Some Quantum-Mechanical Divergences of a Simple Field

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Abstract
In order to study the sources of quantum-mechanical divergences, an elementary model, consisting of a one-dimensional vibrating string elastically coupled to a harmonic oscillator, is studied in detail. Following a preliminary discussion of the classical eigenfunctions of the coupled system for both a finite and an infinitely long string, the system is quantized by the standard methods of boson field theory. The permissible experiments upon the system are investigated from the viewpoint of regarding the measuring apparatus as providing initial conditions for the system. It is shown that certain infinities are of mathematical origin, and arise from breakdown of perturbation theory. Others are physical, and are the result of prescribing impossible requirements for the measuring apparatus. Cut-off methods which remove the infinities are presented.