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

Abstract

We have developed a proper path integral formalism consistent with the deformed version of the quantum mechanics that contains a maximum observable length scale at the order of the cosmological particle horizon, existing in cosmology. We have started by modifying 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. Thereafter, as a most simple case, we have built up the explicit form of the free particle propagator. The modifications to the free particle propagator show some non-trivial effects in this case. Our formalism can be applied to analyze the quantum properties and observations emerging in the context of cosmology.