(Received May 16, 1949)
The mathematical formalism presented here owes its origin to the recent development of quantum electrodynamics. The theory of Tomonaga and Schwinger, which deals with the reaction of the interacting fields in a completely relativistic way, has led us to a deeper insight into the nature of the interaction of the elementary particles. This reaction of the fields manifests itself in such phenomena like the self-energy of the fields and the modification of various physical quantities, and may be considered as effects of the zero-point fluctuation of the quantized fields. Mathematically, it means that a quantity composed of q-number wave functions has a non-vanishing expectation value even in the lowest state. Thus, for instance, the product of two quantized waves ψ*, ψ' describing the electron field, or A, A' describing the radiation field, will not have in general a zero expectation value even if there are no electrons or no photons (vacuum). In order to avoid this zero-point fluctuation, we usually decompose field quantities into creation and annihilation operators, and rearrange the products of these operators so that they become sums of two terms, the one giving no fluctuation, while the other corresponding to pure fluctuation effect. Then the latter may be subtracted nuless it gives rise to any physically observable effect. This was the case for the zero-point energy of the radiation field and the total energy of the completely filled negative states in the hole theory of Dirac elecron since they did not depend on the field variables. The self-energy of the electron arises from the one-body fluctuation in the essentially two body operators of the interaction energy of the form ψ*ψψ*'ψ'. In this case the fluctuation term turns out bilinear in ψ* and ψ, so that it may be amalgamated to the originally existing mass term, thus making only a mass renormalization which is physically undetectable. In some cases, however, the fluctuation terms do really cause observable effects such as the anomalous interaction of the electron with external field.
The above mentioned separation of the fluctuation terms may be carried out in any quantity by the prescription that we rearrange the field variables in such a way that the creation operators (marked with asterisk) stand always to the left of the annihilation operators (without asterisk). For exchanging any two variables into correct order we make use of the comutation relation between them. We are therefore allowed to regard every quantity to be a well-ordered function of the field variables which may be written as
F = ∑f(a*, b*, …)·g(a, b, …),
where each of the set a*, b*, … and a, b, … is commutative or anti-commutative according to the statistics. The well-ordered character is preserved on addition of two such quantities, but unfortunately it is not on multiplication. Hence a rearrangement will be necessary after every operation such as simple multiplication of two quantities, transformation of representation, and making commutation relation. To give a clear perspective of this rearrangement process we shall introduce in the subsequent sections some new algebraical concepts.
URL : http://ptp.ipap.jp/link?PTP/4/331/
DOI : 10.1143/PTP.4.331