  The earlier treatments of Lorentz covariant harmonic oscillator have brought
to light various difficulties, such as reconciling Lorentz symmetry with the
full Fock space, and divergence issues with their functional representations.
We present here a full solution avoiding those problems. The complete set of
Fock states is obtained, together with the corresponding explicit wavefunction
and their inner product integrals free from any divergence problem and the
Lorentz symmetry fully maintained without additional constraints imposed. By a
simple choice of the pseudo-unitary representation of the underlying symmetry
group, motivated from the perspective of the Minkowski spacetime as a
representation for the Lorentz group, we obtain the natural non-unitary Fock
space picture commonly considered though not formulated and presented in the
careful details given here. From a direct derivation of the appropriate basis
state wavefunctions of the finite-dimensional irreducible representations of
the Lorentz symmetry, the relation between the latter and the Fock state
wavefunctions is also explicitly shown. Moreover, the full picture including
the states with non-positive norm may give consistent physics picture as a
version of Lorentz covariant quantum mechanics. Probability interpretation for
the usual von Neumann measurements is not a problem as all wavefunctions
restricted to a definite value for the `time' variable are just like those of
the usual time independent quantum mechanics. A further understanding from a
perspective of the dynamics from the symplectic geometry of the phase space is
shortly discussed.
