Different ways to the Aharonov-Bohm effect and elucidation of its physical meaning

We compare different ways to the Aharonov-Bohm (AB) phase for an electrically charged pointlike particle in an electromagnetic field and disclose three mathematically equivalent representations for the AB phase, which explicitly demonstrate its Lorentz invariance and gauge invariance. Based on these results, we obtain a linear dependence of the wave vector of a moving charged particle on the vector composition of mechanical and electromagnetic momenta, which sheds new light on the physical origin of the AB effect.