Covariance in fractional calculus

Based on the requirement of covariance, we propose a new approach for generalizing fractional calculus in multi-dimensional space. As a first application, we calculate an approximation for the ground state energy of the fractional two-dimensional harmonic oscillator using the Ritz variational principle. In lowest order perturbation theory, a new analytic result is derived for the energy level spectrum of the one-dimensional fractional harmonic oscillator.