Previous attempts at constructing a classical field theory of the electron are reviewed and it is shown that hitherto it has not been possible to combine the two conditions: gauge-invariance and freedom from singularities. It is pointed out that this can be done by the introduction of new matter functions similar to the wave functions of quantum theory. A simple possibility of this type is considered and is found to lead to equations admitting electron-like solutions. The electron turns out to have a negative mass (but perhaps this difficulty will disappear when the equations are quantized). The theory, although classical, offers in principle a possibility of accounting for the Sommerfeld fine-structure constant. There are also solutions corresponding to excited states of the electron and to heavier particles, including particles with zero charge.

• Received 6 October 1938

DOI:https://doi.org/10.1103/PhysRev.55.94