in ordinary three-dimensional space, all three coordinates being treated on an equal footing; what sort of theory would one get by applying Hamilton's method to such a variational principle in four-dimensional space, xl, x~, .v~, x~ (with x4=ict), treating all four coordinates on an equal footing? The result is a general and completely relativistic theory of de Broglie waves, which Synge calls "relativistic geometrical mechanics."
After a short Introduction (Chapter I) in which the general status and setting of the theory are described, the general theory of rays and waves in space-time is developed in Chapter II. The theory up to this point is strictly a "ray" theory, there being no "phase" involved; nevertheless, Synge shows that there is a fundamental distinction between ray velocity and wave-packet velocity, and that the connecting formula is identical in form with the familiar formula between phase velocity and group velocity.
Chapter III treats the geometrical mechanics of a particle, both free and in a given field, and here are introduced de Broglie waves. Chapter IV is on Primitive Quantization; the concepts of phase, frequency and wavelength are introduced, as in passing from geometrical to physical optics. The quantization is carried out by the introduction of a scalar wave equation (in four-space) which involves the Planck constant. It is then shown that the action along a ray between events (points xl, x~, .rs, x,) of the same phase is precisely Planck's constant h. The theory is applied to the problem of a central force-field and the hydrogen-like atom, and to the Zeeman effect. Chapter V gives a generalization of the theory to N-dimensional space, and this generalization is used to discuss the two-body problem.
C. DOMENICALI