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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. |
