Does anyone take the Wightman axioms seriously? Mainly with respect to quantum gravity or gauge theores, abelian or non-abelian? Anyone doing any research on axiomatization of QFTs in some way?
The question sounds like the following question sounds to a classical physicist: Does anyone take Lagrangian mechanics seriously?
Wightman axioms describe qft how it should be within a certain paradigm, assuming that some fundamental difficulties affecting perturbative qft can be solved in some way (but without suggesting any solution). It just presents the final theory, I mean, that including interactions, as it should be in that view. There is no guaratee however that it is a correct and complete picture of the world. In particular because, the mentioned difficulties could be a clue, and probably are, of new physics like string theory or other structures relevant at very high energy or very small scales. Moreover the description of gauge theories within Wigthman formulation is by no means straigthforward. Nevertheless this approach stands as a mathematically solid framework where proofs of physically fundamental statements of qft have been rigorously built up. I mean, for instance, spin statistic theorem, cpt theorem and so on. However, it does not mean that these results would not arise from other formulations based on different physics. I think that Wightman axioms can be viewed as Lagrangian mechanics with respect to "real" classical physics. Lagrangian formulation is a model where some important relationships between crucial notions can be analysed, I think of the interplay between conserved quantities and symmetries for instance. On the other hand, it is however clear that Lagrangian formulation is too physically naive, since for instance it does not properly consider forces due to friction that reveal the existence of another level of reality (I mean thermodynamics and microscopic physics... It assumes that physical objects are pictured by differential geometry disregarding the discrete microphysical structures...). The core of Garding Wightman Streater‘s formulation has produced other formulations of qft that insist on the notion of local field. A textbook on those ideas is Haag's one. These ideas have been implemented to develop qft in curved spacetime, with application to black hole physics in particular and, recently, to cosmology. I belong to that community of mathematical physicists. The uv renormalized procedure has ben completely reformulated in curved spacetime into a generally covariant framework without assuming the existence of a preferred vacuum in view of the absence of Poincare' symmetry.
Wightman axioms are nothing more than the general features recognized to be common in the practise of relativistic quantum field theory. Even a practitioner uses Wightman axioms all the time, even if he does not care about mathematical rigor and tends to hide this fact. However, it is clear you are asking about Algebraic Quantum Field Theory.
The main goal of AQFT is formalizing the properties of quantum fields. Strictly speaking, this task does not belong to the realm of physics, but to that of mathematical physics. It is common lore of physicists to ignore mathematical details when they are handeling a working but ill-defined theory, an example being the raising of the Dirac delta function (actually, a distribution) and its use in applications long before the invention of generalized functions by Sobolev and Schwartz. Hence, a physicist it is not interested in study a system of axioms and their consequences; all he needs are a system of rules thanks to which he's able to perform actual computations. The word "rule" means a standard presctiption that in peculiar situations can be modified or neglected, when useful. Of course, when possibile, it is often convenient take advantage of a polished mathematical framework to do calculations.
Mathematicians can't accept a such state of things, so they are concerned with foundations, existence problems and so on. This is part of their work and has and its own value, even in the case physicists won't exploit it. However, sometimes it happens that mathematical rigor, careful definitions and all sort of mathematical complications are physical necessities. In the case of QFT this arises when one consider the measurability in principle of the electromagnetic field. It was soon realized by Bohr and Rosenfeld that quantum fields have to be operator-valued distributions, in order to make sense as representatives of physical quantities. Since this fact concernes measurability, it turns in a physical problem. So physicists can't ignore the complicated nature of these objects, they can try to live with it and fact all sort of elaborated recipes have been conceived in order to escape such a difficult coexistence.
Nevertheless, there was (and in part, there is) an active community that tried to face these formal problems, generated by physical reasons and stimulated by the sake of a comprehensive logical treatment of the matter. Big names in this community are that of Wightman himself, Rudolf Haag, Araki, Borchers, Sergio Doplicher, Robert Powers, Fredenhaghen, Daniel Kastler, Buchholz, Ostenwaald, Baumann etc... Many general results have been founded and a good number of paradox of QFT have been solved thanks to this more careful treatment. Among them (in broad terms):
- the existence of the free fields;
- the PCT theorem;
- the leading role of the Principle of Locality;
- general connections between spin and statics;
- a weak form of a Noether theorem for quantum fields;
- general forms for the Goldstone theorem;
- the relations between fields, observable and gauge groups;
- any compact group is a gauge group;
- the Reeh-Schlieder theorem (informally, density of states generated by the action of the local observables on the vacuum in the Hilbert space of the theory)
and so on. Summarizing, a good host of result has been obtained. Of course, this is not all the story: many other things are missing, first of all no one knows whether interaction fields exists in a rigorous sense. A theory without interaction can't be physical, so AQFT is always regarded as a sort of oddity by physicists. Nevertheless, the development of AQFT has stimulated many other field of research, both in mathematics and in physics. Nowadays, trend has been changed and a minor fervour survives around these topics. People who worked in the field in the past usually continue to contribute but a smaller number of new researchers joins this path, being attracted in other areas, such as Quantum Gravity, strings, particle physics, phenomenology etc...
You've heard the opinion of a mathematical physicst, i.e. of a person who works in a department of mathematics and ventures into physics, namely V.M. Here's my 2 cent view as a physicist. In one of his famous lectures (see Deser, vol. II), Rudolf Haag remembers: "The following story was reported to me. A few years ago Klaus Hepp gave some lectures in the Brandeis summer school. At some stage he praised the beauty of axiomatic field theory. Next day he found the note on the blackboard: "Axiom 1: Axiomatic Field Theory is beautiful in an empty sort of way.""
Surely, for mathematicians is a bread to eat, since they continuously reformulate what the physicists have given them as physical intuition in more and more abstract ways. However, to compare Lagrangian theory of classical physics with Wightman's axioms and say they are on the same footing is a major overstatement. Indeed, in Lagrangian mechanics there is a dynamical evolution principle, whereas in Wightman's axioms there is none. To see how dire the situation is, is like having all Newton's laws WITHOUT THE SECOND, i.e. F=MA. Or, if one goes quantum, it's like having all the postulates of non-relativistic QM without the Schrodinger equation or Heisenberg's equations of motion. All you can calculate is obviously only static problems, like the celebrated spin-statistics and cpt theorems in AQFT.
Moreover, to talk about frictional forces not being "covered" by Lagrangian mechanics or thermodynamics is mathematician's talk. THERE ARE NO FUNDAMENTAL FRICTIONAL FORCES TO BEGIN WITH. All there is is disipativive dynamics for a given system in interaction with some other system. If you take the two combined everything is conserved! Friction and thermodynamics are PHENOMENOLOGICAL THEORIES, not FUNDAMENTAL ONES! Take a "friction" problem for example. All interaction is electromagnetic in essence between 2 bodies that slide against each other, and you can really come up with a microscopic model BASED ON LAGRANGIAN THEORY and compare it with experiment.
Coming back to AQFT, besides static problems, you can really calculate nothing. No number that can be compared with experiment. Even the description of scattering in the celebrated Haag-Ruelle scattering theory is NOT ENTIRELY FORMULATED IN THE HEISENBERG PICTURE of AQFT, and AQFT is notorious for claiming to be using a description entirely within the Heiseinberg picture.
Of course, it's entirely up to you who's one of the two views you pick: physicist Rudolf Haag's or that of a math department person as V.M.'s ...