The relativistic oscillator algebra revisited in the quantum groups formalism

We revisit the harmonic oscillator algebra in an indefinite metric by reinterpreting consistently the time-like components in a way compatible with a positive-definite metric. We show that, despite its unusual features, the relativistic oscillator algebra can be derived starting from the Euclidean q-oscillator algebra. The consistency of this isomorphism is examined at different levels including the possible implication on the dynamics of the two formalisms by means of their respective Hamiltonians.