Path integral action of a particle with the generalized uncertainty principle and correspondence with noncommutativity

Abstract

The existence of an observer independent minimum length scale can lead to the modification of the Heisenberg uncertainty principle to the generalized uncertainty principle. This in turn would be responsible for the modification of the Hamiltonian describing a non-relativistic particle moving in the presence of an arbitrary potential. In this work we carry out a path integral formulation to compute the transition amplitude for this particle. The formalism yields the action of such a particle in an arbitrary potential. Interestingly, the action indicates that there is an upper bound to the velocity that a particle can have which depends on the generalized uncertainty principle parameter. We then compute explicitly the propagator of a free particle and particle moving in a harmonic oscilltor potential using the path integral representation of the transition amplitude. We observe that there exists a curious connection between the transition amplitude of the free particle in the generalized uncertainty priciple framework with the corresponding result in noncommutative space found from the path integral formulation in \cite{sgprl}. From the harmonic oscillator result for the transition amplitude, we calculate the ground state energy of the harmonic oscillator. The result shows that the ground state energy of the harmonic oscillator in the framework of the Heisenberg uncertainty principle gets augmented by the presence of the generalized uncertainty principle and also depends on the mass of the particle. We also demonstrate that the result agrees with that obatined using the operatorial approach.