Where does quantum field theory fit into the big picture? (Q2724955)

In this essay, Arthur Jaffe takes a look back onto mathematical physics in the twentieth century from his standpoint as an outstanding constructive quantum field theorist.NEWLINENEWLINE The artificial fissure between mathematics and physics that came to pass with the advent of quantum field theory is critically reviewed. This rupture is now healed thanks to the enormous efforts and progresses that have taken place in mathematical and theoretical physics and today we ``marvel at the `unreasonable effectiveness of theoretical physics in mathematics','' as Jaffe poses the converse to Wigner's famous statement.NEWLINENEWLINE After briefly sketching the mathematical problems of field quantization that led to the need for renormalization, he goes on to describe the development of constructive quantum field theory in the period 1960 - 1990. First successes were made in space-time dimension \(2\) with the construction of relativistic, covariant quantum fields obeying a quartic field equation, yielding examples for the Wightman axioms, as well as for Haag-Kastler axioms of algebraic quantum fields, Haag-Ruelle axioms for scattering theory, and the Osterwalder-Schrader axioms for Euclidean fields. After these achievements, models in three space-time dimensions came within reach, and even Yang-Mills theory in \(3\) and \(4\) dimensions could be studied. Now, deep physical questions could be answered, or at least posed in a rigorous form: Existence of phase transitions, bound states, and scaling limits, the low-energy behavior and supersymmetric theories are only some keywords.NEWLINENEWLINE One of the most important method in these treatments is that of phase cell localization, based on the heuristical reasoning that correlations between separated cells in phase space should decay exponentially in many cases (for example if the theory has a mass gap). The method proceeds by integrating out (in the functional integral) the contributions of large phase cells, leading to an effective interactions. If organized inductively, this defines the phase cell expansion. Using it, Glimm and Jaffe were able to show, among other things, the positivity of the Hamiltonian in \(\phi^4_3\) theory.NEWLINENEWLINE Finally, the author takes a look at a different thread of thought towards quantization, Connes' noncommutative geometry. This fascinating field is still emergent, yet it has already produced remarkable physical results in Connes' reformulation of the standard model (extensively reviewed in 1993 Zbl 0798.58007; 1996, Zbl 0856.46044; 1996, Zbl 0867.58003). One other notable, more recent development is Connes' and Kreimer's applications of Hopf algebras to Feynman graphs and renormalization (1998, Zbl 0932.16038; 1998, Zbl 1041.81087; 1999, Zbl 1041.81086; 1999, Zbl 1049.81048; 2000, Zbl 1032.81026; 2000, Zbl 0986.16015; 2001, Zbl 0995.81077; 2001, Zbl 1042.81059).NEWLINENEWLINE Further, Jaffe describes some of his own recent results: He defined a noncommutative geometric invariant which, in the \(N=2\) supersymmetric case exhibits a \(\text{SL}(2,\mathbb Z)\) symmetry. This hidden symmetry, ordinarily associated with conformal field theories, shows startling connections to the elliptic genus (2001; Zbl 0979.58010).NEWLINENEWLINE The treatise concludes with some optimistic remarks on the future of the mathematical approach to quantum physics, in particular along the lines of novel geometrical approaches and noncommutative geometry.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00052].