  This paper introduces a systematic algorithm for deriving a new unitary
representation of the Lorentz algebra ($so(1,3)$) and an irreducible unitary
representation of the extended (anti) de-Sitter algebra ($so(2,4)$) on
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$. This representation is
equivalent to a representation on $\mathcal{L}^{2}(\mathcal{R}^{3})$, and the
corresponding similarity transformation is identified. An explicit
representation in terms of differential operators is given, and it is shown
that the inner product is Lorentz invariant. Ensuring Lorentz covariance
demands a modification of the Heisenberg algebra, recognized as a phase space
algebra at the interface of gravitational and quantum realms (IGQR), which we
consider subordinate to Lorentz covariance. It is also demonstrated that time
evolution can be cast in a manifestly covariant form. Each mass sector of the
Hilbert space carries a representation of the Lorentz algebra, and the (anti)
de-Sitter algebra on each mass sector contracts to the Poincare algebra in the
flat configuration and momentum space limits. Finally, we show that
three-dimensional fuzzy space also carries a unitary representation of these
algebras, algebraically equivalent to the
$\mathcal{L}^{2}(\mathcal{R}^{3},\frac{1}{r})$ representation but not
necessarily equivalent as representations. Several outstanding issues are
identified for future exploration.
