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We present in the article the formulation of a version of Lorentz covariantquantum mechanics based on a group theoretical construction from aHeisenberg-Weyl symmetry with position and momentum operators transforming asMinkowski four-vectors under the Lorentz symmetry. The basic representation isidentified as a coherent state representation, essentially an irreduciblecomponent of the regular representation, with the matching representation of anextension of the group $C^*$-algebra giving the algebra of observables. The keyfeature of the formulation is that it is not unitary but pseudo-unitary,exactly in the same sense as the Minkowski spacetime representation. Thelanguage of pseudo-Hermitian quantum mechanics is adopted for a clearillustration of the aspect, with a metric operator obtained as really themanifestation of the Minkowski metric on the space of the state vectors.Explicit wavefunction description is given without any restriction of thevariable domains, yet with a finite integral inner product. The associatedcovariant harmonic oscillator Fock state basis has all the standard propertiesin exact analog to those of a harmonic oscillator with Euclidean position andmomentum operators of any `dimension'. Galilean limit of the Lorentz symmetryand the classical limit are retrieved rigorously through appropriate symmetrycontractions of the algebra and its representation, including the dynamicsdescribed through the symmetry of the phase space.

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and $c \rightarrow \infty $ Limit