On the Extension of the Casimir Trick, I : On the Definition of Density Matrix and its Reduction

概要

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When a Fermion is described by a plane wave, quantities which are to be compared with experimental data can be computed easily by using the well-known Casimir trick. But, when the influence of interaction of the Fermion with another field is taken into account, the calculation becomes very difficult. As the Casimir trick can not be used, complicate calculations have hitherto been necessary. It is the purpose of this paper to give a method of calculation in which the density matrix, instead of the Casimir operator, is used to avoid the intricate method above mentioned in the case of a Fermion moving in an arbitrary central field. This density matrix will be represented as a sum of direct products of ρ- and σ-operators with numerical coefficients. It is also verified that, for a free Fermion, the density matrix defined in this paper is reduced exactly to the usual Casimir operator(projection operator). The method of calculation proposed in this paper will be not only favourable to treat such various problems as β-decay, Bremsstrahlung and so on where the influence of a central potential must be considered, but also useful to solve the problem for free Fermions by the method of partial waves.
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