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The harmonic oscillator transition kernel is the position-space unitary time evolution amplitudeFor $0<T<\pi /\omega$,Here $\hslash$ is the reduced Planck constant. The interval fixes the initial square-root branch; extension beyond focusing times requires its continued phase.

In the path integral, write a path as the classical oscillator path plus a fluctuation vanishing at both endpoints. The quadratic action splits exactly into the classical action and the endpoint-independent fluctuation action. The Dirichlet oscillator determinant ratio gives the prefactor, while the classical boundary action gives the exponent. ^[ref]

As $\omega \to 0$ with $T$ fixed, the expression reduces to the free-particle propagatorAt $T=n\pi /\omega$, the coordinate kernel is distributional because classical paths refocus. The Feynman i-epsilon prescription fixes the continuation through these caustics. Integrating the kernel against an initial wavefunction gives its later wavefunction.

Riccardo Rattazzi developed a path-integral account of quantum mechanics that connects interference, operator evolution and statistical methods. His notes derive the path integral from quantum time evolution, work out the free-particle propagator and treat Gaussian fluctuations.^[ref]

The weight $e^{iS/\hslash }$, with $S$ the action functional, is an amplitude with a phase, not a classical probability assigned to each path. Combining amplitudes before taking probabilities produces interference. Analytic continuation to Euclidean time then connects suitable quantum calculations with statistical-mechanical formulations, under the required convergence and boundary assumptions.