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

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.