Question

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?

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

Comments

I would vote for changing the 1 million for proving Yang Mills consistent to Wightman axioms to "1 million for finding a good axiomatization for QFT"... Wightman axioms look utterly incomplete. You put on hold, you lose! What do you want me to say? Please think before you down-vote?!

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

---

@user33923 Not sure if you are meaning me, but I gave you an upvote and voted to leave open ;-). Cheers

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Dilaton

---

Ok, so the aspects I would like to underline are 1. There is no valid axiomatization of Quantum Field Theories (including gauge) so 2. There are no obvious ways of generalizing (gauge) QFTs because usually generalizations are constructed from quitting or deforming axioms 3. are there generalizations of QFT other than dimensional obtained in the same random manner? If yes, which are they? 4. As gauge theories are so widely used and there is no (co)homology axiom there I may wonder if (co)homological generalizations could not be an option? Why not?

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

---

@user33923 Maybe you should update your question accordingly. I think to say if they are taken seriously is a little bit harsh. Maybe inspirehep.net/record/1094230 is an interesting read.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user ungerade

---

at least a good starting point, thanks!

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

---

Ok, I see ;-). Maybe you could edit what you told me in the comment unto the question, as you presently have 3 close votes hanging around your neck here ... :-/. I voted to leave open, as I think this is interesting too.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Dilaton

---

Is beeing part of an US$1,000,000 problem serious enough ;)? en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user ungerade

Answer 1

The question sounds like this, for 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.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user V. Moretti

Comments

Great, balanced answer, +1

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Zoltan Zimboras

---

(of course, the rest is important but...) First two lines, to me, answer the question:)

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user c.p.

---

I also like this answer, although I don't mark it as "accepted answer"... I will comment on this a bit later...

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

---

Ok, first: Lagrangeans or Hamiltonians are choices. You can use lots of other functions with the same effects. From this point of view one should not take them extremely seriously either. Next, a set of axioms may or may not be unique and in some cases you can replace some axioms with some theorems and derive the axioms as theorems from them. So, no, you should not take axioms extremely "serious" either.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

---

Actually, there are non Lagrangian or Hamiltonian in W. axioms. Yes it is true that "a set of axioms may or may not be unique and in some cases you can replace some axioms with some theorems and derive the axioms as theorems from them", but it is true for every theoretical construction in theoretical physics. That is the way of life: We should not take axioms for anything extremely "serious". It is already a miracle that something so technically complicated like renormalization in QFT or GR, or QM in general give some physically sensible result.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user V. Moretti

---

"The question sounds like this, for a classical physicist: Does anyone take Lagrangian mechanics seriously?" Lagrangian formulation is used in actual physics. But does anyone ever use a Wightman-axiomatizable QFT for anything? So far as I know, Wightman axioms do not apply to renormalizable QFT, which excludes the whole of particle physics. But maybe some "Wightman-able" theory is used in condensed matter?

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Mitchell Porter

---

Well, strictly speaking, certain fundamental tools in QFT such as the LSZ reduction formula (in its generalized form including composite fields given by Hepp) and the Kallén-Lehmann integral representation were originally derived from the Wightman axioms. Steinmann also formulated an approach to renormalized perturbative QFT starting from the Wightman axioms. Even if such results are used (as it happens) in contexts beyond their original scope, the Wightman axioms provided the mindset for their inception.

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

---

Hi Pedro, I was about writing the same things, but I do not think it is worth continuing a discussion like this as it is mostly based on personal convictions...

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user V. Moretti

---

I know Valter, these are muddy waters. There is a lot more I thought about this, but I don't know if it's worth it to share here...

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

---

I would be interested to hear other opinions, as muddy as they might be

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user user33923

Answer 2

The Wightman axioms are nearly irrelevant for those physicists who are satisfied with the nonrigorous level of arguments typical for most of theoretical physics. But they cannot be ignored by anyone doing mathematical physics, i.e., physics on a mathematically rigorous level, as they are universally believed to capture the minimal properties observable fields must have.

Note that the Wightman axioms capture only observable fields, hence cannot be applied directly to gauge fields (for which only the associated curvature fields are observable).

There is a fairly dedicated group of people from all over the world working on Algebraic and Axiomatic Quantum Field Theory, primarily in the framework of local quantum physics, closely related to the Wightman setting. See

Local Quantum Physics Crossroads, http://www.lqp2.org/users

Next month there is a conference in Vienna about the topic: http://www.esi.ac.at/activities/events/2014/algebraic-quantum-field-theory-its-status-and-its-future

Records of a conference in 2009 celebrating 50 years of Algebraic Quantum Field Theory can be found at http://www.uni-math.gwdg.de/aqft/

The Stanford Encyclopedia of Philosophy has a long article http://plato.stanford.edu/entries/quantum-field-theory/ about the relevance of the subject.