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Hilbert's 6th Problem and Axiomatic Quantum Field Theory
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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 field 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 first 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 field theories are axiomatizations in this sense.

The systematic part of the paper isolates and comments on some characteristic features of the axiomatic quantum field theories, elements of which are recalled briefly in Section 4. The features will be of two types: general-methodological and specific-physical. The first category includes what we call here 'intended non-categoricity" and "large internal room," discussed in Section 5; the specific-physical we discuss (in Section 6) are "ontological silence" and "causal completeness." "Intended non-categoricity" is the intention of the mathematical physicist to design the axioms in such a way that they are weak enough in the sense that sufficiently many physically different models of the axioms exist. "Large internal room" is the feature that the axioms are strong enough to entail a large number of physically relevant consequences independent of any model. "Ontological silence" refers to the feature that the notion of "field" and "particle" vanish from the axiomatic quantum field theories as fundamental entities. "Causal completeness" refers to the feature that the theory is compatible with the causal structure of the relativistic spacetime that underlies the theory. It will be argued that these general features of axiomatization are related to deep problems in quantum field theory, some of which are still open. Most of the results in axiomatic quantum field theory that are relevant for these features are very technical but we suppress all technicalities in this paper. References are mainly given to review papers containing references to original research articles where the relevant results are published in full detail. Axiomatic quantum field theory is sometimes referred to as "constructive quantum field theory," emphasizing that the main task of mathematical physics—especially at this stage in the development of quantum field theory—is the construction of more and more realistic models of the axioms, models that describe specific physical fields. An up to date review of constructive quantum field theory is given in the paper (Summers 2011).

The claim that axiomatic quantum field theories are instances of axiomatizations of physical theories in the spirit of Hilbert's 6th problem is not new: Wightman (1983) surveys some developments in axiomatic quantum field theories with the intention of showing them cases of axiomatization as suggested by Hilbert. The present paper's addition to Wightman's analysis is the attempt to link Hilbert's 1900 idea to the 1927 specification of the axiomatic approach to physical theories (earlier papers called this latter axiomatization "opportunistic soft axiomatization" [Rédei and Stöltzner 2006, Rédei 2005]) and reflection on the more general features of the axiomatization.

2. Hilbert's Sixth Problem

The modern history of the axiomatic approach in physics starts with Hilbert's 6th problem: In his famous lecture delivered at the International Congress of Mathematicians (Paris, 1900), Hilbert formulated the program of axiomatizing physical theories:

The investigations on the foundations of geometry suggests the problem: To treat in the same manner, by means of axioms those physical sciences in which mathematics plays an important part; in...

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