  We derive an explicit manifestly covariant expression for the most general
positive-definite and Lorentz-invariant inner product on the space of solutions
of the Klein-Gordon equation. This expression involves a one-parameter family
of conserved current densities J_a^\mu, with a\in(-1,1), that are analogous to
the chiral current density for spin half fields. The conservation of J_a^\mu is
related to a global gauge symmetry of the Klein-Gordon fields whose gauge group
is U(1) for rational a and the multiplicative group of positive real numbers
for irrational a. We show that the associated gauge symmetry is responsible for
the conservation of the total probability of the localization of the field in
space. This provides a simple resolution of the paradoxical situation resulting
from the fact that the probability current density for free scalar fields is
neither covariant nor conserved. Furthermore, we discuss the implications of
our approach for free real scalar fields offering a direct proof of the
uniqueness of the relativistically invariant positive-definite inner product on
the space of real Klein-Gordon fields. We also explore an extension of our
results to scalar fields minimally coupled to an electromagnetic field.
