Quantum geodesic flow on the integer lattice line

TL;DR

This paper explores quantum geodesic flow on an infinite discrete line using noncommutative geometry, revealing that discretization induces complex amplitudes and suggesting a link to the quantum nature of spacetime at the Planck scale.

Contribution

It applies noncommutative geometric formalism to model quantum geodesic flow on a lattice, highlighting effects of discretization on wave function properties.

Findings

Discretization causes real amplitudes to become complex.

Quantum effects may originate from spacetime discreteness.

Wave interference can be traced to noncommutative geometry.

Abstract

We use a recent formalism of quantum geodesics in noncommutative geometry to construct geodesic flow on the infinite chain $\cdots\bullet$--$\bullet$--$\bullet\cdots$. We find that noncommutative effects due to the discretisation of the line cause an initially real geodesic flow amplitude $\psi$ (for which the density is $|\psi|^2$) to become complex. This has been noted also for other quantum geometries and suggests that the complex nature of the wave function in quantum mechanics (and the interference effects that follow) may have its origin in a quantum/discrete nature of spacetime at the Planck scale.