By Hans Günter Dosch

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Jordan to quantize a classical field theory (the second quantization) were crowned by success only two years after the quantization of classical mechanics. However, it took another 20 years before the powerful tool of renormalized perturbation theory was developed for performing quantitative calculations. The only case in which the quantization of fields has been completely solved is that of free fields, that is, fields without any interactions. At first glance, this seems of little interest, but nevertheless, important features of a quantum field theory can be studied for this case.

This has consequences: if the interaction does not change under rotations, that is, if it is invariant under the rotational group, then the experimental results are invariant too. This means that if a particle with spin 12 is involved, it does not matter whether its spin orientation is + 12 or − 12 (↑ or ↓). The symmetry properties of a system consisting of two or more particles are of special importance for our further analysis. Let us begin with a system of two particles with spin 12 , for which both spins have the same orientation (↑↑).

One could tolerate the violation of the locality principle in quantum mechanics, but in the course of the history of science it has turned out that it is generally worthwhile to be conservative and not to surrender physical princi- ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 24 1. 9. The positive and negative energy states of the Dirac equation. In addition to the well-established transitions between states of positive energy (solid arrows) one expects transitions of states with positive energy into those with negative energy and also transitions between states of negative energy (dashed arrows).