Deformed Heisenberg algebra with minimal length and equivalence principle

TL;DR

This paper explores how a deformed Heisenberg algebra, motivated by string theory and quantum gravity, can incorporate a minimal length scale while preserving the equivalence principle through analysis of classical motion and kinetic energy.

Contribution

It demonstrates that the equivalence principle can be maintained in a deformed phase space with minimal length by considering the effective deformation parameters of composite bodies.

Findings

The motion of a composite body's center of mass depends on an effective deformation parameter.

The equivalence principle is recoverable in deformed space when kinetic energy is independent of composition.

GUP can be reconciled with the equivalence principle through analysis of classical motion in deformed phase space.

Abstract

Studies in string theory and quantum gravity lead to the Generalized Uncertainty Principle (GUP) and suggest the existence of a fundamental minimal length which, as was established, can be obtained within the deformed Heisenberg algebra. The first look on the classical motion of bodies in a space with corresponding deformed Poisson brackets in a uniform gravitational field can give an impression that bodies of different mass fall in different ways and thus the equivalence principle is violated. Analyzing the kinetic energy of a composite body we find that the motion of its center of mass in the deformed space depends on some effective parameter of deformation. It gives a possibility to recover the equivalence principle in the space with deformed Poisson brackets. and thus GUP is reconciled with the equivalence principle. We also show that the independence of kinetic energy on composition leads to the recovering of the equivalence principle in the space with deformed Poisson brackets.