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Functional integral transition elements of a massless oscillator

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.