Title: 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.