Yesterday Peter Woit received some notes from Bert Schroer about Lagrangian path integral and algebraic quantum field theory. As expected, Schroer tries to dismiss all approaches to QFT except his own. Lubos Motl was really quick in pointing out the technical mistakes and wrong statements about the path integrals results in Schroer's notes so I will no repeat them here. To those interested in more details see the discussion on the spinning top and the hydrogen atom in Kleinert's book. As Lubos points out,
every known well-defined problem in any quantum mechanical system with a classical limit that is solvable using other methods is also solvable using path integrals. And of course, one obtains the right results. Moreover, path integrals are superior in their treatment of non-perturbative corrections (instantons) and gauge symmetries (Faddeev-Popov ghosts). They have become the canonical way to define the Lorentz-covariant prescription for perturbative amplitudes in string theory. Attempts to remove path integrals from physics in 2006 are plain ludicrous.
I would add that AQFT, the approach advocated by Schroer, is in such a precarious state that it can say nothing about non-abelian gauge theories, instantons or other non-perturbative effects, or even loop corrections to the standard model. The effects of loop corrections were measured in several accelerators and the agreement with experiment is excelent.
While Schroer statements are largely ignored and not taken seriously in the USA and Europe, here in Brazil the situation is dramatically distinct. Being a retired German professor living in Brazil he has an aura of a great scientist which was nourished by his Brazilian subordinates. His statements are used by physicists that do not support research in particle physics and sometimes they become an obstacle to the development of the area. Besides that we also have troubles with misguided students. What a student can say when he is told by Schroer or his followers that path integrals are not mathematically well defined and should not be used (like in the notes just released)? Or that the phi to the fourth scalar theory is trivial (inducing the idea that QFT is trivial as well)? Or that the perturbative series of QED is not convergent (casting doubts on the vality of QED and not saying that it is part of the electroweak theory)? Or when they hear statements that supersymmetry and string theory are just rubbish and you should not waste time on them (when string theory is the main research topic today and supersymmetry is expected to be found at LHC)? We have to spend a lot of energy teaching these students and other colleagues what is the real situation in the area. Just to show the prestige of Schroer in Brazil, he is giving a colloquium in the main Brazilian university next Thursday. He is going to talk about the history of QFT and I wonder how many people will get a false image of the area. So what can we do? We just keep working honestly and trying to tell the truth to the students and colleagues. Next week we will have a series of 4 lectures on AdS/CFT given by Carlos Nunez from Swansea University. This is the best way to promote good research in science. Being sociable and civilized even with those that do not accept the modern world and its new ideas.
UPDATE: I received an e-mail from João Barata asking about the subordinates mentioned in the text. Clearly João is not one of them since his work is independent and he has his own opinions. He works and has interest in several topics, including AQFT, but he is wise in recognizing the advantages and limitations of each approach. I never had you in mind João!
UPDATE: Today I met Bert in my Institute and we had a very friendly conversation. He was much less radical than in his writings. In a long talk we both agreed that the path integral and AQFT has its advantages and limitations. And that some aspects of field theory may be better described by one formalism than the other. All in all he was much more moderate. He was also worried because some people may think that we have personal problems. That is not true.