as a wave equation, as has been our practice up to now, we have ignored a number of fundamental difficulties. The equation possesses negative energy solutions and, for particles at rest, solutions with negative rest mass. The kinetic energy in these states is negative; the particle moves in the opposite direction to one occupying the usual state of positive energy. Thus, a particle carrying the charge of an electron is repelled by the field of a proton. (The matrix β with the negative matrix elements β₃₃ and β₄₄ multiplies m and the kinetic energy, but not the potential term eΦ in Eq. (9.1.9).) States such as these are not realized in nature. The main problem, of course, is their negative energy, which lies below the smallest energy for states with positive rest energy. Thus, one would expect radiative transitions, accompanied by the emission of light quanta, from positive energy into negative energy states. Positive energy states would be unstable due to the infinite number of negative-energy states into which they could fall by emitting light – unless, that is, all of these latter states were occupied. It is not possible to exclude these states simply by arguing that they are not realized in nature. The positive energy states alone do not represent a complete set of solutions. The physical consequence of this is the following: When an external perturbation, e.g., due to a measurement, causes an electron to enter a certain state, this will in general be a combination of positive and negative energy states. In particular, when the electron is confined to a region that is smaller than its Compton wavelength, the negative energy states will contribute significantly.