A consistent approach to the path integral formalism of quantum mechanics based on the maximum length uncertainty

TL;DR

This paper develops a path integral formalism for quantum mechanics incorporating a maximum length scale, revealing non-trivial modifications to particle propagators with potential cosmological significance.

Contribution

It introduces a consistent path integral approach based on a deformed quantum algebra that accounts for a maximum observable length scale.

Findings

Modified classical equations of motion due to the length scale

Deformed quantum algebra representation

Explicit free particle propagator with non-trivial effects

Abstract

We have developed a proper path integral formalism consistent with the deformed version of the quantum mechanics which contains a maximum observable length scale at the order of the Cosmological particle horizon, existing in cosmology. First, we have presented the modifications to the classical mechanics which shows non-minimal effects on the equation of motion of a particle. Next, we have provided representation of the deformed quantum mechanical algebra. With this algebra in hand, we have calculated the general form of the path integral propagator in this deformed background. Finally, as a most simple case, we have built up the explicit form of the free particle propagator. The modifications to the free particle propagator shows some non-trivial effects in this case, which can have some important significance.