This paper is the first of a series, concerned with formal aspects of the theory of elementary particles. It introduces a classical approach to the interaction representation of Tomonaga and Schwinger, based upon a generalization of analytical dynamics due to Weiss. General rules are deduced for constructing the interaction Hamiltonian function, and it is explained how the consistency of the representation may be established. The classical part of the theory is developed in detail. I n t r o d

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... c t io n One of the current problems in physics is the analysis of the elementary particles. Recent developments in this subject suggest that the 'real' particles investigated in the laboratory must be extremely complicated, since even the properties of electrons can only be obtained approximately, after long and difficult calculation. Although the results of present quantum electrodynamics may not be correct, it would be surprising if those of a future theory were very much simpler, and one may conclude that these particles can only be elementary in a relative sense; something more fundamental must lie behind them. It must be possible to find a theory from which all the properties of elementary particles can be deduced, including, for instance, their mass-ratios and charges, and in which divergences never appear. On the quantum theory of the elementary 'particles. I 125 n the quantum theory of the elementary 'particles. I Vol. 2 0 4 . A. 9 R e f e r e n c e s Belinfante, F.