  The massless harmonic oscillator is a rare example of a system whose Feynman
path integral can be explicitly computed and receives its main contributions
from regions of the functional space that are far from the classical and
semiclassical configurations near the stationary point of the action. The
functional average $\langle q_m^2 \rangle$ of the square of the coordinate at a
time $t_m$ which is intermediate between the initial and final time gives a
measure of the amplitude of quantum fluctuations with respect to the classical
path. This average, or "transition element", is divergent in the massless
limit, signaling a quantum runaway. We show that the divergence is not due to
the continuum limit and formulate the conjecture that the divergent
contributions come from regions where the action $S$ is constant and therefore
the interference factor $e^{-iS/\hbar}$ does not oscillate. For most systems
these regions have zero functional measure and thus give a null contribution to
the path integral, but this is not the case for the massless oscillator. We
study the simplest functional subspace with constant action, namely the one
with $S=0$, which is connected to the classical solutions but extends to
infinity, like an hyperplane through the origin; this subspace turns out to be
infinite-dimensional. Some possible applications and developments are
mentioned.
