Question

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 2014-04-02 05:35 (UCT), posted by SE-user V. Moretti

Comments

"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 2014-04-02 05:35 (UCT), posted by SE-user Mitchell Porter

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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 2014-04-02 05:35 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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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 2014-04-02 05:35 (UCT), posted by SE-user V. Moretti

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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 2014-04-02 05:35 (UCT), posted by SE-user Pedro Lauridsen Ribeiro

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

This post imported from StackExchange Physics at 2014-04-02 05:35 (UCT), posted by SE-user user33923

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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 2014-04-02 05:35 (UCT), posted by SE-user user33923

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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 2014-04-02 05:35 (UCT), posted by SE-user user33923