Relativistic Invariance and the Square-Root Klein-Gordon Equation

Abstract

Although the usual operator invariance requirements and corresponding commutation conditions encountered in the study of the invariance of relativistic wave equations (and other equations of physics) are sufficient conditions for invariance, they are by no means necessary. More general conditions are given and illustrated with the square‐root Klein‐Gordon equation. A new proof is thereby given of the Lorentz invariance of this equation. The methods developed are extended to cover the presence of external fields, and it is proved that the usual gauge invariant modification of the relativistic Hamiltonian of a spinless particle which takes into account the presence of an external electromagnetic field leads, in the quantum mechanical case, to an equation which does not admit the proper Lorentz group. This theorem and its generalization are discussed in connection with Dirac's statement that the square‐root equation cannot be extended to include interaction without losing Lorentz invariance.