  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 at 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. The
issues that arise at each of these different levels are thoroughly analyzed.
