Mechanical Model for Quantum Field Theory

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Abstract
We consider two harmonic oscillators, coupled during a finite time. Initial and final states can be defined unambiguously, and if the duration of the coupling is sufficiently short, the S matrix can be computed explicitly. The coupling gx2y is investigated in detail, for complex values of g. It is found that, along the real axis, the S matrix behaves smoothly as a function of g and tends to the unit matrix for g → 0, as it should. However, the S matrix has a line of essential singularities along the imaginary g axis, including the origin, so that it cannot be expanded into powers of g. If such an expansion is sought by means of a perturbation procedure, it is found that each term of the series is finite (no need of renormalization), but the series as a whole diverges.

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