Quantization of spacetime based on a spacetime interval operator

Motivated by both concepts of Adler’s recent work on utilizing Clifford algebra as the linear line element $ds=⟨{\gamma }_{\mu }⟩d{X}^{\mu }$ 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 $ds={\gamma }_{\mu }⟨\lambda {\gamma }^{\mu }⟩$, where $\lambda$ is the characteristic length of the theory. We name this new operator the “spacetime interval operator” and argue that it can be regarded as a natural extension to the one-forms in the $U(\mathfrak{s}u(2))$ noncommutative geometry. By treating Fourier momentum as the particle momentum, the generalized uncertainty principle of the $U(\mathfrak{s}u(2))$ noncommutative 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.