Hilbert's 6th Problem and Axiomatic Quantum Field Theory

Abstract

��� 1. The Main Claims This paper has two parts, a historical and a systematic. In the historical part it is argued that the two major axiomatic approaches to relativistic quantum aeld theory, the Wightman and Haag-Kastler axiomatizations, are realizations of the program of axiomatization of physical theories announced by Hilbert in his 6th of the 23 problems discussed in his famous 1900 Paris lecture on open problems in mathematics, if axiomatizing physical theories is interpreted in a soft and opportunistic sense suggested in 1927 by Hilbert, Nordheim, and von Neumann. To show this, Section 2 recalls arst some of Hilbert’s views about the axiomatic approach to physical theories as these were formulated in his 6th problem. It will be seen that Hilbert did not specify in his Paris lecture what precisely the axiomatic approach in physics is supposed to be. It will be seen in Section 3 that he was more explicit in 1927 however: The joint paper (Hilbert et al. 1927) addresses the issue of the axiomatic approach in physics explicitly and embraces a soft notion of axiomatization, which is assumed to be carried out in an opportunistic way only in physics. Axiomatic aeld theories are axiomatizations in this sense. The systematic part of the paper isolates and comments on some characteristic features of the axiomatic quantum aeld theories, elements of which are recalled brieoy in Section 4. The features will be of two types: general-methodological and speciac-physical. The arst category includes what we call here ‘intended non-categoricity” and “large internal room,” discussed in Section 5; the speciac-physical we discuss (in Section 6) are “ontological silence” and “causal completeness.” “Intended Research supported in part by the Hungarian Scientiac Research Found (OTKA). Contract number: 100715