Interferometric discrepancy between the non-relativistic solution to the Klein-Gordon and Schr\"odinger wave equations due to their dissimilar phase velocities

Abstract

Adding a constant energy offset leaves classical dynamics unchanged. In quantum mechanics it changes the phase velocity of the wavefunction. The inclusion of the constant rest energy in the Klein-Gordon formulation leads to significantly higher phase velocities compared with the Schr\"odinger equation. The Schr\"odinger equation predicts an attenuation of the wavefunction along one of the paths in a Sagnac interferometer when a beamsplitter's trajectory along that path includes a segment where its speed exceeds the phase velocity of a free particle. Such an attenuation does not occur for electromagnetic waves nor for eigenstates of momentum in the Klein-Gordon equation since the speed of the beamsplitter cannot then exceed the phase velocity of the wave. This attenuation reduces the amplitude without introducing a phase shift, preserving the overall structure of the transmitted wave group. While a Klein-Gordon wave group undergoes three traversals of the beamsplitter that moved, it experiences attenuation equivalent to only a single pass, whereas the Schr\"odinger equation predicts the expected attenuation for three passes. This discrepancy highlights a fundamental incompatibility between the Klein-Gordon equation and the Schr\"odinger equation in a regime where their predictions should converge.