The special theory of relativity results from the postulation of invariance under coordinate transformation of the hyperbolic wave equation $\square \psi \equiv {\nabla }^2\psi -\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial t^2}=0,$ and it is required that all laws of physics (except perhaps general theory of relativity) be invariant under Lorentz transformations. Divergencies in present relativistic field equations may be removed by considering more general wave equations, for example, $-{l_0}^2{\nabla }^4\psi +{\nabla }^2\psi -\frac{1}{c^2}\frac{{\partial }^2\psi }{\partial t^2}=0.\mathrm{}$ This equation introduces a universal length $l_0\approx {10}^{-13\mathrm{}}$ cm as a second invariant and destroys Lorentz invariance except as an approximate invariance. Some theoretical and experimental consequences of this four-dimensional wave equation are discussed.

• Received 20 September 1965

DOI:https://doi.org/10.1103/PhysRev.159.1106