The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

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6.
The choice i=(ti+ti−1)/2 is the only prescription that leads to a time translational invariant propagator.
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