  The discrete phase space continuous time representation of relativistic
quantum mechanics involving a characteristic length $l$ is investigated.
Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for
most sections of the paper. The energy eigenvalue problem for the Planck
oscillator is solved exactly in this framework. Discrete concircular orbits of
constant energy are shown to be circles $S^{1}_{n}$ of radii $2E_n
=\sqrt{2n+1}$ within the discrete (1 + 1)-dimensional phase plane. Moreover,
the time evolution of these orbits sweep out world-sheet like geometrical
entities $S^{1}_{n} \times \mathbb{R} \subset \mathbb{R}^2$ and therefore
appear as closed string-like geometrical configurations. The physical
interpretation for these discrete orbits in phase space as degenerate,
string-like phase cells is shown in a mathematically rigorous way. The
existence of these closed concircular orbits in the arena of discrete phase
space quantum mechanics, known for the non-singular nature of lower order
expansion $S^{\#}$ matrix terms, was known to exist but has not been fully
explored until now. Finally, the discrete partial difference-differential
Klein-Gordon equation is shown to be invariant under the continuous
inhomogeneous orthogonal group $\mathcal{I} [O(3,1)]$ .
