Hilbert's 6th problem and constructive quantum field theory

Miklos Redei

London School of Economics and Political Science

Hilbert formulated the program of axiomatizing physical theories in his 1900 lecture in Paris onopen problems in mathematics; this was the 6th of the 23 open problems. Hilbert’ short formulation ofthe 6th problem does not specify what precisely axiomatization in physics means and how it is relatedto the investigation of the ”foundations” of science; which Hilbert also mentions in his 6th problemas being of prime importance. In his 1926 joint work with von Neumann on foundations of quantum mechanics Hilbert addresses the nature of the axiomatic approach to physical theories in more detail, and the first part of the talk reconstructs the main features of Hilbert’s 1927 concept of axiomatic physics. It will be argued that Hilbert interprets the axiomatic approach in a soft and opportunistic manner, which makes it different from formal axiomatics. The second part of the paper isolates and investigates some characteristic features of the axiomatic (also called: constructive) quantum field theories. It will be argued that the emergence (around 1959-1960) of axiomatic approaches to quantum field theory (Wightman axioms, Haag-Kastler axioms) were realizations of Hilbert’s program formulated in is 6th problem if axiomatization is understood in the spirit of opportunistic soft axiomatic. The features of constructive quantum field theory will be of two types: generalmethodological, which are characteristic of axiomatic approaches to other physical theories as well, and specific-physical related to quantum field theory. The first category includes what will be called ‘intended non-categoricity” and “large internal room”, the specific-physical we discuss 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. The talk claims that achieving the proper balance of these two requirements, which are pulling in opposite directions, is a central problem in every axiomatic approach, and that the “internal room” can be identified with the foundations of science Hilbert mentions in his 6th problem. The “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 features are related to deep problems in quantum field theory, some of which are still open.