Quantization of Spacetime Based on Spacetime Interval Operator

Abstract

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.