An Elementary Derivation of the Harmonic Oscillator Propagator
  The harmonic oscillator propagator is found straightforwardly from the free particle propagator, within the imaginary-time Feynman path integral formalism. The derivation presented here is extremely simple, requiring only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation & creation operators, cumbersome determinant evaluations or any kind of involved algebra. 
L. Moriconi