Journal of Statistical Physics

, Volume 172, Issue 6, pp 1617–1640 | Cite as

Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal

  • Rami Ahmad El-Nabulsi


Fractional path integration and particles trajectories in fractional dimensional space are motivating issues in quantum mechanics and kinetics. In this paper, a fractional path integral characterized by a fractional propagator is developed based on the framework of the fractional action-like variational approach. A fractional generalization of the free particle problem is found, the corresponding fractional Schrödinger equation is derived and a fractional path integral formulation of harmonic oscillators characterized by a perturbed Lagrangian is constructed after reducing the fractional action to an integral action on fractal. The new fractal-like path integral offers a number of motivating features which are discussed and analyzed. The main outcome is connected to the possibility of constructing on a fractal a path integral for the oscillators characterized by modified ground energy. In particular for low-temperature case, the fractional perturbed oscillator is characterized by a free energy larger than the standard value \( E_{0} = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right. \kern-0pt} 2}.\) Such an increase in the ground energy generalizes the uncertainty principle without involving differentiable paths or even invoking new phenomenological theories based on deformed algebra.


Fractional path integral Fractionally harmonic oscillators Modified ground state energy level 



I would like to thank the anonymous reviewers for their careful reading of my manuscript, their several insightful comments and valuable suggestions.


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Authors and Affiliations

  1. 1.Mathematics and Physics DivisionsAthens Institute for Education and ResearchAthensGreece

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