Quantum mechanics working with systems of a finite number of
degrees of freedom is usually considered incompatible with
relativistic kinematics. In particular, the free Klein-Gordon
equation is usually understood as one of the building blocks of a
relativistic quantum field theory rather than as an eligible
quantum-mechanical candidate for a sufficiently consistent
description of a relativistic motion of a single spinless
particle. One of the most immediate reasons lies in the
non-Hermitian (in fact, in the present language,
$\sigma_3-$pseudo-Hermitian) character of the corresponding
one-particle time-evolution generator H (i.e., energy operator,
usually called Feshbach-Villars Hamiltonian). Obviously, some
states (one usually calls them antiparticles) may have negative
norm so that, formally, the metric in Hilbert space does not exist
(or, in the other words, $\sigma_3$ remains indefinite and may
merely be called a pseudo-metric). In the language of physics one
simply cannot avoid a spontaneous creation of indefinitely many
particle-antiparticle pairs in some sufficiently strong external
fields.

Putting main emphasis on the author's own contribution to the
recent intensive development of the possible alternative physical
interpretation of all the similar pseudo-Hermitian
quantum-mechanical systems (also known as PT- or CPT-symmetric
quantum mechanics, see, e.g., Czechoslovak J. Phys. 54 (2004), pp.
1 - 156 and pp. 1005 - 1148 for complementary reading), this
review tries to throw new light on many of the above-mentioned 75
years old questions. A. Mostafazadeh describes an alternative
interpretation of the free one-particle Klein-Gordon system with
real spectrum (i.e., in the absence of any strong external field).
His proposal (with details available in some of his recent
publications) is based on an introduction of a new scalar product
(= the only formula in his text) reflecting the existence of a
nontrivial positively-definite metric $\eta_+\neq I$ (factorized
as $CP$ in a major portion of the literature) in Hilbert space.
This makes the Feshbach-Villars Hamiltonian H (quasi-)Hermitian
and endows Klein-Gordon states with a new eligible consistent
probabilistic interpretation.


MR2074087
Mostafazadeh, Ali . Is pseudo-Hermitian quantum mechanics an indefinite-metric quantum theory? 
Quantum groups and integrable systems. Czechoslovak J. Phys. 53 (2003), no. 11, 1079--1084.