Functional integral transition elements of a massless oscillator

Abstract

For any dynamical system, the functional average ((q - qdass)2),or "Feynman transition element", gives a measure of the amplitude ofquantum fluctuations with respect to the classical path. This is usuallyof order h or smaller, but for some systems it is not bounded, thussignaling 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 mostsystems these regions have zero functional measure and thus give anull contribution to the path integral, but this is not the case for themassless oscillator. We study the simplest functional subspace withconstant action, namely the one with S = 0, which is connected to theclassical solutions but extends to infinity, like an hyperplane throughthe origin; this subspace turns out to be infinite-dimensional. Somepossible applications and developments are mentioned.