  Motivated by both concepts of R.J. Adler's recent work on utilizing Clifford
algebra as the linear line element $ds = \left\langle \gamma_\mu \right\rangle
dX^\mu $, and the fermionization of the cylindrical worldsheet Polyakov action,
we introduce a new type of spacetime quantization that is fully covariant. The
theory is based on the reinterpretation of Adler's linear line element as $ds =
\gamma_\mu \left\langle \lambda \gamma ^\mu \right\rangle$, where $\lambda$ is
the characteristic length of the theory. We name this new operator as
"spacetime interval operator", and argue that it can be regarded as a natural
extension to the one-forms in the $U(\mathfrak{s}u(2))$ non-commutative
geometry. By treating Fourier momentum as the particle momentum, the
generalized uncertainty principle of the $U(\mathfrak{s}u(2))$ non-commutative
geometry, as an approximation to the generalized uncertainty principle of our
theory, is derived, and is shown to have a lowest order correction term of the
order $p^2$ similar to that of Snyder's. The holography nature of the theory is
demonstrated, and the predicted fuzziness of the geodesic is shown to be much
smaller than conceivable astrophysical bounds.

High Energy Physics - Theory;General Relativity and Quantum Cosmology