What can we learn from the conformal noninvariance of the Klein-Gordon equation?

Abstract: It is well known that the Klein-Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. We show that such a noninvariance provides nontrivial physical insights at different levels, first within the fully relativistic regime, then in the nonrelativistic regime leading to the Schr\"odinger equation, and then within the de Broglie-Bohm causal interpretation of quantum mechanics. The conformal noninvariance of the Klein-Gordon equation coupled to a vector potential is confronted with the conformal invariance of Maxwell's equations in the presence of a charged current. The conformal invariance of the non-minimally coupled Klein-Gordon equation to gravity is then examined in light of the conformal invariance of Maxwell's equations. Finally, the consequence of the noninvariance of the equation on the Aharonov-Bohm effect in curved spacetime is discussed.