A proposal of Quantization in flat space-time with a minimal length present

TL;DR

This paper extends space-time to pseudo-complex coordinates, proposing a quantization framework that introduces non-commuting coordinates and a minimal length scale, blending quantum mechanics with a new algebraic structure.

Contribution

It introduces a pseudo-complex extension of quantum mechanics that incorporates a minimal length and non-commuting coordinates, expanding the theoretical framework.

Findings

Pseudo-complex coordinates lead to non-commuting space-time relations.

The algebraic structure preserves quantum mechanics simplicity while adding complexity.

Example shows classical annihilation replaced by non-zero amplitude in pseudo-complex form.

Abstract

The 4-dimensional space-time is extended to pseudo-complex coordinates. Proposing the standard quantization rules in this extended space, the ones for the 4-dimensional sub-space acquire, as one solution, the commutation relations with non-commuting coordinates. This demonstrates that the algebraic extension keeps the simple structure of Quantum Mechanics, while it also introduces an effective quite involved structure in the 4-dimensional sub-space. The similarities to H. S. Snyder's work, a former proposal to include the effects of a minimal length, are exposed. The first steps to pseudo-complex Quantum Mechanics in 1-dimension are outlined, awaiting still the interpretation of some new emerging structures. As an example, two waves, out of phase by 90 degrees, are added which classically annihilate each other, while in the pseudo-complex description there is a non-zero amplitude.