Prog. Theor. Phys. Vol. 13 No. 6 (1955) pp. 612-626
Products of Improper Operators and the Renormalization Problem of Quantum Field Theory
Institut für Theoretische Physik, Technische Hochschule, Aachen, Germany
Inctituto de Fisica Teorica, Rua Pamplona, 145, São Paulo, Brasil
(Received February 7, 1955)
A theory of products of “improper” operators in quantized field theories is developed. Especially the meaning of products such as δ2, δx-1, SF, DF, ψψ etc. is clarified. It is shown that a theory of products of such quantities is generally neither commutative nor associative. It can be proved that there exists no generalized analysis of any kind which yields perfectly unique products of improper operators. On the basis of this product theory the mathematical foundation of the renormalization formalism is given and it is shown that it is in principle impossible to get uniquely determined values of the proper constants of a quantum theoretical system within the current field theoretical formalism. Finally it is pointed out, how a “self-quantized” field theory may be constructed without reference to matrix representation by the new operators algebra. Every field theory–to be based upon an exact definition of operator products–deviates from the present quantum theory but encloses the renormalization formalism as a limit of it.
DOI : 10.1143/PTP.13.612
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