Ultra-light and strong: the massless harmonic oscillator and its singular path integral

Abstract: In classical mechanics, a light particle bound by a strong elastic force just oscillates at high frequency in the region allowed by its initial position and velocity. In quantum mechanics, instead, the ground state of the particle becomes completely de-localized in the limit $m \to 0$. The harmonic oscillator thus ceases to be a useful microscopic physical model in the limit $m \to 0$, but its Feynman path integral has interesting singularities which make it a prototype of other systems exhibiting a "quantum runaway" from the classical configurations near the minimum of the action. The probability density of the coherent runaway modes can be obtained as the solution of a Fokker-Planck equation associated to the condition $S=S_{min}$. This technique can be applied also to other systems, notably to a dimensional reduction of the Einstein-Hilbert action.