--------------------------------------------------------------- S6c. Functional integrals, Wightman functions, and rigorous QFT --------------------------------------------------------------- QFT assumes the existence of interacting (operator distribution valued) fields Phi(x) with certain properties, which imply the existence of distributions W(x_1,...,x_n)=<0|Phi(x_1)...Phi(x_n)|0>. But the right hand side makes no rigorous sense in traditional QFT as found in most text books, except for free fields. Axiomatic QFT therefore tries to construct the W's - called the Wightman functions - directly such that they have the properties needed to get an S-matrix (Haag-Ruelle theory), whose perturbative expansion can be compared with the nonrigorous mainstream computations. This can be done successfully for many 2D theories and for some 3D theories, but not, so far, in the physically relevant case of 4D. To construct something means to prove its existence as a mathematically well-defined object. Usually this is done by giving a construction as a sort of limit, and proving that the limit is well-defined. (This is different from solving a theory, which means computing numerical properties, often approximately, occasionally - for simple problems - in closed analytic form.) To compare it to something simpler: In mathematics one constructs the Riemann integral of a continuous function over a finite interval by some kind of limit, and later the solution of an initial value problem ordinary differential equations by using this and a fixed point theorem. This shows that each (nice enough) initial value problem is uniquely solvable. But it tells very little of its properties, and in practice no one uses this construction to calculate anything. But it is important as a mathematical tool since it shows that calculus is logically consistent. Such a logical consistence proof of any 4D interacting QFT is presently still missing. Since logical consistency of a theory is important, the first person who finds such a proof will become famous - it means inventing new conceptual tools that can handle this currently intractable problem. Wightman functions are the moments of a linear functional on some algebra generated by field operators, and just as linear functionals on ordinary function spaces are treated in terms of Lebesgue integration theory (and its generalization), so Wightman linear functionals are naturally treated by functional integration. The 'only' problem is that the latter behaves much more poorly from a rigorous point of view than ordinary integration. Wightman functions are the moments of a positive state < . > on noncommutative polynomials in the quantum field Phi, while time-ordered correlation functions are the moments of a complex measure < . > on commutative polynomials in the classical field Phi. In both cases, we have a linear functional, and the linearity gives rise to an interpretation in terms of a functional integral. The exponential kernel in Feynman's path integral formula for the time-ordered correlation functions comes from the analogy between (analytically continued) QFT and statistical mechanics, and the Wightman functions can also be described in a similar analogy, though noncommutativity complicates matters. The main formal reason for this is that a Wick theorem holds both in the commutative and the noncommutative case. For rigorous quantum field theory one essentially avoids the path integral, because it is difficult to give it a rigorous meaning when the action is not quadratic. Instead, one only keeps the notion that an integral is a linear functional, and constructs rigorously useful linear functionals on the relevant algebras of functions or operators. In particular, one can define Gaussian functionals (e.g., using the Wick theorem as a definition, or via coherent states); these correspond exactly to path integrals with a quadratic action. If one looks at a Gaussian functional as a functional on the algebra of fields appearing in the action (without derivatives of fields), one gets - after time-ordering the fields - the traditional path integral view and the time-ordered correlation functions. If one looks at it as a functional on the bigger algebra of fields and their derivatives, one gets - after rewriting the fields in terms of creation and annihilation operators - the canonical quantum field theory view with Wightman functions. The algebra is generated by the operators a(f) and a^*(f), where f has compact support, but normally ordered expressions of the form S = integral dx : L(Phi(x), Nabla Phi(x)) : make sense weakly (i.e., as quadratic forms). The art and difficulty is to find well-defined functionals that formally match the properties of the functionals 'defined' loosely in terms of path integrals. This requires a lot of functional analysis, and has been successfully done only in dimensions d<4. For an overview, see: A.S. Wightman, Hilbert's sixth problem: Mathematical treatment of the axioms of physics, in: Mathematical Developments Arising From Hilbert Problems, edited by F. Browder, (American Mathematical Society, Providence, R.I.) 1976, pp.147-240.