Abstract
For any dynamical system, the functional average ((q - qdass)2),or "Feynman transition element", gives a measure of the amplitude of quantum fluctuations with respect to the classical path. This is usually of order h or smaller, but for some systems it is not bounded, thus signaling a quantum runaway and a considerable increase of complexity. Working with the example of the massless harmonic oscillator, which isformally viable, we test the conjecture that the divergent contributionscome from regions of the functional space where the action S is constantand therefore the interference factor ẽl' does not oscillate. For most systems these regions have zero functional measure and thus give anull contribution to the path integral, but this is not the case for the massless oscillator. We study the simplest functional subspace withconstant 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.