Quantization of Spacetime Based on Spacetime Interval Operator

TL;DR

This paper introduces a covariant spacetime quantization framework using a new spacetime interval operator based on Clifford algebra, leading to a generalized uncertainty principle with minimal length effects and holographic properties.

Contribution

It proposes a novel covariant spacetime quantization method utilizing a spacetime interval operator rooted in Clifford algebra, extending non-commutative geometry concepts.

Findings

Derives a generalized uncertainty principle with a $p^2$ correction term.

Demonstrates the holographic nature of the theory.

Predicts geodesic fuzziness smaller than astrophysical bounds.

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.