  Several models of quantum gravity predict the emergence of a minimal length
at Planck scale. This is commonly taken into consideration by modifying the
Heisenberg Uncertainty Principle into the Generalized Uncertainty Principle. In
this work, we study the implications of a polynomial Generalized Uncertainty
Principle on the harmonic oscillator. We revisit both the analytic and
algebraic methods, deriving the exact form of the generalized Heisenberg
algebra in terms of the new position and momentum operators. We show that the
energy spectrum and eigenfunctions are affected in a non-trivial way.
Furthermore, a new set of ladder operators is derived which factorize the
Hamiltonian exactly. The above formalism is finally exploited to construct a
quantum field theoretic toy model based on the Generalized Uncertainty
Principle.
