Summary
Jauch’s paper with the same title is re-examined by use of the usual quantization formalisms. Schwinger’s action principle can give the commutation relations suggested byJauch, if we choose appropriate canonical independent variables, different from the variational independent variables. Quantization in the interaction representation is also discussed. The interaction betweenJauch’s field and the electromagnetic field is considered.
Riassunto
Per mezzo del normale formalismo di quantizzazione si riesamina il lavoro diJauch con lo stesso titolo. Il principio di azione di Schwinger può dare le relazioni di commutazione proposte daJauch se si scelgono variabili canoniche indipendenti adeguate, differenti dalle variabili indipendenti variazionali. Si discute anche la quantizzazione nella rappresentazione d’interazione. Si esamina l’interazione fra il campo di Jauch e il campo elettromagnetico.
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Footnotes
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Besides, we have the quantization rule which is given byPeierls (Proc. Roy. Soc., A214, 143 (1952). Some erroneous points are found in his proof and are quite essential in our present discussion. A quantity of the Heisenberg representation and one of the interaction representation cannot generally be combined with each other by a unitary transformation, as in equation (3.5) in his paper. This is the very reason why Qx(x, σ) has to be introduced in addition to the quantities in the Heisenberg and interaction representation. As will be shown in section2 (iii) in our paper, the field quantity in the Heisenberg representation is Jauch’s field and the quantity in the interaction representation is the electron-positron field which satisfies a different commutation relation from Jauch’s. This is a typical example contradicting his equation (3.5). The neutral vector field interacting with a spin 1/2 field is also such a case.
(14)Schwinger’s action principle gives only the commutation relation att=t′. See reference (5).
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Takahashi, Y. A note concerning the quantization of spinor fields. Nuovo Cim 1, 414–426 (1955). https://doi.org/10.1007/BF02855170
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DOI: https://doi.org/10.1007/BF02855170